Upload
vuongdiep
View
217
Download
0
Embed Size (px)
Citation preview
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
National Chung-Cheng University
OUTLINE
1. Array Signal Processing Concept
2. Blind Channel Estimation Using Array Signal ProcessingTechniques
3. Principal Component Combiner with Differential Decoding forDPSK Signals
2
National Chung-Cheng University
Array Signal Processing
Concepts
3
National Chung-Cheng University
Direction of Arrival Estimation Problem
Direction of Arrival
s(t)
d
4
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
Blind Channel Estimation
Using Array Signal
Processing Techniques
7
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
Conventional MethodConstrained MethodSubspace MethodTorlak−Xu Method (SF=2)Torlak−Xu Method (SF=3)True Channel Information
18
National Chung-Cheng University
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
Conventional MethodConstrained MethodSubspace MethodTorlak−Xu Method (SF=2)Torlak−Xu Method (SF=3)True Channel Information
19
National Chung-Cheng University
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
SNR is 10 dB, 10 users, 4−ray channel, DBPSK
Conventional MethodConstrained MethodSubspace MethodTorlak−Xu Method (SF=2)Torlak−Xu Method (SF=3)True Channel Information
20
National Chung-Cheng University
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)
SNR is 10 dB, 5 users, 4−ray channel, DBPSK
Conventional MethodConstrained MethodSubspace MethodTorlak−Xu Method (SF=2)Torlak−Xu Method (SF=3)
21
National Chung-Cheng University
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
National Chung-Cheng University
Principal Component
Combiner with Differential
Decoding for DPSK Signals
23
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
National Chung-Cheng University
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
Average Signal−to−Noise Ratio per Bit (dB)
PCC−DDDD−EGC
L=2
L=4
L=8
33
National Chung-Cheng University
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
Average Signal−to−Noise Ratio per Bit (dB)
Binary DPSKQuadrature DPSKOctal DPSK
L=2
L=4
L=8
34
National Chung-Cheng University
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
References
[1] D. Brennan, “Linear diversity combining techniques,” Proceedings of the IEEE,
vol. 91, no. 2, pp. 331-356, February 2003.
[2] P. Comon and G. H. Golub, “Tracking a few extreme singular values and vectors
in signal processing,” Proceedings of the IEEE, vol. 75, no. 8, pp. 1327-1343,
August 1990.
[3] D. Johnson and D. Dudgeon, Array Signal Processing: Concepts and
Techniques. Prentice Hall, 1993.
[4] S. Haykin, Adaptive Filter Theory. 2nd, Ed., Prentice Hall, 1991.
[5] M. Hayes, Statistical Digital Signal Processing and Modeling. Wiley, 1996.
[6] J. Karhunen, “Adaptive algorithm for estimating eigenvectors of
correlation-type matrices,” IEEE Int’l Conf. Acoustics, Speech, and Signal
Processing, vol. 9, pp. 592-595, March 1984.
[7] T.-H. Liu, “Linearly constrained minimum variance filters for blind multiuser
detection,” to appear in the IEEE Trans. Commun.
[8] T.-H. Liu, Y.-C. Li, and C.-C. Chang, “Applications of array signal processing
techniques to code division multiple access systems for blind channel
estimation,” to be presented at the IEEE PIMRC, Beijing, China, Sep. 2003.
[9] J. Proakis, Digital Communications. 4th Ed., McGraw Hill, 2001.
36
REFERENCES REFERENCES
[10] D. Reynolds, X. Wang, and H. V. Poor, “Blind adaptive space-time multiuser
detection with multiple transmitter and receiver antennas,” IEEE Trans. Signal
Proc., vol. 50, no. 6, pp. 1261-1277, Jun. 2002.
[11] M. Simon and M. Alouini, Digital Communication over Fading Channels- A
Unified Approach to Performance Analysis. Wiley, 2000.
[12] M. Torlak and G. Xu, “Blind multiuser channel estimation in asynchronous
CDMA systems,” IEEE Trans. Signal Proc., vol. 45, no. 1, pp. 137-147, Jan.
1997.
[13] M. K. Tsatsanis and Z. Xu, “Performance analysis of minimum variance CDMA
receivers,” IEEE Trans. Signal Proc., vol. 46, no. 11, pp. 3014-3022, Nov. 1998.
[14] S. Verdu, Multiuser Detection. Cambridge, 1998.
[15] X. Wang and H. V. Poor, “Blind equalization and multiuser detection in
dispersive CDMA channels,” IEEE Trans. Commun., vol. 46, pp. 91-103, Nov.
1998.
[16] X. Wang and H. V. Poor, “Blind joint equalization and multiuser detection for
DS-CDMA in unknown correlated noise,” IEEE Trans. Signal Proc., vol. 46,
pp. 886-895, July 1999.
37