automatic modulation classi cation and blind … · automatic modulation classi cation and blind...

Automatic Modulation Classification and Blind Equalization for

Cognitive Radios

Barathram Ramkumar

Dissertation submitted to the Faculty of

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Electrical Engineering

Tamal Bose

Jeffrey H. Reed

Allen B. MacKenzie

Yaling Yang

Christopher W. Zobel

July 28, 2011

Blacksburg, Virginia

Keywords: Automatic Modulation Classification, Blind Equalization, Cognitive Radios

Chapter 2 c©2009 by IEEE

Section 3.4 c©2010 by The Wireless Innovation Forum

All other materials c©by Barathram Ramkumar

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Automatic Modulation Classification and Blind Equalization for Cognitive

Radios

Barathram Ramkumar

(ABSTRACT)

Cognitive Radio (CR) is an emerging wireless communications technology that addresses

the inefficiency of current radio spectrum usage. CR also supports the evolution of existing

wireless applications and the development of new civilian and military applications. In

military and public safety applications, there is no information available about the signal

present in a frequency band and hence there is a need for a CR receiver to identify the

modulation format employed in the signal. The automatic modulation classifier (AMC) is

an important signal processing component that helps the CR in identifying the modulation

format employed in the detected signal. AMC algorithms developed so far can classify only

signals from a single user present in a frequency band. In a typical CR scenario, there is a

possibility that more than one user is present in a frequency band and hence it is necessary

to develop an AMC that can classify signals from multiple users simultaneously. One of the

main objectives of this dissertation is to develop robust multiuser AMC’s for CR. It will be

shown later that multiple antennas are required at the receiver for classifying multiple signals.

The use of multiple antennas at the transmitter and receiver is known as a Multi Input Multi

Output (MIMO) communication system. By using multiple antennas at the receiver, apart

from classifying signals from multiple users, the CR can harness the advantages offered by

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classical MIMO communication techniques like higher data rate, reliability, and an extended

coverage area. While MIMO CR will provide numerous benefits, there are some significant

challenges in applying conventional MIMO theory to CR. In this dissertation, open problems

in applying classical MIMO techniques to a CR scenario are addressed.

A blind equalizer is another important signal processing component that a CR must possess

since there are no training or pilot signals available in many applications. In a typical wireless

communication environment the transmitted signals are subjected to noise and multipath

fading. Multipath fading not only affects the performance of symbol detection by causing

inter symbol interference (ISI) but also affects the performance of the AMC. The equalizer is

a signal processing component that removes ISI from the received signal, thus improving the

symbol detection performance. In a conventional wireless communication system, training

or pilot sequences are usually available for designing the equalizer. When a training sequence

is available, equalizer parameters are adapted by minimizing the well known cost function

called mean square error (MSE). When a training sequence is not available, blind equaliza-

tion algorithms adapt the parameters of the blind equalizer by minimizing cost functions

that exploit the higher order statistics of the received signal. These cost functions are non

convex and hence the blind equalizer has the potential to converge to a local minimum. Con-

vergence to a local minimum not only affects symbol detection performance but also affects

the performance of the AMC. Robust blind equalizers can be designed if the performance

of the AMC is also considered while adapting equalizer parameters. In this dissertation

we also develop Single Input Single Output (SISO) and MIMO blind equalizers where the

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performance of the AMC is also considered while adapting the equalizer parameters.

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Dedicated to my parents, sister and guru

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Acknowledgments

I thank my advisor Dr. Tamal Bose for his guidance and support. It has been a true privilege

to work with a well-reputed advisor at Virginia Tech. His sincere guidance has helped me

shape up my research and career. I hope to collaborate with him in the future. I thank Dr.

Jeffrey H. Reed for being my committee member. His suggestions were helpful in improving

the quality of this dissertation. I am also grateful to all other committee members for their

suggestions and time. I am thankful to my mother and sister for their unconditional love

and support. I am grateful to my father for the sacrifices he made to ensure a high quality

education for me. I am grateful to all my gurus and teachers for their guidance and wisdom.

I thank my uncle Trimbakeshwar for his encouragement and support. I thank my friends

( Mukund, Srinath, Rajagopal, Sampath, Abhishek, Rama Krishnan, C.Karchick, Umesh,

Ajeet and Harpreet) and cousins (Sunder, Sivaram, Hari, Jayashree, Anu, Vidu, Nathan,

Nikhil, Viggu and Chinnu) for their support. I thank Cyndy Graham for helping me with

administrative tasks.

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Contents

1 Introduction, Background and Problem Statement 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Automatic Modulation Classification . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Open Problems in AMC . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Blind SISO Channel Equalization and Estimation . . . . . . . . . . . . . . . 5

1.3.1 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 MIMO Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 MIMO Blind Equalization and Channel Estimation . . . . . . . . . . 11

1.4.2 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Overall Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 16

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2 AMC: Preliminaries and Methodologies 17

2.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Cyclostationarity Based AMC . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Background on Cyclostationary Spectral Analysis . . . . . . . . . . . 21

2.2.2 AMC based on Cyclostationarity . . . . . . . . . . . . . . . . . . . . 32

2.3 Cumulants Based AMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.1 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.2 Effect of Multipath Channel . . . . . . . . . . . . . . . . . . . . . . . 49

2.4 Adjusting the Equalizer Length . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Combined Blind Equalizer and Single User AMC 52

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 AMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.1 Cumulants Based AMC . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.2 Cost function for the Cumulants Based AMC . . . . . . . . . . . . . 58

3.4 Minimum Phase Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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3.4.1 Proposed Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.2 Adapting S(z−1), R(z−1) and D(z−1). . . . . . . . . . . . . . . . . . . 63

3.4.3 Adapting B(z−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.4 AMC Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Mixed Phase Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5.2 Computing the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5.3 Cost Function Related to Symbol Detection . . . . . . . . . . . . . . 72

3.5.4 Overall Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5.5 Decision Feedback Equalizer . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.6.1 Experiment 1 (Minimum Phase Channel) . . . . . . . . . . . . . . . . 76

3.6.2 Experiment 2 (Minimum Phase Rayleigh Channel) . . . . . . . . . . 77

3.6.3 Experiment 3 (Minimum Phase Ricean Channel) . . . . . . . . . . . 79

3.6.4 Experiment 4 (Higher Order QAM’s) . . . . . . . . . . . . . . . . . . 79

3.6.5 Experiment 5 (Mixed Phase Rayleigh Channel) . . . . . . . . . . . . 82

3.6.6 Experiment 6 (Mixed Phase Rician Channel) . . . . . . . . . . . . . . 84

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3.6.7 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Multiuser AMC 89

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Channel Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.1 Channel Model and Assumptions . . . . . . . . . . . . . . . . . . . . 92

4.3 Cumulants Based MAMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Blind Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.1 Adaptive Estimation of A(z−1) . . . . . . . . . . . . . . . . . . . . . 100

4.4.2 Estimation of H(z−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.5 Classification Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.6 Extension to Cyclic Cumulants (CC) . . . . . . . . . . . . . . . . . . . . . . 109

4.6.1 Cyclic Cumulants Features . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6.2 CC Based Multiuser AMC . . . . . . . . . . . . . . . . . . . . . . . . 110


4.7.1 Realistic MIMO Channels . . . . . . . . . . . . . . . . . . . . . . . . 111

4.7.2 Fourth Order Cumulants . . . . . . . . . . . . . . . . . . . . . . . . 112

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4.7.3 Realistic MIMO Channel I: Two-user three-class . . . . . . . . . . . . 114

4.7.4 Realistic MIMO Channel II: Two-user three-class . . . . . . . . . . . 114

4.7.5 Fourth Order Cumulants: Classifying QAM’s . . . . . . . . . . . . . 116

4.7.6 Sixth Order CC: MIMO Flat Fading . . . . . . . . . . . . . . . . . . 117

4.7.7 Sixth Order CC: MIMO Multipath Fading I . . . . . . . . . . . . . . 118

4.7.8 Sixth Order CC: MIMO Multipath Fading II . . . . . . . . . . . . . . 118


4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5 Combined MIMO Blind Equalizer and Multiuser AMC 122

5.1 Background and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2 Cost Function for the Multiuser AMC . . . . . . . . . . . . . . . . . . . . . . 127

5.3 Designing the Matrix Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4 Overall Classification and Equalization Algorithm . . . . . . . . . . . . . . . 130


5.5.1 Multiuser AMC Performance . . . . . . . . . . . . . . . . . . . . . . 131

5.5.2 Symbol Detection Performance . . . . . . . . . . . . . . . . . . . . . 136


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5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6 Conclusion and Future Work 139

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7 Publications 143

7.1 Conference Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2 Journal Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

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List of Figures

1.1 Illustration of multipath communication channel . . . . . . . . . . . . . . . . 6

1.2 FIR channel and equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Illustration of possible scenarios for multiantenna CR . . . . . . . . . . . . . 9

1.4 A MIMO system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Illustration of instantaneous mixture channel. . . . . . . . . . . . . . . . . . 12

2.1 Measurement of SCF using band pass filters . . . . . . . . . . . . . . . . . . 27

2.2 Estimating SCF using FFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Spectral Coherence (SC) function for BPSK . . . . . . . . . . . . . . . . . . 30

2.4 Spectral Coherence (SC) function for QPSK . . . . . . . . . . . . . . . . . . 32

2.5 Cyclic Domain Profile (CDP) for BPSK . . . . . . . . . . . . . . . . . . . . 33

2.6 Cyclic Domain Profile (CDP) for QPSK . . . . . . . . . . . . . . . . . . . . 34

2.7 Block diagram of the AMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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2.8 MAXNET Neural Network structure . . . . . . . . . . . . . . . . . . . . . . 36

2.9 Probability of classification Vs SNR . . . . . . . . . . . . . . . . . . . . . . . 37

2.10 Probability of classification Vs Number of symbols (SNR = 5dB) . . . . . . 38

2.11 Signal classification using HMM. . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.12 Percentage of correct classification vs Number of samples. . . . . . . . . . . . 44

2.13 Hierarchical AMC based on cumulants. . . . . . . . . . . . . . . . . . . . . . 47

2.14 Performance of cumulant based AMC under multipath. . . . . . . . . . . . . 50

2.15 Effect of length of the equalizer on the performance of AMC (5 dB noise). . 51

3.1 Block diagram of the proposed system. . . . . . . . . . . . . . . . . . . . . . 55

3.2 Block diagram of the proposed cognitive receiver. . . . . . . . . . . . . . . . 61



3.5 Performance of the AMC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.6 Symbol error rate (SER) vs SNR (BPSK). . . . . . . . . . . . . . . . . . . . 77

3.7 Performance of the AMC (Minimum phase Rayleigh channel). . . . . . . . . 78

3.8 Performance of the AMC (Minimum phase Ricean channel). . . . . . . . . . 80

3.9 Classifying QAM’s (Fourth order cumulants). . . . . . . . . . . . . . . . . . 81

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3.10 Classifying QAM’s (Sixth order cumulants). . . . . . . . . . . . . . . . . . . 81

3.11 Performance of the AMC (Mixed phase Rayleigh channel). . . . . . . . . . . 83

3.12 Performance of the AMC (Mixed phase Rayleigh channel). . . . . . . . . . . 84

3.13 Symbol detection performance of the proposed receiver. . . . . . . . . . . . . 85

3.14 NMSE vs no of iterations (BPSK). . . . . . . . . . . . . . . . . . . . . . . . 85

3.15 Performance of the AMC (Mixed phase Rician channel). . . . . . . . . . . . 86

4.1 Block diagram of the proposed multiuser AMC. . . . . . . . . . . . . . . . . 92

4.2 Performance of the multiuser AMC BPSK,QPSK(T=5000). . . . . . . 113

4.3 Performance under realistic MIMO channel I(Two-user three-class). . . . . . 115

4.4 Performance under realistic MIMO channel II(Two-user three-class). . . . . 116

4.5 Classification of QAM’s (Two-user three-class problem). . . . . . . . . . . . 117

4.6 Performance of the multiuser AMC(Sixth order CC: MIMO flat fading). . . 118

4.7 Performance of the multiuser AMC (MIMO multipath fading I). . . . . . . . 119

4.8 Performance of the multiuser AMC (MIMO multipath fading II). . . . . . . 120


5.2 Performance of the multiuser AMC (Two-user three-class problem). . . . . 133

5.3 Performance of the multiuser AMC (Two-user three-class problem). . . . . 134

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5.4 Performance of the MAMC (Four-user five-class problem) . . . . . . . . . . 135

5.5 Performance of the MAMC (Realistic multipath channel II). . . . . . . . . . 136

5.6 Symbol detection performance of the proposed system (NMSE Vs SNR). . . 137

5.7 Symbol detection performance of the proposed system (SER Vs SNR). . . . 137

6.1 Block diagram of a multiantenna cognitive transceiver. . . . . . . . . . . . . 142

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List of Tables

2.1 Probability of classification of AMC in the presence of AWGN (SNR = 5dB) 37

2.2 Probability of Classification of CDP Based AMC in the Presence of FIR Chan-

nel (SNR = 5dB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Theoretical Cumulant Values for Some of the Modulation Schemes . . . . . 46

2.4 Confusion Matrix for Cumulant Based AMC in the Presence of AWGN (SNR

= 10dB), N=100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


= 10dB), N=100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


= 10dB), N=500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Theoretical normalized cumulant values . . . . . . . . . . . . . . . . . . . . . 58

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Chapter 1

Introduction, Background and

Problem Statement

1.1 Introduction

Cognitive Radio (CR), originally introduced by Mitola [1], has become a key research area in

communications since the Federal Communications Commission (FCC) published a report

in Nov. 2002 aiming for better utilization of the frequency spectrum in the US [6]. CR is a

promising technology that is capable of achieving better spectrum utilization by opportunis-

tically finding and utilizing unoccupied frequency bands [1]. The important characteristics

of CR are its ability to sense the environment, make decisions based on the observations and

the mission objectives, and learn from past experiences for future decision making.

1

2

The Cognitive Radio Network (CRN) is a network of CR nodes with a cognitive process

that can observe current network conditions, plan, decide, and then act according to those

conditions. The network can learn from these adaptations and use them to make future

decisions while taking into account end-to-end goals [3]. CRN must have the capability

to optimize available resources (e.g. power, bandwidth, etc.) and to adapt each layer of

the protocol stack, including the physical layer, according to the environment. Several

potential applications of CRN are in a) military and public safety where there are needs

for interoperability amongst various standards and guaranteed Quality of Service (QoS)

for secure, reliable, and robust communications, and b) commercial applications where QoS

includes availability of service, plus reliable and fast data transfer [4]. In addition, for military

and public safety applications, the CRs must be capable of performing fixed and on-the-

move communications between highly diverse elements in a very harsh environment, which

is susceptible to jamming attacks and malicious interference [5]. In military applications,

there is no information about the enemy signal and hence the CR receiver needs to identify

the modulation format employed in the signal. Automatic modulation classification (AMC)

is a signal processing component that can identify the modulation format employed in the

received signal. In a typical wireless communication environment, the transmitted signals are

subjected to noise and multipath fading. The multipath channel affects the performance of

receiver symbol detection by causing ISI. The equalizer is a signal processing component that

removes ISI from the received signal and thus improves symbol detection. In a CR scenario,

training or pilot sequences are not available and hence blind equalizers are used to recover the

3

transmitted sequence. Blind equalizers are used to recover the transmitted sequence using

only the received signal with no knowledge of the channel and transmitting sequence. AMC,

a blind channel equalizer, and a blind channel estimator are some of the important signal

processing components a CR must possess in order to realize the previously mentioned QoS.

In this dissertation, some of the open problems in the above mentioned signal processing

components are addressed.

This chapter is organized as follows. In Section 1.2, a brief literature review on AMC

algorithms is provided. Open problems in AMC are also discussed in this section. Section

1.3 reviews the existing literature and open problems in SISO blind channel estimation and

equalization algorithms. Section 1.4 provides a overview of a MIMO communication system

from a CR point of view. Open problems in blind MIMO channel estimation and equalization

are also discussed. Section 1.5 summarizes the overall problem statement of this dissertation.

Finally, the organization of the chapters in this dissertation is provided.

1.2 Automatic Modulation Classification

AMC, as the name suggests, is the automatic recognition of modulated signals present in a

particular frequency band. AMC or a signal classifier is an important component of the CR

to support interoperability amongst various modulation types and standards. AMC has been

an important topic for electronic surveillance over the past two decades [21], especially in

military applications. AMC can play an important role in the security of CR by identifying

4

malicious users. According to [21], there are two categories of AMC: likelihood based and

feature based. Feature based AMCs are widely used because of their easy implementation

and better performance. The feature based AMC consists of two parts: a signal processing

part to extract features from signals and a classifier part to distinguish features. Some of the

widely used features are higher order statistics ([7]-[13]), cyclostationary features ([14]-[20]),

wavelet features ([22],[23]), and signal constellation [24]. For the classifier, Neural Network

(NN), Support Vector Machine (SVM), Hidden Markov Models (HMM), and Clustering

algorithms are commonly used. Due to the popularity of orthogonal frequency division

multiplexing (OFDM), there has been a lot of research in the direction of distinguishing

OFDM signals from single carrier modulated signals. Apart from distinguishing OFDM

from single carrier schemes, they also identify parameters of OFDM such as length of the

cyclic prefix, number of sub carriers, and FFT size [44],[45].

1.2.1 Open Problems in AMC

Research in AMC assumes either SISO or SIMO channels, that is, they assume only a single

transmitting user. However, in a CR scenario, this is not the case. In some applications,

CR must be able to classify signals transmitted by legal users and malicious users at the

same time. Therefore an AMC that can classify signals from multiple users simultaneously

is needed for CR. Thus, one of the objectives of the dissertation is to develop AMC for

a multiuser system. Another open problem is that most of the AMC algorithms in the

literature assume the channel to be Additive White Gaussian Noise (AWGN) and do not

5

consider multipath. Multipath not only affects the performance of receiver symbol detection

but also affects the performance of the AMC. The second objective of this dissertation is to

develop AMC that is robust to multipath channels.

1.3 Blind SISO Channel Equalization and Estimation

A CR uses blind equalizers due to the absence of training or pilot sequences. In a wireless

communication system, the transmitted signal is subjected to noise and multipath effects

which cause distortion and ISI. The equalizer is a signal processing component that is used

to nullify the multipath effects and remove ISI. A typical wireless communication system

with the equalizer is shown in Figure 1.1. The channel and equalizer can be modeled as

a FIR filter and is shown in Figure 1.2. In Figure 1.2, s(n) is the transmitted sequence,

x(n) is the received sequence, y(n) is the recovered sequence, z−1 is the delay operator, ci

(for i = 1 . . . N) are the complex gains of each multipath, and wi (for i = 1 . . . N) are the

weights of the equalizer. Typically, for a non blind equalizer, the weights are adjusted using

a training sequence. Blind equalization is a process by which a transmitted input sequence

is recovered using only the received signal without any knowledge of the training sequence

and channel impulse response. That is, the weights are adjusted without using any training

sequence or channel knowledge. The first SISO blind equalization algorithm was proposed

in [50] and is known as the Sato algorithm. The Sato algorithm was heuristic and lacked

analytical understanding [51]. The Sato algorithm was generalized in [51] and is known as the

6

BGR algorithm. A different generalization of the Sato algorithm was provided by Godard in

[52]. One specific form of Godard’s method is the well-known Constant Modulus Algorithm

(CMA). The CMA algorithm and its variants have been extensively studied in [53],[54].

Other SISO blind equalization algorithms include the stop-and-go algorithm proposed in [55]

and the Bussgang algorithm proposed in [56]. All the above algorithms adapt the equalizer

parameters by minimizing a cost function that is a function of higher order statistics (HOS)

of the received signal.

Figure 1.1: Illustration of multipath communication channel

Blind channel estimation is another problem which is similar to the problem of blind equal-

ization. In blind channel estimation, the channel impulse response is estimated only using

the received signal. These channel estimates are then used to estimate the transmitted se-

quence by using a maximum likelihood (ML) algorithm or differential feed back equalizer

(DFE). SISO blind channel estimation also requires HOS of the received signal. A detailed

survey of SISO blind channel estimation algorithms can be found in [57],[58].

7

Z‐N

Z‐1

Z‐1

Z‐N

Σ

Σ

c1

c2 cN

S(n)

X(n)

Y(n)

w1

w2 wN

Noise

Channel

Equalizer

Figure 1.2: FIR channel and equalizer

1.3.1 Open Problems

As mentioned earlier, adaptive blind equalization typically adapts the equalizer parameter

by minimizing some special cost functions. For non blind equalization due to the availability

of a training sequence, the most widely used cost function is the mean square error (MSE).

