application of array signal processing techniques to...

National Chung-Cheng University

Application of Array Signal ProcessingTechniques to the Design of Rake Receivers

Tsung-Hsien Liu

National Chung Cheng University

Department of Communication Engineering

160, San-Hsing, Ming-Hsiung, Chia-Yi, Taiwan

1


OUTLINE

1. Array Signal Processing Concept

2. Blind Channel Estimation Using Array Signal ProcessingTechniques

3. Principal Component Combiner with Differential Decoding forDPSK Signals

2


Array Signal Processing

Concepts

3


Direction of Arrival Estimation Problem

Direction of Arrival

s(t)

d

4


The Mathematical Model for the DOA Estimation Problem

The signal received at the m-th antenna is

rm(t) =K∑

k=1

sk(t) e−j2πm dλ sin(θk) + nm(t), m = 0, 1, · · · ,M − 1.

where θk is the direction of arrival of user k. It can be written invector form

r0(t)

r1(t)...

rM−1(t)

︸︷︷︸r(t)

=K∑

k=1

1

e−j2π dλ sin(θk)

...

e−j2π(M−1) dλ sin(θk)

︸︷︷︸a(θk)

sk(t) +

n0(t)

n1(t)...

nM−1(t)

︸︷︷︸n(t)

5


Direction of Arrival Estimation Methods fromArray Signal Processing

Conventional Method φ = arg maxaH(φ)Ra(φ)

Constrained Method φ = arg mina(φ)HR−1a(φ)

Subspace Method φ = arg minaH(φ)VnVHn a(φ)

where

a(φ) is the steering vector

R is the sample covariance matrix

Vn contains the vectors in the noise subspace of Rn

6


Blind Channel Estimation

Using Array Signal

Processing Techniques

7


Multipath Fading for the Code Division Multiple Access System

path 0

path 1

path 2

path L-1

t 1

t 2

t L -1

8


Discrete-Time Channel Model for the Desired User

The effective signature vector h4= [h(0) h(1) · · · h(Lc + q − 1)]T

h =

c(0) . . . 0q

c(1) · · · c(0)...

. . .

c(Lc − 1)...

0q . . . c(Lc − 1)

︸︷︷︸C

g(0)

g(1)...

g(q)

︸︷︷︸g

where

0q is a zero vector of dimension q × 1

g(i), i = 0, 1, · · · , q − 1 be the baseband equivalent impulse response

c(0), c(1), · · · , c(Lc − 1) be the spreading code of the desired user

9


Asynchronous CDMA Channel Model at the Receiver

Assume perfect synchronization and chip-rate sampling.

r(n)4=

r(nLc)

r(nLc + 1)...

r(nLc + Lc + q − 1)

= h b(n) + w(n) + i(n) + v(n)

where

w(n) is the intersymbol interference

i(n) is the multiuser interference

v(n) is the additive noise

10


Asynchronous CDMA Channel Model at the Receiver

• The intersymbol interference

w(n)4=

h(Lc)...

h(Lc + q − 1)

0Lc

b(n− 1) +

0Lc

h(0)...

h(q − 1)

b(n + 1)

• The multiuser interference i(n) is comprised of componentscontributed by all other multiusers in the same system. Each ofthe multiusers is modelled in the same manner as the desireduser but with a different delay and a different multipathchannel.

• The additive noise v(n) is assumed to be white Gaussian withcovariance matrix σ2I.

11


The Conventional Method for Blind Channel Estimation

In case that a single user exist and that the delay spread is small,the received signal is reduced to

rconv(n) = h b(n) + v(n)

The solution is

hconv = arg maxf

E{|fH rconv(n)|2} subject to ||f || = 1

= arg maxf

fHRf subject to ||f || = 1

= arg maxf

fHRffHf

12


The Conventional Method for Blind Channel Estimation

The conventional method for blind channel estimation is

gconv,1 = arg maxg

gHCHRCggHCHCg

which indicates that gconv,1 is the normalized generalizedeigenvector of (CHRC,CHC) that is associated with the largesteigenvalue, or

gconv,2 = arg maxg

gHCHRCggHg

which indicates that gconv,2 is the normalized eigenvector ofCHRC that is associated with the largest eigenvalue.

