investigation of carrier frequency o set estimation

Ram Sunil Kanumalli

Investigation of Carrier Frequency OsetEstimation Techniques for Unique Word

(UW) - OFDM Systems

Masterarbeit

zur Erlangung des akademischen GradesDiplomIngenieur

Studium Information Technology

Alpen-Adria-Universität KlagenfurtFakultät für Technische Wissenschaften

Begutachter: Univ.Prof. Dr. Mario HuemerBetreuer 1: Dipl.Ing. (FH) Christian HofbauerBetreuer 2: Univ.Ass. Dipl.Ing. Alexander Onic

Institut für Vernetzte und Eingebettete SystemeEingebettete Systeme und Signalverarbeitung

Klagenfurt, im September 2012

Declaration of Honor

I hereby conrm on my honor that I personally prepared the present academic work andcarried out myself the activities directly involved with it. I also conrm that I have usedno resources other than those declared. All formulations and concepts adopted literallyor in their essential content from printed, unprinted or Internet sources have been citedaccording to the rules of academic work and identied by means of footnotes or otherprecise indications of source.

The support provided during the work, including signicant assistance from my super-visor has been indicated in full. The academic work has not been submitted to any otherexamination authority. The work is submitted in printed and electronic form. I conrmthat the content of the digital version is completely identical to that of the printed version.

I am aware that a false declaration will have legal consequences.

(Unterschrift) (Ort, Datum)

i

Abstract

Unique Word (UW) - Orthogonal Frequency Division Multiplexing (OFDM) is a novelOFDM signalling technique [1], where the usual cyclic prexes are replaced by determin-istic sequences called UW. It has proven to be an appealing alternative to conventionalCP-OFDM because of its predominant features. However, it also has to overcome sev-eral technical challenges. The most well-known issue among those is carrier frequencyoset (CFO) caused due to the mismatch of the oscillator's frequency and the Dopplershifts between the receiver and the transmitter. A small frequency oset will destroy theorthogonality among the subcarriers leading to a signicant degradation in the systemperformance. Hence, it is necessary at the receiver to estimate and compensate the CFOaccurately and keep its impact at a minimum.

One major aim of this work is the investigation of various CFO estimation techniquesfor an UW-OFDM system. A comparison is carried out which evaluates the performanceof time domain against frequency domain based estimators. In the time domain, UWs areutilized for estimating the CFO, whereas in the frequency domain, the data symbols ordedicated pilot tones are used. For the latter, a modied version of the UW-OFDM signalgeneration is derived in this work to enable the insertion of pilot symbols in the frequencydomain. Eects of various parameters like noise, channel characteristics, UW length, andthe amount of CFO on the performance of the estimators are evaluated. The performance ofthe CFO estimators is analyzed by the resulting mean squared error. The signalling conceptof UW-OFDM itself oers a unique opportunity to exploit the inherent redundancy causedby introducing UWs in the time domain by applying appropriate data estimators like e.g.,the LMMSE (Linear Minimum Mean Square Error) estimator or the BLUE (Best LinearUnbiased Estimator). Due to that, the inuence of noise on the frequency domain symbolscan be substantially reduced. Therefore, as a proof, the performance of the frequencydomain estimators utilizing the frequency domain pilot and data symbols after equalizationwith dierent possible equalizers is analyzed and compared.

Additionally, the sensitivity of the UW-OFDM system on the CFO is analyzed andcompared to that of the CP-OFDM scheme. Furthermore, the dependency of the perfor-mance on the specic design of the UW in the presence of CFO is investigated.

ii

Acknowledgments

I take this opportunity to express my deepest gratitude to my professor Dr. Mario Huemerfor allowing me to work on this thesis, providing nancial support, and the wholeheartedcondence he has shown in me. I am also very thankful for his scientic advices and enthu-siastic support. I nd myself very fortunate to have been a part of his group. His impacton me has reached far beyond this thesis, which has helped me to shape my career in avery positive way.

I am highly indebted to my supervisor Mr. Christian Hofbauer for his invaluable guid-ance and for monitoring my progress despite the distance. His eorts and time on revising,discussing, and rewriting the draft manuscripts have resulted in the successful completionof this thesis. His advice and support given throughout this thesis is unforgettable. I havelearnt how to look and analyze a research problem, as a result of countless intermediatediscussions with him. I also take immense pleasure in thanking Mr. Alexander Onic whohas been a source of encouragement and advice. He is the person who helped me in get-ting into the topic and laid foundation of this thesis with his strong support and interest.Finally, yet important, I would like to express my heartfelt thanks to my family for theirendless love and blessings, and to my friends for their help and support throughout myMasters.

iii

Contents

1 Introduction 1

2 Review of UW-OFDM 4

2.1 Generation of UW-OFDM Baseband Signal . . . . . . . . . . . . . . . . . . 52.2 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Multipath Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Additive White Gaussian Noise . . . . . . . . . . . . . . . . . . . . . 10

2.3 Receiver Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Classical Data Estimators . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Linear Bayesian Data Estimators . . . . . . . . . . . . . . . . . . . . 13

3 Carrier Frequency Oset 15

3.1 CFO Eect in Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 CFO Eect in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . 20

4 The Pilot Based UW-OFDM Transceiver Model 22

5 Carrier Frequency Oset Estimation 26

5.1 Acquisition Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Tracking Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.1 Time Domain CFO Estimation . . . . . . . . . . . . . . . . . . . . . 305.2.2 Frequency Domain CFO Estimation . . . . . . . . . . . . . . . . . . 34

6 Results and Discussion 38

6.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2 AWGN Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3 Time-Dispersive Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 526.4 CFO Impact on UW-OFDM and CP-OFDM . . . . . . . . . . . . . . . . . . 63

7 Conclusions 71

Bibliography 74

iv

Chapter 1

Introduction

In recent years, Orthogonal Frequency Division Multiplexing (OFDM) has become an im-portant modulation technique in the eld of telecommunications [2]. The receiver designarchitecture of OFDM is simple, providing a very ecient and simplied way of equalizingthe eects caused by frequency-selective multipath channels [3]. OFDM has been imple-mented in Europe for Digital Audio Broadcasting (DAB) and Digital Video Broadcasting(DVB) systems [4], in Japan for Multimedia Mobile Access Communications (MMAC),and in IEEE 802.11 and 802.16 Wireless standards [5]. Currently, a major research isbeing done on implementing the OFDM technique in cellular mobile communication sys-tems [6]. The most conventional approach to implement the OFDM technique is CyclicPrex OFDM (CP-OFDM), where the last samples of each OFDM symbol are copiedinto the guard interval [7]. Some alternative approaches to the conventional CP-OFDMhave been proposed in [1,8,9]. In [8], a similar structure to CP-OFDM, the known symbolpadded (KSP)-OFDM, where the random CP is replaced by known symbol (KS) sequences,was introduced. In case this known sequence is set to zero, KSP-OFDM coincides withthe zero padded OFDM (ZP-OFDM) [9]. In [1], a new OFDM signalling scheme, wherethe usual cyclic prexes are also replaced by deterministic sequences was proposed. Thisdeterministic sequence is often addressed as Unique Word (UW) and the scheme is referredto as UW-OFDM. The most important dierence between KSP- and UW-OFDM is thefact, that the UW is part of the DFT interval, whereas the KS is not. The insertion ofthe UW within the DFT interval requires to introduce some correlations in the frequencydomain, which can advantageously be exploited by the receiver to improve the bit errorratio (BER) performance [10], whereas the KSP-OFDM does not feature these correlations.

UW-OFDM has proven to be an eective signalling technique because of its many ap-pealing features: 1) The idea of substituting CPs by UWs in OFDM has the advantageof choosing the UW sequence such that it can eectively be used for synchronization (i.e.,timing oset estimation and CFO estimation) and channel parameter estimation purposes,thereby inherently serving as a pilot sequence [11]. 2) The redundancy present in the fre-quency domain can be exploited to reduce the inuence of noise on the data, thus leadingto a substantial improvement over the CP-OFDM in terms of data estimation [1].

Despite these favorable features, it also has to overcome several technical challenges.The most well-known issue among those is CFO caused due to the mismatch of the os-

1

CHAPTER 1. INTRODUCTION 2

cillators and the Doppler shifts between the receiver and the transmitter. Our simulationresults show that, similar to CP-OFDM, the UW-OFDM is also highly sensitive to theCFO. A small frequency oset induces two degrading eects: First of all, the orthogo-nality among the subcarriers is destroyed which leads to Inter Carrier Interference (ICI),resulting in a signicant degradation of the overall BER performance. Secondly, the signalis attenuated and rotated [12]. Therefore, it has become a key challenge at the receiverto estimate and compensate the CFO accurately and minimize its impact. For the pastfteen years there has been a lot of research carried out based on the subject of CFO esti-mation in CP-OFDM. A good overview on various CFO estimation algorithms can be foundin [1322]. The CFO estimation for CP-OFDM is well investigated and many sophisticatedalgorithms have been proposed. However, to the best of my knowledge, CFO estimationin UW-OFDM has not been investigated till now. In order to make the UW-OFDM tobe carried on a success line, it is necessary to focus on the CFO issue. This thesis mainlydeals with the investigation of various algorithms for CFO estimation, which can exploitthe information present in the UW. In addition, a comparison is made between the timedomain based CFO estimation utilizing UWs and frequency domain based CFO estimationutilizing traditional pilot tones. Besides this comparison, the sensitivity of these estimatorson the CFO and the channel characteristics is investigated. Furthermore, a comparison ofUW-OFDM and CP-OFDM regarding their sensitivity to CFO is carried out.

The remaining work is organized as follows: Chapter 2 introduces the UW-OFDMbaseband signal generation and lists the advantages of using this scheme over conventionalCP-OFDM. Furthermore, a system model which includes the transmitter, the channel, andthe receiver signal processing is also described in the same chapter. Chapter 3 presentsthe problem of CFO and also discusses the eects of CFO in time and frequency domain.Chapter 4 describes the modelling of the pilot based transceiver model which is used inthis thesis. Chapter 5 provides an overview of various CFO estimation techniques for UW-OFDM, which are in fact derived from currently available CFO estimation algorithms forCP-OFDM. As the pilot based UW-OFDMmodel will include a pilot sequence in both time(i.e., as UWs) and frequency (i.e., as pilot tones) domain, the time domain (cycliy prex-based) methods as well as the frequency domain (pilot tone aided and decision-directed)methods are discussed. Chapter 6 compares the performance of various CFO estimationtechniques for UW-OFDM that are described in the previous chapter. In addition, thesensitivity of the UW-OFDM system to the CFO, is analyzed and compared to that of theCP-OFDM scheme. Finally, Chapter 7 concludes this work.

CHAPTER 1. INTRODUCTION 3

Notation

The following notations are used throughout this work. Bold face letters with lowercase represent vectors (e.g., x,y, ...), and bold face letters with upper case indicate matri-ces (e.g., X, Y,...). Conjugation, conjugate transposition, and transposition are denotedby (·)∗, (·)H , and (·)T, respectively. Frequency domain variables are represented by plac-ing a tilde above the variable (e.g., x, X), which helps for dierentiating time domainand frequency domain variables. The expectation operator is denoted by E ·. Xi,j de-notes the element in the ith row and jth column of the matrix X. The inverse of thematrix X is denoted by X−1. In and 0n denote an n×n identity and zero matrix, respec-tively. X = diag x1, x2, · · · , xn represents a diagonal matrix with the diagonal elementsx1, x2, · · · , xn. The set of complex numbers is denoted by C. Finally, arg · and | · | denoteargument and magnitude, respectively, while Re · and Im · correspond to the real andimaginary parts of a complex number.

Chapter 2

Review of UW-OFDM

The concept of adopting an UW as an alternative to CP, cf. [23], [11], was rst employedin single-carrier with frequency domain equalization (SC/FDE) systems. Here, the intro-duction of UWs is a straightforward procedure, because the data symbols as well as theUWs are dened in time domain. The idea of substituting CPs by UWs has the advantageof choosing the UW sequence such that it can eciently be used for synchronization (i.e.,timing oset estimation and CFO estimation) and channel parameter estimation purposes,thereby inherently serving as a pilot sequence. Some relevant studies addressing the issuesof CFO tracking and channel estimation using the UW for SC/FDE systems can be foundin [11,2326].

The UW concept was then successfully adopted to OFDM, as recently proposed in [1].However, the introduction of UWs in OFDM is not straightforward, as the data symbolsare dened in frequency domain, whereas the UW is dened in the time domain. A goodcomparison between UW-OFDM and UW-SC/FDE is given in [27]. One advantage ofthe UW-OFDM scheme over conventional CP-OFDM is that the UW itself serves as apilot sequence, thus avoiding the need of pilot carriers. Fig. 2.1 reveals the dierencesbetween the conventional CP-OFDM and the UW-OFDM transmit symbol structures.From Fig. 2.1, some key dierences between an UW and a CP based OFDM system can

Figure 2.1: Transmit symbol structures for (a) CP-OFDM and (b) UW-OFDM [1].

be pointed out:

• The UW lies inside the Discrete Fourier transform (DFT) window, while the CP liesoutside the DFT interval.

4

CHAPTER 2. REVIEW OF UW-OFDM 5

• The CP is based on the transmitted data and thus random. Consequently, it variesfrom one OFDM symbol to the other. The UW, on the other hand, is deterministicand therefore the same for all OFDM symbols.

2.1 Generation of UW-OFDM Baseband Signal

In this section, the generation of an UW-OFDM baseband signal is explained. Similar tothe conventional OFDM systems (e.g., CP-OFDM), a vector of complex QAM/PSK datasymbols d ∈ CNd×1, which are dened in frequency domain, is considered. In general, zerosubcarriers are inserted at the DC position and at the band edges. Let the number of zerosubcarriers to be inserted be Nz. After inserting these zero subcarriers, the OFDM symbolin the frequency domain x ∈ CN×1 can be written as

x = Bd. (2.1)

B ∈ CN×Nd contains zero row vectors at the positions of zero subcarriers, unit row vectorsat the appropriate positions of the data subcarriers, and N represents the DFT windowlength. The time domain OFDM symbol x ∈ CN×1 is calculated as

x = F−1N x, (2.2)

where FN represents the N-point DFTmatrix with [FN ]k,l = e−j2πklN and k, l = 0, 1, · · · , N−

1. The above explained procedure is valid for any OFDM system. However, in UW-OFDM,a deterministic sequence called UW xu ∈ CNu×1 is introduced at the end of each time do-

main OFDM symbol. This can be formulated as x =[xTd xT

u

]T. Here, xd ∈ C(N−Nu)×1

denotes the vector containing the random time domain samples aected by the data d.Fig. 2.2 illustrates the generation of an UW-OFDM symbol. In the following, this genera-tion process is explained in detail.