Because of the lack of a training sequence, blind equalization algorithms use cost functions

that implicitly utilize the HOS of the received signal. These cost functions are generally non

linear and have many local minima. The convergence of these algorithms highly depends

on the initial setting of the equalizer. Since the cost function is non-MSE, good symbol

detection performance is not always guaranteed. Due to the convergence of the algorithm to

a local minimum, not only symbol detection performance is affected, but the performance

of the AMC, which is an integral part of the CR, is also affected. Robust blind equalizers

can be designed if the performance of the AMC is also considered while adapting equalizer

parameters.

8

One of the open problems is to design a robust blind equalizer that enhances both the

performance of the AMC and symbol detection. This can be achieved by formulating a cost

function that also incorporates the performance of the AMC. This cost function will differ

for different kinds of feature based AMC’s. The parameters of the blind equalizer are then

adapted so that this new cost function is minimized. Thus some of the main objectives of

the dissertation with respect to SISO blind equalization are to:

• Design new blind equalizer architectures that can improve the performance of both

symbol detection and AMC.

• Formulate cost functions that are related to the performance of some of the widely

used feature based AMC’s.

• Develop algorithms that adapt the parameters of the new equalizer such that the cost

function is maximized.

1.4 MIMO Communication

With the decreasing cost of RF components and advancing RF technologies, the use of mul-

tiple antennas for both transmission and reception has gained a lot of attention. It will be

shown later that multiple antennas are used at the receiver for classifying signals from multi-

ple users. The use of multiple antennas at both the transmitter and receiver is referred to as

MIMO communications [62]. Different ways by which a multiantenna CR can communicate

9

with other radios in the network are illustrated in Figure 1.3. MIMO communications offer

increased system reliability, higher data rates, and an increased coverage area [63]. MIMO

communication techniques can be broadly classified into three categories: Harnessing spa-

tial diversity for reliable communications, beamforming for direction location and focusing

the power in a particular direction for increasing the range, and spatial multiplexing for

increasing the data rate. The multiantenna CR can use any one of these techniques or a

combination of a few techniques for communicating with other radios. By using multiple

MIMOCR 1

MIMOCR 2

Single userSpatial Multiplexing

Tx Rx

Multiuser SpatialMultiplexing

Multiuser TransmitBeamforming

TxRx

Transmit diversityschemes

Tx

Receiver diversityschemes

Rx

Malicioususer

Counter jammingusing Beamforming

ReceiverBeamforming

Rx Tx

Figure 1.3: Illustration of possible scenarios for multiantenna CR

antennas at the receiver, the CR can harness the flexibility and advantages offered by clas-

sical MIMO schemes apart from classifying signals from multiple users. A CR employing

MIMO communication techniques can effectively optimize resources and achieve a high data

rate. Even though MIMO is an attractive option, there are several shortcomings in applying

MIMO concepts to CRs. One of the important shortcomings of applying classical MIMO

theory to CRs is the channel model [62]. Classical MIMO theory is based on the following

10

channel model (Figure 1.4):

y(i) = Hs(i) + w(i) i = 0, 1, 2, . . . (1.1)

where y(i) is a (m × 1) received signal, s(i) is a (l × 1) transmitted signal, w(i) is white

Gaussian noise, and H is a (m × l) matrix whose entries are scalar random values. Since

y1(t) =h11s1(t)+ h12s2(t)+…+h1LsL(t)+w(t)

yM(t) =hM1s1(t)+ hM2s2(t)+…+hMLsL(t)+w(t)

s1(t)

sL(t)

Figure 1.4: A MIMO system.

H is a matrix of scalar random variables, classical MIMO theory assumes multipath to be

negligible, that is, there is no frequency selective fading [62]. The scalar channel in (1.1) is

also known as an instantaneous mixture channel. This assumption is not only inaccurate

for CRN but even for cellular MIMO deployments. However, in WiMAX and other cellular

MIMO deployments, the OFDM modulation scheme is used. OFDM converts a frequency

selective channel to a flat fading channel and hence the assumption in (1.1) holds. Also, in a

cellular MIMO system, the channel matrix H is estimated using known pilot sequences [69].

One of the important characteristics of CR is interoperability, that is, CR devices must be

able to communicate with a wide range of other radio devices which use different modulation

11

schemes other than OFDM and hence the model in equation (1.1) may not hold. The more

appropriate channel model for the CR is

y(i) = H(z−1)s(i) + w(i), i = 0, 1, 2, . . . (1.2)

where y(i), s(i), and w(i) are the same as in (1.1) and H(z−1) is the transfer function

operator given by

H(z−1) =

nA∑k=0

Hkz−k

with Hk , k ≥ 0 is an m× l matrix sequence called the system impulse response, and z−1

is the unit delay operator. Note that classical MIMO theory cannot be applied to the model

given by (1.2). The solution to this is to use MIMO blind equalization and channel estimation

techniques to compensate for the multipath. The reason for using blind equalization is that

the pilot signals are not usually available in a CR environment. The MIMO blind equalizer

converts a multipath channel into an instantaneous mixture channel model (similar to (1.1))

which is illustrated in Figure 1.5. Classical MIMO techniques can now be applied to this

instantaneous mixture channel H0.

1.4.1 MIMO Blind Equalization and Channel Estimation

In multiuser communications, a source signal undergoes a convolutive distortion between

its symbols and the channel impulse response and a mixture distortion from other source

signals. These distortions are referred to as an intersymbol interference (ISI) and interuser

interference (IUI), respectively. The MIMO channel in (1.2) effectively models the IUI and

12

H

H0

H(Z 1)Proposed

MIMO blindequalizer

S(i) X(i)

S(i)X(i)

X(i)S(i)

Channel model for classical MIMO theory

Instantaneous mixture model

Figure 1.5: Illustration of instantaneous mixture channel.

ISI. The purpose of the MIMO blind equalizer is to remove ISI and IUI without the knowledge

of the channel impulse response and use of a training sequence. Normally the task of blind

equalization involves estimation of the channel impulse response. Using only the second

order statistics (SOS) of the received signal, the convolutive channel given by (1.2) can

be converted to a instantaneous mixture channel given by (1.1). MIMO equalization and

channel estimation algorithms using second order statistics (SOS) can be broadly classified

into three categories: the whitening approach, linear prediction approach, and subspace

approach. In the whitening approach, the coefficients of the inverse filter are estimated using

the correlation of the received signal, which is further used to calculate the channel impulse

response. A minimum mean square error (MMSE) equalizer is then designed to estimate

the instantaneous mixture of the transmitted symbol sequence. In the linear prediction

approach, the channel is assumed to be an auto regressive (AR) process and therefore the

coefficients of the predictor filter are estimated using the correlation of the received signal.

The channel impulse response is then calculated using the predictor coefficients, which is

13

then used to design the MMSE equalizer. The subspace approach usually involves fractional

sampling of the received signal and requires knowledge about the order of the channel. All

of the above approaches involve block processing of data and hence cannot efficiently track

time varying channels.

1.4.2 Open Problems

These batch processing algorithms are not suitable for CR, because CR must have the ca-

pability to track time varying channels and adjust the transmission and reception of data

accordingly. Therefore a computationally efficient MIMO blind equalizer and channel es-

timator that can track changes in the channel for every sample of data is needed. The

MIMO Constant Modulus Algorithm (CMA) is one such equalizer which updates for every

sample of data, but it works only for a certain class of signals [89]. The MIMO multipath

channel shown in (1.2) not only affects the performance of MIMO symbol detection but also

affects the performance of mutliuser AMC. Since multiuser AMC is an integral part of a

multiantenna CR receiver, a robust MIMO blind equalizer can be built if the performance

of the multiuser AMC is also considered while adapting the parameters of the MIMO blind

equalizer. Specifically one of the open problems is to develop a MIMO blind equalizer that

improves the performance of both symbol detection and multiuser AMC. Thus, some of the

main objectives of the dissertation with respect to MIMO blind equalization are to:

• Develop a MIMO blind equalizer architecture that can improve the performance of

14

both multiuser AMC and symbol detection

• Formulate a cost function that is related to the performance of the proposed multiuser

AMC

• Adapt the parameters of the MIMO blind equalizer such that the formulated cost

function is minimized

• The MIMO blind equalization and channel estimation algorithm must be adaptive,

that is, it should have the ability to track time varying channels

1.5 Overall Problem Statement

The objective of this dissertation is to develop a transceiver for Cognitive Radio (CR) for

secure, reliable, and robust communications which will benefit both commercial and military

applications. The proposed transceiver will have the following special characteristics apart

from the usual radio characteristics:

• Ability to track time varying SISO and MIMO channels

• Ability to classify multiple users in the frequency band

• Ability to classify signals under severe multipath channels

The following tasks needs to be accomplished in order to achieve the above objectives:

15

• Develop a multiuser Automatic Modulation Classification (AMC) which can classify

signals from multiple users.

• The multiuser AMC needs to be developed by exploiting different features of the re-

ceived signal. Some of the features that will be considered are fourth order cumulant,

fourth order cyclic cumulant, and higher order cyclic cumulants.

• Develop SISO blind equalizer architectures that can improve the performance of both

symbol detection and AMC. Also, the SISO blind equalizer should track time varying

channels.

• Formulate cost functions that are related to the performance of some of the widely

used feature based single user AMC’s.

• Develop algorithms that adapt the parameters of the new SISO blind equalizer such

that the cost function is minimized.

• Formulate cost functions for the newly developed multiuser AMC.

• Develop an adaptive MIMO blind equalizer and channel estimators that can track time

varying channels. The blind equalizer needs to be designed in such a way that both

the symbol detection performance and multiuser AMC performance are improved.

In this dissertation we address the above mentioned tasks.

16

1.6 Organization of the Dissertation

This dissertation is organized as follows. In Chapter 2, we discuss two feature based single

user AMCs. Performance degradation of these AMCs when subjected to a multipath channel

is illustrated. In Chapter 3, SISO blind equalization algorithms that improve the performance

of both single user AMC and symbol detection are presented. In Chapter 4, we present the

multiuser AMC based on cumulants and cyclic cumulants. In Chapter 5, we present the

MIMO blind equalizer that improves the performance of both multiuser AMC and multiuser

symbol detection.

Chapter 2

AMC: Preliminaries and

Methodologies

Reprinted, with permission from, B.Ramkumar, Automatic modulation classification for

cognitive radios using cyclic feature detection, IEEE circuits and systems, June 2009.

Automatic Modulation Classification (AMC) is the automatic recognition of the modulation

format of a sensed signal. For an intelligent receiver, AMC is the intermediate step between

signal detection and demodulation [21]. AMC plays an important role in civilian and military

applications, especially in dynamic spectrum management and interference identification.

It has also been an important topic for electronic surveillance for over two decades [21],

primarily in military applications. With the growing popularity of software defined radios

and cognitive radios, AMC is becoming an important technology for commercial applications.

17

18

AMC is often a difficult task when there is no a priori information about the signal, including,

signal power, carrier frequency and timing parameters.

In this chapter we provide the basic preliminaries and methodologies for automatic mod-

ulation classification. We begin with a short survey of the broad classes of modulation

classification algorithms. The main focus of this chapter is on feature based AMC’s. We

illustrate in detail two specific feature based AMC’s: cyclostationarity based and cumulants

based AMC. The cyclostationarity based AMC is a good example of how feature extract-

ing algorithms can be used with classifiers such as Neural Networks (NN), Hidden Markov

Models (HMM), Support Vector Machines (SVM), etc. The effect of multipath channel on

these feature based AMC’s is also illustrated in this chapter.

This chapter is organized as follows. In Section 2.1 we provide a brief survey of AMC

algorithms in literature. In Section 2.2 we present the cyclostationarity based AMC. Clas-

sification algorithms such as NN and HMM are also briefly explained in this section. In

Section 2.2 fourth order cumulant based AMC is presented. The effect of multipath on the

performance of this AMC is also presented. One of the important parameters of the blind

equalizer is the filter length. The dependence of AMC performance on this parameter is

illustrated using simulations in Section 2.4.

19

2.1 Literature Survey

Automatic modulation classification research goes back at least two decades. A large number

of modulation classification methods have been developed. According to [21],they have been

traditionally grouped into two broad categories, likelihood-based and feature-based methods.

The second category is much more frequently represented.

One of the classic modulation classification approaches and its first broad category is the

maximum likelihood technique where the classification is treated as a multiple-hypothesis

testing problem [29]-[31]. The probability density function (PDF) of the observed waveform,

conditioned on the embedded modulated signal, contains the information required for clas-

sification. Depending on the model chosen for the unknown quantities like amplitude and

phase, three variations of the likelihood method are possible: average likelihood ratio test

(ALRT), generalized likelihood ratio test (GLRT) and hybrid likelihood ratio test (HLRT).

Feature based methods form the larger group of modulation classification algorithms [21].

These groups of algorithms uses signal features such as signal statistics [32]-[33], higher order

signal statistics (moments, cumulants, kurtosis) [7]-[13], Wavelet Transform (WT) [22]-[23],

spectral features [34], signal constellations [35], zero-crossings [36], multi-fractals [37] and

the Radon transform [38] to distinguish amongst the various modulation types and constel-

lations.

Some modulation classification algorithms are based on the principle of signal cyclostation-

arity [14]-[20]. This technique also falls under the category of feature-based methods. This

20

type of algorithm can be applied to linear modulation classification and to low SNR signals

[14]. Many signals can be modeled as cyclostationary rather than wide-sense stationary, due

to their underlying periodicities. For such processes, both their mean and autocorrelation are

periodic. A spectral correlation function (SCF) can be obtained from the Fourier transform

of the cyclic autocorrelation. A maximum value of normalized SCF over all cycle frequencies

gives the cycle frequency domain profile (CDP). Several modulation schemes have unique

CDP patterns, which can be used as a discriminator in the classification process. By uti-

lizing higher order cyclic cumulants a wide variety of modulated signals can be classified

[8]. However, one of the disadvantages of this method is the large amount of data required

to estimate these statistics. Some of the new trends in modulation classification based on

the emerging wireless technologies include multi antenna inputs and adaptive Orthogonal

Frequency Division Multiplexing (OFDM) [44], [45]. From the previous discussion it can be

seen that there exist numerous algorithms for AMC. The problem is that no single algorithm

can effectively classify all modulation types. Choosing a particular AMC greatly depends

on the scenario at hand.

2.2 Cyclostationarity Based AMC

Most modulated signals exhibit the property of cyclostationarity that can be exploited for

the purpose of classification. In this section, AMC that is based on exploiting the cy-

clostationarity property of the modulated signals is discussed. As mentioned earlier, the

21

cyclostationarity based AMC is a good example of how feature extracting algorithms can be

used with classifiers.

2.2.1 Background on Cyclostationary Spectral Analysis

Many man made signals encountered in practice have parameters that vary periodically

with time [42], [43]. Examples include radar signals and periodic keying of amplitude,

phase or frequency in digital communication systems. In conventional signal receivers, these

periodicities are usually not explored for extracting information or extracting parameters.

Performance of signal processing can be improved in many cases by considering these hidden

periodicities. This requires the underlying random signal to be modeled as cyclostationary.

In this section a systematic tutorial on cyclostationarity based signal processing is presented.

Hidden periodicity and quadratic time invariant transformation (QTI)

Consider a signal x(t), which is a finite strength additive sinusoidal wave with frequency α

and phase θ given by [41]

x(t) = a cos(2παt+ θ). (2.1)

The Fourier coefficient is defined as

Mαx =

⟨x(t)ej2πt

⟩(2.2)

22

where

〈.〉 = limT→∞

1

T

∫ −T/2T/2

(.) dt.

The Fourier coefficient of (2.1) is given by

Mαx =

1

2aejθ. (2.3)

The Power spectral density (PSD) of (2.1) has a spectral line at f = −α and at f = α and

is given by

PSD = |Mαx |

2 [δ(f − α) + δ(f + α)]] , (2.4)

where δ(.) is the impulse function. It is said that such a signal exhibits first order periodicity.

In other words, a signal whose PSD has spectral lines is said to exhibit first order periodicity.

Now consider the signal

x(t) = cos(2παt+ θ) + n(t), (2.5)

where n(t) is a random signal. If the sine wave is weak compared to the random signal, the

periodicity may not be observable, hence it is called hidden periodicity. However, the PSD

of the signal (2.5) shows a spectral line, by which the hidden periodicity can be detected.

There are signals which have hidden periodicity that do not give rise to spectral lines in the

PSD, but can be converted into a first order periodic signal by a nonlinear time-invariant

transformation. The hidden periodicity that can be converted to first order periodicity by

quadratic transformation of the signal is called second order periodicity.

23

A transformation of x(t) to y(t) is called QTI if and only if there exists a kernel k(., .) such

that y(t) can be expressed as [41]

y(t) =

∫ ∞−∞

∫ ∞−∞

k(t− u, t− v)x(u)x(v)dudv (2.6)

or

y(t) =

∫ ∞−∞

∫ ∞−∞

k(u, v)x(t− u)x(t− v)dudv.

A QTI is stable if and only if

∫ ∞−∞

∫ ∞−∞

k(u, v)dudv <∞.

Definition: A time series x(t) contains second-order periodicity with frequency α if and

only if there exists a stable QTI transformation of x(t) to y(t) such that y(t) consist of

first-order periodicity with frequency α, that is, y(t) exhibits spectral lines at f = ±α.

Cyclic Autocorrelation function

By substituting (2.6) into (2.2) it can be shown that x(t) contains second order periodicity

with frequency α 6= 0 if and only if [42], [43]

Rαx = lim

T→∞

1

T

∫ T/2

−T/2x(t+

τ

2)x(t− τ

2)e−i2παtdt (2.7)

exists and is not identically zero as a function of τ . Rαx in (2.7) is known as the limit cyclic

autocorrelation (also called cyclic autocorrelation). When α = 0, it can be seen from (2.7)

that Rαx turns out to be the conventional limit autocorrelation Rx.

24

Probabilistic interpretation

A probabilistic phenomenon with second order periodicity can be modeled as a cyclosta-

tionary stochastic process. A process x(t) is said to be cyclostationary in the wide sense

if its mean and auto correlation function are periodic with period T0. The probabilistic

autocorrelation function defined as

Rx(t, τ) = Ex(t+

τ

2)x(t− τ

2)

(2.8)

must be periodic in the variable t i.e.

Rx(t+ T0, τ) = Rx(t, τ). (2.9)

Since the autocorrelation function is periodic it can be expressed as a Fourier series [41]

Rx(t, τ) =∑α

Rαx(τ)ei2παt, (2.10)

where α = m/T0 and m is an integer. The Fourier coefficient can be obtained by

Rαx(τ) = lim

T→∞

1

T

∫ T/2

−T/2Rx(t, τ)ei2παtdt. (2.11)

Rαx(τ) is known as the probabilistic cyclic autocorrelation function. If the empirical cyclic au-

tocorrelation function Rαx(τ), (2.7) , and probabilistic cyclic autocorrelation function Rα

x(τ),

(2.11) , are equal, then the process is said to be cycloergodic.

Cross covariance correlation coefficient

Another interpretation of cyclic autocorrelation is obtained by factoring ei2παt in (2.11) as

Rαx(τ) =

⟨[x(t+ τ/2)e−i2πα(t+τ/2)

] [x(t− τ/2)ei2πα(t−τ/2)

]⟩. (2.12)

25

Rαx(τ) can now be written as conventional cross correlation function as

Rαx(τ) = 〈[u(t+ τ/2)] [v∗(t− τ/2)]〉 , (2.13)

where u(t) = x(t)e−iπαt and v(t) = x(t)e+iπαt. This interpretation of Rαx(τ) gives an appro-

priate normalization for Rαx(τ) as explained below.

If x(t) does not have any finite-strength frequency component at f = ±α/2, the mean values

of u(t) and v(t) are zero. Under the above assumption, Rαx(τ) = Ruv(τ) is actually a temporal

cross covariance [42], [43] Kuv(τ) . That is,

Kuv(τ) = 〈[u(t+ τ/2)− 〈u(t+ τ/2)〉] [v(t− τ/2)− 〈v(t− τ/2)〉]〉 (2.14)

= 〈[u(t+ τ/2)] [v∗(t− τ/2)]〉 = Ruv(τ).

An appropriate normalization for temporal cross covariance is the geometric mean of the

two corresponding variances. Therefore, the temporal cross covariance correlation coefficient

can be defined as [42]

Kuv(τ)

[Ku(0)Kv(0)]1/2=Rαx(τ)

Rx(0)= γαx (τ). (2.15)

Spectral Correlation Density or Spectral Correlation Function (SCF or SCD)

Function

From the Wiener-Khintchine theorem we know that PSD (Sx(f)) is equal to the Fourier

transform of the autocorrelation function

Sx(f) =

∫ ∞−∞

Rx(τ)e−i2πfτdτ. (2.16)

26

Similarly, the SCD is the Fourier transform of the cyclic autocorrelation function [40] and is

given by

Sαx (f) =

∫ ∞−∞

Rαx(τ)e−i2πfτdτ. (2.17)

Equation (2.17) is known as cyclic Wiener relation. The conventional Wiener-Khintchine

relation (2.16) is a special case of (2.17) when α = 0. In this section, we will discuss how to

estimate SCD from a time series.