13


The Constrained Method for Blind Channel Estimation

The constrained method for blind channel estimation is

gcapon,1 = arg ming

gHCHR−1CggHCHCg

which indicates that gcapon,1 is the normalized generalizedeigenvector of (CHR−1C,CHC) that is associated with thesmallest eigenvalue. A different selection of the constrained methodfor g is

gcapon,2 = arg ming

gHCHR−1CggHg

which indicates that gcapon,2 is the normalized eigenvector ofCHR−1C that is associated with the smallest eigenvalue.

14


The Subspace Method for Blind Channel Estimation

The subspace method for blind channel estimation is

gsub,1 = arg ming

gHCHEnEHn Cg

gHCHCg

which indicates that gsub,1 is the normalized generalizedeigenvector of (CHEnEH

n C,CHC) that is associated with thesmallest eigenvalue. Similarly, we can select g as

gsub,2 = arg ming

gHCHEnEHn Cg

gHg

which indicates that gsub,2 is the normalized eigenvector ofCHEnEH

n C that is associated with the smallest eigenvalue.

15


Simulation Environment

• Pulse shaping function: root-raised cosine function with rollofffactor 0.5

• Gold code of length 31

• Four-ray multipath channel with relative path gains 0, -3, -6,and -9 dB

• Differential BPSK signals

• Sample covariance is estimated by R =∑100

n=1 r(n)rH(n)/100

• Number of Monte Carlo runs: 1000

• Minimum mean-squared error detector f = EsD−1s EH

s C gwhere

R =[Es En

] Ds

Dn

[Es En

]H

16


Comparisons of Blind Channel Estimation Methods Under the4-ray Channel for Different Signal-to-Noise Ratios

2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

Bit

Err

or R

ate

Signal to Noise Ratio (dB)

10 users, 4−ray channel, DBPSK

Conventional MethodConstrained MethodSubspace MethodTorlak−Xu Method (SF=2)Torlak−Xu Method (SF=3)True Channel Information

17


Comparisons of Blind Channel Estimation Methods Under the4-ray Channel for Different Number of Interferers

2 4 6 8 10 12 1410

−4

10−3

10−2

10−1

Bit

Err

or R

ate

Number of Users

SNR is 10, 4−ray channel, DBPSK


18


Comparisons of Blind Channel Estimation Methods Under the4-ray Channel for a Strong Interferer at Different Interference

to the Desired User Power Ratio

2 4 6 8 10 12 14 16 18 2010

−2

10−1

100

Bit

Err

or R

ate

Interferer to the Desired User Power Ratio (dB)

SNR is 10 dB, 10 users, 4−ray channel, DBPSK


19


Comparisons of Blind Channel Estimation Methods UnderDifferent Number of Multipath Components

2 4 6 8 10 12 14 16 18 2010

−2

10−1

100

Bit

Err

or R

ate

Number of multipath components of the desired user



20


MSE Comparisons of Blind Channel Estimation Methods UnderDifferent Intercell Interference to the Desired Signal Ratio

−20 −15 −10 −5 0 5 10 15 2010

−3

10−2

10−1

100

101

Mea

n S

quar

ed E

rror

Intercell Interferer to the Desired User Power Ratio (dB)


Conventional MethodConstrained MethodSubspace MethodTorlak−Xu Method (SF=2)Torlak−Xu Method (SF=3)

21


Comparisons of the 3 Blind Channel Estimation Methods

Conventional Constrained Subspace

Method Method Method

Performance Poor Good Good

Near-Far

ResistanceNo Yes Yes

Robust to

Intercell

Interference

No Yes No

Complexity 0 O((Lc + q)2) O((Lc + q)3)

Adaptation

ComplexityLow Medium High

22


Principal Component

Combiner with Differential

Decoding for DPSK Signals

23


Problem of Interest

The signal received at the i-th diversity channel is

ri(k) =√Esci ejφ(k) + vi(k), i = 1, 2, · · · , L

whereci is the channel gain

φ(k) = φ(k − 1) +4φ(k) carries the M -ary DPSK information

vi(k) is the zero-mean white Gaussian noise with variance N0

The above model can be written in vector form

r(k) =√Esc ejφ(k) + v(k)

24


The Differential Decoding with Equal Gain Combiner

Equal Gain Combiner

Differential Detector

r 1 (k)

Z -1 ( )*

r 2 (k)