To generate an UW-OFDM symbol with the desired properties, it is shown in [28] thata two step approach is benecial, or else the symbol energy will almost explode.Step 1: Generate an OFDM time domain symbol with a zero UW such that x =[xTd 0T

]T, where x can be obtained from the equation x = F−1

N x. Note that the vector xhere is dierent from the one dened in (2.1). The construction of x is explained shortlybelow.Step 2: Now, add the desired UW sequence xu to the vector x to obtain the nal UW-

OFDM time domain symbol x′

= x +[0T xT

u

]T.

In the following, the implementation of step 1 is explained in detail: In order to producea zero UW in the time domain, along with the data subcarriers and the zero subcarriers anew set of subcarriers called redundant subcarriers r = CNr×1 is introduced in frequencydomain. Furthermore, the number of redundant subcarriers Nr is set to Nu (i.e., the lengthof the UW xu). As a result, to solve the system of equations x = F−1

N x, the number ofdata carriers Nd has to be reduced by Nr, such that N = Nd +Nr +Nz. One idea wouldbe to take the rst Nd subcarriers and load it with data symbols, and then take the next


QAM Mapping

Serial/Parallel

Redundant

Subcarriers

Calculation, T

Assemble

OFDM Symbol, P, B

IFFT

Parallel/Serial

ADD

Unique Word

d~

d~

r~

x~

x

ux

x′

0

0

⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅Binary data

Figure 2.2: Block diagram of the time-discrete baseband representation of an UW-OFDMtransmitter.

Nr subcarriers and load it with redundant symbols. The frequency domain symbol wouldthen be constructed as

x = B

[dr

]. (2.3)

However, this in not recommended for practical systems. It turns out that these redundantsubcarriers should be placed on optimal locations over the available bandwidth (for alloca-tion, refer to [1]), or else the energy contribution of the redundant subcarriers on the meansymbol energy is very high compared to the data symbols, which is not desired. In order todistribute the data and the redundant subcarriers to their respective locations, a permuta-tion matrix P ∈ C(Nd+Nr)×(Nd+Nr) is introduced. The matrix P is selected in such a waythat the mean energy of the redundant subcarriers is minimum. Therefore, P optimallypermutes the data and the redundant subcarrier positions, cf. [1]. An optimal permutationmatrix for a specic setup will be given in chapter 6. After the distribution of the dataand the redundant subcarriers, the zero carriers are inserted with B ∈ CN×(Nd+Nr). Thiscan be seen in the following equation:

x = BP

[dr

]. (2.4)

For a better understanding, examples of generating P and B matrices are shown. For thatan exemplary system setup with a DFT size of N = 16, UW length Nu = 4, Nd = 8 datasubcarriers, Nr = Nu = 4 redundant subcarriers, and Nz = 4 zero subcarriers is chosen.One zero subcarrier is positioned at DC, and the others at the band edges. The datasymbol vector d is dened as

d =[d0 d1 d2 d3 d4 d5 d6 d7

]T, (2.5)

and the redundant symbols vector r as

r =[r0 r1 r2 r3

]T. (2.6)

Choosing the indices of the redundant subcarriers to be 1, 5, 10, 12, and inserting the


zero subcarriers at DC (i.e., 0) and at the band edges (i.e., 7, 8, 9), the frequency domainOFDM symbol x can be written as

x =[

0 r0 d0 d1 d2 r1 d3 0 0 0 r2 d4 r3 d5 d6 d7

]T. (2.7)

Consequently, (2.7) can easily be obtained by appropriate matrices P and B, respectively.As a rst step, P ∈ C12×12 shall permute the positions of the data and the redundantsubcarriers such that

P

[dr

]=[r0 d0 d1 d2 r1 d3 r2 d4 r3 d5 d6 d7

]T. (2.8)

P can easily be determined and (2.8) can be written as

r0

d0

d1

d2

r1

d3

r2

d4

r3

d5

d6

d7

=

0 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0

︸︷︷︸

P12×12

d0

d1

d2

d3

d4

d5

d6

d7

r0

r1

r2

r3

. (2.9)

The next step is to insert the zero subcarriers by applying a matrix B ∈ C16×12 such that

x = B[r0 d0 d1 d2 r1 d3 r2 d4 r3 d5 d6 d7

]T. (2.10)


With an appropriate matrix B, (2.10) immediately follows to

0r0

d0

d1

d2

r1

d3

000r2

d4

r3

d5

d6

d7

︸︷︷︸x16×1

=

0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

︸︷︷︸

B16×12

r0

d0

d1

d2

r1

d3

r2

d4

r3

d5

d6

d7

. (2.11)

Up till now it has only been established that redundant subcarriers are introduced inorder to fulll

F−1N BP

[dr

]=

[xd0

]. (2.12)

Equation (2.12) represents the frequency and the time domain relation of the UW-OFDMsymbol with zero UW. The redundant subcarrier symbols have to be determined now suchthat the zero UW is produced in the time domain. For that let M ∈ CN×(Nd+Nr) bedened as

M =

M11 | M12

−− −− −−M21 | M22

= F−1N BP, (2.13)

whereas M11 ∈ C(N−Nr)×Nd , M12 ∈ C(N−Nr)×Nr , M21 ∈ CNr×Nd , and M22 ∈ CNr×Nr(note that Nr = Nu). Considering (2.12) and (2.13), this leads to

M

[dr

]=

M11 | M12

−− −− −−M21 | M22

[ dr

]=

[xd0

]. (2.14)

From (2.14) it immediately follows that

M21d + M22r = 0 (2.15)

and consequentlyr = −M−1

22 M21d. (2.16)


Dening T = −M−122 M21 (T ∈ CNr×Nd), (2.16) can then be written as

r = Td. (2.17)

From (2.17) it becomes obvious that the vector r is just a linear function of the data d.Therefore, (2.12) can be expressed as

x = F−1N BP

[dr

]= F−1

N B

[IT

]d =

[xd0

]. (2.18)

It is worth to mention that the data subcarriers in combination with the redundantsubcarriers form a systematic complex valued codeword c ∈ C(Nd+Nr)×1 given by

c = P

[dr

]= P

[IT

]d = Gd, (2.19)

whereas G ∈ C(Nd+Nr)×Nd can be interpreted as the code generator matrix. G introducescorrelations among the frequency domain vector x, which can advantageously be used toimprove the bit error rate (BER) performance, cf. [10].

In order to achieve the nal OFDM transmit signal, the time domain UW xu is addedto the zero-UW OFDM symbol x obtained in (2.18). Hence, the transmit UW-OFDMsymbol x

′follows to

x′

= x +[0T xT

u

]T. (2.20)

The time domain vector x′can also be written as

x′

= F−1N (x + xu) , (2.21)

where xu ∈ CN×1 represents the frequency domain version of the time domain UW xudened as

xu = FN

[0T xT

u

]T. (2.22)

From this it can be noticed that, although the UW in time domain is appended at theend of the data, in frequency domain it spreads across the entire spectrum, inuencing thedata d as well.By using (2.18), (2.19), and (2.21), the time domain UW-OFDM baseband signal x

′can

be formulated asx′

= F−1N

(BGd + xu

). (2.23)

In the next sections, the system model including the channel and the receiver side isdescribed in detail.


2.2 Channel Model

In a time dispersive environment, the receiving antenna receives multiple copies of thetransmitted signal from dierent transmission paths, where each path shows dierent prop-agation delays and attenuations. The two fundamental parameters of a multipath channelare the coherence time and the coherence bandwidth, which are inversely proportionalto the Doppler spread and the delay spread, respectively [29]. Depending on the symbolperiod and the bandwidth of a signal, the fading is categorized into four types. Whenthe signal bandwidth is considerably larger than the coherence bandwidth of the channel,the channel is considered as frequency-selective, otherwise, as frequency-at. If the burstperiod is larger than the coherence time of the channel, then the channel is time-varying,otherwise time-invariant. In this thesis, the channel is always assumed to be frequency-selective and time-invariant.

2.2.1 Multipath Propagation

The multipath channel is modelled as a Finite Impulse Response (FIR) lter with theChannel Impulse Response (CIR)

h = [h0 h1 h2 ... hNk−1]T, (2.24)

where h ∈ CNk×1. Each delay tap hk is modeled as a complex Gaussian random variablehaving zero mean. All the taps are considered to be statistically independent and havingan exponential decaying power prole [30]. In order to completely eliminate Inter SymbolInterference (ISI), the length of the channel Nk and the UW length Nu must satisfy thecondition Nk ≤ Nu. The CIR is assumed to be constant for one OFDM burst (typicallya burst consists of a preamble and several OFDM symbols). This assumption seems rea-sonable, because in many applications that employ OFDM, the coherence time is largecompared to the chosen OFDM burst duration.

2.2.2 Additive White Gaussian Noise

In addition to the multipath, the transmitted signal is also aected by Additive WhiteGaussian Noise (AWGN). Each element in the noise vector n ∈ CN×1 is assumed to bea zero-mean complex Gaussian random variable, and all the samples are considered to beuncorrelated resulting in a covariance matrix σ2

nI.

On considering the multipath and AWGN eects, the received time domain OFDMsignal yr ∈ CN×1 at the receiver can be modelled as

yr = Hx′+ n, (2.25)

where H ∈ CN×N represents the channel matrix. Due to the UW, a cyclic structure isintroduced, which transforms the linear convolution of the channel with the transmittedsignal to a circular convolution. Therefore, the channel matrix H turns out to be a cyclic


convolution matrix and is given by

H =

h0 0 · · · 0 hNk−1 · · · h1

h1 h0. . . 0

. . ....

... h1. . .

.... . . hNk−1

hNk−1...

. . . h0 0 0

0 hNk−1 h1 h0. . .

......

. . .. . . . . .

. . .. . . 0

0 · · · 0 hNk−1 · · · h1 h0

. (2.26)

Note that the matrix H is a circulant matrix which is dened by its rst column[h0 h1 · · · hNk−1 0 · · · 0

]T.

Equation (2.25) holds only if there is no frequency or timing oset, thus representing anidealized system. However, in practical cases these eects will appear and will cause severedegradation of the OFDM system performance. These impairments will be investigated indetail in the next chapter.

2.3 Receiver Model

On considering (2.21) and (2.25), the received OFDM symbol yr follows to

yr = HF−1N (x + xu) + n. (2.27)

A well known property of the circulant matrix is that any circulant matrix can be diagonal-ized by the pre and post multiplication with the DFT and the IDFT matrices, respectively.Let us dene a matrix H ∈ CN×N such that

H = F−1N HFN , (2.28)

where H is a diagonal matrix whose diagonal elements represent the sampled frequency

response of the channel (i.e., H = diag

FN

[hT 0T

]T). By substituting H in (2.27),

yr can be rewritten as

yr = F−1N HFNF−1

N (x + xu) + n

= F−1N H (x + xu) + n. (2.29)

Applying the DFT on yr, the received frequency domain symbol yr ∈ CN×1 follows to

yr = FNF−1N H (x + xu) + FNn (2.30)

= H (x + xu) + FNn. (2.31)

By using (2.23), (2.31) can be written as

yr = H(BGd + xu

)+ FNn. (2.32)


It can be seen that the vector yr contains zero subcarriers, which can be excluded fromfurther processing. So, the vector yr can be downsized from N × 1 to (Nd +Nr) × 1 bymultiplying yr with BT, leading to yd ∈ C(Nd+Nr)×1 dened as

yd = BTyr. (2.33)

Thus, (2.32) can be written as

yd = HdGd + HdBTxu + v, (2.34)

where Hd ∈ C(Nd+Nr)×(Nd+Nr) is a downsized version of H dened as Hd = BTHB. Thematrix Hd contains the channel frequency response coecients only at the data and at theredundant subcarrier positions. v ∈ C(Nd+Nr)×1 is a noise vector dened as v = BTFNnwith zero mean and a covariance matrix Nσ2

nI.

From the above equation it can be observed that the received symbol yd still containsthe inuence of HdB

Txu. Since the UW is known to the receiver, its inuence can easily beremoved by just subtracting HdB

Txu from yd, assuming that the channel or an estimateof the channel is known. The resulting vector y turns out to be a linear model, and isgiven by

y = HdGd + v. (2.35)

Based on this linear model, several linear estimators are introduced in the next sections.Whether the presented equalizers have much practical aspect or not, they all give anillustrative view on how they exploit the correlations in the frequency domain introducedby the matrix G. More detailed information can be found in [10]. Fig. 2.3 depicts thebasic block diagram of the baseband receiver model for UW-OFDM.

Serial/Parallel

⋅⋅⋅⋅⋅⋅

⋅⋅⋅Binary data

FFT

⋅⋅⋅

Exclude zero

subcarriers, BT

Subtract UW

Influence

Linear Data

Estimation, E

Parallel/Serial

QAM Demapping

⋅⋅⋅ry

ry

ry~

dy~ y~ d

~

Figure 2.3: Baseband Receiver Model for UW-OFDM.

2.3.1 Classical Data Estimators

Let us consider a linear unbiased estimator of the formd = Ey, (2.36)

where E ∈ CNd×(Nd+Nr) represents the equalizer. As the estimator is assumed to beunbiased, it has to satisfy the relation

E[d]

= E [Ey] = EHdGd = d. (2.37)

This leads to the following Zero Forcing (ZF) criterion for linear equalizers:

EHdG = I. (2.38)


Channel Inversion Equalizer

A straightforward linear zero forcing equalizer that can be applied to the UW-OFDMsystem is the Channel Inversion (CI) equalizer given by

ECI =[I 0

]P−1H−1

d . (2.39)

This diagonal matrix equalizer just inverts the channel frequency response, and does notexploit the correlations among the subcarriers. Note that the CI equalizer is the opti-mum data estimator in CP-OFDM, as the frequency domain data in CP-OFDM does nothave any correlations among them. In contrast, the UW-OFDM has correlations in fre-quency domain, but the CI equalizer does not exploit these, resulting in a sub-optimumZF solution.

Best Linear Unbiased Estimator

Based on the linear model given in (2.35), an optimum ZF equalizer, the Best LinearUnbiased Estimator (BLUE) [31], follows to

EBLUE =(GHHH

d HdG)−1

GHHHd . (2.40)

Since the noise vector v is assumed to be Gaussian with a covariance matrix

Cvv = E[vvH

]= Nσ2

nI, (2.41)

the BLUE is also the Minimum Variance Unbiased (MVU) estimator whose error covariancematrix is given by

Cee = Nσ2n

(GHHH

d HdG)−1

. (2.42)

Note that, for UW-OFDM EBLUE is a full matrix, whereas for CP-OFDM it is a diago-nal matrix, which is equivalent to the CI equalizer. In contrast to the CI equalizer, itsimplementation leads to a substantially higher computational complexity, as it requires aNd×Nd matrix inversion. In [10], a complexity reduced version of this estimator is derived.