Method 1 In order to estimate the power in a frequency band, we simply pass the signal

x(t) into a narrow band pass filter and measure the average power of the output. By passing

the signal into a series of contiguous narrow disjoint band pass filters, and measuring the

average power, we can estimate the signal’s PSD. That is, at any particular frequency f , the

PSD of x(t) is given by [39].

Sx(f) = limB→0

1

B

⟨∣∣∣hfB(t)⊗ x(t)∣∣∣2⟩ , (2.18)

where hfB(t) is the impulse response of an ideal band pass filter with center frequency f and

bandwidth B. For estimating the SCD, we pass the frequency translated signals u(t) and

v(t) (refer to (2.13)) through same set of bandpass filters and then measure the temporal

correlation of the filtered signals. The block diagram of this method is shown in Figure 2.1.

The estimated SCD is given by the equation [39]

Sx(f) = limB→0

1

B

⟨∣∣∣hfB(t)⊗ u(t)∣∣∣ ∣∣∣hfB(t)⊗ v(t)

∣∣∣∗⟩ . (2.19)

27

)( fSx

)(tu

)(tv

)(tx

tje

2

tje

2

BPF

BPF

(.)T

Figure 2.1: Measurement of SCF using band pass filters

Method 2 Using the third interpretation of cyclic auto correlation (refer to (2.14)), one can

show that [39]

Sαx (f) = lim∆f→∞

lim∆t→∞

1

∆t

∫ ∆t/2

−∆t/2

∆fX1/∆f (t, f + α/2)X∗1/∆f (t, f − α/2)dt, (2.20)

where X1/∆f (t, v) is called the short time Fourier transform of the signal x(t) given by

X1/∆f (t, v) =

∫ t+1/∆f

t−1/∆f

x(u)e−j2πvudu. (2.21)

Equation (2.20) is the correlation of two temporally smoothed spectral components at fre-

quencies f − α/2 and f + α/2. Another way of expressing (2.20) is

Sαx (f) = lim∆f→∞

lim∆t→∞

1

∆f

∫ f+∆f/2

f−∆f/2

1

∆tX∆t(t, f + α/2)X∗∆t(t, f − α/2)df, (2.22)

where X∆t(t, v) is defined by (2.21) by replacing 1/∆f with ∆t .

For a real time signal it is difficult to evaluate (2.20) and (2.21). So we use cyclic periodogram

defined as

SαxT (f) =1

TXT (t, f + α/2)X∗T (t, f − α/2), (2.23)

28

where XT (t, v) is defined by (2.21) by replacing 1/∆f with T . The cyclic periodogram is

the Fourier transform of the cyclic correlogram defined as

RαxT (t, τ) =

1

T

∫ t−(T+|τ |/2)

t+(T−|τ |/2)

x(u+ τ/2)x(u− τ/2)e−j2πvudu. (2.24)

Additionally, the spectrally smoothed cyclic periodogram is defined by

Sαx∆t(t, f)∆f =1

∆f

∫ f+∆f/2

f−∆f/2

Sαx∆t(t, f)dv. (2.25)

It is shown in [39] that SCD can be estimated by increasing the observation length ∆t and

reducing the size of the smoothing window ∆f ,

Sαx (f) = lim∆f→0

lim∆t→∞

Sαx∆t(t, f)∆f . (2.26)

Spectral Coherence function

The SCD is a cross correlation between two frequency components separated by f−α/2 and

f +α/2. If x(t) contains no spectral components at f = ±α/2, then the SCF is actually the

covariance of the two spectral components. Therefore, an appropriate normalization is the

geometric mean of the corresponding variances given by

Su(f) = Sx(f + α/2) and Sv(f) = Sx(f − α/2).

The Spectral coherence (SC) function is defined as

Cαx (f) =

Sαx (f)

[Su(f)Sv(f)]1/2=

Sαx (f)

[Su(f + α/2)Sv(f − α/2)]1/2. (2.27)

The magnitude of the SC lies between 0 and 1.

29

Discrete implementation of SCF

Equation (2.26) can be implemented efficiently in the discrete domain with the use of FFT.

Discrete-frequency smoothening method is widely used and is given by

Sαx∆t(t, f)∆f =1

M

v=(M−1)/2∑v=−(M−1)/2

1

∆tX∆t(t, f + α/2 + vFs)X

∗∆t(t, f − α/2 + vFs), (2.28)

where

X∆t(t, f) =N−1∑k=0

a∆t(kTs)x(t− kTs)e−j2πf(t−kTs). (2.29)

In (2.29) X∆t(t, f) is the sliding DFT, a∆t is the data tapering window, ∆f = Mfs is the

width of the spectral smoothening interval, Fs = 1/NTs is the sampling frequency, and

N is the number of samples in the data segment of length ∆t . The block diagram of

implementation is shown in Figure 2.2.

)2

( fX

)2

( fX

Calculatethe N-point

FFT

Correlationand

SmootheningSCFx(t) X(f) and

Shift to obtain

Figure 2.2: Estimating SCF using FFT.

Examples of discrete SC: SC is computed for BPSK and QPSK modulation schemes. The

number of samples for the FFT was T = 500. For generating the plots a smoothening method

30

proposed in [18] was used. The formula used is

SαxT (f) =1

N

k=N∑k=1

SαxT (tk, f). (2.30)

For example if N=100 then the total number of samples is 100 × T . This method helps

to reduce the number of required samples in FFT. If N is increased, the erratic behavior in

SC is reduced and hence cyclic features can be distinguished. A square root raised cosine

pulse was used for generating this plot. Figure 2.3 and Figure 2.4 show the SC functions for

BPSK and QPSK, respectively. The MATLAB pseudocode for the estimation of SCF and

SC is given below.

−0.5

0

0.5

00.2

0.40.6

0.81

0

0.05

0.1

0.15

0.2

α/fs

f/fs

SC

F

Figure 2.3: Spectral Coherence (SC) function for BPSK

31

MATLAB pseudocode for estimating SCF and SC

Step 1 Divide the incoming modulated signal into N frames. If the total signal has ∆t

samples, then each frame has T = ∆tN

samples.

Step 2 Take the Fourier transform of each frame using FFT function in MATLAB.

Step 3 Shift the FFT of each frame by +α2

and −α2

and multiply them i.e.,

SαxT (f)∆t∆f = 1TXT (f + α

2)X∗T (f − α

2).

Step 4 Take the average value of all the N frames to obtain SαxT (f)∆f .

Step 5 Perform frequency smoothening by passing SαxT (f)∆f into a moving average filter to

obtain SαxT (f).

Step 6 Repeat the operation from step 2 for each value of alpha to obtain SCF.

Step 7 Normalize the SCF according to equation (2.27) to obtain SC.

32

−0.5

0

0.5

00.2

0.40.6

0.81

0

0.05

0.1

0.15

0.2

α/fs

f/fs

SC

F

Figure 2.4: Spectral Coherence (SC) function for QPSK

2.2.2 AMC based on Cyclostationarity

Cyclostationarity-based AMC explores the sensed signal’s SC for modulation signal classifi-

cation. Using SC requires large amounts of data and hence one of the solution is to use only

the highest values in the SC. These highest values in SC are called Cyclic Domain Profile

(CDP) or α− profile which is defined as [18]

I(α) = maxf |Cαx (f)| . (2.31)

The CDP for BPSK and QPSK signals used for generating the SC function (Figure 2.3 and

Figure 2.4) are shown Figure 2.5 and Figure 2.6. From Figure 2.5, it can be seen that the

CDP for BPSK has three distinct peaks. The peak in the center corresponds to the carrier

frequency (Fc) and the remaining peaks are related to the symbol rate (Fsym and Fc +Fsym)

of the transmitted sequence. From Figure 2.6, it can seen that the CDP for QPSK has

33

only one distinct peak that corresponds to the symbol rate (Fsym). The reason for this is

that QPSK is a balanced modulation scheme i.e., it has balanced inphase and quadrature

components. The block diagram of the AMC is shown Figure 2.7. For pattern matching,

Neural Networks and Hidden Markov model are employed in [18] and [14], respectively.

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

α/fs

CD

P

Figure 2.5: Cyclic Domain Profile (CDP) for BPSK

Neural Network based AMC

Neural Networks trained using the Cyclic Domain Profiles (CDP) are used for signal classifi-

cation due to its pattern matching capabilities. Neural Networks (NN) have been motivated

by the recognition that the brain computes in a different manner from the conventional dig-

ital computer [28]. The brain is made up of basic constituents called neurons. The basic

definition of NN from [27] is

34

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

α/fs

CD

P

Figure 2.6: Cyclic Domain Profile (CDP) for QPSK

SCcreation

CDP Extraction

PatternMatching

X[n]

Figure 2.7: Block diagram of the AMC

35

A NN is a parallel distributed processor that has a natural propensity for storing experienced

knowledge and making it available for use. The two main aspects of NN are

1. Knowledge is acquired by the network through a learning process.

2. Interneuron connection strengths known as synaptic weights are used to store knowl-

edge.

Based on the interconnections of the neuron, there are four basic classes of NN structure,

single-layer feed forward networks, multilayer feed forward networks, recurrent networks,

and lattice structures [27]. One of the widely used algorithms for training is the Back-

Propagation (BP) algorithm [27]. In BP, weights are adjusted during the training process in

such a way that the error between desired output and the actual output is reduced. There

are other methods of learning such as Hebbian Learning, Competitive Learning, Boltzmann

Learning and Reinforcement Learning. NN are widely used for pattern matching due to their

simple implementation.

In [18], [16] The MAXNET structure shown in Figure 2.8 is used for classification. In the

MAXNET structure each feed forward network has two hidden layers with 5 neurons in each

layer, and the activation function used is tanh(x) . The network is trained using the back

propagation algorithm with an initial learning rate of µ =0.05 and a momentum constant of

α =0.7. The input to the feed forward network is the 200 point α− profile and the output

varies between [-1, 1]. The function of the MAXNET structure is to choose the highest value

36

among all the feed forward networks, i.e.

z = argmax[Yi]. (2.32)

BPSK

QPSK

FSK

MSK

MAXNET

max(Y1,Y2,Y3,Y4)

Y1

Y2

Y3

Y4

-profile

Figure 2.8: MAXNET Neural Network structure

By training the neural network with different realizations of the signal allows it to extract

features such as carrier and keying-rate features of the signal. When the neural network

is trained with a variety of signal realizations with different SNRs, the network performs

exceptionally, even at low SNR levels. This suggests that the network will be able to detect

spread spectrum signals [18].

Performance Analysis

For the simulations, we assumed the signal’s carrier, pulse shape, pulse width and bandwidth

to be known. AWGN channel of SNR 5 dB is considered and Monte Carlo simulation of 1,000

37

Table 2.1: Probability of classification of AMC in the presence of AWGN (SNR = 5dB)

BPSK QPSK/QAM FSK MSK

BPSK 0.999 - - -

QPSK/QAM - 0.997 - 0.02

FSK - 0.02 0.987 -

MSK - - - 0.99

trials was performed and the results are shown in Table 2.1. Figure 2.9 shows the performance

of the classifier under different SNR. The peaks in the SCF are more pronounced when the

length of the signal observed is longer. The probability of classification given a certain

number of observed symbols is shown in Figure 2.10. In Figure 2.10 the SNR was fixed at

5 dB and Monte Carlo simulation was performed for 1000 trials. It is shown in [18] that by

training the NN for various levels of SNR, performance of the AMC improves.

−10 −8 −6 −4 −2 0 2 4 6 8 100.65

0.7

0.75

0.8

0.85

0.9

0.95

1

SNR (dB)

Pro

babi

lity

of c

orre

ct c

lass

ifica

tion

BPSKQPSKFSKMSK

Figure 2.9: Probability of classification Vs SNR

38

0 50 100 150 200 250 300 3500.75

0.8

0.85

0.9

0.95

1

no of samples

prob

abili

ty o

f cor

rect

cla

ssifi

catio

n

BPSKQPSKFSKMSK

Figure 2.10: Probability of classification Vs Number of symbols (SNR = 5dB)

The performance of the above designed classifier in the presence of the multipath channel

is analyzed. The multipath channel is modelled to be a 8-tap FIR filter. Monte Carlo

simulation is performed on each output and the average probability of classification for each

modulation scheme is presented in Table 2.2.

The simulation results indicate that AMC provides inconsistent results in the presence of a

multipath fading channel for a particular modulation scheme and hence the probability of

correct classification decreases.

HMM based classification

In [14], discrete HMM is used for classifying the CDP. Signal detection using CDP is discussed

first because it helps in the discretization of the CDP. In signal detection we assume that

a rough estimate of bandwidth is known. The crest factor (CF) is used for signal detection

39

Table 2.2: Probability of Classification of CDP Based AMC in the Presence of FIR Channel

(SNR = 5dB)

BPSK QPSK FSK MSK

BPSK 0.41 0.20 - 0.39

QPSK 0.32 0.31 - 0.35

FSK - 0.14 0.72 0.14

MSK 0.62 - - 0.38

and extraction from the CDP [14], which is a dimensionless quantity. The CF of a waveform

is equal to the peak amplitude of a waveform divided by its RMS value. When peaks are

known, this is a simple single cycle detector [14]. For signal detection, threshold values are

calculated first when no signal is present, i.e. only in the presence of AWGN we have

CTH =max(I(α))√(∑α=0N I2(α)

)/N

. (2.33)

If the CF is greater than CTH we declare the signal is present. For feature extraction, all CDP

peaks greater than CTH are encoded as 1 and the others are encoded as 0. This generated

binary feature vector is fed into the HMM signal classifier.

40

HMM as a classifier

A discrete sequence or process S[k] is a Markov process if the future of the process given the

present is independent of the past, that is

P (S[t+ 1] = j|S[t] = i, S[t− 1] = k, S[t− 2] = l, . . .) = P (S[t+ 1] = j|S[t] = i). (2.34)

The above equation is known as a Markov property. A Markov model is a stochastic model

of a system capable of being in finite states 1, 2, . . . , S. Also from the Markov property, one

can derive the probability of arriving at the next state by adding up all the probabilities of

the ways of arriving at that state, therefore [95]

P (S[t+ 1] = j) = P (S[t+ 1] = j|S[t] = 1)P (S[t] = 1)

+P (S[t+ 1] = j|S[t] = 2)P (S[t] = 2) . . . (2.35)

+P (S[t+ 1] = j|S[t] = S)P (S[t] = S).

The above equation can be expressed in matrix notation. Let

P [t] =

P (S[t] = 1)

P (S[t] = 2)

...

P (S[t] = S)

41

be the vector of probabilities for each state, and let the matrix A contain the transition

probabilities

A =

P (1|1) P (1|2) . . . P (1|S)

P (2|1) P (2|2) . . . P (2|S)

...

P (S|1) P (S|2) . . . P (S|S)

.

Thus one can write the probabilistic update equation as [95]

P [t+ 1] = AP [t] with P [0] = π.

The particular value of the state at time t is given by s[t]. In each state at time t, a random

variable v[t] ∈ Rm is selected according to a pmf fV |S(v[t]|S[t] = i). The variable v[t]

is observed, but the underlying state is not known, and such a process is called a hidden

Markov model.

From the above discussion, one can see that a HMM contains the following elements: N , the

number of states in the model (these states may be hidden and therefore not observable),

M , the number of distinct observations in the state (the observed signals correspond to a

physical output of the system to be modeled), the state transition probability distribution

P = aij where

aij = P [S(t+ 1) = i|S(t) = j]

and B = bj(k), the observation symbol probability distribution in state j where

bj(k) = P [vk at t|S(t) = j] 1 ≤ j ≤ N and 1 ≤ j ≤M,

42

and the initial state distribution π. For convenience, a compact notation for HMM is used

i.e.λ = (P,B, π). These parameters can be estimated using the Baum-Welch algorithm

(BWA), which is another form of the expectation-maximization (EM) algorithm for HMMs.

Due to the need for an online estimation in real world applications, one uses a modified

version of the BWA, called as the forward-only BWA (FO-BWA), or a block-orthogonal

BWA that can estimate HMM parameters in real time. For the case of binary sequences,

the probability of generating the observation sequence given the model, can be written

mathematically as

P (yT1 /λ) = πB(y1)PB(y2) . . . PB(yT )1

Because of the significantly long data size, one uses the logarithm of P (yT1 /λ), usually known

as log-likelihood.

Signal classification

If the CDP based detector declares that a signal exists, then this signal goes through the

signal classification stage. For training purposes, ideal binary feature vectors are generated

using CDPs for various signal types. The feature vectors are fed into the HMM for learning

process that uses the Baum-Welch algorithm. The Baum-Welch algorithm produces hidden

Markov models, λ = (P,B, π) , based on each training sequence (signal type). After training,

the unknown incoming signal is used to find its likelihood using each HMM generated in the

training phase. The likelihood values hence generated are compared with the likelihood of

the original sequence and the closest match is selected as the signal type. A simplified block

43

diagram of signal classification is shown in Figure 2.11.

)|( 1OP

)|( 2OP

)|( vOP

2

v

1

))|(max(arg vOP

FeatureExtraction

SelectMaximum

ProbabilityComputation for



Figure 2.11: Signal classification using HMM.

Performance analysis

To analyze the performance of this AMC, Monte Carlo simulations were performed for sig-

nal classification. The HMMs in Figure 2.11 were trained with ideal feature vectors for each

signal type. Different incoming signals with SNR of -3dB are observed with varying obser-

vation lengths to obtain the percentage of successful classification. The result is summarized

in Figure 2.12. Note that the percentage of correct signal classification (for each signal type)

44

reaches 100% when we increase the observation length to 300 blocks. MATLAB code for the

Baum-Welch algorithm and the block-orthogonal variation of Baum-Welch algorithm, can

be found in [95].

50 100 150 200 250 30010

20

30

40

50

60

70

80

90

100

number of samples

perc

enta

ge o

f cor

rect

cla

ssifi

catio

n

BPSKQPSKFSKMSKSB−AM

Figure 2.12: Percentage of correct classification vs Number of samples.

2.3 Cumulants Based AMC

In this section, AMC based on the fourth order cumulant of the received signal is presented.

The idea of using the fourth order cummulant for classification was first proposed in [7].

Preliminaries

For a complex-valued stationary random process y(n), second-order moments can be defined

in two different ways as

C20 = E[y2(n)] and C21 = E[|y(n)|2]. (2.36)

45

Similarly, fourth order cumulants can be written in three ways [7]

C40 = cumm[y(n), y(n), y(n), y(n)]

C41 = cumm[y(n), y(n), y(n), y∗(n)] (2.37)

C42 = cumm[y(n), y(n), y∗(n), y∗(n)]

where

cumm(w, x, y, z) = E(wxyz)− E(wx)E(yz)− E(wy)E(xz)− E(wz)E(xy). (2.38)

The cumulants in (2.36) and (2.37) can be estimated from the sample estimates of the

corresponding moments. By assuming zero mean, we have

C20 =1

N

N∑n=1

y2(n),

C21 =1

N

N∑n=1

|y(n)|2. (2.39)

Similarly, for the fourth-order cumulants

C40 =1

N

N∑n=1

y4(n)− 3C220,

C41 =1

N

N∑n=1

y3(n)y∗(n)− 3C20C21, (2.40)

C42 =1

N

N∑n=1

|y(n)|2 − |C20|2 − 2C221.

The cumulant value for each modulation scheme is unique and hence can be used as a feature

for modulation classification. The theoretical cumulant values for some of the modulation

schemes are tabulated in Table 2.3. Detailed tabulation can be found in [7]. Based on the

46

values of C42 and C40, the hierarchical modulation scheme similar to the one shown in Figure

2.13 is proposed in [7].

Table 2.3: Theoretical Cumulant Values for Some of the Modulation Schemes

BPSK QPSK PAM(4) PAM8 QAM16 QAM64

C40 -2 -1 -1.36 -1.2381 -0.68 -0.6191

C42 - 2 -1 -1.36 -1.2381 -0.68 -0.6191

2.3.1 Simulation Example

In this section the performance of the cumulant based AMC is demonstrated using simula-

tions. For our simulation, the four class problem from [7] is considered, that is

Ω4 = BPSK,PAM(4), QAM(4, 4), PSK(8)

For the above four class problem |C40| was used to make decisions. The decision rule con-

sidered was |C40| < 0.34 ⇒ PSK(8), 0.34 ≤ |C40| < 1.02 ⇒ QAM(4, 4) , 1.02 ≤ |C40| <

1.68 ⇒ PAM(4), and 1.68 ≤ |C40| ⇒ BPSK. The channel was considered to be a simple

10 dB AWGN. Table 2.4, Table 2.5, and Table 2.6 show the confusion matrix for the number

of samples N = 100, 250, and 500 respectively. It can be seen from the table that one can

get better classification by increasing the number of samples. Also from the discussion, it

can be seen that the cumulant based AMC can classify higher order modulations.