Z -1 ( )*

r L (k)

Z -1 ( )*

Decision

25


Principal Component Combiner with Differential Decoding

r 2 (k)

r 1 (k)

w 1 *

r L (k)

w L *

w 2 *

Principal Component Combiner

Z -1

y(k) z(k) Decision

( )*

Differential Detector

26


Principal Component Combiner with Differential Decoding

The PCC-DD determine the combining weight as

wP = arg maxw

E{‖ wHr(k) ‖2}‖ w ‖2

= arg maxw

wHRrw‖ w ‖2

where the covariance matrix is

Rr = EsccH + N0I

Theoretically,

wP = ejθ c/ ‖ c ‖

27


BEP of Binary DPSK Signals over Rayleigh Fading Channels

Recall that the BEP of the binary DPSK signal over AWGNchannel is

P2(γs) =12

e−γs

where the instantaneous SNR per symbol

γs4= Es ‖ c ‖2 /N0

Recall that the instantaneous SNR is Chi-Squared distributed withdensity function

p(γs) =1

(L− 1)! γLc

γL−1s e−γs/γc

where the average SNR per channel is defined by

γc4=Es

N0E{|ci|2}

28


BEP of Binary DPSK Signals over Rayleigh Fading Channels

The exact BEP of binary DPSK signals for the PCC-DD is

P2 =∫ ∞

0

P2(γs)p(γs) dγs

=12

1(1 + γc)L

∫ ∞

0

γL−1s e−γs/(γc/(1+γc))

(L− 1)!(γc/(1 + γc))Ldγs

=12

1(1 + γc)L

Note that the last equality follows from the fact that the integral inthe second line is equal to unity, because integral of the probabilitydensity function of a chi-square distributed random variable isunity.

29


BEP of M -ary DPSK Signals over Rayleigh Fading Channels

The approximate BEP of M -ary DPSK signals for the PCC-DD is

PM ≈ 2log2 M

(1− µ

2

)L L−1∑

k=0

L− 1 + k

k

(1 + µ

2

)k

where

µ =

√12 sin2 π

M γc

1 + 12 sin2 π

M γc

30


Adaptive Calculation of the Principal Component Combining Weight

Use the rank-1 subspace tracker, e.g., Karhunen’s method

% Initialization: u(0) = [1, 0, · · · , 0]T

% λ(0) = 1, β = 0.99

% As time increases, k = 1, 2, 3, · · · Complexity

ξ = rH(k)u(k − 1) L

α = ξ/β/λ(k − 1) 2

u(k) = u(k − 1) + α r(k) L

u(k) = u(k)/ ‖ u(k) ‖ 2L

λ(k) = βλ(k − 1) + |ξ|2 2

% Go to k = k + 1

31


BEP Comparisons of 2-DPSK Signals for PCC-DD and DD-EGCover Rayleigh Fading Channels

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

Bit

Err

or P

roba

bilit

y

Average Signal−to−Noise Ratio per Bit (dB)

PCC−DDDD−EGC

L=2

L=4

L=8

32


BEP Comparisons of 4-DPSK Signals for PCC-DD and DD-EGCover Rayleigh Fading Channels

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

Bit

Err

or P

roba

bilit

y


PCC−DDDD−EGC

L=2

L=4

L=8

33


BEP Comparisons of 2-, 4-, and 8-DPSK Signals for PCC-DDover Rayleigh Fading Channels

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

Bit

Err

or P

roba

bilit

y


Binary DPSKQuadrature DPSKOctal DPSK

L=2

L=4

L=8

34


Conclusions

• The exact BEP of binary DPSK signals for PCC-DD isavailable.

• The approximate BEPs of M -ary DPSK (M ≥ 4) signals forPCC-DD are derived.

• For binary DPSK signals, the PCC-DD outperforms theDD-EGC by 0.9, 1.5, and 2.3 dB when the number of diversitychannels is 2, 4, and 8, respectively.

• For M -ary DPSK (M ≥ 4) signals, the PCC-DD outperformsthe DD-EGC when the average SNR per bit is low, butperforms approximately the same as the DD-EGC when theaverage SNR is high.

• The proposed PCC-DD needs 6L complex-valuedmultiplications in each iteration to update its principlecombining weight vector.

35

REFERENCES REFERENCES

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36

REFERENCES REFERENCES

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