2.3.2 Linear Bayesian Data Estimators

All before mentioned estimators are based on classical data estimation where the elementsof a data vector are assumed to be deterministic but unknown constants. The LinearMinimum Mean Square Error (LMMSE) estimator is based on the Bayesian approach,where the data vector is assumed to be a realization of a random vector [31]. The theoryof Bayesian estimators yields another equalizer for UW-OFDM. Considering the modelin (2.35) as a Bayesian linear model, where d is the realization of a random vector withcovariance matrix Cdd = σ2

dI and v is a noise vector with zero mean and a covariancematrix Cvv = Nσ2

nI, the LMMSE equalizer is given by

ELMMSE = WH−1d . (2.43)

Here, W is a Wiener smoothing matrix given by

W = GH

(GGH +

Nσ2n

σ2d

(HHd Hd

)−1)−1

. (2.44)


ELMMSE can be interpreted as applying the CI equalization H−1d at rst, followed by a

Wiener smoothing operation W. The Wiener smoothing operation exploits the correlationsamong the subcarriers introduced at the transmitter side and minimizes the inuence ofthe noise on the received data. The estimator in (2.43) can also be written as

ELMMSE =

(GHHH

d HdG +Nσ2

n

σ2d

I

)−1

GHHHd , (2.45)

with the error covariance matrix expressed as

Cee = Nσ2n

(GHHH

d HdG +Nσ2

n

σ2d

I

)−1

. (2.46)

The estimator dened in the latter equation requires less computational complexity com-pared to the former one. It is obvious that at higher SNRs (i.e., σ2

n → 0), ELMMSE

is equivalent to EBLUE. Comparing the three equalizers described in (2.39), (2.40) and(2.45) in terms of computational complexity, the following order holds:

ELMMSE > EBLUE > ECI. (2.47)

Similarly, when compared in terms of performance, they have again the same order:

ELMMSE > EBLUE > ECI. (2.48)

Chapter 3

Carrier Frequency Oset

A well-known and critical issue of all OFDM systems is the presence of carrier frequencyoset (CFO). In this chapter the causes and eects of CFO are analyzed in detail. Besidesthe CFO, there is a need of timing synchronization at the receiver, as the arrival time ofan OFDM symbol is unknown. The main goal in the timing synchronization is to detectthe instant at which the rst sample of an OFDM symbol is received so as to place theDFT window accordingly. In this thesis, only the frequency oset is considered, therefore,an ideal timing synchronization is assumed.

In an OFDM system, the data is transmitted on orthogonal subcarriers in parallelthrough the channel. However, all the subcarriers are orthogonal if and only if the trans-mitter and the receiver are operating at the same frequency. In this case the subchannelspectral component is zero at the frequency positions of all other subchannels. Fig. 3.1exemplarily shows the individual spectrum of 4 orthogonal subcarriers, whereas each sub-carrier is loaded with a data symbol. Any mismatch of the oscillators between the trans-

0 50 100 150 200 250 300 350 400 4500

0.5

1

1.5

Frequency

|s(f

)|2

Figure 3.1: Frequency response of 4 orthogonal subcarriers.

mitter and the receiver, or a Doppler eect caused due to the movement of receiver ortransmitter, will lead to CFO. The causes of a frequency drifting in the oscillators canbe due to temperature, humidity, electromagnetic interference, aging, and pressure. Usu-ally, the oscillator stability is measured in parts-per-million (ppm) (1ppm = 10−6). It is

15

CHAPTER 3. CARRIER FREQUENCY OFFSET 16

common that the stability of a typical mobile phone crystal varies between 2 ppm and 12ppm [32], which is equivalent to a frequency shift of 4 to 24 kHz for a carrier frequencyof 2 GHz. The frequency oset due to the Doppler eect mainly depends on the directionof the movement of the receiver/transmitter with respect to the direction of the arrival ofthe incoming signal and the velocity of the transmitter/receiver. The frequency shift fdintroduced by the receiver movement is given by

fd =v

cfccos(θ), (3.1)

where v is the velocity of the receiver, c is the speed of light, fc is the carrier frequency,and θ represents the angle of arrival. For example, a mobile receiver moving towards thetransmitter with a speed of 100 km/h will experience a frequency shift of 185 Hz at acarrier frequency of 2 GHz. The frequency shift in a typical mobile environment variesbetween 1 Hz to a few hundreds of Hz. Note that the amount of CFO due to the Dopplerspread is very low compared to that of a mismatch between the oscillators.

Even a small frequency oset δf induces two degrading eects on the data:

• The orthogonality among the subcarriers is destroyed which leads to Inter CarrierInterference (ICI), resulting in a signicant degradation of the overall BER perfor-mance.

• The signal is attenuated and rotated.

In the presence of a frequency oset, the receiver cannot sample exactly at the centerfrequency of the subcarriers. Fig. 3.2 clearly visualizes the two main eects of CFO.Therefore, it has become a key challenge at the receiver to estimate the CFO accuratelyand keep its impact at a minimum.

0 50 100 150 200 250 300 350 400 4500

0.5

1

1.5

Frequency

|s(f

)|2

Frequency offset (δf )

Figure 3.2: Inter Carrier Interference caused due to the CFO.

The CFO could be many times larger than the intercarrier spacing. It is thus dividedinto an integer part and a fractional part. The integer part only leads to a shift of thesubcarriers by this integer number of positions. Hence, there will be no ICI, and theorthogonality among the subcarriers is preserved. The fractional part, however, causesICI which destroys the orthogonality among the subcarriers. It was shown in [19] that


the OFDM system performance degrades signicantly when the oset exceeds 4 - 5 % ofthe intercarrier spacing, and it was also shown that the SNR degradation is quadraticlyrelated to the relative oset. The following sections describe the eect of CFO in time andfrequency domain, respectively.

3.1 CFO Eect in Time Domain

Assuming that a frequency oset δf is present in the system (for the moment multipathand AWGN is neglected), due to this oset, the subcarrier frequencies are shifted by δf .This oset leads to a phase rotation of 2πδf t in time domain. This can be written as

y′(t) = x

′(t) ej2πδf t (3.2)

where x′(t) and y

′(t) are the transmitted and the received time continuous signal, respec-

tively. Thus, the received baseband spectrum will lie around δf instead of lying aroundf = 0 (i.e., the DC position).

Let ε be a relative frequency oset which is dened as the actual frequency oset δf asa fraction of the intercarrier spacing fic given by

ε =δffic. (3.3)

The intercarrier spacing fic can be expressed as

fic =fsN

=1

NTs=

1

TDFT, (3.4)

where fs and Ts denote the sampling frequency and the sampling time, respectively. Byusing the relation in (3.4), ε can be rewritten as

ε = δfNTs. (3.5)

In discrete time domain, the phase rotation factor e2πδf t can be expressed as e2πδfnTs withn = 0, 1, · · · , N − 1, when considering only one OFDM symbol. With (3.5), the discreterepresentation of the phase rotation factor is given by

ej2πδfnTs = ej2πεnN n = 0, 1, · · · , N − 1. (3.6)

Thus, the phase rotation for one OFDM symbol can conveniently be written in matrixform as

Λ′

= diag

[1 e

j2πεN · · · e

j2πε(N−1)N

]T

=

1 0 · · · 0

0 ej2πεN

. . ....

.... . .

. . . 0

0 · · · 0 ej2πε(N−1)

N

. (3.7)


Λ′ ∈ CN×N represents the phase rotation matrix that models the frequency oset eect on

one OFDM symbol. For simplicity, the phase accumulated by the previous OFDM symbolsis ignored (i.e., the oset matrix Λ

′includes only the phase accumulated by the current

OFDM symbol).

In order to derive a general oset matrix for an UW-OFDM symbol, the phase accu-mulated by the previous OFDM symbols and the extra appended UW (usually a UW isappended in front of every UW-OFDM burst to maintain the cyclic structure, cf. Fig. 2.1)should also be taken into account, leading to

Λl = ej2πε[N(l−1)+Nu]

N

1 0 · · · 0

0 ej2πεN

. . ....

.... . .

. . . 0

0 · · · 0 ej2πε(N−1)

N

, (3.8)

where l indicates the index of the lth transmitted OFDM symbol. Each diagonal elementof Λl ∈ CN×N represents the corresponding phase rotation factor for each sample of thetransmitted OFDM symbol.

Fig. 3.3 exemplarily illustrates the phase rotation of the UW xu for dierent values ofCFO. Here, all the samples of the UW are set to ones (i.e., xu = [1, 1, · · · , 1]T) and theUW shown in the gure belongs to the rst OFDM symbol. It can be observed that theUW is rotating along the unit circle in the presence of CFO, and when the CFO is absent,the samples of the UW lie at the point (1,0). It can also be noticed from the same gurethat the rst sample of the UW lies at various points (i.e., at dierent phases) for dierentCFOs. The reason can be explained as follows. The phase (or frequency oset) of the nth

sample can be calculated as the sum of the phase accumulated by the previous (n − 1)samples and the current phase of the nth sample. This can be expressed as

argnth sample

= e

2πε(n−1)N + e

2πεN . (3.9)

As the UW is at the end of an OFDM symbol, at the time of the rst sample of the UWthere is already a phase accumulated by the preceding data samples and this accumulatedphase increases with the increase in ε. This makes the rst sample of the UW start atdierent points for dierent ε values.

Fig. 3.4 reveals that there is a linear phase dierence between the transmitted timedomain signal and the signal inuenced by CFO. Furthermore, it can also be noticed thatthe slopes are increasing with increasing CFO.

By taking into account all the above described CFO eects, the lth received time domainOFDM signal zl with zero UW can be modelled as

zl = ΛlF−1N xl (3.10)


−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Imag

inar

y P

art

data1data2data3ε = 0ε = 0.1ε = 0.2ε = 0.4ε = 0.6

Figure 3.3: Phase rotation of the UW for dierent CFOs (ε = 0.1, 0.2, 0.4, and 0.6).

0 50 100 150 200 250 3000

0.5

1

1.5

Time [samples]

Pha

se [r

ad]

ε = 0.01

ε = 0.02

ε = 0.05

Figure 3.4: Eect of CFO on the phase of the time domain signal.


3.2 CFO Eect in Frequency Domain

After performing the DFT operation on (3.10), the lth received frequency domain OFDMsymbol zl can be written as

zl = F−1N ΛlF

−1N xl. (3.11)

By considering the properties of the circulant matrix discussed in the previous chap-ter, (3.11) follows to

zl = F−1N ΛlFN xl (3.12)

= Λlxl, (3.13)

where Λl represents the frequency domain version of the time domain oset matrix Λl.Note that in contrast to Λl, Λl is no longer a diagonal matrix, consequently, interferencesare introduced among the subcarriers by the non-diagonal elements along with a phaseshift caused by the diagonal elements.

The following derivation elaborates the eects of CFO in frequency domain. Con-sidering that the lth received time domain UW-OFDM symbol zl contains the sampleszl [0] , zl [1] , · · · , zl [N − 1], applying the DFT on zl yields the frequency domain OFDMsymbol zl with the samples zl [0] , zl [1] , · · · , zl [N − 1]. Mathematically, this can bewritten as

zl [k] = DFT zl [n] =

N−1∑n=0

zl [n] e−j2πkn

N

=

N−1∑n=0

1

N

N−1∑m=0

xl [m] ej2π(m+ε)n

N ej2πε[N(l−1)+Nu]

N e−j2πkn

N , (3.14)

where zl is the IDFT of the vector xl and l is the index of the received OFDM symbol.On reordering (3.14), it follows to

zl [k] =1

Nej2πε[N(l−1)+Nu]

N

N−1∑m=0

xl [m]N−1∑n=0

ej2π(m+ε−k)n

N

=1


N

N−1∑n=0

ej2πεnN

N−1∑m=0

xl [m] ej2π(m−k)n

N

=1


N xl [k]

N−1∑n=0

ej2πεnN +

1


N

N−1∑m=0,m6=k

xl [m]N−1∑n=0

ej2π(m−k+ε)n

N . (3.15)

It can be shown that [33]

zl [k] =sin(πε)

N sin(πεN )ejπε(N−1+2[N(l−1)+Nu])

N xl [k] + Il [k] . (3.16)


I [k] denotes the ICI on the kth subcarrier caused by the remaining subcarriers and is givenby

Il [k] =1


N

N−1∑m=0,m 6=k

xl [m]N−1∑n=0

ej2π(m−k+ε)n

N . (3.17)

This shows that the orthogonality of the subcarriers is lost in the presence of CFO, leadingto the ICI. Fig. 3.5 illustrates this eect for ε = 0.5. The rst term of (3.16) represents theamplitude and the phase distortion of the kth subcarrier caused due to the CFO. It can beobserved that this distortions become severe with increasing CFO.

0 50 100 150 200 250 300 350 400 4500

0.5

1

1.5

Frequency

|s(f

)|2

ε = 0.5

Figure 3.5: Inter Carrier Interference caused due to the CFO.

Chapter 4

The Pilot Based UW-OFDM

Transceiver Model

The traditional UW-OFDM signalling structure does not incorporate pilot subcarriers inthe frequency domain, as it is assumed that the UW will take over their estimation tasks.However, as one aim of this work is the comparison of time domain versus frequency do-main based estimators, without loss of generality, the traditional scheme has been slightlyadapted. Thus, the altered UW-OFDM structure will include a pilot sequence in bothtime (i.e., as UWs) and frequency (i.e., as pilot tones) domain. In practice, of course onlythe UW or the frequency domain pilots will exist at one time.

When briey recalling the traditional approach, an UW-OFDM symbol can be formu-

lated as x′

=[xTd xT

u

]T, where xd denotes a vector containing random values aected by

the data, and xu represents the UW vector which is added at the end of each OFDM sym-

bol. x′is constructed in two steps: 1) An OFDM symbol with a zero UW x =

[xTd 0T

]Tis generated. 2) The UW sequence xu is added to the vector x and the nal UW-OFDM

symbol follows to x′

= x +[0T xT

u

]T.

The point where the modied scheme diers from the conventional UW-OFDM is thegeneration of the vector x. The pilot subcarriers p ∈ CNp×1 are accommodated in the fre-quency domain by introducing them along with the data subcarriers d, and furthermore,to produce a zero UW in the time domain, redundant subcarriers r are introduced in thefrequency domain. The number of redundant subcarriers Nr is set to Nu. As a result,to solve the system of equations x = F−1

N x, the current number of data subcarriers Nd

should be reduced by Np +Nr, so that both the pilots and the redundant subcarriers canbe accommodated. In this work the pilot subcarriers are distributed according to the IEEE802.11a WLAN standard [34]. Moreover, the redundant subcarriers should be placed onoptimal locations over the available bandwidth (for details refer to Section 2.1). Note that,the optimal redundant subcarrier positions here in this modied scheme are dierent tothe ones in the traditional scheme, because the pilot carriers have occupied some of thetraditional redundant subcarrier positions.