47

C42

BPSK PAM PSK(>2) QAM

QAM(4) … QAM(>4)

PSK(>4) PSK(4)

PSK(4) … PSK( )

C42C40

|C40|

Figure 2.13: Hierarchical AMC based on cumulants.

Table 2.4: Confusion Matrix for Cumulant Based AMC in the Presence of AWGN (SNR =

10dB), N=100.

BPSK QAM(4,4) PAM(4) PSK(8)

BPSK 0.983 0.017 - -

QAM(4,4) - 0.970 0.030 -

PAM(4) - 0.038 0.940 0.022

PSK(8) - - 0.038 0.962

48


10dB), N=100.


BPSK 0.996 0.007 - -

QAM(4,4) - 1 - -

PAM(4) - 0.002 0.995 0.003

PSK(8) - - - 1


10dB), N=500.


BPSK 1.000 - - -

QAM(4,4) - 1.000 - -

PAM(4) - - 1.000 -

PSK(8) - - - 1.000

49

2.3.2 Effect of Multipath Channel

In this section we briefly discuss the effect of the multipath channel on the cumulant value of

the received signal for a single user case. The received signal subjected to multipath fading

is given by

y(n) =L−1∑k=0

h(k)x(n− k) + g(n) (2.41)

where y(n) is the received signal, x(n) is the transmitted signal, g(n) is the additive noise,

and h(n) are the fading coefficients for each multipath. The C40y and C21y values are given

by

C40y =L−1∑k=0

|h(k)|4C40x, (2.42)

and

C21y =L−1∑k=0

|h(k)|2C21x + σ2g . (2.43)

The normalized fourth order cumulant C21y is then given by

C40y =C40y

(C21y − σ2g)

2= βC40x, (2.44)

where

β =

∑L−1l=0 |h(l)|4∑L−1l=0 |h(l)|2

2 . (2.45)

Since β < 1 [7], the effect of the multipath channel is to drive the actual cumulant value of

the transmitted signal toward zero and hence one cannot distinguish the modulation scheme.

50

Figure 2.14 shows the performance degradation of the AMC under a multipath channel. For

Figure 2.14 the same four class problem Ω4 = BPSK,PAM(4), QAM(4, 4), PSK(8) is

considered. It can be seen from Figure 2.14 that the multipath channel severely affects the

performance of the cumulant based AMC.

−10 −5 0 5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

AWGNMultipath

Figure 2.14: Performance of cumulant based AMC under multipath.

2.4 Adjusting the Equalizer Length

The performance of the AMC is now analyzed by adding a CMA blind equalizer. Choosing

the length of the equalizer is a difficult task when there is no information about the channel.

Here we vary the length of the equalizer according to the performance of the AMC. Monte

Carlo simulations are performed and results are shown in Figure 2.15. It can be seen from

Figure 2.15 that the performance of the AMC depends on the length of the equalizer. This

experiment basically illustrates the dependence of the AMC performance on the parameters

of the blind equalizer.

51

Figure 2.15: Effect of length of the equalizer on the performance of AMC (5 dB noise).

2.5 Conclusion

In this chapter we discussed two feature based AMC’s. The performance degradation of

the AMC’s in the presence of a multipath channel was illustrated. We also illustrated

using simulations the dependence of the AMC performance on the parameters of the blind

equalizer.

Chapter 3

Combined Blind Equalizer and Single

User AMC

3.1 Introduction

In a typical wireless communication environment, the transmitted signals are subjected to

noise and multipath fading. Multipath fading affects symbol detection by causing Inter

Symbol Interference (ISI). Multipath fading not only affects the performance of symbol

detection by causing ISI, but also affects the performance of the AMC. The performance

degradation of the AMC’s due to multipath channel was also illustrated in the previous

chapter.

Adaptive blind equalizers are used to remove ISI when there is no training sequence and chan-

52

53

nel knowledge available. Since there is no training sequence available, the blind equalization

algorithms adapt the weights of the equalizer by minimizing some special cost functions that

are non mean square error (MSE). Some of the well known blind equalization algorithms

are Sato [50], Godart [52], Bussgang, and Shalvi-Eeinsten [59]. Detailed literature on single

input single output (SISO) blind equalizers can be found in [48]-[49]. Since the cost functions

are non quadratic, the weights of the adaptive blind equalizer have the potential to converge

to an undesirable local minimum. The convergence of the blind equalizer to an undesired

local minimum not only affects the symbol detection performance but also the performance

of the AMC.

The objective of a blind equalizer is to remove ISI, but its impact on the AMC [70] has to

be evaluated as well. Also it was shown in the previous chapter, the dependence of AMC

performance on blind equalizer parameters. In a cognitive radio scenario it is preferable to

design a blind equalizer which not only removes ISI but also improves the performance of

the AMC. Two approaches in this direction are found in the literature. The first method is

proposed in [70], where performance of the cumulants based AMC is improved by estimating

the channel using fourth order statistics. Also in [70], the performance of the AMC is

improved but there is no improvement in symbol detection. The other method proposed

in [71] is the same as the one proposed in [70], except that a higher order statistics (HOS)

based blind equalizer is added to the received signal and a switching mechanism is proposed

based on which the AMC chooses between a raw signal and an equalized signal. There is

no improvement in the performance of the AMC due to the switching mechanism and blind

54

equalizer, but the switching mechanism makes sure that there is no performance degradation

in the AMC due to the blind equalizer.

In this chapter, we propose novel cognitive receivers where the performance of the AMC

is also considered, while designing the blind equalizer and thus eliminating the need for

switching. The proposed approach involves formulating cost functions that are related to

the performance of AMC and the performance of symbol detection and then adapting the

equalizer parameters such that these cost functions are maximized. The proposed approach

thus improves both signal detection and AMC performance. In this chapter we propose

novel cognitive receivers for two different multipath channel conditions: minimum phase

channel and mixed phase channel. For the minimum phase channel, the proposed receiver

architecture is an adaptation of the blind equalizer presented in [86], [87]. The reason for

choosing this architecture is that it offers two fold diversity for AMC decision making, that

is, the AMC makes a decision based on two estimated cumulant values. Because of this

diversity, the performance of the AMC is better than those of [70] and [71]. For the mixed

phase channel, we propose two different receiver architectures. In the first architecture,

the equalizer considered is a simple FIR filter. In the second architecture, the equalizer

considered is a modified version of a decision feedback equalizer. In both the architectures,

the parameters of the equalizer are adapted using modified stop and go adaptation rules.

This chapter is organized as follows: In Section 3.2, we provide a block diagram description

of the proposed system along with channel model and assumptions. In Section 3.3, we briefly

describe nth order cumulants based AMC. The cost function related to the performance of this

55

AMC is also formulated in this section. In Section 3.4, the proposed receiver architecture for

minimum phase channels is described. In Section 3.5, the proposed receiver architectures for

mixed phase channels is described. Simulation results are presented in Section 3.6, followed

by the conclusion.

Notation: (.)∗ stands for the complex conjugate, (.)H denotes the conjugate transpose, and

(.)T is the transpose operation. Also, E(.) stands for the expectation operation and z−1 is

the unit delay operator in time domain.

3.2 Problem Statement

AMC

H(z-1)Blind

Equalizer

SymbolDetection

Blind Adaptive Algorithm

v(k)

y(k)s(k)

r(k)

Figure 3.1: Block diagram of the proposed system.

The block diagram of a typical intelligent receiver is shown in Figure 3.1. In the figure, s(k)

is the complex baseband transmitted signal and H(z−1) is the channel transfer function.

56

The multipath channel is modelled as a FIR filter given by

H(z−1) = 1 + h(1)z−1 + . . .+ h(L)z−L (3.1)

where z−1 is the unit delay operator and h(i) (for i = 1, . . . L) are the impulse response

coefficients. The received signal r(k) which is subjected to multipath fading is given by

r(k) = H(z−1)s(k) + v(k) (3.2)

where v(k) is the additive white noise. The received signal is then fed to the adaptive blind

equalizer. The equalizer output y(k) is used for both AMC and symbol detection. Typically

blind equalization algorithms adapt the parameters of the blind equalizer by minimizing

the cost function that is related to symbol detection performance. Since the output of the

blind equalizer is used for both AMC and symbol detection, it is necessary to consider the

performance of the AMC also while adapting the equalizer parameters. In order to do so, two

cost functions are formulated such that one is related to the AMC performance and the other

one is related to symbol detection performance. Then adaptive algorithms are developed to

adapt the parameters of the blind equalizer such that both the cost functions are maximized.

In rest of the chapter, we propose cognitive receiver architectures for two different multipath

channels. The cost function related to the performance of nth order cumulants based AMC

is formulated. Adaptive algorithms to adapt the parameters of the blind equalizers in the

proposed receiver architectures are developed.

57

3.3 AMC

As mentioned earlier, nth order cumulant based AMCs are widely used because of their

ability to classify multiple modulation schemes and easy implementation. We first briefly

describe the nth order cumulant based AMC from [7] - [11]. We then propose a cost function

that is related to the performance of the nth order cumulant based AMC.

3.3.1 Cumulants Based AMC

In this section, we present the basic theory behind nth order cumulant based AMC. For a

complex random signal v(k), the nth order moment is defined as

Rv(n,m)(τ) = E

[n∏j=1

v(∗)j(τj)

](3.3)

where n is the order, m is the number of conjugate factors, and τ = [τ1, . . . , τn] is the delay

vector. The nth order cumulant function is defined as [9], [10]

Cv(n,m)(τ) =∑Pn

F (p)

p∏j=1

Rv(nj ,mj)(τ) (3.4)

where the sum is over distinct partitions of the indexed set 1, 2 . . . n and F (p) = (−1)p−1(p−

1)!. The normalized nth order cumulants values are defined as

Cv(n,m)(τ) =Cv(n,m)(τ)[C2v(2,1)(0)

]n/2 for n = 4, 6, . . . . (3.5)

Theoretical normalized cumulant values for some of the modulation schemes are shown in

Table 1. Detailed tabulation can be found in [7], [9]. From Table 1 it can be seen that the

58

Table 3.1: Theoretical normalized cumulant values

(n=4,m=0,τ=0) (n=6,m=1,τ=0)

BPSK -2 16

QPSK 1 -4

QAM(16) -0.68 2.08

PSK(8) 0 0

normalized cumulants values are unique for each modulation scheme and hence are used as

a feature for classification.

3.3.2 Cost function for the Cumulants Based AMC

In this subsection we derive the cost function J1 that is related to the performance of nth

order cumulant based AMC. In order to do so, we need to analyze the effect of the multipath

channel on normalized nth order cumulant features. The following properties of the nth order

cumulant features are used to analyse the effect of multipath.

Property 1 Additive: Let x(k) and y(k) be two independent random processes. If z(k) =

x(k) + y(k), then the nth order cumulant value of z(k) is the sum of those for x(k) and y(k).

That is

Cz(n,m)(τ) = Cx(n,m)(τ) + Cy(n,m)(τ). (3.6)

Property 2 Scaling property: Let x = ay. Then the nth order cumulant value of x is |a|n

59

times the nth order cumulant value of y.

Using the scaling and additive properties of cumulants, the normalized cumulants of the

received signal r(k) (refer to (3.2)) is given by

Cr(n,m)(τ) =γ

∆n/2Cs(n,m)(τ) (3.7)

where Cs(n,m)(τ) is the normalized cumulant value of the transmitted sequence s(i),

γ =L−1∑k=0

|h(k)|n, and ∆ =L−1∑k=0

|h(k)|2. (3.8)

It can be easily shown that

Ω =γ

∆n/2< 1. (3.9)

Since Ω < 1, the magnitude of the normalized cumulants of the received signal r(k) is driven

toward zero. The multipath channel basically clusters all the normalized cumulant features

around zero. This clustering makes it hard for the classifier to distinguish the features. For

this reason, we propose the following cost function:

J1 = (Cy(n,m)(τ))2. (3.10)

The above cost function maximizes the magnitude of the normalized cumulant values of the

signals so that the classifier can distinguish between the features.

3.4 Minimum Phase Channels

Reprinted, with permission from, B.Ramkumar, T. Bose, and M. Radenkovic, Robust au-

tomatic modulation classification and blind equalization: A novel cognitive approach, The

60

wireless innovation forum, December 2010.

In this section we present the blind equalizer architecture for minimum phase channels. That

is, we make the following assumption about the channel transfer function H(z−1).

Assumption A31 The channel H(z−1) is a minimum phase polynomial, i.e., it has no zeros

in |z| ≥ 1.

Assumption A31 implies that the energy in the direct component of the received signal is

more when compared to the energy in the delayed multipath component. As mentioned

earlier, the proposed architecture is an adaptation of the blind equalizer presented in [86],

[87]. The reason for choosing this architecture is that it offers two fold diversity for AMC

decision making which will be shown later in this section. Because of this diversity, the

performance of the AMC is better than those of [70] and [71]. This section is organized as

follows: First we briefly describe the proposed CR receiver architecture. Then we develop

algorithms to adapt the parameters of the blind equalizer in the proposed receiver. Finally,

we propose the fusion rule for AMC decision making.

3.4.1 Proposed Architecture

The block diagram of the proposed receiver is shown in Figure 3.2. From Figure 3.2 it can

be seen that the received signal r(i) is branched out into two signals x1(i) and x2(i) where

x1(i) = r(i) and (3.11)

x2(i) = B(z−1)r(i).

61

)( 1zB)(

)()(

1

11

2

zD

zSzF

)(

)()(

1

11

1

zD

zRzF

)(1 ix

)(2 ix

)1(1 ix

)1( iy )1( ie

+

+

- +

)(ir

Cumulant Estimation (For AMC)

AMC Decision Maker

p1 (eqn: 33)

p2 (eqn: 34)

Figure 3.2: Block diagram of the proposed cognitive receiver.

The polynomial B(z−1) can be any arbitrary polynomial such that

degree(B(z−1) ≥ 1.

Let the polynomial B(z−1) be defined as

B(z−1) = b0 + b1z−1 + . . .+ b(L1−1)z

−(L1−1). (3.12)

The polynomial B(z−1) basically induces a non common factor in the two branches, so that

the Recursive Extended Least Square (RELS) algorithm from [86], [87] can be applied. Even

though B(z−1) can be any arbitrary polynomial, it is a necessary polynomial required for the

convergence of the RELS algorithm. The signals x1(i) and x2(i) are further passed through

filter F1(z−1) and F2(z−1) respectively, where

F1(z−1) =R(z−1)

D(z−1)and F2(z−1) =

S(z−1)

D(z−1). (3.13)

62

The coefficients of these filters are adapted by minimizing the cost function that is related

to the symbol detection performance. In order to do so we consider the well known cost

function known as step ahead prediction error given by

J2 = E(|x1(i+ 1)− y(i+ 1)|2), (3.14)

where y(i) = F1(z−1)x1(i) + F2(z−1)x2(i) and the prediction error e(i + 1) provides the

equalized symbol sequence for symbol detection. The filters F1(z−1) and F2(z−1) are also

known as prediction error filters. The recursive algorithm for estimating R(z−1),S(z−1) and

D(z−1) is presented in the next subsection.

Another important component in Figure 3.2 is the AMC. As mentioned earlier, the nth

order cumulant of a received signal is used for classification. Since B(z−1) is an arbitrary

polynomial, we adapt it in such a way that the performance of the AMC is improved. For

the AMC based on the nth order cumulant, we adapt B(z−1) by minimizing the cost function

that was proposed in the previous section (refer to equation (3.10)). For a different feature

based AMC, an appropriate cost function must be chosen accordingly.

From Figure 3.2 it can seen that the AMC makes decisions by fusing p1 and p2, which are

functions of Cx1(n,m) and Cx2(n,m) respectively. Appropriate functions for p1 and p2 and the

fusion rule are derived in subsection 3.4.4.

63

3.4.2 Adapting S(z−1), R(z−1) and D(z−1).

As mentioned in the previous section, the polynomials S(z−1), R(z−1) and D(z−1) are

adapted by minimizing (3.14). From Figure 3.2 it can be seen that

x2(i) = H(z−1)B(z−1)s(i) (3.15)

x1(i) = H(z−1)s(i). (3.16)

The recursive algorithm for updating B(z−1) is discussed in the next subsection. In this sub-

section we considerB(z−1) to be an arbitrary polynomial with the condition degree(B(z−1)) ≥

1. Now

y(i+ 1) =R(z−1)H(z−1)

D(z−1)s(i) +

S(z−1)B(z−1)H(z−1)

D(z−1)s(i) (3.17)

and

x1(i+ 1) = s(i+ 1) + [z(H(z−1)− 1)]s(i). (3.18)

It can shown from (3.17) and (3.18) that

x1(i+ 1)− y(i+ 1) = Q(i) + s(i+ 1) (3.19)

where

Q(i) = [(R(z−1) + S(z−1)B(z−1))H(z−1)

D(z−1)− z(H(z−1)− 1)]s(i). (3.20)

From (3.19) and (3.20) it can be seen that the cost function (3.14) is minimum when Q(i) = 0.

Therefore setting (3.20) to zero we get

D(z−1) = H(z−1) (3.21)

64

and

(R(z−1) + S(z−1)B(z−1)) = z(H(z−1)− 1). (3.22)

Note: It should be noted that the channel impulse response can be estimated from (3.19) .

This information can be used to calculate Ω (refer to (3.9)).

Since the polynomial H(z−1) is not known, it is not possible to solve the above equations.

For degree(B(z−1)) ≥ 1, the whole system can be viewed as a special case of the SIMO

blind equalizer in [86]. Hence we can modify the recursive algorithm in [86] for estimating

the unknown polynomials. Let

R(z−1) = r0 + r1z−1 + . . .+ rN1z

−N1

S(z−1) = s0 + s1z−1 + . . .+ sN2z

−N2 (3.23)

where N1, N2 ≥ max(L1, L). Define

φ(i)T = [x1(i), . . . , x1(i−N1), x2(i), . . . ,

x2(i−N2),−y(i), . . . ,−y(i−N3)], N3 ≥ L (3.24)

and

θ = [r0, r1, . . . , rN1, s0, s1, . . . , sN1, h0, . . . , hL, 0, . . . , 0] (3.25)

where the number of zeros at the end is the difference between the chosen N3 and the

unknown L. From (3.24) and (3.25) we have

y(i+ 1) = θHφ(i). (3.26)

65

The value of θ is estimated using the following Recursive Extended Least Squares (RELS)

algorithm:

θ(i+ 1) = θ(i) + p(i)φ(i)ε(i+ 1)∗ (3.27)

ε(i+ 1) = x1(i+ 1)− θ(i)Hφ(i) (3.28)

p(i) =1

λp(i− 1)− 1

λ

p(i− 1)φ(i)φ(i)Hp(i− 1)

λ+ φ(i)Hp(i− 1)φ(i), 0 < λ ≤ 1 (3.29)

p(0) = p0I, p0 > I.

Since the above algorithm is a special case of the algorithm in [86], the convergence property

derived in [86] applies here. One of the important properties is that for λ = 1 under

assumption A1 and degree(B(z−1)) ≥ 1 the a’posteriori prediction error converges to a

scalar version of the symbol sequence, i.e.,

limn→∞

1

n

n∑i=1

[x1(i+ 1)− y(i+ 1)− s(i+ 1)]2 = 0 (3.30)

3.4.3 Adapting B(z−1)

As mentioned earlier, B(z−1) is adapted by minimizing (3.10). It can be seen that (3.10) is

non quadratic and we use a gradient search method to find the coefficients of B(z−1). Let

W = [b0, b1, · · · , bL1 ]T be the vector of coefficients of B(z−1). The gradient search algorithm

[88] for updating W is stated as follows. Let Wk denote the coefficient vector during the

iteration k = 0, 1, 2, . . ..

66

• Step 1: For k = 0, initialize W0 to a random value.

• Step 2: For k = 1, 2, . . . calculate the output of the filter

x2(n) =

m=L1∑m=0

Wk−1(m)r(n−m) (3.31)

• Step 3: Update the coefficient vector using the following equation

Wk = Wk−1 − µ∂J1

∂W Wk−1

(3.32)

where µ is the step size.

• Step 4: If |J1(Wk)−J1(Wk−1)|J1(Wk−1)

< ζ terminate the iteration and go to step 5. If not, repeat

step 2, where ζ is chosen to be a small number less than one.

• Step 5: Calculate the equalized output using Wk.

The equalized signal x2(n) has a higher cumulant value but does not guarantee good signal

to interference noise ratio (SINR). The reason is that the cost function J1 is non quadratic

and the gradient decent algorithm converges to a local minimum [88]. The low SINR of x2(n)

is not a concern because x2(n) is used only for the AMC and not for symbol detection. The

coefficients of B(z−1) are updated for every batch of data, whereas the other polynomials

are updated for every sample. The forgetting factor in the recursion (3.27)-(3.29) is used to

track the slowly varying B(z−1) polynomial.