To place the data, pilots, and redundant subcarriers on their respective locations, a per-

22

CHAPTER 4. THE PILOT BASED UW-OFDM TRANSCEIVER MODEL 23

mutation matrix P ∈ C(Nd+Nr+Np)×(Nd+Nr+Np) is introduced. The matrix P is optimizedin such a way that the energy contribution of the redundant subcarriers is minimum.After the distribution of d, r and p, the usual zero subcarriers are inserted with B ∈CN×(Nd+Nr+Np). The number of zero subcarriers Nz is calculated as N − (Nd +Nr +Np).The time domain and the frequency domain relationship of the modied UW-OFDM sym-bol with zero UW can be formulated as

F−1N BP

drp

=

[xd0

]. (4.1)

The redundant subcarriers r have now to be determined from (4.1) such that a zero UWis produced at the IDFT output. Dening the matrix M ∈ CN×(Nd+Nr+Np) as

M = F−1N BP =

[M11 M12 M13

M21 M22 M23

], (4.2)

whereas M11 ∈ C(N−Nr)×Nd , M12 ∈ C(N−Nr)×Nr , M13 ∈ C(N−Nr)×Np , M21 ∈ CNr×Nd ,M22 ∈ CNr×Nr , and M23 ∈ CNr×Np are appropriate sized sub-matrices. Using (4.1) and(4.2), the following relation is obtained:

r = −M−122 M21d−M−1

22 M23p. (4.3)

Note that, the redundant subcarriers in this approach depend on both the data and thepilot subcarriers, whereas in the traditional UW-OFDM system, the redundant subcarriersonly depend on the data. Dening the matrices T1 = −M−1

22 M21 and T2 = −M−122 M23

(T1 ∈ CNr×Nd and T2 ∈ CNr×Np), the above equation can conveniently be written as

r = T1d + T2p. (4.4)

It can be observed from (4.4) that r results from a linear combination of d and p. Using(4.1) and (4.4), this leads to

F−1N BP

IT1

0

d +

0T2

I

p

=

[xd0

]. (4.5)

The above equation can be rewritten as

F−1N B

[P1 P2 P3

] IT1

0

d +[P1 P2 P3

] 0T2

I

p

=

[xd0

], (4.6)

where P1 ∈ C(Nd+Nr+Np)×Nd , P2 ∈ C(Nd+Nr+Np)×Nr , and P3 ∈ C(Nd+Nr+Np)×Np are thesub-matrices constructed from the permutation matrix P. From (4.1) and (4.6) it followsthat

P

drp

=[P1 + P2T1 + P30

]d +

[P10 + P2T2 + P3

]p. (4.7)


By introducing the matrices G ∈ C(Nd+Nr+Np)×Nd and Gp ∈ C(Nd+Nr+Np)×Np dened asG =

[P1 + P2T1 + P30

]and Gp =

[P10 + P2T2 + P3

], (4.7) can be formulated as

c = P

drp

= Gd + Gpp, (4.8)

where c ∈ C(Nd+Nr+Np)×1 is considered as a complex valued code word. G and Gp cannow be interpreted as code generator matrices for the code word c.Therefore, the nal transmit symbol x

′(assuming without loss of generality that the UW

has been set to zero) can be written as

x′

= F−1N B

(Gd + Gpp

). (4.9)

Similar to (2.27), after experiencing multipath and AWGN, the received symbol yr ∈ CN×1

can be modelled asyr = HF−1

N B(Gd + Gpp

)+ n, (4.10)

where H ∈ CN×N and n ∈ CN×1 represent the circulant channel matrix and the noisevector, respectively. On performing the DFT operation on yr, the frequency domainsymbol yr can be written as

yr = HB(Gd + Gpp

)+ FNn. (4.11)

As yr contains zero subcarriers, they can be excluded from further processing, leading toa downsized vector yd = BTyr with yd ∈ C(Nd+Nr+Np)×1:

yd = HdGd + HdGpp + v. (4.12)

Here, Hd ∈ C(Nd+Nr+Np)×1 is a downsized version of H dened as Hd = BTHB. Thematrix Hd contains the channel frequency response coecients only at the data, pilotsand at the redundant subcarrier positions. v ∈ C(Nd+Nr+Np)×1 is a noise vector denedas v = BTFNn with zero mean and a covariance matrix Nσ2

nI. From the above equationit can be noticed that the received symbol yd contains the known portion HdGpp (if thechannel Hd is assumed to be known), and this portion can be subtracted to obtain thelinear model

y = yd − HdGpp (4.13)

= HdGd + v (4.14)

Based on this linear model, several linear estimators were introduced in the previous chap-ter (see Section 2.3), and the same procedures can be applied here for the data estimation.

By observing the result in (4.13) it can be concluded that this model is not practical forpilot based estimation, as it completely excludes them. However, for the CFO estimationalong with the data, the pilot symbols are also needed at the receiver side, and therefore,


a modied model is introduced where the data as well the pilots are taken into account.For this purpose it is better to treat pilots like random data (although they are in factdeterministic) and estimate the pilots along with the data. Then these estimated pilotsare used for the CFO estimation. For that reason, it is more convenient to introduce agenerator matrix G′ constructed from G and Gp such that the BLUE and the LMMSEequalizers discussed in Section 2.3 can be applied in a straightforward manner. Let thematrix G

′be given by

G′

=[

G Gp

]. (4.15)

Therefore, the received symbol yp ∈ C(Nd+Nr+Np)×1 containing both the pilots and thedata symbols as well as the redundant symbols is given in the form of the linear model

yp = HdG′[

dp

]+ v. (4.16)

Considering a linear unbiased estimator of the form[ dp]

= E′yp, (4.17)

where E′ ∈ C(Nd+Np)×(Nd+Nr+Np) represents the equalizer, the BLUE equalizer follows to

E′BLUE =

((G′)H

HHd HdG

′)−1 (

G′)H

HHd , (4.18)

with the error covariance matrix

Cee = Nσ2n

((G′)H

HHd Hd

(G′))−1

. (4.19)

Similar to (2.45), the LMMSE equalizer is given by

E′LMMSE =

((G′)H

HHd HdG

′+Nσ2

n

σ2d

I

)−1 (G′)H

HHd , (4.20)

with the covariance matrix expressed as

Cee = Nσ2n

((G′)H

HHd HdG

′+Nσ2

n

σ2d

I

)−1

. (4.21)

The CI equalizer described in (2.39) can be applied to this model as well. After the

equalization process the CFO can be estimated out of the obtained vector[

dT pT

]which will be discussed in the next chapter.

Chapter 5

Carrier Frequency Oset Estimation

It is a well known fact that OFDM systems are highly sensitive to CFO. A demodulation ofan OFDM signal in the presence of CFO leads to the loss of orthogonality between the sub-carriers causing ICI, thus leading to a signicant degradation in the BER performance [12].Therefore, an accurate frequency oset estimation and compensation is necessary to over-come this degradation. Numerous papers have been published so far on the subject of CFOestimation in CP-OFDM systems. However, to the best of my knowledge, CFO estimationin UW-OFDM has not been investigated till now. This chapter gives an overview of variousCFO estimation techniques for UW-OFDM, which are in fact derived from current CFOestimation techniques for CP-OFDM. In a real system, CFO estimation is usually carriedout in two phases [35]. First, there is an acquisition phase, where a rough CFO estimate iscomputed to align the local oscillator frequency to the received carrier frequency. This op-eration is normally carried out for each received OFDM burst. Then, in the second step, ane CFO estimate is computed, which is often referred to as tracking phase. The splittingof the estimation in two phases allows the design of the whole synchronization task witha high degree of freedom, thereby, at each phase a separate optimized algorithm can beused. This means that, during acquisition phase an optimized algorithm can be utilizedto cover a large acquisition range while neglecting the task in tracking phase, whereas inthe tracking phase the algorithm can be optimized to exhibit high tracking performance,as the large acquisition range is no longer required in this stage. As one aim of this thesisis to utilize the UWs for the CFO estimation, from here on, the focus is restricted to thene CFO estimation techniques. However, a brief introduction to the acquisition phase iscovered in the following section.

5.1 Acquisition Phase

In the acquisition phase a coarse CFO estimation is carried out. Furthermore, the coarseCFO, in general, is estimated in two steps [18]. In the rst step, the fractional part of theCFO is estimated and corrected, and in the second step, the integer CFO is estimated andcorrected. The entire acquisition process is normally done by utilizing a preamble thatis sent at the beginning of each burst (a burst is composed of a group of several OFDMsymbols), cf. [34]. The preamble consists of training sequences that are repeated in smallparts and will experience the same eects of the channel, but they dier in the phase shift

26

CHAPTER 5. CARRIER FREQUENCY OFFSET ESTIMATION 27

due to the frequency oset. Therefore, the oset can conveniently be estimated by mea-suring the phase dierence between two successive parts in frequency domain. This ideawas rst proposed by Moose in [14]. He used two identical OFDM training symbols andestimated the CFO out of them. However, the main drawback of this technique is that theacquisition range is limited to one half of the subcarrier spacing due to the discontinuityof the arctangent function used in the algorithm. It means that, if suppose ε is very closeto +0.5 or -0.5, then, due to the discontinuity of the arctangent and the noise, the esti-mated value might jump to -0.5 or +0.5, respectively, thus making the estimator a biasedone, and therefore useless in practice. In the same paper, he also described a strategy toincrease the acquisition range by using shorter training symbols. The range will be thenincreased on the one hand, but the estimates will have a larger variance compared to theones estimated with longer training symbols on the other hand. The reason for that is thereduced number of samples the average is built over. Another method that extends theacquisition range with reduced computational complexity and with improved accuracy wasproposed by Schmidl & Cox in [18]. Relevant literature similar to the above ideas can befound in [3640].

On considering (3.10), the lth received time domain UW-OFDM symbol zl having anUW xu can be written as

zl = ΛlF−1N (x + xu) (5.1)

= Λlx′, (5.2)

where x′is the transmitted signal and Λl is the frequency oset matrix as dened in (3.8).

To compensate the eect of CFO, let us dene a de-rotating matrix Γ which is given by

Γl = e−j2πε[N(l−1)+Nu]

N

1 0 · · · 0

0 e−j2πεN

. . ....

.... . .

. . . 0

0 · · · 0 e−j2πε(N−1)

N

, (5.3)

where ε is the estimated oset obtained through the coarse estimation. By de-rotating thevector zl with Γl, the coarse CFO can be compensated and the resulting vector rl ∈ CN×1

follows torl = Γlzl = ΓlΛlx

′. (5.4)

Assuming that the estimated oset is equal to the true oset (i.e., ε = ε), then ΓlΛl willbecome an identity matrix. Hence, the eect of the oset will be completely eliminated.However, in practice, there is always some residual oset left in the signal rl. This is wherethe tracking phase comes into play. The following section explains the tracking procedurefollowed by some CFO estimation techniques.

5.2 Tracking Phase

After the coarse frequency synchronization, some residual CFO will probably remain in thesignal. This residual CFO will vary in the presence of time-varying Doppler shifts. This


oset, in general, is in the order of 4 to 5 % of the intercarrier spacing, and this inducesICI and a phase rotation of data symbols (see Fig. 5.4). If this small oset is not compen-sated accurately, after a few OFDM symbols, the QAM/QPSK data symbols will rotateand may fall in a dierent quadrant of the constellation diagram, leading to a degradedsystem performance. Therefore, the oset should be estimated (tracked) continously andthe received data corrected accordingly. The tracking methods that are currently avail-able in the literature for CP-OFDM are broadly classied into three categories: those thatbase the estimation on the signal's statistical properties (e.g., cyclic prex-based (CPB)methods), those that base the estimation on pilot tone information (e.g., pilot tone-aided(PTA) methods), and decision-directed (DD)/data-carrier aided schemes. In traditionalUW-OFDM, since the UW is dened in the time domain, CPB methods can be appliedto this scheme. However, as the modied model incorporates pilot tones in the frequencydomain, PTA techniques and DD schemes are also discussed in this thesis.Before going into the details of these techniques, it is very important to notice that the CFOestimation algorithms for UW-OFDM are derived from currently available CFO estima-tion techniques for the CP-OFDM. As a consequence, only the CFO estimation techniquesfor the UW-OFDM are described in this chapter without having a pre-discussion of thealgorithm for CP-OFDM.

The following explains the basic tracking procedure that is being followed in an OFDMsystem [19]. Assume that the received signal rl obtained from (5.4) has some residualcarrier frequency oset ∆ε (i.e., ε − ε). In reality, this error may vary from symbol tosymbol. Therefore, this error has to be tracked and compensated continuously to avoidICI. Usually the tracking operation is done on a symbol-by-symbol basis, thus leading toa closed-loop structure as shown in Fig. 5.1 [19]. Here, rl(k) are the received time-domain

Loop Filter

)(krl )(kvl

kj l

e

επ ˆ2 ∆−

Further

Processsing

Error

Generator

leNCO

lε∆

Figure 5.1: Recursive-loop structure for tracking the CFO [19].

samples of the lth OFDM symbol. The error signal el provides the information on ∆ε, andis fed to the loop lter to calculate the frequency error estimate using the iterative function

∆εl = ∆εl−1 + αel, (5.5)

where ∆εl and ∆εl−1 are the estimated frequency errors of the lth and the (l−1)th OFDMsymbol, respectively. α is the parameter that controls the behavior of the loop. The pa-


rameter α has to be chosen properly such that a reasonable tradeo exists between theaccuracy and the convergence speed in the steady state.

The estimate ∆εl calculated from the loop lter (i.e., ∆εl = ∆εl−1 + αel) is fed to the

Numerically Controlled Oscillator (NCO) which generates an exponential term e−j2π∆εlk

N

for the kth sample (time instant) of the lth OFDM symbol. Therefore, the de-rotatingmatrix Υl for the l

th OFDM symbol rl is given by

Υl = e−j2π∆εl[N(l−1)+Nu]

N

1 0 · · · 0

0 e−j2π∆εl

N. . .

......

. . .. . . 0

0 · · · 0 e−j2π∆εl(N−1)

N

. (5.6)

Most of the available frequency tracking algorithms in the literature will track the osetbased on the above explained closed loop structure. They dier from each another in thecalculation of the error signal el. If el is calculated from the time domain samples (e.g.the CP) of the OFDM symbol, then the estimation belongs to the time-domain trackingscheme, whereas if el is calculated from the samples at the DFT output (i.e., pilot symbolsor data symbols) then it is called a frequency domain tracking algorithm. In case of CPBmethods, estimation is done based on the correlation of the CP and the corresponding lastsamples of each OFDM symbol (in UW-OFDM, it is the correlation between two succes-sive UWs). However, the cyclic prex is usually corrupted by the transmission channel,therefore, accurate estimates are dicult to obtain in real world scenarios. In PTA algo-rithms, pilot symbols are placed periodically on particular subcarriers and the estimationis done by correlating the received pilot symbols with their transmitted values. In mostcommunication standards, the pilot subcarriers are utilized for channel estimation and forsynchronization purposes as well. Furthermore, the data subcarriers are also potentialcandidates to improve the CFO estimation at the cost of some increased complexity. Inthese techniques, the estimation is done by comparing the received data symbols with thesliced values obtained from the de-mapper. Therefore, a larger number of estimates areobtained to average over.Note that, in this thesis, it is assumed that the CFO does not vary from symbol to symboland the estimation is always carried out from the rst two OFDM symbols, this means,only one iteration is performed. Therefore, the tracking scheme is not implemented in thisthesis. Fig. 5.2 illustrates the signal processing steps involved for the CFO estimation in anUW-OFDM baseband receiver. Note that, the gure depicts both the time and frequencydomain estimation procedures simultaneously, but in reality, either the time domain or thefrequency domain estimation procedure is carried out at one time. The following sectionaddresses the time domain CFO estimation algorithms followed by the frequency domainestimation techniques.


S/PCFO

CorrectionDFT

Equalizat

ion

Substract

UW

influence

Post

processing

Coarse CFO

Estimation

Channel

Estimation

Preamble

Extract

Data

Extract

Pilots

Fine CFO

Estimation/TrackingExtract UW

IDFT

Decision Directed

Methods

Pilot Tone Based

MethodsTime Domain

Methods

Figure 5.2: CFO estimation in an UW-OFDM baseband receiver.

5.2.1 Time Domain CFO Estimation

This subsection describes the time domain estimation algorithm for UW-OFDM, bearingin mind that this technique is originally derived for CP-OFDM. The algorithm that wasproposed in [13] is found to be the basis for many currently available CPB algorithms.They proposed a Maximum Likelihood (ML) estimation procedure for estimating the fre-quency error ∆ε and the timing error θ as well. In this method a log-likelihood function isconsidered from which the frequency oset estimate ∆ε is derived. A brief description ofthe algorithm will be given in the following.

For a while assume that the received signal s was left with some residual oset ∆ε.For simplicity, consider that the channel is an AWGN channel. Therefore, the vectors ∈ C(N+Nu)×1 can be modelled as

s = Ωq + n, (5.7)

where q =[xTu x′T

]Tis the transmitted signal containing the OFDM symbol x

′preceded

by the UW belonging to the previous OFDM symbol. Ω ∈ C(N+Nu)×(N+Nu) is the osetmatrix modelling the residual oset ∆ε and n represents the noise vector. The length ofthe guard interval is set to Nu. Here, every sample in the transmitted signal q is assumedto be an independent, identically distributed random variable (i.i.d.). According to thecentral limit theorem, it can thus be assumed that the transmitted signal q is a complexGaussian random process. However, this process is not perfectly white, because due to the


presence of the UW, there will denitely be a correlation between the two UWs present inthe signal q. Therefore, the received signal s is also not a white process. Additionally, scontains the information about the frequency oset (for instance, assume perfect timingsynchronization i.e., θ = θ). By exploiting this information, the frequency oset can beestimated as will be discussed in the following.

It can be noticed that, in the signal s, every sample of the rst UW is correlated withits corresponding sample in the next UW, whereas the rest of the samples are uncorrelatedwith each other. This can be written as [13]

E s∗(k)s(k +m) =

σ2s + σ2

n : m = 0σ2sej2π∆ε : m = N

0 : otherwise,

where σ2s and σ

2n represent the signal power and the noise power, respectively, and k refers

to the sample index within s. It can be observed from the above equation that the phasedierence between any two corresponding samples of the two successive UWs in s is 2π∆ε,and the samples in each sample pair are separated by the length N . Let the probabilitydensity function (PDF) of the received signal s for a given frequency oset ∆ε be denedas f (s|∆ε). Therefore, the log-likelihood function Ψ(∆ε) can be obtained by taking thelogarithm of the above PDF, leading to

Ψ(∆ε) = log f (s|∆ε) . (5.8)

The goal is to nd an estimate ∆ε such that the log-likelihood function is maximized. Forthat, the derivative of the likelihood function with respect to ∆ε is set to zero, and thisresults a maximum point which is ∆ε. In [13], ∆ε is shown to be

∆ε =1

2πarg

Nu−1∑k=0

s∗(k)s(k +N)

, (5.9)

where k refers to the index of the samples belong to the rst UW and s(k) represents asample of the symbol s. Here, arg · is with respect to the averaging of the correlationsbetween the samples which areN samples apart (i.e., between the two UWs). Fig. 5.3 showsthe structure of the frequency oset estimator. Since the argument function only returnsvalues in between (−π, π), therefore, it can be observed from (5.9) that the estimation rangeis limited to −1

2 < ∆ε < 12 . As the maximum deviation of the frequency oset in tracking

mode will never exceed half of the subcarrier spacing, this algorithm can asymptoticallybe used in tracking mode. The simulation results in [13] have proven this.

The performance of the estimator mainly depends on two factors, one is the length of theUW Nu and the other is the correlation coecient ρ between the corresponding samplesof two UWs. It can be noticed that as the length of the UW increases (i.e., the numberof samples), the estimation accuracy is improved because averaging is done over more


Moving

Sum

Filterπ21

( )∗⋅

Z

)(ks

⋅arg

( )ks +

)(* ks

ε∆

K = 0 to u-1

Figure 5.3: Frequency oset estimation structure as proposed in [13].

samples. The correlation coecient ρ is given by [13]

ρ =cov (s(k)s(k +N))√

var (s(k)) var (s(k +N))

=

∣∣∣∣∣∣∣∣E s∗(k)s(k +N)√

E|s(k)|2

E|s(k +N)|2

∣∣∣∣∣∣∣∣

=σ2s

σ2s + σ2

n

=SNR

SNR + 1. (5.10)

From the above equation, it can be recognized that the correlation between the samples(the samples corresponding to the UW) is corrupted in the presence of noise. Hence, theestimator works well at high SNR values in the AWGN channel. However, in the presenceof multipath (i.e., a time dispersive channel) the situation becomes worse, as the estimatoris not optimal for these channels. The multipath introduces correlations among all thesamples of s, thus violating the assumption of independence. Therefore, the performanceof the estimator will degrade signicantly in multipath channels, which is not desired.However, the huge advantage of the UW-OFDM is that the UW can be chosen by thedesigner. Therefore, the UW can be chosen such that the correlation properties are onlypartly destroyed in the presence of a multipath channel, which can be an advantage re-garding timing oset estimation. However, in the case of CFO estimation, the estimatorperformance is independent of the choice of the UW.


In matrix/vector notation, (5.9) can be written as

∆ε =1

2πarg

Nu−1∑k=0

s∗u1(k)su2(k)

=1

2πargsHu1su2

, (5.11)

where su1 ∈ CNu×1 and su2 ∈ CNu×1 are two UWs that belong to the vector s. Thevariance of the above estimator under AWGN is given by [23]

σ2∆ε =

1

Nu (2πN)2

1

SNR. (5.12)

Another unique feature of UW-OFDM is, as the UW is the same for all the OFDMsymbols, in contrast to CP-OFDM, the autocorrelation can be performed between anytwo non-subsequent UWs separated by Nsy OFDM symbols. The maximum likelihoodestimator of ∆ε follows to [23]

∆ε =1

2πNsyargsHu1su2

, (5.13)

with the error variance given by

σ2∆ε =

1

Nu (2πNsyN)2

1

SNR. (5.14)

This shows that the error variance for the latter estimator is low compared to the onebefore. This is because the former one has to estimate a small phase for a given SNR,while the latter one has to estimate a larger phase (because the distance between two nonsubsequent UWs is always larger than for Nsy = 1) having the same SNR, thereby reduc-ing the error variance of the estimator. This means that the latter estimator performswell for a slow varying frequency oset, leading to a very accurate estimation. However,the estimation range of this new estimator is decreased to −1

2Nsy< ∆ε < 1

2Nsy. For fast

varying osets, this scheme maintains the possibility to perform exactly the same as theold one (i.e., Nsy = 1). Note that, a vector of estimates can be obtained by performingthe estimation with dierent Nsy. To get a better accuracy, averaging over the dierentestimates can also be done if necessary.

Once the estimate is obtained, the time domain signal rl (from (5.4)) is de-rotatedwith this estimate using the matrix Υl ∈ CN×N . Therefore, the corrected signal vl can beobtained as

vl = Υlrl

= ΥlΩlx′. (5.15)

Ωl represents the downsized version of the matrix Ω, excluding the rows corresponding toprevious UW. If the residual frequency oset is perfectly estimated (i.e., ∆ε = ∆ε), thenthe matrix ΥlΩl will become an identity matrix, thus the oset is compensated completely.


5.2.2 Frequency Domain CFO Estimation

The most conventional approach to track the CFO in frequency domain is with the useof pilot subcarriers that are usually transmitted along with the data subcarriers in anOFDM symbol. These pilot subcarriers typically serve also for synchronization and chan-nel estimation purposes. The principles of CFO estimation using pilot carriers have beenpresented in [35], [14], and [41]. Besides these PTA techniques, Decision Directed (DD)methods are also resulting in a robust and a considerable CFO estimation. These methodsare based on comparing the received frequency domain data symbols with their correspond-ing sliced version symbols. Some good literature on DD based CFO estimation techniquescan be found in [4245]. The following subsections detail the PTA and DD approachesindividually.

Pilot Tone Aided CFO Estimation

The pilot subcarriers p are transmitted in every OFDM symbol and are usually distributeduniformly over the entire bandwidth. This kind of distribution is recommendable especiallyin frequency selective fading environments. In an OFDM system, the presence of CFO willintroduce a phase shift on each frequency domain data symbol along with the ICI. In [35], aMaximum likelihood estimation algorithm was introduced where the CFO is estimated bymeasuring the phase shift between two corresponding pilots in subsequent OFDM symbols.Let, pl,n and pl+1,n be the pilot symbols transmitted in the lth and the (l + 1)th OFDMsymbol on the nth subcarrier. Then, the CFO ∆ε can be estimated as [35]

ε =N

N +Nu

1

2πarg

∑n∈I

(zl,nz

∗l+1,n

) (pl,np

∗l+1,n

), (5.16)

where zl,n and zl+1,n are the pilot symbols received in the lth and the (l + 1)th OFDMsymbols, respectively, and Nu represents the length of the guard interval. The vector Icontains the indices of the pilot subcarriers p. The same algorithm can be applied to theUW-OFDM as well. However, in this thesis, the CFO estimation in frequency domain iscarried using only the rst UW-OFDM symbol. Therefore, ε can be estimated using thephase shift introduced on the pilot symbols p of the rst UW-OFDM symbol.The lth received frequency domain symbol zl [k] in the presence of oset, AWGN, andmultipath for an UW-OFDM symbol with zero UW can be written as

zl [k] =sin(π∆ε)

N sin(π∆εN )

ejπ∆ε(N−1+2[N(l−1)+Nu])

N H [k] xl [k] + Il [k] + vl [k] , (5.17)

where H [k] and vl [k] represent the frequency response of the channel and the noise onthe kth subcarrier of the lth OFDM symbol, respectively. Il [k] denotes the ICI on the kth

subcarrier caused by the other subcarriers and is given by

Il [k] =1

Nej2π∆ε[N(l−1)+Nu]

N

N−1∑m=0,m 6=k

H [m] xl [m]

N−1∑n=0

ej2π(m−k+∆ε)n

N . (5.18)


When considering only the rst UW-OFDM symbol (i.e., l = 1), (5.17) and (5.18) reduceto

z [k] =sin(π∆ε)

N sin(π∆εN )

ejπ∆ε(N−1+2Nu)

N H [k] x [k] + I [k] + v [k] (5.19)

and

I [k] =1

Nej2π∆εNu

N

N−1∑m=0,m 6=k

H [m] x [m]N−1∑n=0

ej2π(m−k+∆ε)n

N . (5.20)

When neglecting the eects of ICI and noise, the received symbol z [k] can be seen as theproduct of the channel frequency response H [k] and the transmitted symbol x [k], atten-uated, and rotated by some phase. Therefore, the frequency oset can be estimated bysimply calculating the phase of the received symbols, thus turning into a phase estimationproblem. However, the presence of a multipath channel will also aect the phase of thereceived symbol. Therefore, the data symbols as well as the pilot carriers should be equal-ized rst before utilizing them for estimation purposes. The received symbol z containingboth the pilots and the data symbols is given in the form of the linear model

z = ΛHdG′[

dp

]+ v, (5.21)

where Hd is a downsized version of H, excluding the rows corresponding to the zerosubcarriers. The equalization can be performed with a linear estimator of the form[

dp]

= E′z, (5.22)

where E′can be any linear equalizer. Here, in this thesis, three particular equalizers are

considered which are described in (4.18), (4.20), and (2.39).After the equalization process the pilot symbols are extracted from the estimated vector[

dT pT

]T. Assuming that perfect channel knowledge is available at the receiver (i.e.,

ˆHd = Hd), the CFO can be estimated out of these estimated pilot symbols p by

∆ε =N

π(N − 1 + 2Nu)arg

∑n∈I

(p∗ [n] ˆp [n]

). (5.23)

The above equation in matrix/vector notation can be written as

∆ε =N


pH ˆp. (5.24)

As the pilots are known to the receiver, the phase ambiguity can be resolved perfectly. Inreality, the receiver does not have perfect channel knowledge, thus limiting the performanceof the estimator.The above estimator has given equal weightage to all the pilot subcarriers, and this worksabsolutely well in at fading environments (or wideband channels). However, in frequency


selective fading, weighting the pilots equally is not an optimum combination. The estima-tion accuracy can be increased by weighting each coecient dierently by following somecriterion. One popular way is to weight the estimates with the inverse of the correspondingelements in the error covariance matrix of the estimator. The estimator with weightingfactors is given by

∆ε =N


∑n∈I

(p∗ [n] w [n] ˆp [n]

)(5.25)

For example, if the receiver uses a CI equalizer, then, after equalization there will be ahigh inuence of noise on the subchannels experiencing higher channel attenuations. Thismeans, that pilot subcarriers with higher channel attenuation will be given less weightagecompared to that of a pilot with lower channel attenuation. This procedure is calledselection combining. The pilots are weighted with the squared channel response coecients,and the CFO can be estimated as

∆ε =N


∑n∈I

(p∗ [n] abs

∣∣∣H [n]∣∣∣2 ˆp [n]

)(5.26)

Considering a matrix Hp ∈ CNp×Np , whose diagonal elements represent the correspondingchannel responses on the pilot subcarriers, then the CFO estimate ε in matrix/vector formfollows to

∆ε =N


pHHHp Hp

ˆp. (5.27)

In case of BLUE and MMSE equalization, the weighting coecients for the pilots are givenby the inverse of the corresponding diagonal elements in the error covariance matrices givenin (4.19) and (4.21), respectively.

Decision Directed CFO Estimation

Like as the pilot subcarriers, the data symbols also experience a phase rotation in the pres-ence of CFO, and this phase shift is the same for all subcarriers (when excluding randomphase shifts due to the ICI). Hence, this phase shift can be utilized to improve the CFOestimation [42, 43, 45]. The estimation procedure using the data symbols is similar to thepilot tone based approaches, except the fact that in the DD approach the data symbols aresliced to the nearest QPSK/QAM symbol in the constellation diagram by tentative/harddecisions, which might lead to the errors in the estimate when wrong decisions are made.