67

3.4.4 AMC Decision Making

The decision about the modulation scheme is made by fusing the cumulant value calculated

from two sources. From equations (3.21) and (3.25) it can be seen that the channel impulse

response can be estimated using the recursion (3.27)-(3.29) apart from achieving equalization.

From the estimated impulse response D(z−1), the value of Ω can be estimated using (3.8).

Let Ω be the estimated value of Ω, then

p1 =1

Ω|Cx1(n,m)|. (3.33)

p2 = |Cx2(n,m)|. (3.34)

Since the channel tends to drive the cumulant value of a transmitted signal to zero, the

natural choice for the fusion rule is

pf = max(p1, p2). (3.35)

The performance of the AMC in the proposed receiver is enhanced because of the above

fusion rule and higher cumulant value of the signal x2. Both symbol detection performance

and AMC performance for the proposed receiver is analyzed in Section 3.6.

3.5 Mixed Phase Channels

In this section we present the CR receiver architecture for mixed phase channels. That is,

we make no assumption about the channel transfer function H(z−1). The block diagram of

68

AMC

H(z-1) W(z-1)Symbol

Detection

Blind Adaptive Algorithm

v(k)y(k)s(k)

r(k)


the proposed system is shown in Figure 3.3. In Figure 3.3 the equalizer W (z−1) is modeled

as a FIR filter given by

W (z−1) = w0 + . . .+ w(L1−1)z−(L1−1), (3.36)

where z−1 is the unit delay operator and wi (for i = 1 . . . (L1 − 1)) are the weights of

the equalizer. Denote the weight vector of the equalizer as w(k) = [w0, . . . , wL1 ] and the

regressor vector as r(k) = [r(k), . . . , r(k−L1)], then the output y(k) is given by w(k)r(k)T .

The equalizer output y(k) is used for both AMC and symbol detection. As mentioned

earlier, the equalizer weights are adapted using a modified version of stop and go adaptation

rules. In the following subsection, the background theory on stop and go adaptation rules is

presented.

Note: When the channel is minimum phase, the receiver architecture presented in the

previous section (Refer to Figure 3.3) offers better performance when compared to the one

proposed in this section. However the receiver architecture in Figure 3.3 cannot be applied

69

to mixed phase channels.

3.5.1 Background

Most blind equalization algorithms are designed as stochastic gradient schemes for updating

the weight vector by minimizing cost functions that are non-MSE. These cost functions are

chosen such that the symbol detection performance is improved. Let the cost function be

defined as

J(w(k)) = E Φ(y(k)) = E

Φ(w(k)r(k)T ), (3.37)

where Φ(y(k)) is a nonlinear function of the equalizer output y(k). Then the well known

stochastic gradient decent algorithm for updating weights is given by

w(k + 1) = w(k)− µ∂Φ(y(k))

∂w(k)(3.38)

= w(k)− µΦ′(y(k))

where µ is the step size and Φ′(y(k)) is the partial derivative of Φ(y(k)) with respect to

w(k). Since the cost functions are non-quadratic, the weights have the potential to converge

to a local minimum. From (3.4) it can be seen that the convergence of the blind equalizer

depends on the gradient direction, and more specifically, the sign of the gradient Φ′(y(k)).

Since the output of the equalizer y(k) is used for both symbol detection and AMC, the

convergence of the blind equalizer can be improved if the performance of the AMC is also

considered while adapting equalizer weights. In order to do so, we consider the stop and go

adaptation rules proposed in [60]. In the stop and go methodology, two cost functions are

70

considered for adapting the equalizer weights. For each sample of the received signal, the

equalizer weights are updated if the signs of the gradients of the two cost functions agree.

Let us define the two cost functions as

J1(w(k)) = E Φ1(y(k)) = E

Φ1(w(k)r(k)T )

(3.39)

and

J2(w(k)) = E Φ2(y(k)) = E

Φ2(w(k)r(k)T ), (3.40)

where Φ1(y(k)) and Φ2(y(k)) are nonlinear functions of the equalizer output y(k). Then the

stop and go adaptation rule is given by

w(k + 1) =

w(k)− µΦ

′1(y(k)), for sgn[Φ

′1(y)] = sgn[Φ

′2(y)]

w(k), for sgn[Φ′1(y)] 6= sgn[Φ

′2(y)]

(3.41)

So far in literature, both the cost functions (J1 and J2) are related to the symbol detection

performance. Here we choose the cost functions such that one of them is related to symbol

detection performance and the other is related to the performance of the cumulants based

AMC. This insures that the performance of the AMC is not affected due to the blind equal-

izer. This method also eliminates the need for switching that was used in [71]. The cost

function for the nth order cumulants based AMC was proposed in Section II (3.10). For the

symbol detection performance we consider the cost function proposed in [18], which is briefly

explained in subsection 3.5.3. For the cost function J1, we need to calculate the stochastic

gradient function Φ′1(y(n)) in order to use the stop and go adaptation rule in (3.41). In the

following subsection we derive the expression for the stochastic gradient.

71

3.5.2 Computing the Gradient

It should be noted that the cost function (3.10) is non quadratic and nonlinear. Since only

the sign of the gradient is required, we compute an approximate function for the gradient.

By substituting (3.5) in (3.10), the cost function becomes

J1 =

(Cy(n,m)(k, τ)

Cy(2,1)(0)

)2

. (3.42)

Now the gradient ∂J1/∂w is given by

∂J1

∂w= J1(w)[

1

Cy(n,m)(k, τ)

∂Cy(n,m)(k, τ)

∂w∗(3.43)

+1

Cy(m,n)(k, τ)

∂Cy(m,n)(k, τ)

∂w− m+ n

Cy(2,1)(0)

∂Cy(2,1)(0)

∂w∗].

By substituting the expression for cumulants in the above equation and replacing the expec-

tation operation by a sample estimate we obtain the expression for the stochastic gradient.

Here we present the stochastic gradient function for some specific cases that were used for

the simulations.

Case 1. n = 4, m = 0 and τ = 0 (Fourth order cumulants)

∂J1

∂w=y4(k)[y∗(k)y(k)− 1]

y∗(k)r(k)H = ψ1(y(k))r(k)H (3.44)

Case 2. n = 6, m = 1 and τ = 0 (Sixth order cumulants)

∂J1

∂w= y5(k)y∗(k)[y7(k) + 5y∗7(k)− 6y5(k)y∗4(k)]

1

y∗4(k)y4(k)r(k)H (3.45)

= ψ1(y(k))r(k)H

where r(k) in the above equations is the (1× L1) regression vector.

72

3.5.3 Cost Function Related to Symbol Detection

As mentioned earlier, one of the cost functions is chosen such that the symbol detection per-

formance is improved. For this receiver architecture, we consider the cost function proposed

in [18], which is also known as the Bussgang algorithm. The cost function is the maximum

a posteriori (MAP) estimate of the transmitted sequence. The adaptive Bussgang algorithm

is a special case of the stochastic gradient descend algorithm and is given by

w(k) = w(k − 1)− µψ2(y(k))r(k)H (3.46)

= w(k − 1)− µ[f(y(k))− y(k)]r(k)H

where ψ2(y(k))r(k)H is the stochastic gradient and f(y(k)) is a nonlinear function. One

of the widely used nonlinear functions is the tanh() function. The reason for choosing the

Bussgang algorithm is that all the existing HOS based blind equalization algorithms can be

viewed as a special case of the Bussgang algorithm. A detailed explanation of the above

algorithm can be found in [88].

3.5.4 Overall Algorithm

The algorithm to adapt the weights of the equalizer is obtained by substituting the gradient

functions derived in this section in (3.41). The overall adaptive algorithm is given by

w(k + 1) =

w(k)− µψ1(y(k))r(k)H , for sgn[ψ1(y)] = sgn[ψ2(y)]

w(k), for sgn[ψ1(y)] 6= sgn[ψ2(y)]

(3.47)

73

where ψ2(y(k)) is given by (3.46) and ψ1(y(k)) depends on order of the cumulants based

AMC (refer to (3.44) and (3.45) for specific cases).

3.5.5 Decision Feedback Equalizer

In this subsection we propose a CR receiver with a nonlinear equalizer architecture known

as the decision feedback equalizer (DFE). The proposed CR receiver architecture in this

subsection is similar to the one proposed previously (refer to Figure 3.3) except that there

is an additional feedback filter. Compared to the receiver proposed before, this DFE based

receiver offers better symbol detection performance when the channel impulse response is

long. However the performance of AMC for both the receivers will be the same. The block

diagram of the CR receiver with DFE is shown in Figure 3.4.

SymbolDetection

Feedback FilterB(z-1)

Feedforward FilterF(z-1)

y(k)

AMC

r(k)

y(k)

ŝ(k)+

-


From Figure 3.4 it can be seen that the equalizer has two filters. The first one is a linear

filter in the direct path known as feedforward filter. The second filter feedbacks the decision

74

made by the symbol detection block and hence is called a feedback filter. The feedback filter

uses the previous decisions made by the symbol detector to reduce ISI and thus improves

symbol detection performance. Let F (z−1) and B(z−1) denote the transfer functions of the

feedforward and feedback filters respectively. Both the filters are modeled as FIR filters

given by

F (z−1) = f0 + . . .+ f(L−1)z−(L−1). (3.48)

and

B(z−1) = b0 + . . .+ b(L1−1)z−(L1−1). (3.49)

Denote the weight vector of the feedforward filter as f(k) = [f0, . . . , fL] and the feedforward

regressor vector as r(k) = [r(k), . . . , r(k − L)], then the output y(k) is given by f(k)r(k)T .

The weights of the feedforward filter are adapted such that both AMC performance and

symbol detection performance are improved. In order to do so, we use the modified stop and

go adaptation rule proposed in the previous subsection (refer to (3.47)). Now denote the

weight vector of the feedback filter as b(k) = [b0, . . . , bL1] and the feedback regressor vector

as s(k) = [s(k), . . . , s(k−L1)], then the output of the filter y(k) is given by b(k)s(k)T . The

weights of the feedback filter are adapted by minimizing the following cost function

J3(b(k)) = E

[y(k)− y(k)]2. (3.50)

The above cost function minimizes the ISI and thus improves symbol detection. By com-

puting the gradient of the above cost function and setting it to zero we obtain the following

75

stochastic gradient algorithm to update the weights of the feedback filter

b(k) = b(k − 1)− µ[y(k)− y(k)]s(k)H , (3.51)

where µ is the step size. As mentioned earlier the CR receiver proposed in this subsection

offers better symbol detection performance. The reason for this improved performance is the

feedback filter which is adapted by minimizing (3.50).

3.6 Performance Analysis

In this section, we analyze the performance of the proposed CR receiver architectures using

Monte Carlo simulations. Similar to [71], both AMC performance and symbol detection

performance are analysed. For the AMC performance analysis, the probability of correct

classification Pcc is considered as a performance measure. Suppose there are K possible

modulation schemes defined by the following K class problem

ω = d1, . . . , dK (3.52)

Then the probability of correct classification Pcc is defined as

Pcc =K∑i=1

P (di|di)P (di) (3.53)

where P (di) is the probability that the particular modulation scheme is transmitted and

P (di|di) is the correct classification probability when modulation scheme di has been trans-

mitted. For the simulation we assume P (di) = 1K,∀i, where all scenarios are equally probable.

For the Monte Carlo simulations 1000 trials were considered.

76

3.6.1 Experiment 1 (Minimum Phase Channel)

In this experiment we consider the channel to be a minimum phase multipath channel. The

channel is modeled as a 4-tap FIR filter such that there are no zeros on or outside the unit

circle. Since the receiver is modelled as minimum phase, the equalizer architecture that was

proposed in Section 3.4 is considered. In order to analyse the performance of the AMC, the

following AMC four class problem is considered

ω = BPSK,QPSK,QAM(16), PSK(8) . (3.54)

Fourth order cumulant (n=4 in (3.5)) was considered as a feature for classification. Figure

3.5 shows the probability of correct classification versus signal-to-noise ratio. In Figure 3.5,

−10 −5 0 5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2pc3pc4

Figure 3.5: Performance of the AMC.

Pc1 is the performance of the AMC in the presence of an AWGN channel (ideal condition).

Pc2 is the performance of the proposed system and Pc3 is the performance of the system

proposed in [70] and [71]. Pc4 is the performance of the AMC in the presence of a multipath

77

channel with no channel estimation or equalization and hence it is the worst. Pc2 is better

than Pc3 because AMC performance is also considered while adapting equalizer weights in

the proposed system. For analysing the performance of symbol detection, the same 4-tap

FIR channel was considered. Symbol error rates (SER) before and after equalization are

presented in Figure 3.6. From the figure it can be seen that the proposed system offers

good symbol detection performance. Also from the simulation results it can be seen that the

performance of symbol detection and AMC are simultaneously improved.

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SNR(dB)

SE

R

After equalizarionBefore equalization

Figure 3.6: Symbol error rate (SER) vs SNR (BPSK).

3.6.2 Experiment 2 (Minimum Phase Rayleigh Channel)

In this experiment we analyse the performance of the AMC in the performance of realistic

minimum phase Rayleigh channel. The same four class problem from the previous experi-

ment is considered hare. Rayleigh distribution is commonly used to describe statistical time

varying envelope of an individual multipath components [90]. We consider here the following

78

three tap multipath channel

H(z−1) = α0ejφ0 + α1e

jφ1z−1 + α2ejφ2z−2, (3.55)

where α0, α1 and α2 are independent and Rayleigh distributed, φ0, φ1 and φ2 are independent

and uniformly distributed over [0,2π]. In order to make sure the channel is minimum phase

we arrange the chosen multipath gains in ascending order with direct component having

the highest gain. Fourth order cumulant (n=4 in (3.5)) was considered as a feature for

classification. Figure 3.5 shows the probability of correct classification versus signal-to-noise

ratio. In Figure 3.7, Pc1 is the performance of the system proposed in [70] and [71] and Pc2 is

the performance of the proposed system. Pc2 is better than Pc1 because AMC performance

is also considered while adapting equalizer weights in the proposed system. From the results

it can be seen that, the proposed system performs well under Rayleigh fading channel.

−5 0 5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2

Figure 3.7: Performance of the AMC (Minimum phase Rayleigh channel).

79

3.6.3 Experiment 3 (Minimum Phase Ricean Channel)

In this experiment we analyse the performance of the AMC in the performance of realistic

minimum phase Ricean channel. The same four class problem from experiment 1 is consid-

ered hare. Ricean distribution is commonly used to describe statistical time varying envelope

of an individual multipath components which has a dominant line of sight component. We

consider here the following three tap multipath channel

H(z−1) = α0ejφ0 + α1e

jφ1z−1 + α2ejφ2z−2, (3.56)

where α0, α1 and α2 are independent and Ricean distributed, φ0, φ1 and φ2 are independent

and uniformly distributed over [0,2π]. Fourth order cumulant (n=4 in (3.5)) was considered

as a feature for classification. Figure 3.6 shows the probability of correct classification versus

signal-to-noise ratio. In Figure 3.8, Pc1 and Pc2 have the same meaning as the previous

experiment. Pc2 is better than Pc1 because AMC performance is also considered while

adapting equalizer weights in the proposed system. From the results it can be seen that, the

proposed system performs well under Ricean fading channel.

3.6.4 Experiment 4 (Higher Order QAM’s)

In this experiment we analyse the performance of the AMC in classifying higher order QAM’s.

The channel is modelled as a minimum phase Rayleigh channel. In order to analyse the

performance of the AMC, the following AMC four class problem is considered

ω = BPSK,QAM(4), QAM(16), QAM(64) . (3.57)

80

−5 0 5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2

Figure 3.8: Performance of the AMC (Minimum phase Ricean channel).

Figure 3.7 shows the probability of correct classification versus signal-to-noise ratio when

fourth order cumulant (n=4 in (3.5)) was considered as a feature for classification. In Figure

3.9, Pc1 and Pc2 have the same meaning as the previous experiment. It can be seen from

the figure that even though Pc2 is better than Pc1, the performance of the AMC is not

good. The reason for this poor performance is that, fourth order cumulant features have

poor discriminatory capability in classifying higher order QAM’s.

The experiment is repeated using sixth order cumulant (n=6 in (3.5)) features and the

results are shown in Figure 3.10. In Figure 3.10, Pc1 and Pc2 have the same meaning as the

previous experiment. From the figure it can be seen that sixth order cumulants can classify

QAM’s better but requires more samples to estimate it.

81

−5 0 5 10 15 200.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2

Figure 3.9: Classifying QAM’s (Fourth order cumulants).

−5 0 5 10 15 200.7

0.75

0.8

0.85

0.9

0.95

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2

Figure 3.10: Classifying QAM’s (Sixth order cumulants).

82

3.6.5 Experiment 5 (Mixed Phase Rayleigh Channel)

In this experiment we consider the channel to be a mixed phase multipath channel. The

channel is modeled as a three mixed phased Rayleigh channel. Since the channel is modeled

as mixed phase, the equalizer architecture that was proposed in Section 3.5 is considered.

As mentioned earlier, nth order cumulants based AMC is considered in this paper. For this

experiment, we consider two specific cases with n = 4 and n = 6 respectively. For both cases

we consider the following four class problem

ω = BPSK,QPSK,QAM(16), PSK(8) . (3.58)

Case 1(Fourth order cumulants)

For this case, we consider a fourth order cumulant feature with n = 4, m = 0 and τ = 0

(refer to (3.5)). The number of samples used to estimate the cumulant features was T1 =

1,000. The Bussgang cost function was considered for the symbol detection performance.

The stochastic gradient of the AMC (Ψ1(y)) cost function for this case is given by (3.44).

The performance of the AMC for the proposed system is shown in Figure 3.11. In Figure

3.11, Pc1 denotes the performance of the AMC using the switching equalizer proposed in

[71], and Pc2 denotes the performance of the AMC using the proposed equalizer.

83

−5 0 5 10 15 200.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2

Figure 3.11: Performance of the AMC (Mixed phase Rayleigh channel).

Case 2(Sixth order cumulants):

For this case, we consider a sixth order cumulant feature with n = 6, m = 1, and τ = 0

(refer to (3.5)). The number of samples used to estimate the cumulant features was T1 =

3000. The stochastic gradient of the AMC (Ψ1(y)) cost function for this case is given by

(3.45). The performance of the AMC for the proposed system is shown in Figure 3.12. In

Figure 3.12, Pc1 and Pc2 have the same meaning as that of Figure 3.11.

From Figure 3.11 and Figure 3.12, it can be seen that the proposed system performs better

than the switching equalizer in [70]. The reason is that AMC performance is also considered

while adapting the weights. In order to analyze the symbol detection performance, SER and

steady state normalized mean square error (NMSE) are considered as performance measures.

The SER Vs SNR after equalization is presented in Figure 3.13. In Figure 3.13, p2 is the

84

−5 0 5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2

Figure 3.12: Performance of the AMC (Mixed phase Rayleigh channel).

symbol detection performance of the receiver architecture with a linear equalizer (refer to

Figure 3.3) and p1 is the symbol detection performance of the receiver architecture with a

DFE (refer to Figure 3.4). From the figure it can be seen that the DFE based receiver offers

better symbol detection performance. The reason for this better performance is the feedback

filter in DFE. The convergence of the NMSE is shown in Figure 3.14. In can be seen that

when higher order cumulants are used for AMC the convergence is slower. The reason for

this is that for higher order cumulants the stochastic gradient Ψ1(y) has higher variance.

3.6.6 Experiment 6 (Mixed Phase Rician Channel)

In this experiment we consider the channel to be a mixed phase multipath channel. The

channel is modeled as a three mixed phased Rician channel. Since the channel is modeled as

85

−5 0 5 10 1510

−3

10−2

10−1

100

SNR(dB)

SE

R

p1p2

Figure 3.13: Symbol detection performance of the proposed receiver.

0 1000 2000 3000 4000 50000

2

4

6

8

10

12

14

16

18

20

no of iterations

MS

E

(n=6)

(n=4)

Figure 3.14: NMSE vs no of iterations (BPSK).

86

mixed phase, the equalizer architecture that was proposed in Section 3.5 is considered. We

consider fourth order cumulant with n = 4, m = 0 and τ = 0 (refer to (3.5)) as a feature for

classification. The stochastic gradient of the AMC (Ψ1(y)) cost function for this case is given

by (3.44). The Bussgang cost function was considered for the symbol detection performance.

The performance of the AMC for the proposed system is shown in Figure 3.15. In Figure

3.15, Pc1 and Pc2 have the same meaning as that of Figure 3.11. It can be seen from the

figure that the proposed system performs well under Ricien fading.

−5 0 5 10 15 200.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pc1pc2

Figure 3.15: Performance of the AMC (Mixed phase Rician channel).