After performing equalization, the estimated data symbol vectorˆd is extracted out

of the equalized vector. As the receiver does not have the knowledge on the sent data

d, slicing is performed onˆd. It means that each symbol in

ˆd is mapped to the nearest

symbol in the constellation diagram, assuming that each received symbol is originated from

the corresponding mapped alphabet. Dening the sliced vector asˆdsl, the CFO can be

estimated by measuring the angle betweenˆdsl and

ˆd given by

∆ε =N


ˆdHsl

ˆd. (5.28)


In case of selection combining, the weighting coecients are calculated in a similar fashiondescribed in the previous section. In reality, there might be a chance of making wrongdecisions, if the received symbols are near to the decision boundaries of the constellationdiagram. However, if the number of data symbols is large enough, then the impact ofwrong decisions can be compensated.

Note that, in the presence of CFO, the phase of the received frequency domain sym-bols will increase from symbol to symbol. Therefore, it is necessary to compensate thisincremental phase, before the estimation is performed with the current symbol. If notcompensated, then, after a few OFDM symbols, the data symbols will cross the decisionboundaries of the constellation and the majority of data symbols will be sliced to a wrongQPSK/QAM symbol, leading to a signicant error in the estimation. This eect can beclearly observed from Fig. 5.4. This incremental phase can be compensated by using thetracking procedure discussed in the previous section. After estimating the angle from thecurrent symbol, this angle is added to the angle obtained from the previous symbol to geta new total rotation angle for the next symbol. In this way, the probability of makingwrong decisions will always be small, providing an accurate estimate.In the next chapter the performance of various CFO estimation procedures is evaluatedand discussed.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Real Part

Imag

inar

y P

art

1st OFDM symbol2nd OFDM symbol3rd OFDM symbol

Figure 5.4: Constellation diagram of the received data symbols of the rst three consecutiveOFDM symbols for ∆ε = 0.05 and at σ2

n = 0.

Chapter 6

Results and Discussion

In this chapter, the performance of dierent CFO estimation algorithms for UW-OFDMin terms of Eb/N0 and length of training sequence is analyzed. In addition, a comparisonis made between the time domain and the frequency domain CFO estimation techniques.Besides this comparison, the sensitivity of these estimators on the CFO and the channelcharacteristics is investigated. The sensitivity of the UW-OFDM system on the CFO isanalyzed and compared to that of the CP-OFDM scheme.

6.1 Simulation Setup

All the simulations are performed with the pilot based transceiver model, where the UWas well as the pilots are introduced at the same time. This transceiver model is consideredonly for a comparison of time domain versus frequency domain based estimators. In prac-tice, of course only the UW or the frequency domain pilots will exist at one time. The totalnumber of subcarriers in each OFDM symbol is set to 64, where 32 are data subcarriers,16 are redundant subcarriers, 4 are pilot subcarriers, and the remaining 12 are chosen tobe zero subcarriers. All the data subcarriers are modulated with a 4-QAM alphabet. Thefour pilots are assigned with the values 1, -1, 1, and 1, respectively. The length of theUW is set to 16. The positions of the redundant subcarriers are calculated according tothe optimization procedure described in [1]. For the modied scheme (i.e., the pilot basedmodel) the optimized positions are 3, 7, 11, 14, 18, 21, 25, 27, 39, 41, 45, 48, 52, 56, 60, 63.The pilot subcarrier indices in the modied scheme are selected to be 8, 22, 44, 58. Withthese indices it is trivial to construct the permutation matrix P. All the remaining param-eters are chosen like in the 802.11a standard [34]. This scheme serves as a reference systemwhen comparing UW-OFDM against conventional CP-OFDM. An OFDM burst havingtwo UW-OFDM symbols is considered for one iteration run, and in total 5000 iterationsare performed at each Eb/N0 point, completing one simulation run. The performance ofthe estimator is evaluated by means of its mean squared error (MSE). In case of multipath,the Finite Impulse Response (FIR) of the channel is modelled as in (2.24), where each delaytap is modelled as a complex Gaussian random variable having zero mean. All the tapsare considered to be statistically independent and having an exponential decaying powerprole. The channel delay spread is chosen to be 100ns and the total length of the channel

38

CHAPTER 6. RESULTS AND DISCUSSION 39

is shortened to the UW length Nu. Table 6.1 summarizes the most important parametersused in this simulation setup.

Table 6.1: Important parameters used in this simulation setup.

Parameter Number

N 64

Nd 32

Nr 16

Np 4

Nz 12

Nu 16

Pilots (p) 1,1,-1,1 (BPSK alphabet)

Data (d) 4-QAM alphabet

fs 20 MHz

DFT period 3.2 µs

Gaurd duration 800 ns

Total OFDM symbol duration 4 µs

Subcarrier spacing 312.5 kHz

In the simulations, 3 CFO estimation techniques utilizing UW, pilots, and data symbolsare investigated and analyzed. However, it seems clearer to discuss the results in AWGNand multipath channels separately. The following section rst discusses the results inAWGN environment followed by the multipath channel case.

6.2 AWGN Environment

Before diving into the nal estimation results, let us rst analyze the behavior of twosuccessive UWs in the presence of CFO. The UW is chosen to be xu = 1. Fig. 6.1depicts the phase rotation of the UWs belonging to the rst and the second OFDM symbol,respectively, for ∆ε = 0.05. Note that the phase shift between each sample in the rstUW to its corresponding sample in the subsequent UW is 2π∆ε, as they are N samplesapart. The CFO can easily be calculated by taking the phase dierence between eachcorresponding sample pair of two successive UWs and then averaging over the measuredphase dierences. However, in the presence of noise, the correlation properties between thetwo UWs are partly lost (i.e., the phase dierence for all the sample pairs is not same, theyvary around the mean value). By averaging the phase dierence over all the samples, theinuence of the noise can be reduced. Fig. 6.2 shows two subsequent UWs in the presenceof noise and CFO. It can be observed that the UWs are spread around the unit circle, andthis leads to the eect that the corresponding samples of the two UWs have dierent phasedierences instead of having an equal phase dierence of 2π∆ε.

Fig. 6.3 shows the performance of the time domain estimator (i.e., using UWs) with∆ε = 0.05. It can be clearly observed that the estimator performance increases with


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increasing Eb/N0 values. This is because at higher Eb/N0 values, the correlation betweenthe corresponding sample pairs in the subsequent UWs is higher compared to the casefor lower Eb/N0 values (see (5.10)). Note that the estimation is carried out using twosuccessive UWs belonging to the rst two OFDM symbols.

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Figure 6.3: Performance of the time domain CFO estimator for ∆ε = 0.05 in an AWGNchannel.

Fig. 6.4 shows the error between the true and the estimated CFO at Eb/N0 = 15 dBfor ∆ε = 0.05. As can be seen from the gure, the error follows a Gaussian distributionhaving zero mean and a standard deviation of 0.004.

The MSE curves of the estimator for various Eb/N0 values (Eb/N0 = 10, 15, and 20dB) over dierent UW lengths are plotted in Fig. 6.5. Note that there is a signicantimprovement (i.e., with a larger slope) in the performance of the estimator until a certainUW length, and after that length the performance is improved very slowly (i.e., with alesser slope). However, the estimator performance will continuously be improved with theincrease of the length of the UW (i.e., averaging reduces the MSE).

Fig. 6.6 proves that the time domain estimator performance is almost independent ofthe value of ∆ε, provided that the oset is in the estimation range (−0.5 to 0.5) andadditionally, for values close to the borders of the estimation range, the Eb/N0 value ishigh. In other words, the estimator performs exactly the same for all osets within thisrange. It can be observed from the gure that the estimator performance is identical for∆ε = 0.01 and 0.4 for all Eb/N0 values, whereas for ∆ε = 0.49, the estimator performanceis worse at low Eb/N0 range while performing the same in the high Eb/N0 region. This


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Figure 6.4: The error distribution of the estimated CFO for ∆ε = 0.05 at Eb/N0 = 15 dBin an AWGN channel .

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Figure 6.5: Performance of the time domain estimator for dierent UW lengths at ∆ε =0.05 in an AWGN channel.


is because at ∆ε = 0.49, due to the noise, the phase between each corresponding samplesof the two UWs might exceeds π, leading to a discontinuity in the arctan function, thusresulting in a heavy error in the estimation. However, for high Eb/N0 values, as the noiseeect is minimum, the estimator works very well because the value 0.49 is still in theestimation range. It can also be noticed that the estimator performance is completelyworse for ∆ε = 0.51 due to the fact that ∆ε is out of the estimation range, and theperformance will never improve, even in a noise free region (i.e., σ2

n = 0).

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Figure 6.6: Performance of the time domain estimator for dierent CFOs in an AWGNchannel.

In UW-OFDM, as the UW is the same for all the OFDM symbols, the autocorrelationcan be performed between any two non-subsequent UWs which are separated by severalOFDM symbols. Fig. 6.7 shows the performance of the estimator using two non-subsequentUWs with ∆ε = 0.05. Here, for this particular simulation, a burst consisting of 4 OFDMsymbols is considered which contradicts the usual case where the simulation is performedwith a burst having only two OFDM symbols. The estimation is performed with dierentUW pairs (UW1,UW2), (UW1,UW3), (UW1,UW4), where UW1, UW2, UW3, and UW4belong to the 4 OFDM symbols, respectively. Fig. 6.7 reveals that there is a signicantimprovement of the performance of the estimator when the distance between the UWpair increases, thereby improving the estimation accuracy. In CP-OFDM, the maximumpossible distance is one OFDM symbol, which is the same as estimating with the pair(UW1, UW2) in UW-OFDM. However, if the distance between the UWs increases, theestimation range decreases ( −1

2Nsy< ∆ε < 1

2Nsy, where Nsy is the number of OFDM symbols

between the considered UWs). It means that the estimation range for the pair (UW1,UW4)is limited to −0.17 < ∆ε < 0.17. Fig. 6.8 shows the performance of the estimator with theabove mentioned UW pairs for ∆ε = 0.3. It can be observed that the estimation with the


UW pairs (UW1,UW3) and (UW1,UW4) fails because ∆ε is out of the estimation range,whereas with the UW pair (UW1,UW2) the estimation is good as ∆ε is below 0.5.

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Figure 6.7: Performance of the time domain estimator using two non-subsequent UWs for∆ε = 0.05 in an AWGN channel.

In the following the focus is laid on the frequency domain estimation techniques, namelyPTA and DD schemes. The pilot based methods are analyzed rst followed by the DDapproaches. The simulations are performed with the pilot based model having zero UW.At the receiver, the pilots are estimated like they were data symbols by applying anylinear equalizer at rst (e.g., the equalizers introduced in chapter 4), and then the CFOis estimated out of these estimated pilots using (5.23). The performance of the pilotbased estimator after applying the LMMSE equalizer (using (4.20)) for ∆ε = 0.01 and∆ε = 0.05 is shown in Fig. 6.9. The weighting average is done by weighting the estimateswith the inverse of the corresponding diagonal elements of the error covariance matrixgiven in (4.21). It can be noticed that weighting the estimates has no additional benetcompared to the simple average because the error covariance matrix for an AWGN channelis close to a scaled identity matrix. It can be seen that the MSE curves for ∆ε = 0.05approaches an error oor at high Eb/N0 values. This is due to the result of ICI on the pilotsubcarriers with the increase in CFO. Note that this error oor raises with the increase inthe oset.

Fig. 6.10 compares the performance of CFO estimators for ∆ε = 0.01 after equalizationwith CI, BLUE, or LMMSE. It can be noticed that the estimator with the CI equalizerperforms worst compared to the others due to the fact that it completely ignores theinformation present on the redundant subcarriers. The LMMSE performance is optimum


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Figure 6.8: Performance of the time domain estimator using two non-subsequent UWs for∆ε = 0.3 in an AWGN channel.

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Figure 6.9: Performance of the PTA estimator after equalization with the LMMSE equal-izer for ∆ε = 0.01 and ∆ε = 0.05 in an AWGN channel.


among all the three estimators as it employs a Wiener smoother, which reduces the noisesignicantly on the subcarriers. Therefore, the noise inuence on the pilots gets reduced,thus leading to a better accuracy in the estimate. This can be observed in the gureparticularly in the low Eb/N0 region. However, at high Eb/N0 values the LMMSE performs

identical to the BLUE, as the term Nσ2n

σ2d

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Figure 6.10: Performance of the PTA estimator after equalization with CI, BLUE, andLMMSE equalizers for ∆ε = 0.01 in an AWGN channel.

Fig. 6.11 plots the MSE curves of the CFO estimators for ∆ε = 0.05 after equalizationwith CI, BLUE, or LMMSE. Clearly, all the estimators reach an error oor after a certainEb/N0 point because of the eect of ICI induced due to the CFO on the pilot subcarriers.Here, the LMMSE outperforms the CI and BLUE particularly in low Eb/N0 regions.

In the DD approach, the CFO is estimated by measuring the phase dierence betweenthe received data symbols and its corresponding sliced symbols. The mean power of thedata symbols is normalized to unity and a total number of 32 data symbols are presentin each OFDM symbol. Fig. 6.12 illustrates the performance of the estimators with CI,LMMSE, and BLUE equalizers for ∆ε = 0.01. In addition, it also compares the perfor-mance between the estimators utilizing PTA and DD schemes. Clearly, the estimatorswith the DD scheme follow the same trend like as the ones with the PTA approach. TheLMMSE outperforms the CI and BLUE equalizers. However, at high Eb/N0 values, theBLUE approaches the LMMSE performance. It can be observed from the gure that theMSE curves of the data aided estimators always lie below the MSE curves of the PTAscheme. This is because of the fact that an averaging is done over 32 data symbols in


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Figure 6.11: Performance of the PTA estimator after equalization with CI, BLUE, andLMMSE equalizers for ∆ε = 0.05 in an AWGN channel.

the DD scheme, whereas in the PTA approach only 4 symbols (in this case pilots) areincluded in the estimation. Although the decisions in the DD scheme are hard decisions,the estimator works very well because for ∆ε = 0.01, the phase rotation experienced by thedata symbols is very small, therefore the probability of each data symbol to fall in anotherdecision region is very low. Consequently, most of the data symbols are demapped to theircorresponding right symbols in the constellation diagram. However, when ∆ε = 0.05 asshown in Fig. 6.13, the MSE curves of all the DD estimators, regardless of their underlyingequalizer, reached an early error oor compared to that of when ∆ε = 0.01. The reason issimply because for ∆ε = 0.05 the phase rotation as well as the ICI on the data symbolsis large and the probability that the data symbols fall in another decision region is high,thus making most of the data symbols to be sliced to a wrong plane, leading to errors inthe estimates. Therefore, in DD schemes the error oor is due to the phase ambiguity ofthe data symbols as well as the ICI. In contrast, for the PTA approach, as the transmittedsymbols are known to the receiver, the phase ambiguity is resolved, and the error oor isobtained only due to the ICI.

The estimates obtained from both the PTA and DD schemes can be combined to getan estimate that outperforms both individual schemes. Assuming that ∆εp and ∆εd arethe estimates from the PTA and DD schemes, respectively, then, the combined estimate∆εc can be obtained as

∆εc = α∆εp + (1− α)∆εd, (6.1)

where α is an appropriate weighting factor and is chosen such that α2MSE ∆εp +


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Figure 6.12: Performance of the frequency domain estimators after equalization with CI,BLUE, and LMMSE equalizers for ∆ε = 0.01 in an AWGN channel.

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Figure 6.13: Performance of the frequency domain estimators after equalization with CI,BLUE, or LMMSE equalizers for ∆ε = 0.05 in an AWGN channel.


(1− α)2 MSE ∆εd is minimized. The α value is calculated as

α =MSE ∆εd

MSE ∆εp+ MSE ∆εd. (6.2)

Fig. 6.14 and Fig. 6.15 depict the MSE curves of the individual as well as the combinationalschemes for ∆ε = 0.01 and ∆ε = 0.05, respectively. It can be observed that the combina-tional estimator clearly outperforms both individual schemes. Note that the performancegain of this combined estimator is lower for ∆ε = 0.01 compared to ∆ε = 0.05.

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Figure 6.14: Performance of the combined estimator (PTA and DD) with LMMSE equalizerwhen ∆ε = 0.01 in an AWGN channel.

In the following, a comparison between time domain and frequency domain estimators isillustrated and analyzed. To fairly compare the time domain estimation with the frequencydomain estimation, the energy of the UW is scaled such that it is equal to the energy ofthe pilots by using Parseveals theorem, leading to the relation

xHuxu =

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N. (6.3)

Fig. 6.16 compares the time domain estimation (i.e., UWs) and the frequency domain es-timation (i.e., using pilots and data) for ∆ε = 0.01. It can be noticed that the frequencydomain estimators (pilot based as well as data aided) outperform the time domain esti-mators until a certain Eb/N0 range. This is because frequency domain estimators reachan error oor due to the ICI caused by ∆ε, whereas the time domain estimator improvescontinuously with the increase in Eb/N0. One can observe that after a certain Eb/N0


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Figure 6.15: Performance of the combined estimator (PTA and DD) with LMMSE equalizerwhen ∆ε = 0.05 in an AWGN channel.

value, the time domain estimator outperforms the frequency domain estimators becauseof its continuous improvement. It can be veried from Fig. 6.17 and Fig. 6.16 that thetime domain estimator is independent of the specic value ∆ε while the performance ofthe frequency domain estimators heavily depend on the values of ∆ε, as the estimators arehighly inuenced by the ICI.


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Figure 6.16: Performance comparison between time domain and frequency domain CFOestimators for ∆ε = 0.01 in an AWGN channel.

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Figure 6.17: Performance comparison between time domain and frequency domain CFOestimators for ∆ε = 0.05 in an AWGN channel.


6.3 Time-Dispersive Environment

This section rst analyzes the behavior of UWs in a multipath environment followed by adiscussion on CFO estimation results in the time and frequency domain. Fig. 6.18 presentstwo subsequent UWs after propagating through a multipath channel in the presence ofCFO. The CIR is modeled as in (2.24), consisting of 16 taps, and it is assumed thatthe CIR is static for the entire OFDM burst. The UWs are chosen to be xu = 1. Itcan be observed from the gure that, although the UWs are completely corrupted by themultipath channel, the phase dierence between the samples of the UWs caused due to theCFO is clearly visible. However, the UWs are now more complexly correlated with eachother than a simple pairwise correlation seen in the AWGN channel case, thus makingthe situation more dicult. Another interesting fact is that the UW is inuenced by theCIR, and therefore the UW contains the entire channel information. This information canadvantageously be used for channel estimation purposes [46].

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Since the UW is part of the DFT window, it can be equalized at the receiver similarto the data symbols. Therefore, by exploiting this benet, the UWs are equalized rst byapplying the CI equalizer, and then the equalized UWs can be used for CFO estimation.However, in practice, the equalization is not perfect in the presence of CFO, which will beanalyzed in the following part. Fig. 6.19 shows two subsequent UWs before and after theequalization for a small and a large frequency oset. It can be easily noticed that for asmall oset the UWs are almost perfectly equalized (i.e., the UWs lie on the unit circle),and thus the UWs are only left with the phase rotation caused by CFO. However, in case


of a large oset, after the equalization the samples of the UWs are still spread around theunit circle, although there is no noise introduced in the system. Hence, this is just becauseof the imperfect equalization.The CFO estimation in the time domain is carried out by comparing the phase dierence

Figure 6.19: Two subsequent UWs before and after the equalization when (a) ∆ε = 0.01(b) ∆ε = 0.05 and σ2

n = 0.

between the corresponding sample pairs in two successive UW-OFDM symbols. Fig. 6.20and Fig. 6.21 depict the MSE curves of the time domain estimator in a multipath channelenvironment for ∆ε = 0.01 and ∆ε = 0.05, respectively. For comparison purposes, theAWGN MSE curve is also plotted as reference. To show the potential of the estimator, theaveraging is done over 5000 dierent multipath channels. The estimator is derived underthe assumption that the received time domain samples, which are lying outside the UW,are uncorrelated with each other. However, the presence of multipath introduces corre-lations among them, thus violating this assumption. Therefore, the performance of theestimator degrades signicantly in a multipath channel compared to that of the AWGNchannel. One can observe that the MSE curve of the estimator using unequalized UWsreaches an error oor in multipath, because after a certain Eb/N0 value, the inuence ofthe noise on the estimator is negligible, and the MSE will not decrease further beyondthis Eb/N0 point. Since the UW is part of the DFT window, the UW can be equalizedby applying the CI equalizer on the received OFDM symbol. However, Fig. 6.20 showsthat the performance of the estimator using the equalized UWs is worse compared to thatof the unequalized UWs, except for low Eb/N0 region. This is probably due to the rea-son that the CI equalizer is no longer considered to be a valid equalizer in the presence ofCFO. Therefore, it can be concluded that, on equalizing the multipath channel with the CIequalizer, the performance of the estimator will degrade signicantly at low Eb/N0 values.An interesting observation that can also be made is that the estimator performance withthe unequalized UWs reaches an error oor after a certain Eb/N0, whereas the estimatorperformance with the equalized UWs goes on improving with the increase in the Eb/N0. Itis due to the fact that as Eb/N0 increases, the inuence of the noise on the data decreases,and the CI equalizer will work reasonable in this region.


Fig. 6.22 shows that the performance of the estimator is more dependent on the con-crete amount of the CFO compared to the AWGN case. It can also be noticed that when∆ε exceeds 0.25, the estimator performance starts degrading (especially at low Eb/N0 val-ues) and at ∆ε = 0.51, the estimator performs completely worse as ∆ε is out of range.

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Figure 6.20: Performance comparison of the time domain estimator between an AWGNand a multipath environment for ∆ε = 0.01.

Fig. 6.23 shows the performance of the estimator using two non-subsequent UWs when∆ε = 0.01. This reveals that there is a signicant improvement in the performance, if thedistance between the correlating UWs increases on the one hand, but the estimation rangedecreases on the other hand.

In frequency domain, the CFO can be estimated using PTA or DD schemes. As theCFO estimation procedure using these techniques was already discussed in the previoussection (i.e., for the AWGN channel case), the focus is devoted to analyze the results. Thesimulations are performed with the pilot based model having zero UW. Fig. 6.24 shows theperformance of the estimator after applying the LMMSE equalizer when ∆ε = 0.01. Theweighting of the estimates is done with the inverse of the corresponding diagonal elementsin the error covariance matrix given in (4.21). It is clearly visible from the gure thatweighting the estimates has a huge performance gain over the simple average with regardto the MSE. This is because as the channel is frequency selective, the pilots that fall intodeep spectral notches are heavily inuenced and attenuated by the noise and channel,respectively, therefore, the estimates from those pilots should be weighted less. Fig. 6.25plots the MSE curve of the PTA estimator with the LMMSE equalizer when ∆ε = 0.05.


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Figure 6.21: Performance comparison of the time domain estimator between an AWGNand a multipath environment for ∆ε = 0.05.

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Figure 6.22: Performance of the time domain estimator for dierent CFOs in a multipathenvironment.


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Figure 6.23: Performance of the time domain estimator using two non-subsequent UWs for∆ε = 0.01 in a multipath environment.

It can be observed that the curves have an early error oor compared to the curves when∆ε = 0.01. This is due to the fact that the estimator performance is limited by the ICIcaused due to the CFO. The larger the CFO, the more intense the ICI, and the lower isthe performance of the estimator. It can also be noticed from Fig. 6.25 that after a certainEb/N0 value, the weighted average scheme performs worse than the simple average. Thisis because from that Eb/N0 value the inuence of noise will become negligible compared tothe inuence of ICI. However, the weighting coecients are still extracted from the errorcovariance matrix (see (4.21)) which is no longer a useful means as the elements in thematrix will not consider the ICI.

Fig. 6.26 and 6.27 illustrate the performance of the PTA based CFO estimator withCI, LMMSE, and BLUE equalizers when ∆ε = 0.01 and ∆ε = 0.05, respectively. Bothresults show that the estimator with the CI equalizer has the worst performance, becauseit completely ignores the correlations among the subcarriers, whereas the LMMSE andBLUE equalizers perform identical at all shown Eb/N0 values. The MSE curves of all theestimators at ∆ε = 0.05 exhibit an early error oor compared to that of the MSE curveswhen ∆ε = 0.01 because of the intense ICI induced due to the large CFO.

Fig. 6.28 and 6.29 compare the performance of the PTA estimators with LMMSE equal-ization in both AWGN and multipath channels for ∆ε = 0.01 and ∆ε = 0.05, respectively.It can be noted that the equalizer performs very well in mitigating the multipath eects inthe presence of CFO, and thus the MSE curve for the multipath case almost approaches


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Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.24: Performance of the PTA estimator with the LMMSE equalizer for ∆ε = 0.01in a multipath environment.

0 5 10 15 20 25 30 35 40 4510

−5

10−4

10−3

10−2

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.25: Performance of the PTA estimator with the LMMSE equalizer for ∆ε = 0.05in a multipath environment.


0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.26: Performance of the PTA estimator after equalization with CI, BLUE, andLMMSE equalizers for ∆ε = 0.01 in a multipath environment.

0 5 10 15 20 25 3010

−4

10−3

10−2

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.27: Performance of the PTA estimator after equalization with CI, BLUE, andLMMSE equalizers for ∆ε = 0.05 in a multipath environment.


the AWGN curve. Similar tendencies can be observed with the BLUE as well as with theCI equalizer (not shown in this work).

0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror

AWGNMultipath

Figure 6.28: Performance comparison of the PTA estimator with LMMSE equalizer be-tween an AWGN and a multipath environment for ∆ε = 0.01.

The performance of DD schemes in multipath channels are illustrated in Fig. 6.30and 6.31 for ∆ε = 0.01 and ∆ε = 0.05, respectively. In the same gures the MSE curves ofa PTA estimator is also plotted for comparison. It can be observed that the MSE curvesof the data aided estimators always lie below the MSE curves of the PTA scheme becausean averaging is done over more symbols. As the equalizers are applied before the CFOestimation, the multipath eect on the data symbols is almost eliminated, resulting in MSEcurves very similar to the AWGN case (see Fig. 6.12 and Fig. 6.13). For ∆ε = 0.05, theMSE curves of the DD scheme reach an early error oor compared to that of ∆ε = 0.01,because of more wrong decisions made due to the rotation of the received symbols due toCFO and also with the increased ICI.

The following illustrates the time domain versus frequency domain estimation methodsin multipath channels for dierent CFOs. Fig. 6.32 and 6.33 plot the MSE curves of thetime domain and frequency domain estimators for ∆ε = 0.01 and ∆ε = 0.05, respectively.Note that, for comparison purposes, the UW energy is scaled such that it has the sameenergy as the pilots. As a result, the energy of the UW used in these simulations is very lesscompared to that of the UW used in the previous simulations (see Fig. 6.20 and Fig. 6.21).It can be noticed from Fig. 6.32 and 6.33 that, unlike in AWGN, the time domain method isworse compared to the frequency domain estimation methods. Although the performance


0 5 10 15 20 25 3010

−4

10−3

10−2

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror

AWGNMultipath

Figure 6.29: Performance comparison of the PTA estimator with LMMSE equalizer be-tween an AWGN and a multipath environment for ∆ε = 0.05.

0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.30: Performance of the frequency domain estimators after equalization with CI,BLUE, and LMMSE equalizers for ∆ε = 0.01 in a multipath environment.


0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.31: Performance of the frequency domain estimators after equalization with CI,BLUE, and LMMSE equalizers for ∆ε = 0.05 in a multipath environment.

of the time domain estimation method is independent of the amount of CFO, whereas theperformance of frequency domain estimation methods heavily depend on the CFO, the timedomain estimator has an early error oor in multipath channels compared to frequencydomain estimators. The reason is obvious. In frequency domain, there is a possibilityto remove the inuence of the multipath channel on the data, from which ∆ε has to beestimated, but in time domain the estimation is still carried out based on the multipathcorrupted UW because the equalization of UWs with the CI equalizer is not performingwell.


0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.32: Performance comparison between time domain and frequency domain CFOestimators for ∆ε = 0.01 in a multipath environment.

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

Eb/N

0 [dB]

Mea

n sq

uare

d er

ror


Figure 6.33: Performance comparison between time domain and frequency domain CFOestimators for ∆ε = 0.05 in a multipath environment.


6.4 CFO Impact on UW-OFDM and CP-OFDM

This section analyzes the sensitivity of the UW-OFDM system on the CFO, and alsocompares it with that of the CP-OFDM system. An OFDM burst having 20 OFDM sym-bols is considered for one iteration run, and in total 2000 iterations (i.e., 2000 bursts)are performed for each ε value, completing one simulation run. For all the simulations, atraditional UW-OFDM system is considered, and the CP-OFDM system is built like asin the 802.11a standard [34]. It is assumed that ε is constant for the entire burst. Thephase rotation factor due to the CFO is assumed to start from zero at the beginning ofeach OFDM burst, and each sample in the burst is multiplied with its corresponding phase

rotation factor ej2πεnN , where n represents the index of the sample. The tracking mechanism

discussed in the Section 5.2 is implemented so that the phase accumulation due to the CFOfrom one OFDM symbol to the other OFDM symbol is compensated. Briey summarized,the tracking mechanism is implemented in the following way. The phase rotation of thereceived pilot symbols caused due to the CFO is estimated from the current OFDM sym-bol. Then, all the data symbols in the subsequent OFDM symbols including the currentsymbol is de-rotated by the estimated phase angle in the frequency domain. Thereafter,again the phase of the received pilots from the next OFDM symbol is estimated and thesubsequent symbols are corrected. This process is repeated until the estimation reachesthe last OFDM symbol in the burst. After the correction process, the data symbols arethen estimated out of the corrected OFDM symbols based on the linear model describedin (4.16). The performance of the systems is analyzed by the resulting mean squared error.