3.6.7 Summary of Results

In experiment 1, we analysed the performance of the receiver architecture proposed in Figure

3.2. From Figures 3.5 and 3.6 it can be seen that the proposed receiver improves the per-

87

formance of both symbol detection performance and AMC performance. The performance

of the AMC using the proposed architecture is better the performance of the AMC using

the switching equalizer proposed in [71] (refer to curves labelled pc2 and pc3 in Figure 3.5).

The reason for this improvement is the two fold diversity in AMC decision making offered

by the proposed architecture and the design of the B(z−1) filter (refer to section 3.4.3).

In experiments 2 and 3, we analysed the performance of the AMC under minimum phase

Rayleigh and Ricean channels. From Figures 3.7 and 3.8 it can be seen that the performance

of the AMC using the proposed architecture is better the performance of the AMC using the

switching equalizer proposed in [71] for the same reasons explained above. In experiment 4,

we analysed the performance of the AMC in classifying QAM’s. From Figure 3.9 it can be

seen that the AMC is not good in classifying QAM’s. The reason for this is that fourth order

cumulant features were used as a feature for classification. Fourth order cumulant features

are not capable of discriminating between QAM’s. From Figure 3.10 it can be seen that the

performance of AMC in classifying QAM’s is improved when sixth order cumulant features

were used. In experiments 5 and 6 we analyse the performance of the AMC under mixed

phase Rayleigh and Reician channels. Since the channel was mixed phase, receiver architec-

ture proposed in Figure 3.3 was used. Both forth order and sixth order cumulant features

were considered. In both cases AMC performance was better than the AMC proposed in [70].

However from Figures 3.11 and 3.12 it can be seen that sixth order cumulant feature offer

better classification compared to fourth order cumulants for the reasons explained before.

Also the convergence of the equalizer while using sixth order cumulant features was slower

88

(refer to Figure 3.14) because of the higher variance of the stochastic gradient. It should be

noted that when the channel is minimum phase, the receiver architecture presented in Figure

3.2 offers better performance when compared to the one proposed in Figure 3.3. However

the receiver architecture in Figure 3.2 cannot be applied to mixed phase channels.

3.7 Conclusion

In this chapter we proposed CR receivers where the performance of the AMC is also consid-

ered while adapting parameters of the blind equalizer. The proposed receivers thus enhance

the performance of both the AMC and symbol detection. nth order cumulant based AMC

was considered in this paper. The receivers were proposed for both minimum phase and

mixed phase multipath channel conditions. The performance of the proposed CR receivers

were analysed using simulations and yielded promising results.

Chapter 4

Multiuser AMC

4.1 Introduction

AMC in literature is mostly developed for classifying the signal transmitted by a single

user. Multiuser AMC, as the name suggests, simultaneously classifies signals transmitted

by multiple users. This chapter presents nth order cumulant and cyclic cumulant based

multiuser AMC. The idea of multiuser AMC using the fourth order cumulant based approach

is recently proposed in [46]. However, it assumes that the number of transmitting users is

known and all the users transmit at the same power over an AWGN channel, which is not

true in general in a cognitive radio setup. Also, the method in [46] does not identify the

exact modulation schemes used by the transmitting users but rather identifies the possible

family of modulation schemes that might be present in a frequency band.

89

90

In this chapter, a novel multiuser AMC based on normalized nth order cumulant and cyclic

cumulant has been proposed that can identify the exact modulation schemes used by multiple

transmitting users in a frequency band. The proposed multiuser AMC is developed for more

realistic multipath fading environments and no assumption on the transmission power of the

users is made. In the proposed multiuser AMC, multiple antennas for reception are used

whereas only a single receiving antenna was used in [46]. By using multiple antennas at

the receiver, the CR can identify the number of transmitting users which is generally not

possible while using a single antenna receiver. Also, by using multiple antennas, the CR

can harness the flexibility offered by traditional MIMO communication schemes apart from

classifying the signals from multiple users.

The normalized cumulant value based single user AMC was first proposed in [7]. The mul-

tipath channel drives the cumulant value of the transmitted signal to zero [7] and hence

severely affects the performance of the cumulants based AMC. In [70], [71] a robust cumu-

lant based single user AMC was developed for multipath fading channels. The approach in

[70] involves estimating the multipath channel and using the estimated channel information

to improve the performance of the AMC. The proposed multiuser AMC for a multipath

channel was motivated by the works reported in [70], [71] for single user AMCs. As shown in

a later section, the cumulant based multiuser AMC requires the knowledge of the multiuser

channel impulse response. However, channel knowledge or a pilot sequence for estimating

the channel is not available in a CR scenario. Therefore, one needs to estimate the channel

blindly. In blind channel estimation, the channel impulse response is estimated using only

91

the received data sequence with no knowledge of the transmitted or pilot sequence. Most of

the blind multiuser channel identification algorithms reported in the literature ([73]-[84] and

references therein) are batch processing algorithms. A high computational overhead involved

in computing the inverses of a large correlation matrix as a part of these algorithms is not

suited for CRs in rapidly varying channel conditions. To overcome this challenge, a recursive

channel estimation scheme that does not require taking inverses of a large correlation matrix

is proposed.

The block diagram representation of the proposed multiuser AMC is shown in Figure 4.1. It

consists of two major blocks: a signal processing block and a classifier block. In the signal

processing block, the normalized cumulant of the received signal and the multiuser channel

impulse response are estimated. Using this information, the normalized cumulant value of

each transmitting user is then estimated. These estimated cumulant values are finally fed to

the classification unit to identify the modulation schemes employed by the users. Detailed

explanations of all the components in the block diagram are presented in subsequent sections.

The chapter is organized as follows. In Section 4.2 the theory behind the nth order cumulant

based multiuser AMC is presented. The channel model and the assumptions made are

also presented in this section. In Section 4.3 the new recursive multiuser channel estimation

algorithm is presented. In Section 4.4 the final multiuser classification algorithm is presented.

In Section 4.5 extension of the nth order cumulant based multiuser AMC to cyclic cumulants

is presented. Simulation results are presented in Section 4.6 followed by the conclusion.

92

Classifie

r

ReceiverCumulantFeatureExtraction

(m x 1) receivedsignal

Blind ChannelEstimation

AMCDecision

Estimate thecumulant values of

the transmitting users

Signal Processing Block

Figure 4.1: Block diagram of the proposed multiuser AMC.

4.2 Channel Model and Preliminaries

In this section the underlying theory behind the proposed cumulant based multiuser AMC

is provided. We begin our discussion by presenting the channel model and the assumptions

made in this work.

4.2.1 Channel Model and Assumptions

In order to classify the signal from multiple users simultaneously a receiver should have

multiple antennas. Let l be the number of transmitting users and m be the number of

receiving antennas, and it is required that m > l. The above condition is required for the

blind estimation of the multiuser channel. Usually in a CR scenario, l is not known but there

are methods available in the literature for estimating l using multiple receiving antennas (see

for example [93]).

93

The multipath channel between the jth user and ith receiving antenna is denoted as hij(z−1)

and is given by

hij(z−1) = hij(0) + hij(1)z−1 + . . .+ hij(L)z−L, (4.1)

where L is the number of multipath components, z−1 is the unit delay operator, and hij(k)

(for k = 1, . . . , L) is the fading coefficient of the corresponding multipaths. The overall

system can now be represented by the following model

y(i) = x(i) + w(i), i = 0, 1, 2, . . . (4.2)

x(i) = H(z−1)s(i),

where s(i) is the l×1 transmission vector whose elements sk(i) (k = 1, 2 . . . l) denote the kth

transmitting user, y(i) is the m × 1 reception vector whose elements yk(i) (k = 1, 2 . . .m)

denote the received signal at the kth receiving antenna, w(i) denotes the m× 1 noise vector

and H(z−1) is given by

H(z−1) =

h11(z−1) . . . h1l(z

−1)

.... . .

...

hm1(z−1) . . . hml(z−1)

. (4.3)

Another representation of H(z−1) used in this chapter is

H(z−1) =L∑k=0

Hkz−k (4.4)

where Hk (for k = 1, 2 . . . L) is the m× l scalar matrix. This is also known as a MIMO FIR

channel. We make the following assumptions regarding the system model (4.3).

94

Assumption A41: Rank[H(z−1)] = l, for all complex z 6= 0, i.e. H(z−1) is irreducible.

Assumption A42: s(k) is zero mean, spatially independent and temporally white i.e,

E[s(k)s∗(k + i)] =

Il i = 0

O i 6= 0

, (4.5)

Non identity correlation matrices are absorbed into H(z−1), i.e., the transmission power of

the users can be different.

Assumption A43: w(k) is zero-mean Gaussian with

E[w(k)w∗(k + i)] =

σ2wIm i = 0

O i 6= 0

, (4.6)

where O in (4.5) and (4.6) is a zero matrix of appropriate dimension and σ2w is the noise

variance.

According to [78], assumption A41 is verified with probability one for any practical MIMO

wireless channel with reasonable spatial diversity and hence for our CR scenario this as-

sumption is valid. Assumption A42 implies that signals transmitted by two different users

are uncorrelated. Assumption A43 implies that that the noise vector is uncorrelated and

variance σ2w is known. In general σ2

w is not known but there exists a lot of methods for

estimating it (see for example [83], [84]).

95

4.3 Cumulants Based MAMC

In this section we present the basic theory behind higher order cumulants based multiuser

AMC. For a complex random signal v(k), the nth order moment is defined as

Rv(n,m)(k, τ) = E

[n∏j=1

v(∗)j(k + τj)

](4.7)


vector. In the above expression when n = 2 and m = 1 it becomes the standard auto

correlation function. The nth order cumulant function is defined as [9]

Cv(n,m)(k, τ) =∑Pn

F (p)

p∏j=1

Rv(nj ,mj)(k, τ) (4.8)

where the sum is over distinct partitions of the indexed set 1, 2 . . . n and F (p) = (−1)p−1(p−

1)!. For example, in the above expression when n = 4 and m = 0 we get the expression for

one of the fourth order cumulants given by

Cv40(k) = E[x4(k)]− 3E[x2(k)]2. (4.9)

The following are some of the properties of nth order cumulants that makes it an ideal

candidate for MAMC.

Property 1 Additive: Let x(k) and y(k) be two independent random processes. If z(k) =

x(k) + y(k), then the nth order cumulant value of z(k) is the sum of those for x(k) and y(k).

That is

Cz(n,m)(τ) = Cx(n,m)(τ) + Cy(n,m)(τ). (4.10)

96

Property 2 Scaling property: Let x = ay. Then the nth order cumulant value of x is |a|n

times the cumulant value of y.

In this paper we consider the following feature for classification

Cv(n,m)(τ) =Cv(n,m)(τ)[C2v(2,1)

]n/2 for n = 4, 6, . . . . (4.11)

The above feature is only the normalized version of the nth order cumulant. As mentioned

earlier multiple antennas are used for reception. Since multiple receiving antennas are used,

the received signal at the ith receiving antenna due to multiple transmitting users is given

by

yi(n) = hi1(z−1)s1(n) + . . .+ hil(z−1)sl(n) (4.12)

+wi(n).

Using the Properties 1 and 2, the value of the nth order cumulant of yi is given by

Cyi(n,m)(τ) = Cs1(n,m)(τ)γi1 + . . . (4.13)

+Csl(n,m)(τ)γil,

where

γij =L−1∑k=0

|hij(k)|n. (4.14)

Similarly, the second order cumulant for yi is given by

Cyi(2,1) = Cs1(2,1)ρi1 + . . .+ Csl(2,1)ρil + σ2w, (4.15)

97

where

ρij =L−1∑k=0

|hij(k)|2. (4.16)

Assumption A42 implies Csi(2,1) = 1 (for i = 1, . . . , l), i.e., transmitted signals are of unit

energy. It should be noted that non unit energy signals are converted to unit energy by

absorbing the scaling factor into the channel matrix H(z−1). Thus (4.15) can be written as:

Cyi(2,1) = ρi1 + . . .+ ρil + σ2w (4.17)

= ∆i + σ2w.

Then the normalized nth order cumulant of yi is given by

Cyi(n,m)(τ) =Cyi(n,m)(τ)

(Cyi(2,1) − σ2w)n/2

= (4.18)

=l∑

j=1

γij

∆n/2i

Csj(n,m)(τ).

Extending the above equation to all receiving antennasCy1(n,m)(τ)

...

Cym(n,m)(τ)

= (4.19)

=

γ11

∆n/21

. . . γ1l

∆n/21

.... . .

...

γm1

∆n/2m

. . . γml

∆n/2m

Cs1(n,m)(τ)

...

Csl(n,m)(τ)

.or

~Cy(n,m)(τ) = Bc~Cs(n,m)(τ). (4.20)

98

The cumulant value of the signal transmitted by different users can be obtained by solving

(4.20). The extracted cumulant features are then used for classification. The overall block

diagram of the MAMC is shown in Figure 4.1. The solution to (4.20) is given by

~Cs(n,m)(τ) = (BHc Bc)

−1BHc~Cy(n,m)(τ). (4.21)

In order to compute the Bc matrix, we require knowledge of the channel matrix H(z−1). In

a CR scenario, H(z−1) is not known and needs to be estimated blindly. In the following

section we discuss the blind estimation of H(z−1).

4.4 Blind Channel Estimation

Blind MIMO channel identification involves the use of second order statistics (SOS) and

higher order statistics (HOS). Blind MIMO channel identification algorithms in the literature

that use SOS can be broadly classified into three categories: whitening approach ([82]-[84]

and references there in), linear prediction ([77]-[80] and references there in) and subspace

approach ([73]-[76] and references there in). All the above methods are block processing

algorithms which involve computing the inverse of large correlation matrices. In this paper

we propose new MIMO FIR identification scheme which is computationally effective. The

proposed scheme is recursive and hence can track time varying channels. The proposed

algorithm is developed on the basic results from [82], [83].

When assumption A41 holds, there exists a finite degree left-inverse G(z−1) (not necessarily

99

unique) of H(z−1) [82], [83], such that

G(z−1)H(z−1) = Il, (4.22)

where G(z−1) is the l ×m matrix polynomial given by

G(z−1) =

nG∑k=0

Gkz−k, nG ≥ (2l − 1)L− 1. (4.23)

From (4.2) and (4.22) it can be seen that

G(z−1)x(i) = s(i). (4.24)

Also (4.2) can be expressed as

x(i) = H0s(i) + [H(z−1)−H0]s(i) (4.25)

Substituting s(i) from (4.24) in the second term on the RHS of (4.25) we obtain

x(i) = H0s(i) + [H(z−1)−H0]G(z−1)x(i)

or

A(z−1)x(i) = H0s(i) (4.26)

where

A(z−1) = Im − [H(z−1)−H0]G(z−1).

For future reference we write the matrix polynomial A(z−1) in the form

A(z−1) = Im +

nA∑k=1

Akz−k. (4.27)

100

It can be shown that nA ≥ 2lL− 1. Observe that from (4.2) and (4.26), we obtain

A(z−1)[y(i)− w(i)] = H0s(i). (4.28)

From (4.28), we can see that x(i) is an output of a Auto Regressive (AR) process whose

input is H0s(i). Also H0 is known as the instantaneous mixture channel and H0s(i) is the

instantaneous mixture of the transmitted sequence. Since (4.28) can be viewed as an AR

process, the polynomial A(z−1) can be estimated by minimizing the one step ahead prediction

error. In the following subsection we present a recursive algorithm to estimate the predictor

polynomial A(z−1). The algorithm was developed as a part of the MIMO blind equalizer in

[85]. In this paper we present the algorithm from a channel estimation point of view. The

theorems and proofs on convergence of the proposed recursive algorithm is similar to the

algorithm in [85] and hence not repeated. Once A(z−1) is estimated we can estimate the

FIR MIMO channel H(z−1) by solving the following equation.

A(z−1)H(z−1) = H0. (4.29)

The above equation can be easily obtained from (4.2) and (4.28). Later in this section we

discuss the method to solve (4.29) so that H(z−1) can be estimated.

4.4.1 Adaptive Estimation of A(z−1)

Define

θ∗ = [A1, . . . , AnA] (4.30)

101

The following algorithm is proposed to adaptively estimate θ∗ for i ≥ 1.

θ(i) = θ(i− 1)

+p(i)ϕ(i− 1)[y(i)∗ − ϕ(i− 1)∗θ(i− 1)]

+p(i)σ2w[(i− 1)θ(i− 1)− (i− 2)θ(i− 2)], (4.31)

where

ϕ(i− 1)T = [−y(i− 1)T , . . . ,−y(i− nA)T ], (4.32)

p(i) = p(i− 1)

−p(i− 1)ϕ(i− 1)ϕ(i− 1)∗p(i− 1)

1 + ϕ(i− 1)∗p(i− 1)ϕ(i− 1). (4.33)

and σw is an estimate of σw from assumption A3. Initial θ(0) is an arbitrary vector of finite

norm, and p(0) is an arbitrary positive definite matrix. The typical choice is p(0) = p0I ,

where p0 is a positive scalar. Without loss of generality we assume that y(k) = 0, x(k) = 0

and w(k) = 0 for k < 0. In the following, we give the heuristics behind the algorithm

(4.31)-(4.33). Note that (4.28) can be written in the form

x(i) = θ∗ϕx(i− 1) +H0s(i), (4.34)

where θ∗ is defined by (4.30), while

ϕx(i− 1)T = [−x(i− 1)T , . . . ,−x(i− nA)T ]. (4.35)

The minimum mean-square estimate of θ is obtained by minimizing the following cost func-

tion

J1 = E[(x(i)− θ∗ϕx(i− 1))(x(i)∗ − ϕx(i− 1)∗θ)]. (4.36)

102

Setting to zero the gradient of J1 with respect to θ∗ gives

E[ϕx(i− 1)x(i)∗] = E[ϕx(i− 1)ϕx(i− 1)∗]θ. (4.37)

Define

ϕw(i− 1)T = [−w(i− 1)T , . . . ,−w(i− nA)T ]. (4.38)

It is not difficult to see that by combining (4.2), (4.35) and (4.38), the vector ϕ(i) given by

(4.32) satisfies

ϕ(i) = ϕx(i) + ϕw(i) (4.39)

By using (4.39) and assumption A3, one can derive

E[ϕx(i− 1)ϕx(i− 1)∗] = E[ϕ(i− 1)ϕ(i− 1)∗]

−σ2wImnA

. (4.40)

Since by the assumption A3, w(i) and x(i) are independent sequences, we have

E[ϕw(i− 1)x(i)∗] = 0.

Also by virtue of the fact that w(i) is temporally white (see eqn (4.5)), it follows that

E[ϕ(i− 1)w(i)∗] = 0 (a.s.). By using the last two equations along with (4.39) we obtain

E [ϕx(i− 1)x(i)∗] = E [(ϕx(i− 1) + ϕw(i− 1))x(i)∗]

= E [ϕ(i− 1)x(i)∗]

= E[ϕ(i− 1)(x(i) + w(i))∗]

= E[ϕ(i− 1)y(i)∗]. (4.41)

103

Then by substituting (4.40) and (4.41) into (4.37) we get

E[ϕ(i− 1)y(i)∗] = E[ϕ(i− 1)ϕ(i− 1)∗

−σ2wImnA

]θ. (4.42)

Replacing expectations in the previous equation with sample averages, one can obtain

1

i

i∑k=1

ϕ(k − 1)y(k)∗ =1

i

i∑k=1

ϕ(k − 1)ϕ(k − 1)∗θ(i)

−σ2wθ(i).

or

i∑k=1

ϕ(k − 1)y(k)∗ =i∑

k=1

ϕ(k − 1)ϕ(k − 1)∗θ(i)

−σ2wiθ(i). (4.43)

where θ in (4.42) is replaced with θ(i) to signify the fact that it is the estimate derived based

on the observations up to sample time i. If in (4.43) i is replaced with i− 1, we have

i−1∑k=1

ϕ(k − 1)y(k)∗ =i−1∑k=1

ϕ(k − 1)ϕ(k − 1)∗θ(i− 1)

−(i− 1)σ2wθ(i− 1). (4.44)

Subtracting (4.44) from (4.43) yields

ϕ(i− 1)y(i)∗ = p(i)−1θ(i)− p(i− 1)−1θ(i− 1)

−σ2w[iθ(i)− (i− 1)θ(i− 1)], (4.45)

where

p(i)−1 :=i∑

k=1

ϕ(k − 1)ϕ(k − 1)∗ (4.46)

104

clearly

p(i)−1 = p(i− 1)−1 + ϕ(i− 1)ϕ(i− 1)∗. (4.47)

Then substituting (4.47) in (4.45) yields

ϕ(i− 1)y(i)∗ = p(i)−1[θ(i)− θ(i− 1)]

+ϕ(i− 1)ϕ(i− 1)∗θ(i− 1)

−σ2w[iθ(i)− (i− 1)θ(i− 1)]. (4.48)

At this point of algorithm construction we assume that asymptotically θ(i) ∼= θ(i− 1), and

in the last term on the RHS of (4.48), time sample index i is replaced with i − 1. We thus

obtain

ϕ(i− 1)y(i)∗ = p(i)−1[θ(i)− θ(i− 1)]

+ϕ(i− 1)ϕ(i− 1)∗θ(i− 1)

−σ2w[(i− 1)θ(i− 1)− (i− 2)θ(i− 2)]. (4.49)

From the above, (4.31) directly follows by replacing the unknown σw with its a-priori estimate

σw. Equation (4.33) is obtained by using the matrix inversion lemma in (4.47). In the

following subsection we describe a method to estimate H(z−1) using the estimated predictor

polynomial A(z−1).