Consider that a frequency oset ε is introduced in both systems. For the momentassume that the channel is a distortion and a noise free channel. An UW-OFDM systemhaving zero UW is considered for the following simulation results. Fig. 6.34 shows the

MSE curves of the frequency domain symbols (i.e., E

∣∣∣x− ˆx∣∣∣2) of UW-OFDM and CP-

OFDM systems plotted over an ε range of 0 to 1. It can be observed that the MSE ofthe frequency domain symbols in both systems increases with increasing ε because of theincreasing ICI inuence. It is important to notice that the UW-OFDM system exhibitsless sensitivity to the CFO compared to the CP-OFDM scheme, and this turns out tobe a considerable advantage of the UW-OFDM scheme. Note that in UW-OFDM, thefrequency domain symbol x is composed of data subcarriers (d), redundant subcarriers(r), and pilot subcarriers (p), while for CP-OFDM the frequency domain symbol x iscomposed of only the data and pilot subcarriers. The MSE curve of the data symbols afterperforming the CI equalizer coincides exactly with that of the LMMSE equalizer. Notethat, the CI equalizer for a distortion free channel is an Identity matrix. This shows thatthe equalization with the LMMSE equalizer has no additional advantage due to the factthat this equalizer will only account for the multipath propagation and AWGN, but willnot combat over the CFO.

It is also interesting to see the sensitivity of both systems within a short range of CFO,which is more realistic in case of a ne CFO estimation. Fig. 6.35 shows the MSE curvesof the frequency domain symbols (i.e., x) of UW-OFDM and CP-OFDM system for an εrange of 0 to 0.1. It can be observed that even in this range, the UW-OFDM is less sensitive


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

CFO (ε)

Mea

n sq

uare

d er

ror

CP−OFDM − CI equalizerCP−OFDM data − CI equalizerUW−OFDM − CI equalizerUW−OFDM data − CI equalizerUW−OFDM redundant − CI equalizerUW−OFDM data − LMMSE equalizer

Figure 6.34: MSE curves of the frequency domain symbols of UW-OFDM with zero UWand CP-OFDM for an ε range of 0 to 1 in a distortion and noise free channel.

compared to the CP-OFDM system. It can also be noticed that the error contribution ofthe redundant subcarriers (r) on the MSE of x is low compared to the error contributionof the data subcarriers (d).

The following illustrates the behavior of the UW-OFDM system in the presence ofCFO, when having dierent UW sequences (other than zeros). Here, two particular UWsequences xu1 = 1 and xu2 [k] ∈ CN (0, 1) are considered for the simulations. The energyof the xu2 is scaled such that it has the same energy as the xu1. Fig. 6.36 and Fig. 6.37depict the MSE curves of the frequency domain symbols of UW-OFDM and CP-OFDMsystems in the presence of CFO, when utilizing xu1 and xu2, respectively. It can be observedthat in contrast to the zero UW, in case of a CI equalizer, the frequency domain symbols ofUW-OFDM system with the above UW sequences exhibit very high MSE compared to theCP-OFDM scheme because of the subtraction of a "wrong" UW. In the presence of CFO,the UW inuence can be removed completely only if the CFO inuenced UW (i.e., Λxu1

or Λxu2) is subtracted instead of subtracting the UW itself. However, the subtraction ofthe CFO inuenced UW is impossible as ε is unknown to the receiver. An interesting factis that, if the data (d) are equalized with the LMMSE equalizer, the MSE of the equalizeddata exactly coincides with that of the UW-OFDM system having zero UW, thus makingthe MSE of the equalized data to be independent of the chosen UW sequence. This canbe observed in Fig. 6.38.

The following illustrates the sensitivity of UW-OFDM and CP-OFDM systems to ε in


0 0.02 0.04 0.06 0.08 0.10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

CFO (ε)

Mea

n sq

uare

d er

ror


Figure 6.35: MSE curves of the frequency domain symbols of UW-OFDM with zero UWand CP-OFDM for an ε range of 0 to 0.1 in a distortion and noise free channel.

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

CFO (ε)

Mea

n sq

uare

d er

ror


Figure 6.36: MSE curves of the frequency domain symbols of UW-OFDM with xu1 andCP-OFDM for an ε range of 0 to 1 in a distortion and noise free channel.


0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

CFO (ε)

Mea

n sq

uare

d er

ror


Figure 6.37: MSE curves of the frequency domain symbols of UW-OFDM with xu2 andCP-OFDM for an ε range of 0 to 1 in a distortion and noise free channel.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

CFO (ε)

Mea

n sq

uare

d er

ror

CP−OFDM data − CI equalizerUW−OFDM data − zero UW − LMMSE equalizerUW−OFDM data − x

u1 − LMMSE equalizer

UW−OFDM data − xu2

− LMMSE equalizer

Figure 6.38: MSE curves of the estimated data symbols of UW-OFDM and CP-OFDM foran ε range of 0 to 1 in a distortion and noise free channel.


a multipath environment. The UW is set to zero. Fig. 6.39 presents the MSE curves of thefrequency domain symbols of UW-OFDM and CP-OFDM in a multipath channel. It canbe observed that the MSE of the frequency domain symbols of an UW-OFDM system isless compared to that of the CP-OFDM scheme, thus making the UW-OFDM system lesssusceptible to the CFO in a multipath environment. Note that the data symbols that areequalized with an LMMSE equalizer result in a lower MSE compared to the case of a CIequalizer. Therefore, in the presence of CFO and a multipath channel, the LMMSE equal-izer performs optimum out of all the equalizers (i.e., CI, BLUE, and LMMSE) discussedin this work.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

CFO (ε)

Mea

n sq

uare

d er

ror


Figure 6.39: MSE curves of the frequency domain symbols of UW-OFDM with zero UWand CP-OFDM in a multipath environment (without noise).

Fig. 6.40 and Fig. 6.41 compare the MSE curves of the frequency domain symbols ofUW-OFDM and CP-OFDM in a multipath and a no multipath case, for an ε range of 0to 1 and 0 to 0.1, respectively. It can be observed that, the LMMSE equalizer removesthe inuence of the multipath channel very well in the presence of CFO, and as a resultthe MSE curve of the data symbols in a multipath environment approaches almost theno multipath case. However, in CP-OFDM the CI equalizer cannot perform well in thepresence of CFO, leading to a huge dierence between the MSE curves of the data symbolsfor a multipath and a no multipath case.

Fig. 6.42 shows the MSE curves of the frequency domain symbols of UW-OFDM uti-lizing xu1 as UW and CP-OFDM in a multipath environment. Similar to the no multipath(i.e., distortion and noise free) case, the MSE of the frequency domain symbols of UW-


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

CFO (ε)

Mea

n sq

uare

d er

ror

CP−OFDM data − CI equalizer − multipathCP−OFDM data − CI equalizer − no multipathUW−OFDM data − LMMSE equalizer − multipathUW−OFDM data − LMMSE equalizer − no multipath

Figure 6.40: MSE curves of the frequency domain symbols of UW-OFDM with zero UWand CP-OFDM in a multipath (without noise) and a distortion and noise free case.

0 0.02 0.04 0.06 0.08 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

CFO (ε)

Mea

n sq

uare

d er

ror

CP−OFDM data − CI equalizer − multipathCP−OFDM data − CI equalizer − no multipathUW−OFDM data − LMMSE equalizer − multipathUW−OFDM data − LMMSE equalizer − no multipath

Figure 6.41: MSE curves of the frequency domain symbols of UW-OFDM with zero UWand CP-OFDM in a multipath (without noise) and a distortion and noise free case.


OFDM is very high compared to that of the CP-OFDM scheme. It is important to notethat, even the MSE of the data symbols equalized with LMMSE equalizer is high comparedto the CP-OFDM scheme. This means that, in the presence of CFO and multipath propa-gation, the UW-OFDM system with an UW sequence (other than zeros) is more sensitiveto CFO compared to the CP-OFDM scheme. Fig. 6.43 and Fig. 6.44 depict the MSE ofthe equalized data symbols of UW-OFDM with dierent UW sequences, for an ε range of0 to 1 and 0 to 0.1, respectively. It reveals that the performance of the LMMSE equalizeris no longer independent of the UW sequence. It entirely depends on the UW sequence.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

CFO (ε)

Mea

n sq

uare

d er

ror

CP−OFDM data − CI equalizerUW−OFDM − CI equalizerUW−OFDM data − CI equalizerUW−OFDM redundant − CI equalizerUW−OFDM data − LMMSE equalizer

Figure 6.42: MSE curves of the frequency domain symbols of UW-OFDM with xu1 andCP-OFDM in a multipath environment (without noise).


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

CFO (ε)

Mea

n sq

uare

d er

ror




− LMMSE equalizer

Figure 6.43: MSE curves of the equalized data symbols of UW-OFDM with dierent UWsequences and CP-OFDM in a multipath environment (without noise).

0 0.02 0.04 0.06 0.08 0.10

5

10

15

20

25

CFO (ε)

Mea

n sq

uare

d er

ror




− LMMSE equalizer

Figure 6.44: MSE curves of the equalized data symbols of UW-OFDM and CP-OFDM ina multipath environment (without noise).

Chapter 7

Conclusions

The main aim of this work is to investigate the Carrier Frequency Oset (CFO) issue inan UW-OFDM system. For that, eects of CFO in time and frequency domain have beendiscussed. An introduction of a small CFO leads to a phase rotation of samples in timedomain, whereas in frequency domain, CFO leads to Inter Carrier Interference (ICI) alongwith an amplitude distortion and phase rotation of the data symbols. One major goal ofthis work is to evaluate the performance of time domain against frequency domain basedCFO estimators. In case of frequency domain estimators, pilot tones are used for the es-timation. Therefore, for enabling the insertion of pilot tones in an UW-OFDM symbol, amodied transmit signal structure has been introduced.

CFO estimation is normally carried out in two phases, namely an acquisition phase(coarse CFO estimation) and a tracking phase (ne CFO estimation). However, this thesisonly focuses on the investigation of algorithms for the ne CFO estimation. In the timedomain, a Maximum Likelihood (ML) CFO estimation algorithm utilizing the correlationbetween two successive UWs has been presented. This algorithm was derived under theassumption that the signal is only eected by the AWGN. The performance of the esti-mator mainly depends on two factors, one is the length of the UW, and the other is thecorrelation coecient between the corresponding samples of two UWs. Note that, the per-formance of the estimator is independent of the shape of the UW and also, the performanceimproves signicantly with the increase of the UW length, as an averaging is done overmore samples. Simulation results proved that the time domain estimator performance isindependent of the value of the CFO in both AWGN and multipath environments, providedthat the CFO is in the estimation range (-0.5 and +0.5). As the estimator was not derivedunder the multipath assumption, the performance of the estimator degrades signicantlyin a multipath channel compared to that of the AWGN channel. Since the UW is partof the DFT window, the UW can be equalized by applying the Channel Inversion (CI)equalizer on the received OFDM symbol. The simulation results proved that the estimatorusing the equalized UWs is worse compared to that of the unequalized UWs. It seems thata CI equalizer is no longer an appropriate equalizer in the presence of CFO in a multipathenvironment.

In contrast to CP-OFDM, UW-OFDM has the unique feature that the guard inter-val and thus the UW is the same for all OFDM symbols. Hence, the estimation can be

71

CHAPTER 7. CONCLUSIONS 72

performed utilizing two non-subsequent UWs separated by several OFDM symbols. Asa result, the estimator error variance decreases signicantly. This is because the formerone (using two successive UWs) has to estimate a small phase for a given Eb/N0 value,while the latter one has to estimate a larger phase (because the UWs are now non subse-quent) having the same Eb/N0 value, thereby reducing the error variance of the estimator.However, the estimation range of this new estimator is decreased with an increase in thedistance between the two UWs.

In frequency domain, a Pilot Tone Aided (PTA) technique, where the CFO is estimatedusing the phase shift introduced on the received pilot symbols, has been presented. At thereceiver, the pilots are estimated like they were data symbols by applying a linear equalizerat rst, and then the CFO is estimated out of these estimated pilots. In this work, theLMMSE, the BLUE, and the CI equalizers are considered. Besides this PTA technique, aDecision Directed (DD) method, in which the CFO is estimated by comparing the receivedfrequency domain data symbols with their corresponding sliced version symbols, has alsobeen investigated. The data symbols are also equalized rst before the estimation is carriedout. Since the slicing of the data symbols is done by tentative/hard decisions, this mightlead to errors in the estimate when wrong decisions are made. The estimator accuracy canbe increased by weighting each estimate dierently with the inverse of the correspondingelements in the error covariance matrix of the estimator.

It turns out that the LMMSE and the BLUE equalizer perform almost identically asthey exploit redundancy introduced among the frequency domain data in a similar way,whereas the CI equalizer has performed worse as it does not exploit the redundancy. Theequalizers perform very well in mitigating the multipath eects in the presence of CFO,hence the MSE curves for the multipath case almost approach the AWGN curves. Besidesthat, simulations show that the MSE curves of the DD estimators always lie below the MSEcurves of the PTA scheme. In general, the performance of all frequency domain estimatorsdegrades signicantly and reaches an early error oor with an increasing CFO. This is theresult of an increasing ICI on the data symbols with an increasing CFO.

Comparing time and frequency domain estimation methods with each other, the fol-lowing conclusions can be drawn: The performance of the time domain estimation methodis independent of the amount of CFO, while the performance of the frequency domainestimation methods heavily depend on the CFO. In case of a multipath environment, thetime domain estimator is worse compared to the frequency domain estimation methods, asin time domain the estimation has been carried out based on the multipath corrupted UW(as the equalizer even worsens the results), whereas in frequency domain the equalizersmitigate the eect of the multipath channel very well.

Furthermore, the sensitivity of an UW-OFDM system on the CFO has been analyzed.It is shown that, for an UW-OFDM with zero UW, the LMMSE and CI equalizers performalmost the same in both distortion free and multipath (no noise) environments. However,when a UW other than a zero word is considered, the CI equalizer performs completelyworse, while the LMMSE still performs well although the performance depends on the UWsequence as well as the channel characteristics. Additionally, simulation results proved

CHAPTER 7. CONCLUSIONS 73

that, in a distortion free environment, the sensitivity of the UW-OFDM scheme is inde-pendent of the UW sequence, and also less sensitive compared to the CP-OFDM scheme,whereas in a multipath environment, the sensitivity of the UW-OFDM scheme entirelydepends on the UW sequence.

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