105

4.4.2 Estimation of H(z−1)

Once the predictor polynomial A(z−1) is estimated, H(z−1) can be found by solving (4.29).

Another way of expressing (4.29) is

(HH0 H0)HH

0 [A(z−1)H(z−1)] = Il (4.50)

G(z−1)H(z−1) = Il.

It can be seen that the above equation is similar to (4.22) and the solution to the above

equation is provided in [78] and can be expressed as

Hi = [Rxx(i) +

nA∑p=1

Rxx(i+ p)Ap]H#H

0 (4.51)

for i = 1, 2 . . . L,

where H#0 = (HH

0 H0)HH0 and Rxx(p) is the signal correlation matrix at lag p. The method

for estimating Rxx(p) is provided in [83] and is given by

Rxx(p) = Ryy(p)− σ2wIm, (4.52)

where

Ryy(p) = E[y(i+ p)yH(i)]. (4.53)

The noise variance σ2w is assumed to be known in this paper. To estimate H0 (for (4.51)),

we consider the following equation:

A(z−1)x(i) = H0s(i). (4.54)

106

The above equation is known as the instantaneous mixture model and H0 can estimated

using any Blind Source Separation (BSS) algorithm ([91], [92]). BSS algorithms uses HOS

and estimate H0 up to a scaling and permutation ambiguity that is

H0 = DH0P, (4.55)

where H0 is the estimate of H0, D is the m×m diagonal scaling matrix and P is the l × l

permutation matrix. The permutation matrix P has the following properties

P = PH , and PPH = Il. (4.56)

From (4.51) and (4.55) it can be seen that all Hk (for k = 1, . . . L) are subjected to permu-

tation and scaling ambiguity. Therefore the estimated MIMO FIR channel is subjected to

scaling and permutation ambiguities and is given by

H(z−1) = DH(z−1)P, (4.57)

where H(z−1) is the estimate of H(z−1). In the following section we will show how to do

multiuser classification using the channel matrix estimate H(z−1) and the theory developed

in Section 4.3.

4.5 Classification Algorithm

In this section we present the step by step procedure for performing multiuser AMC. The

multiuser AMC is obtained by applying the estimated channel in Section 4.4 to the theory

developed in Section 4.3.

107

Step 1 Initialization: Given the received data y(i), pick the length of the predictor polyno-

mial nA. Since the channel order is not known, choose a large value nA so that the system

is over modeled. Estimate the noise variance σ2w using the method proposed in [83].

Step 2: Estimate the predictor polynomial A(z−1) using the adaptive equations (4.31)-

(4.33). The recursive algorithm is carried out even after the predictor coefficients have

converged so that it can track changes in the environment.

Step 3: Estimate the channel H(z−1) using (4.51). The estimated channel denoted by

H(z−1) is subjected to scaling and permutation ambiguity (refer to (4.57)).

Step 4: Calculate the Bc matrix in (4.20) using the estimated channel H(z−1). It should

be noted that the Bc matrix is not affected by scaling ambiguity in H(z−1) but is affected

by permutation ambiguity that is

Bc = BcP (4.58)

where Bc is the estimated Bc matrix.

Step 5: The cumulant features of all the transmitted sequences is obtained from (4.19)

using the estimated Bc matrix. Substituting (4.39) in (4.19) and using the properties of the

permutation matrix we get

~Cs(n,m)(τ) = P ~Cs(n,m)(τ) (4.59)

where ~Cs(n,m)(τ) is the estimated cumulant feature vector. The above equation indicates that

the extracted features are subjected to permutation ambiguity. Therefore we can classify

108

signals of multiple users up to a permutation ambiguity, i.e., we can identify the modulation

schemes of all the users in a frequency band but cannot determine which modulation scheme

a particular user is using. This permutation ambiguity can be easily resolved in a CR scenario

where some knowledge about the primary or licensed user is usually available.

Step 6: This is the final step where we classify the signals from multiple users using the

estimated cumulant feature vector ~Cs(n,m)(τ). We propose two methods to classify multiuser

signals using ~Cs(n,m)(τ).

Shortest Distance Method: Suppose there are M hypothesis or modulation schemes whose

cumulant values are µ1 . . . µM and l users. Then there are L1 = M l possible (l × 1)

feature combinations denoted as D = d1, . . . , dL1. We can find which feature combination

is transmitted by finding the feature which has the shortest distance to the estimated feature

vector ~Cs(n,m)(τ), that is

r = arg[ mini=1,...,L1

|| ~Cs(m,n)(τ)− di||] (4.60)

where ||(.)|| is the two norm of the vector.

Threshold method: In this method we classify each element of the (l × 1) vector ~Cs(n,m)(τ)

separately. We first arrange all the hypotheses or modulation schemes in ascending order of

their cumulant values, that is, µ1 < µ2 . . . < µM . Assuming each element ~Csi(n,m)(τ) (for

i = 1 . . . l) to be a Gaussian distribution with some mean µk and variance σ2 (equal variance

for all hypotheses) we come up with the following simple decision rule. Choose hypotheses

k for the ith element if (µk+µk−1)

2< ~Csi(n,m)(τ) < (µk+µk+1)

2with µ0 = −∞ and µM+1 =∞.

109

Step 7: Monitor the coefficients of the predictor polynomial A(z−1) which are adapted

recursively. If the channel conditions change drastically, then coefficients of A(z−1) change,

and hence we need to repeat Step 3 to estimate the new channel impulse response.

4.6 Extension to Cyclic Cumulants (CC)

The nth order cumulant based multiuser AMC presented in the previous section can be easily

extended to cyclic cumulants (CC). The reason for this is, CC exhibit the same additive and

scaling property as cumulants. In this section we briefly explain CC based multiuser AMC.

4.6.1 Cyclic Cumulants Features

For a complex random signal v(k), the nth order moment is defined as

Rv(n,m)(k, τ) = E

[n∏j=1

v(∗)j(k + τj)

](4.61)


vector. The nth order cumulant function is defined as [10]

Cv(n,m)(k, τ) =∑Pn

K(p)

p∏j=1

Rv(nj ,mj)(k, τ) (4.62)

where the sum is over distinct partitions of the indexed set 1, 2 . . . n andK(p) = (−1)p−1(p−

1)!. For a communication signal, the nth order cummulant functions exhibit periodicities and

110

hence can be expanded into a Fourier series,

Cv(n,m)(k, τ) =∑β

cβv(n,m)(τ)e(i2πβt) (4.63)

where cβv(n,m)(τ) is called the nth order CC and β is the nth order cyclic frequency [9]. The

following are some of the properties of cyclic cumulants that makes it an ideal candidate for

multiuser AMC. The normalized nth order CC is give by

Cβv(n,m)(τ) =

Cβv(n,m)(τ)[C2v(2,1)

]n/2 for n = 4, 6, . . . . (4.64)

4.6.2 CC Based Multiuser AMC

The relationship between the CC values of the l transmitting users and the CC values of the

m received signal is given by Cβy1(n,m)(τ)

...

Cβym(n,m)(τ)

= (4.65)

=

γ11

∆n/21

. . . γ1l

∆n/21

.... . .

...

γm1

∆n/2m

. . . γml

∆n/2m

Cβs1(n,m)(τ)

...

Cβsl(n,m)(τ)

.or

~Cβy(n,m)(τ) = Bc

~Cβs(n,m)(τ). (4.66)

where γij and ∆i are given by (4.14) and (4.16). The classification algorithm is similar to

that of nth order cumulant based MAMC, except that ~Cβs(n,m)(τ) is used as feature instead

111

~Cs(n,m)(τ).


In this section we demonstrate the performance of the proposed algorithm using computer

simulation. The performance measure considered is probability of correct classification

Pcc. Suppose that there are l users and M modulation schemes which are denoted as

Ω = Ω1, . . . ,ΩM. Then there are L1 = M l possible transmission scenarios denoted as

D = d1, . . . , dL1. The probability of correct classification Pcc is defined as

Pcc =

L1∑i=1

P (di|di)P (di) (4.67)

where P (di) is the probability that the particular transmission scenario occurs and P (di|di)

is the correct classification probability when scenario di has been transmitted. For this sim-

ulation we assume P (di) = 1L1, ∀i, where all scenarios are equally probable. Three different

experiments are performed and the results are summarized below. In all the experiments

the signal-to-noise ratio (SNR) is defined as

SNR =

∑mi=1 E(|xi|2)∑mi=1 E(|wi|2)

. (4.68)

4.7.1 Realistic MIMO Channels

For some of the experiments to follow we consider realistic MIMO multipath channels from

[62]. We assume that the receiving antennas are uniformly spaced. The m× l scalar impulse

112

response matrix Hk (for k = 1 . . . L) (refer to (4.4)) is chosen as follows

Hk = R12rHgrv (4.69)

where Hgrv is a (m× l) matrix whose elements are independent Gaussian random variables

and Rr is a m×m matrix given by

Rr = E[y(k)y(k)T ]. (4.70)

The elements of the correlation matrix depend on the spacing between the antennas and the

distribution of the angle of arrival. In order to simulate various channel conditions, we vary

the distance between the antennas and the distribution of the angle of arrival.

4.7.2 Fourth Order Cumulants

In this experiment we consider l = 2 transmitting users and m = 3 receiving antennas.

This is a common scenario for CR in commercial applications, where CR needs to identify

whether a primary user or malicious user is present in a frequency band apart from the

secondary user. Two modulation schemes are considered for this experiment and they are

Ω = BPSK,QPSK. Since two modulation schemes are considered, there are four possible

scenarios which are

D = [(BPSK,BPSK), (BPSK,QPSK), (QPSK,BPSK), (QPSK,QPSK)]

Each entry of the 3×2 channel matrix is considered to be a three tap FIR filter whose coeffi-

cients are chosen randomly. For the Pcc calculation, permutation ambiguity is tolerated. The

113

shortest distance method is considered for classifying the extracted features. For estimating

the cumulant values and channel impulse response T = 5000 samples are considered. For

the Monte Carlo simulation, 2000 trials were considered and the results are summarized in

Figure 4.2. In Figure 4.2 the curve labelled Pcc1 shows the performance of the AMC when

0 5 10 15 20 250.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

Pcc1Pcc2Pcc3

Figure 4.2: Performance of the multiuser AMC BPSK,QPSK(T=5000).

perfect knowledge of the channel is available. The curve labeled Pcc2 illustrates the perfor-

mance of the AMC using the proposed blind channel estimation scheme. The curve labeled

Pcc3 shows the performance of the AMC when no channel information is available, that is,

we do classification by calculating the normalized cumulant of the received signal with out

any further processing. Figure 4.2 shows that the proposed algorithm performs satisfactorily

under multipath fading channels.

114

4.7.3 Realistic MIMO Channel I: Two-user three-class

In this experiment we consider two-user three-class problem. Three modulation schemes

are considered for this experiment and they are Ω = BPSK,QAM(4), PSK(8). Since

three modulation schemes are considered, there are eight possible scenarios. Fourth order

cumulants was considered as a feature for classification. For the channel we assume the m×1

received antennas are uniformly spaced and the distance between each antenna is λ/2 (λ is

the wavelength). We assume the angle of arrival to be uniformly distributed over [0,2π].

Since the antennas are uniformly spaced and the angle of arrival is uniformly distributed,

the elements of the correlation matrix in (4.70) are given by

E[yi(k)yi+d(k)] = Jo(πd) (for d = 0 . . .m) (4.71)

where Jo is the zero order Bessel function. The Monte Carlo simulation results for this case

are shown in Figure 4.3. In Figure 4.3, Pcc1, Pcc2 and Pcc3 have the same meaning as Figure

4.2. Figure 4.3 shows that the proposed algorithm performs satisfactorily under realistic

MIMO multipath fading channel.

4.7.4 Realistic MIMO Channel II: Two-user three-class

In this experiment we consider two-user three-class problem. Three modulation schemes

are considered for this experiment and they are Ω = BPSK,QAM(16), PSK(8). Since


cumulants was considered as a feature for classification. For the channel we assume the

115

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

Pcc1Pcc2Pcc3

Figure 4.3: Performance under realistic MIMO channel I(Two-user three-class).

m× 1 received antennas are uniformly spaced and the distance between each antenna is λ/2

(λ is the wavelength). We assume the angle of arrival to be Gaussian distributed with mean

π/4 and variance 5. Since the antennas are uniformly spaced and the angle of arrival is

Gaussian distributed, the elements of the correlation matrix in (4.70) are given by

E[yi(k)yi+d(k)] = exp[1

2√

2(πdσ)2] (for d = 0 . . .m) (4.72)

where σ is the variance expressed in radians. The Monte Carlo simulation results for this

case are shown in Figure 4.4. In Figure 4.4, Pcc1 and Pcc2 have the same meaning as Figure

4.3. Figure 4.4 shows that the proposed algorithm performs satisfactorily under realistic

MIMO multipath fading channel.

116

0 5 10 15 20 250.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

Pcc1Pcc2

Figure 4.4: Performance under realistic MIMO channel II(Two-user three-class).

4.7.5 Fourth Order Cumulants: Classifying QAM’s

In this experiment we consider two-user three-class problem. Three modulation schemes are

considered for this experiment and they are Ω = QAM(4), QAM(16), QAM(64). Since


cumulants was considered as a feature for classification. For the channel we assume the

realistic MIMO channel from the previous experiment. The Monte Carlo simulation results

for this case are shown in Figures 4.5. In Figure 4.4, Pcc1 and Pcc2 have the same meaning

as Figure 4.3. It can be seen from the figure that the fourth order cumulant based multiuser

AMC performs poorly in classifying QAM’s. The reason for poor performance is that, the

theoretical fourth order cumulant values for QAM’s are close to each other and hence the

classifier is not able to distinguish it. For this reason we consider higher order cumulant and

117

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

Pcc1Pcc2Pcc3

Figure 4.5: Classification of QAM’s (Two-user three-class problem).

cyclic cumulant features.

4.7.6 Sixth Order CC: MIMO Flat Fading

In this experiment we consider a four-user five-class problem. The modulation scheme con-

sidered are Ω = BPSK,QAM(4), QAM(16), PSK(8), PSK(32). For the simulations we

consider CC of order six and zero delay vector (τ = 0). The channel considered was a re-

alistic flat fading channel with no multipath. The number of samples used for estimating

the CC is varied and the results are shown in Figure 4.6. From Figure 4.6 it can be seen

that the proposed AMC performs satisfactorily at low SNR. Also, the performance improves

when more number of samples are used to estimate the CC.

118

−4 −2 0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

T = 80x103

T = 50x103

T = 20x103

Figure 4.6: Performance of the multiuser AMC(Sixth order CC: MIMO flat fading).

4.7.7 Sixth Order CC: MIMO Multipath Fading I

In this experiment we consider the same four-user five-class problem. The channel considered

was multipath fading channel. Each entry of the channel matrix H(z−1) is modeled as a

realistic three tap MIMO FIR channel similar to the one considered in section 4.7.3 . The

results of the Monte Carlo simulations are shown in Figure 4.7. From Figure 4.7 it can be

seen that the proposed algorithm performs satisfactorily under multipath fading channels.

4.7.8 Sixth Order CC: MIMO Multipath Fading II

In this experiment we consider the same four-user five-class problem. The channel considered

was multipath fading channel. Each entry of the channel matrix H(z−1) is modeled as a

119

−4 −2 0 2 4 6 8 100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

multipath (25x103)

multipath (50x103)

Figure 4.7: Performance of the multiuser AMC (MIMO multipath fading I).

realistic three tap MIMO FIR channel similar to the one considered in section 4.7.4. The

results of the Monte Carlo simulations are shown in Figure 4.8. From Figure 4.8 it can be

seen that the proposed algorithm performs satisfactorily under multipath fading channels.


The performance of the proposed multiuser AMC was analysed using different modulation

schemes and realistic channel conditions. The channel conditions are varied by changing the

distance between the antennas and the distribution of the angle of arrival. For the initial four

experiments fourth order cumulants where considered as a feature for classification. From

Figures 4.2 and 4.4 it can be seen that the proposed multiuser AMC offers good performance

(achieves 85% correct classification at 10dB SNR)in classifying two user three class problem

120

−4 −2 0 2 4 6 8 100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

multipath (25x103)

multipath (50x103)

Figure 4.8: Performance of the multiuser AMC (MIMO multipath fading II).

under realistic multipath channel conditions. However it can be seen from Figure 4.5 that

the fourth order cumulant based multiuser AMC is not good in classifying higher order

QAM’s (achieves only 60% correct classification at 10dB SNR). For this reason we consider

higher order cumulant and cyclic cumulant features. Figure 4.6 illustrate the performance

of four user five class problem under realistic MIMO flat fading channel using sixth order

cyclic cumulants. The muliuser AMC achieves 95% correct classification at 0dB SNR The

performance is good due to the absence of multipath. The performance of the AMC under

different realistic multipath channel conditions for the same problem is shown in Figures 4.7

and 4.8. From the figures it can be seen that the cyclic cumulants based multiuser AMC

offers very good performance (achieves 85% correct classification at 0dB SNR). However

more samples are used to estimate the higher order cumulant features. The performance

121

of the multiuser AMC under multipath channels can be further enhanced by designing a

appropriate MIMO blind equalizer.

4.8 Conclusion

A novel cumulant and cyclic cumulant based multiuser AMC for fading channels was pro-

posed. The proposed multiuser AMC does not require any prior knowledge about the channel

and hence is suitable for CR applications. A computationally efficient blind multiuser chan-

nel estimation scheme, which forms an integral part of multiuser AMC is also proposed. The

channel estimation scheme is adaptive and hence can track rapid changes in the environment.

Simulations were performed under various scenarios and the proposed multiuser AMC yields

promising results.

Chapter 5

Combined MIMO Blind Equalizer and

Multiuser AMC

Due to the presence of multiple signals in a frequency band, any transmitted signal is sub-

jected to inter user interference (IUI). Also, the transmitted signals are subjected to inter

symbol interference (ISI) due to multipath fading. Since there is no training sequence avail-

able in a CR scenario, MIMO blind equalizers are used to remove IUI and ISI. Both second

order statistics (SOS) and higher order statistics (HOS) of the received signal are required

to achieve MIMO blind equalization. Since HOS are used, MIMO blind equalizers have the

potential to converge to a local minimum. Convergence of MIMO blind equalizer to local

minimum not only affects symbol detection performance but also the performance of the

multiuser AMC. Typically, blind equalizers are designed to improve the symbol detection

performance. In a CR, AMC is an important component and hence it is better to design

122

123

a blind equalizer that improves the performance of both AMC and symbol detection. Two

works in this direction are found in the literature. However, both works consider only a single

user AMC and single input single output (SISO) blind equalizer. The first work is in [70],

where a robust switching SISO blind equalizer is proposed that improves the performance

of single user AMC. In the second work [72], the weights of the SISO blind equalizer are

adapted in such a way that performance of the cumulants based single user is improved.

In this chapter we propose a MIMO blind equalizer that improves the performance of both

multiuser symbol detection and multiuser AMC that was proposed in the previous chapter.

In order to do so, we design a cost function that is related to the performance of the multiuser

AMC and then choose the parameters of the blind equalizer such that the cost function is

maximized. The overall block diagram of the proposed CR receiver is shown in Figure 5.1.

In the figure, we design the MIMO blind equalizer G(z−1) by considering the performance

of both symbol detection and multiuser AMC. For designing the blind equalizer we also use

the MIMO channel estimates provided by the multiuser AMC.

The chapter is organized as follows. In Section 5.2, we provide the channel assumptions

and background theory. In Section 5.3, the cost function related to the performance of

the MAMC is developed. In Section 5.4, we present the step by step procedure to design

the MIMO blind equalizer. Simulation results are presented in Section 5.5, followed by the

conclusion.

124

Blind Equalizer

G(z 1)

MultiuserAMC

SymbolDetection

Blind EqualizerDesign

Antennas

Proposed Cognitive Radio Receiver

User 1

User 2

User l

rm

r1


5.1 Background and Theory

As mentioned earlier, multiple receiving antennas are used for classifying signals from mul-

tiple users. Let l be the number of transmitting users and m be the number of receiving

antennas and it is required that m > l. Usually in a CR scenario, l is not known and needs to

be estimated using algorithms like the one proposed in [93]. The multipath channel between

the jth user and ith receiving antenna is denoted as hij(z−1) and is given by

hij(z−1) = hij(0) + hij(1)z−1 + . . .+ hij(L)z−L, (5.1)

where L is the number of multipath components, z−1 is the unit delay operator and hij(k)

(for k = 1, . . . , L) is the fading coefficients of the corresponding multipaths. The overall

system can now be represented by the following model

y(i) = x(i) + w(i), i = 0, 1, 2, . . . (5.2)

x(i) = H(z−1)s(i),

125

where s(i) is the l×1 transmission vector whose elements sk(i) (k = 1, 2 . . . l) denote the kth

transmitting user, y(i) is the m × 1 reception vector whose elements yk(i) (k = 1, 2 . . .m)

denote the received signal at the kth receiving antenna, w(i) denotes the m× 1 noise vector

and H(z−1) is given by

H(z−1) =

h11(z−1) . . . h1l(z

−1)

.... . .

...

hm1(z−1) . . . hml(z−1)

. (5.3)

Another representation of H(z−1) used in this paper is

H(z−1) =L∑k=0

Hkz−k (5.4)

where Hk (for k = 1, 2 . . . L) is the m× l scalar matrix. We make the following assumptions

regarding the system model (5.2).

Assumption A51: rank[H(z−1)] = l, for all complex z 6= 0, i.e. H(z−1) is irreducible.

Assumption A51 is valid for any practical wireless channel with reasonable spatial diversity.

Also we assume that the signals transmitted by various users are uncorrelated and each

element of the noise vector w(i) is zero mean white Gaussian with variance σ2w.

MIMO blind equalizers are used to recover the transmitted signal vector s(i) using only the

received signal vector y(i) with no training sequence and knowledge of the channel transfer

function H(z−1). As mentioned earlier, in this paper we design a blind equalizer that takes

into consideration the performance of the multiuser AMC. In order to do so, we consider the

following theorem from [82].

126

Theorem 1:[82] For the system given in (5.2) under Assumption A51 there exists (l ×m)

polynomial matrix G(z−1) (not unique) such that

G(z−1)H(z−1) = Il. (5.5)

Since G(z−1) is not unique, we can choose G(z−1) such that both symbol detection perfor-

mance and multiuser AMC performances are improved.

According to [83], G(z−1) in (5.5) can be factorized as follows

G(z−1) = G2(z−1)G1(z−1), (5.6)

where G2(z−1) is a l×m polynomial matrix and G1(z−1) is an arbitrary m×m polynomial

matrix with the condition det[G1(z−1)] 6= 0, for |z| ≥ 1. Since G1(z−1) is an arbitrary poly-

nomial matrix, we design G1(z−1) such that the multiuser AMC performance is improved.

To do so, we first construct a cost function Jamc which is related to the performance of the

multiuser AMC. We then choose the parameters of G1(z−1) such that Jamc is maximized. The

overall design of G1(z−1) can be viewed as the following constrained optimization problem

maxG1(z−1)

Jamc

s.t. det[G1(z−1)] 6= 0, for |z| ≥ 1 (5.7)

The rest of the paper is about formulating the cost function Jamc and solving for the poly-

nomial matrices G1(z−1) and G2(z−1).

127

5.2 Cost Function for the Multiuser AMC

In this subsection we develop the cost function Jamc for designing blind equalizer polynomials

G1(z−1) and G2(z−1). In order to do so, we need to understand the effect of the MIMO FIR

filter on the normalized cumulant values of the received received signal. From (4.18) one can

see that the normalized cumulant values of each received signal Cyi(n,m) (for i = 1, 2 . . .m)

is a weighted sum of the normalized cumulant values of all the transmitting users. The

weighting coefficients are given by wij =γij∆2

i(for i= 1,2. . . m, j= 1,2. . . l) (refer to (4.18)).

It can be easily shown that

|wij| = |γij∆2i

| < 1 (for i = 1, 2 . . .m, (5.8)

j = 1, 2 . . . l)

Since the magnitude of weighting coefficients are less than one, the magnitude of the nor-

malized cumulant values of the received signals are driven towards zero. The MIMO FIR

channel clusters all the cumulant features around zero. This clustering makes it hard for the

classifier shown in Figure 4.1 to distinguish between the features. Thus the coefficients of the

matrix polynomial G1(z−1) must be chosen in such a way that the features are unclustered.

For this reason we propose the following cost function

Jamc =m∑j=1

|Cx2j(n,m)|, (5.9)

where x2(i) = G1(z−1)y(i) and Cx2j(n,m) is the cumulant value of the jth component in the

vector signal x2(i). The above cost function maximizes the magnitude of the normalized

cumulant values of the signals so that the classifier can distinguish between the features.

128

5.3 Designing the Matrix Polynomials

In this section we propose the algorithm for designing the polynomials G1(z−1) and G2(z−1).

We also present the overall step by step procedure for designing the blind equalizer. The

cost function in (5.9) can be expressed as follows

Jamc =m∑j=1

|Cx2(j)(n,k)| = J1 + . . .+ Jm, (5.10)

where Ji = |Cx2(i)(n,k)| (for i = 1 . . .m). Now we choose G1(z−1) to be the diagonal matrix

given by

G1(z−1) = diagC1(z−1), . . . , Cm(z−1)

, (5.11)

where the elements of diagonal matrix are the FIR filters given by

Cp(z−1) = cp1z

−1 + . . .+ cpL1z−L1 (5.12)

for p = 1 . . .m

where L1 is the length of the filter and cij (for i = 1, . . . ,m,j = 1, . . . , L1) are the filter

weights. Since G1(z−1) is chosen to be a diagonal matrix, the constraint on G1(z−1) (refer

to (3.8)) implies that the FIR filter Cp(z−1) (for p = 1 . . .m) must be minimum phase. That

is the filter must not have any zeros on or outside the unit circle. Let us denote the weight

vector as cp = [cp1, . . . , cpL] (for p = 1, . . . ,m), then we use the following constrained gradient

search technique for updating the weights. Due to the constraint on G1(z−1) we restrict the

search space to the region where the weights form a minimum phase polynomial. Let cp(k)

denote the coefficient vector during the iteration k = 0, 1, 2, . . ..

129

• Step 1: For k = 0 initialize cp(0) to a random value from the search space.

• Step 2: For k = 1, 2, . . . calculate the output of the filter

x2p(n) =L∑

m=0

cp(m)yp(n−m) (5.13)

for p = 1 . . .m

• Step 3: Update the coefficient vector using the following equation

cp(k) = cp(k − 1)− µ∂Jp∂cp

for p = 1 . . .m (5.14)

where µ is step size. The weights are updated only if the new weights lies in the search

space. If not, repeat step 2.

• Step 4: If |Jp(cp(k))−Jp(cp(k−1))|Jp(cp(k−1))

< ζ terminate the iteration and go to step 5. If not,

repeat step 2, where ζ is chosen to be a small number less than one.

• Step 5: Calculate the output x2(i) using G1(z−1).

Now the cumulant features of the (m× 1) signal vector x2 are maximized and not clustered

around zero, therefore x2 is given to the MAMC shown in Figure 4.1 for classification. Let

us denote

F (z−1) = G1(z−1)H(z−1) =L+L1−1∑k=0

Fkz−k. (5.15)

It can be seen from Figure 4.1, that a blind MIMO channel estimator forms an integral part

of the multiuser AMC (refer to chapter 4 for a detailed explanation). Since x2(i) is fed to

130

the MAMC, we obtain the estimate of the polynomial F (z−1). Using the estimate of F (z−1),

we design G2(z−1) by solving the following equation

G2(z−1)F (z−1) = Il, (5.16)

where Il is the (l × l) identity matrix. Let us denote G2(z−1) as

G2(z−1) =L2−1∑k=0

G2kz−k, (5.17)

where G2k (for k = 0, 2 . . . (L2− 1)) are the l×m scalar matrix. Now the solution to (5.16)

is given by [82],[83]

[G21 G22 G23 . . . . . .

]=

[Il . . .

]S†, (5.18)

where S† is the pseudo inverse of the S matrix given by

S =

F0 F1 F2 . . . . . .

0 F0 F1 . . . . . .

......

...... . . .

0 0 0 F0 . . .

. (5.19)

5.4 Overall Classification and Equalization Algorithm

In this section we present the step by step implementation of the overall proposed system.

• Step 1: Given the (m × 1) received signal vector y(i) estimate the number of trans-

mitting users l using the method proposed in [93].

131

• Step 2: Choose the length of the matrix polynomials L1 and L2. Since the length of

the channel impulse response is not known, choose a sufficiently large length so that

the system is over modeled.

• Step 3: G1(z−1) is chosen to be a diagonal matrix given by (5.11) and its coefficients

are adapted using the gradient search algorithm given by (5.14).

• Step 4: The signal x2(i) is sent to the MAMC for classification. The multiuser AMC

provides an estimate of the matrix polynomial F (z−1).

• Step 5: Using the estimated F (z−1), design the (l ×m) matrix polynomial G2(z−1)

by solving (5.16). The output of G2(z−1) is used for symbol detection.


In this section, we demonstrate the performance of the proposed MIMO blind equalizer

using Monte Carlo simulation. Since the performance of the MAMC is also considered while

designing the blind equalizer, we analyze the performance of both the MAMC and symbol

detection. For the Monte Carlo simulation, 1,000 trials are considered.

5.5.1 Multiuser AMC Performance

In this subsection we demonstrate the performance of the MAMC using computer simulation.

The performance measure considered is the probability of correct classification Pcc. Suppose

132

that there are l users and M modulation schemes which are denoted as Ω = Ω1, . . . ,ΩM.

Then there are L1 = M l possible transmission scenarios denoted as D = d1, . . . , dL1. The

probability of correct classification Pcc is defined as

Pcc =

L1∑i=1

P (di|di)P (di) (5.20)

where P (di) is the probability that the particular transmission scenario occurs and P (di|di)

is the correct classification probability when scenario di has been transmitted. For the

simulation we assume P (di) = 1L1,∀i, where all scenarios are equally probable.

Two-user three-class problem (Fourth order cumulants)


The 3 × 2 channel matrix H(z−1) is modeled as a realistic three tap MIMO FIR channel

similar to the one considered in section 4.7.3. Three modulation schemes are considered for

this experiment and they are Ω = BPSK,QAM(4), PSK(32). Since three modulation

schemes are considered, there are nine possible scenarios. The Monte Carlo simulation results

are summarized in Figure 5.2. In Figure 5.2, the curve labeled Pcc2 shows the performance of

the multiuser AMC without the proposed blind equalizer. The curve labelled Pcc1 illustrates

the performance of the AMC using the proposed system.

133

−5 0 5 10 15 200.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

Pcc1Pcc2

Figure 5.2: Performance of the multiuser AMC (Two-user three-class problem).

Two-user four-class problem (Fourth order cumulants)


The 3 × 2 channel matrix H(z−1) is modeled as a realistic three tap MIMO FIR channel

similar to the one considered in section 4.7.4. Three modulation schemes are considered

for this experiment and they are Ω = BPSK,QAM(4), QAM(16), PSK(32). Since four

modulation schemes are considered, there are sixtenn possible scenarios. The Monte Carlo

simulation results are summarized in Figure 5.3. In Figure 5.3, the curve labeled Pcc2 shows

the performance of the multiuser AMC without the proposed blind equalizer. The curve

labelled Pcc1 illustrates the performance of the AMC using the proposed system.

134

−5 0 5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

Pcc1Pcc2

Figure 5.3: Performance of the multiuser AMC (Two-user three-class problem).

Four-user five-class problem (Sixth order cumulants)

In this experiment we consider l = 4 transmitting users and m = 5 receiving antennas. Each

entry of the 5×4 channel matrix H(z−1) is modeled as a realistic three tap MIMO FIR chan-

nel similar to the one considered in section 4.7.3. Five modulation schemes are considered

for this experiment and they are Ω = BPSK,QAM(4), QAM(16), PSK(8), PSK(32).

The Monte Carlo simulation results are summarized in Figure 5.4. In Figure 5.4, the curve

labeled Pcc1 shows the performance of the MAMC without the proposed blind equalizer. The

curve labelled Pcc2 illustrates the performance of the AMC using the proposed system.

135

−4 −2 0 2 4 6 8 100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pcc1pcc2

Figure 5.4: Performance of the MAMC (Four-user five-class problem) .

Four-user five-class problem (Realistic channel II)

This problem is the same as the previous one except four modulation schemes are considered.

The modulation schemes considered are Ω = BPSK,QAM(4), QAM(64), PSK(8), PSK(32).

The channel considered was a realistic MIMO multipath channel discussed in the previous

chapter (section 4.7.3). The Monte Carlo simulation results are summarized in Figure 5.5.

In Figure 5.5 the curves labelled Pcc1, and Pcc2 have the same meaning as that of Figure 5.4.

From Figures 5.3 - 5.4, it can be seen that the proposed MIMO blind equalizer enhances the

performance of the MAMC.

136

−4 −2 0 2 4 6 8 10

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

SNR

Pro

babl

ity o

f cor

rect

cla

ssifi

catio

n

pcc1pcc2

Figure 5.5: Performance of the MAMC (Realistic multipath channel II).

5.5.2 Symbol Detection Performance

In order to analyze the symbol detection performance, we consider the same 2-input/3-

output FIR random channel considered in the previous experiment. The normalized mean

square error (NMSE) and symbol error rate (SER) are taken as performance measures. The

simulation results are shown in Figure 5.6 and Figure 5.7. In Figure 5.6 and Figure 5.7 the

curve labeled sd1 illustrates the symbol detection performance of the proposed system. The

curve labeled sd2 illustrates the symbol detection performance of equalizer when the channel

impulse response is known (non-blind equalizer). From the figures it can be seen that the

symbol detection performance of the proposed system is close to that of the non-blind MIMO

equalizer.

137

0 1 2 3 4 5 6 7 8 9 10−12

−11

−10

−9

−8

−7

−6

nmse1nmse2

Figure 5.6: Symbol detection performance of the proposed system (NMSE Vs SNR).

0 2 4 6 8 1010

−5

10−4

10−3

10−2

10−1

100

SNR

SE

R

sd2sd1

Figure 5.7: Symbol detection performance of the proposed system (SER Vs SNR).

138


The proposed MIMO blind equalizer was tested under different scenarios. From the simu-

lation results it can be seen the MIMO blind equalizer improves the performance of both

multiuser AMC and multiuser symbol detection. Irrespective of the kind of channel and

the type of feature used, it can be seen from the simulation results that we get atleast 10%

improvement in performance at 0dB SNR and 15% improvement at higher SNR’s.

5.6 Conclusion

In this chapter we presented a MIMO blind equalizer that improves the performance of

both cumulant based multiuser AMC and symbol detection. The performance of proposed

equalizer was analyzed using computer simulations and yielded promising results.

Chapter 6

Conclusion and Future Work

The focus of this dissertation was to add the following special characteristics to a CR apart

from its usual capabilities: ability to track time varying SISO and MIMO channels, abil-

ity to classify multiple users in the frequency band, and ability to classify signals under

severe multipath channels. The following are some of the important contributions of this

dissertation:

• Developed novel SISO blind equalizers that can improve the performance of both sym-

bol detection and AMC. Blind equalizers are developed for both minimum phase and

mixed phase channel conditions. The blind equalizers are adaptive and hence can track

time varying channel conditions. The performance of the blind equalizer was analysed

using computer simulations under noise and realistic multipath channel conditions.

• A novel multiuser AMC that can simultaneously classify multiple users in the frequency

139

140

band was proposed. The multiuser AMC was based on cumulants and cyclic cumulant

features of the received signal. The proposed multiuser AMC was developed for severe

multipath channels. A novel recursive MIMO channel estimation scheme was proposed

which forms an integral part of the multiuser AMC. The performance of the multiuser

AMC was analysed under realistic channel conditions and noise.

• Developed a MIMO blind equalizer that improves the performance of both multiuser

symbol detection and multiuser AMC. This involved formulating a cost function that

is related to the performance of the newly developed multiuser AMC and adapting the

weights of the MIMO blind equalizer such that the cost function is optimized.

6.1 Future Work

In this section, we provide some insights on future research work. The following are some of

the our future research directions:

• The SISO blind equalizers presented in Chapter 3 was designed to enhance the per-

formance of cumulants based AMC. Cumulants based AMC was considered because of

its ability to classify a wide variety of modulation schemes with easy implementation.

This work can be extended to other feature based AMC’s. This will involve formulating

a cost function that is related to the performance of the chosen AMC and adapting the

weights of the equalizer such that the cost function is optimized. Depending on the

type of feature based AMC, it may be required to use nonlinear optimization techniques

141

like a genetic algorithm (GA).

• Multiuser AMC developed using cumulants and cyclic cumulant features can be ex-

tended to other features which exhibit scaling and additive properties.

• Multiple receiving antennas are used for multiuser classification. By using multiple

antennas at the receiver, the CR can harness the flexibility and advantages offered

by classical MIMO schemes apart from classifying signals from multiple users. The

proposed MIMO blind equalizer converts a multipath channel to a instantaneous mix-

ture channel. Methodologies can be developed to apply classical MIMO schemes like

beam forming, diversity combining, and spatial multiplexing to the instantaneous mix-

ture channel. Specifically, one can develop a multiantenna CR transceiver similar to

the one shown in Figure 5.1 using the signal processing components developed in this

dissertation.

• From Figure 5.1 it can be seen that the central component of a multiantenna CR

transceiver is the cognitive engine (CE). The CE is often referred to as the brain of the

CR. The CE makes decisions according the the current scenario, network objectives,

and past experience. It is necessary to develop a CE that can learn from the signal

processing components developed in this dissertation. The CE should also be able to

adjust the parameters of the proposed signal processing components according to the

mission objectives.

142

Recursive Blind MIMO Equalization and

channel estimation MIMO AMC

Band pass signal

processing

Rx

Cognitive Engine

MIMO Transmission

MIMO Receiver

Band pass signal

processing for transmitter

Tx

Combined blind equalization and AMC

Command signals

Data signals

VBLAST

MMSE

Zero Forcing

Receive Diversity

Beam Forming

Flex

ible

An

ten

na

Arr

ay

Transmit Diversity

Spatial Multiplex

Beam Forming

Policy Engine

Data Base

Optimization AlgorithmsProtocol stack

Artificial Intelligence

Figure 6.1: Block diagram of a multiantenna cognitive transceiver.

Chapter 7

Publications

The work presented in this dissertation is published in the following papers.

7.1 Conference Publications

1. B. Ramkumar and T. Bose, Combined blind equalization and classification of multiple

signals, Proc. 1st International Conference on Pervasive and Embedded Computing

and Communication Systems, pp. 339-344, Mar. 2011

2. B. Ramkumar, T. Bose, M. Radenkovic, and R. Thamvichai, Robust automatic mod-

ulation classification and blind equalization: A novel cognitive approach, Proc. SDR

Wireless Innovation Conference, pp. 108-113, Nov.-Dec. 2010.

3. B. Ramkumar, T. Bose, and M. Radenkovic, Robust cyclic cumulants based multiuser

143

144

automatic modulation classifier for cognitive radios, Proc. SDR Wireless Innovation

Conference, pp. 127-132, Nov.-Dec. 2010.

4. B. Ramkumar, T. Bose, and M. Radenkovic, Robust multiuser automatic modulation

classifier for multipath fading channels, Proc. IEEE DYSPAN, Apr. 2010.

5. B. Ramkumar, T. Bose, and M. Radenkovic, Combined blind equalization and auto-

matic modulation classification for cognitive radios, Proc. IEEE 13th DSP Workshop

and 5th SPE Workshop, pp. 172-177, Jan. 2009.

6. M. S. Radenkovic, T. Bose, and B. Ramkumar, Blind adaptive equalization of MIMO

IIR channels, Software Defined Radio Technical Conference and Product Exposition,

Oct. 2008.

7. B. Ramkumar, T. Bose, J. H. Reed, and M. S. Radenkovic, Combined blind equalization

and automatic modulation classification for cognitive radios Under MIMO environment,

Software Defined Radio Technical Conference and Product Exposition, Oct. 2008.

7.2 Journal Papers

1. B. Ramkumar, T. Bose, and M. S. Radenkovic, Robust multiuser automatic modula-

tion classification and blind equalization, In preparation to be submitted to a signal

processing journal.

145

2. B. Ramkumar, T. Bose, and M. S. Radenkovic, Robust automatic modulation classi-

fication and blind equalization: novel cognitive receivers, Accepted for publication in

Springer Journal on Analog Integrated Circuits and Signal Processing, Nov. 2011.

3. M. S. Radenkovic, T. Bose, and B. Ramkumar, Blind adaptive equalization of MIMO

systems: New recursive algorithms and convergence analysis, IEEE Trans. Circuits

and Systems, Part-I, vol. 57, no. 7, July 2010.

4. B. Ramkumar, Automatic modulation classification for cognitive radios using cyclic

feature detection, IEEE Circuits and Systems Magazine, vol. 9, no. 2, May 2009.

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