Chapter 11ece6590/ch11.pdf · 2009. 11. 12. · 11.2.3 Symbol-spaced versus fractionally-spaced...
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Chapter 11 Channel Equalization March 26, 2008
Transcript of Chapter 11ece6590/ch11.pdf · 2009. 11. 12. · 11.2.3 Symbol-spaced versus fractionally-spaced...
Chapter 11
Channel Equalization
March 26, 2008
Demodulator
coderbit/
symbolModulator
⊕ ⊕
Carrier
recovery
transmit
filter, pT(t)
slicer/
decoder
interference form
other usersnoise
sampler
b[n]
b̂[n]
Timing
recovery
filter, pR(t)
receive/matchedEqualizer
Channel, c(t)
LNA/AGC
11.1 Continuous-Time Channel Model
⊕cBB(t)
s(t) =∞∑
n=−∞
s[n]δ(t− nTb)y(t)
ν(t)
w(t)x̂(t)
11.1 Discrete-Time Channel Model
11.2.1 Symbol-spaced equalizer
⊕c[n]
11.2.2 Fractionally-spaced equalizer
⊕c[n]
︸ ︷︷ ︸
11.2.2 Fractionally-spaced equalizer (continued)
Details of a fractionally-spaced equalizer with tap-spacing(M/L)Tb
⊕
11.2.3 Symbol-spaced versus fractionally-spacedequalizer
A fractionally -spaced equalizer has the following advantagesover its symbol-spaced counterpart:
11.2.3 Symbol-spaced versus fractionally-spacedequalizer
A fractionally -spaced equalizer has the following advantagesover its symbol-spaced counterpart:
I No sensitivity to timing phase.
11.2.3 Symbol-spaced versus fractionally-spacedequalizer
A fractionally -spaced equalizer has the following advantagesover its symbol-spaced counterpart:
I No sensitivity to timing phase.I Superior performance in most cases.
11.2.3 Symbol-spaced versus fractionally-spacedequalizer
A fractionally -spaced equalizer has the following advantagesover its symbol-spaced counterpart:
I No sensitivity to timing phase.I Superior performance in most cases.
Symbol-spaced equalizers, on the other hand, may offer lowercomplexity, in some cases (NOT ALWAYS!).
11.3 Performance Study of Equalizers
This section presents a detailed derivation of equations thatmay be used to evaluate the optimum coefficients ofsymbol-spaced and fractionally-spaced equalizers and therespective minimum mean-square errors (MMSEs).
11.3 Performance Study of Equalizers (continued)
System set-up for study of a symbol-spaced equalizer:
⊕ ⊕
11.3 Performance Study of Equalizers (continued)
System set-up for study of a half symbol-spaced equalizer:
⊕ ⊕↑ 2
11.3.2 Numerical examples
Simulated channels:
c = c1 = [1 zeros(1, 91) 0.4];
c = c2 = [0.5 zeros(1, 60) 1 zeros(1, 123) 0.25];
c = c3 = [1 zeros(1, 67) 0.75 zeros(1, 145) 0.4];
c = c4 = [1 zeros(1, 75) 0.6 zeros(1, 103) 0.2];
11.3.2 Numerical examples
Simulated channels:
c = c1 = [1 zeros(1, 91) 0.4];
c = c2 = [0.5 zeros(1, 60) 1 zeros(1, 123) 0.25];
c = c3 = [1 zeros(1, 67) 0.75 zeros(1, 145) 0.4];
c = c4 = [1 zeros(1, 75) 0.6 zeros(1, 103) 0.2];
A good choice of ∆:
∆ =12
(length of channel + length of equalizer)
11.3.2 Numerical examples
c1 = [1 zeros(1, 91) 0.4];
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9
0.92
0.94
0.96
0.98
1
1.02S
igna
l Pow
er
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−4
10−3
10−2
10−1
Timing Phase
MM
SE
T
b spaced equalizer (N=31)
Tb/2 spaced equalizer (N=31)
Tb/2 spaced equalizer (N=61)
11.3.2 Numerical examples
c2 = [0.5 zeros(1, 60) 1 zeros(1, 123) 0.25];
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.24
1.26
1.28
1.3
1.32
1.34
1.36S
igna
l Pow
er
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−5
10−4
10−3
10−2
10−1
Timing Phase
MM
SE
Tb spaced equalizer (N=31)
Tb/2 spaced equalizer (N=31)
Tb/2 spaced equalizer (N=61)
11.3.2 Numerical examples
c3 = [1 zeros(1, 67) 0.75 zeros(1, 145) 0.4];
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.07
1.08
1.09
1.1
1.11
1.12
1.13
1.14S
igna
l Pow
er
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−4
10−3
10−2
10−1
Timing Phase
MM
SE
T
b spaced equalizer (N=31)
Tb/2 spaced equalizer (N=31)
Tb/2 spaced equalizer (N=61)
11.3.2 Numerical examples
c4 = [1 zeros(1, 75) 0.6 zeros(1, 103) 0.2];
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.55
0.6
0.65
0.7
0.75
0.8S
igna
l Pow
er
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−4
10−3
10−2
10−1
Timing Phase
MM
SE
T
b spaced equalizer (N=31)
Tb/2 spaced equalizer (N=31)
Tb/2 spaced equalizer (N=61)
11.4 Adaptation AlgorithmsSymbol-spaced equalizer:
c1 = [1 zeros(1, 67) 0.75 zeros(1, 145) 0.4];
0 200 400 600 800 100010−4
10−3
10−2
10−1
100
101
No. of Iterations
ξ
NLMSAPLMSRLS
11.4 Adaptation AlgorithmsSymbol-spaced equalizer:
c3 = [1 zeros(1, 67) 0.75 zeros(1, 145) 0.4];
0 200 400 600 800 100010−4
10−3
10−2
10−1
100
101
No. of Iterations
ξ
NLMSAPLMSRLS
11.4 Adaptation Algorithms
0 0.2 0.4 0.6 0.8 110−2
10−1
100
101
fTb
|cB
B(e
j2πf
)|2
Channel c
1Channel c
3
11.4 Adaptation AlgorithmsFractionally-spaced equalizer:
c1 = [1 zeros(1, 67) 0.75 zeros(1, 145) 0.4];
0 200 400 600 800 100010−4
10−3
10−2
10−1
100
101
No. of Iterations
ξ
NLMSAPLMSRLS
11.4 Adaptation AlgorithmsFractionally-spaced equalizer:
c3 = [1 zeros(1, 67) 0.75 zeros(1, 145) 0.4];
0 200 400 600 800 100010−4
10−3
10−2
10−1
100
101
No. of Iterations
ξ
NLMSAPLMSRLS
11.5 Cyclic EqualizationSymbol-spaced equalizer:
⊕
z−1
z−1
z−1
y[n] y[n − 1] y[n − 2] y[n − N ]
z−1
z−1
s[N ]s[2]s[1]s[0]
⊕Adaptation
Algorithm
z−1
for i = 0, 1, 2, · · ·
e[i] = s[i mod N + 1] − wH[i]yi
w[i + 1] = w[i] + 2µe∗[i]yi
end
11.5 Cyclic EqualizationFractionally-spaced equalizer:
⊕
z−1
z−1
z−1
y[n] y[n − 1] y[n − 2] y[n − N ]
s[N ]s[2]s[1]s[0]
⊕Adaptation
Algorithm
z−1
z−2
z−2
z−2
z−2
Iterate after every 2 clock cycles
Pilot Symbols
s[n] =
{
ejπn2/(N+1), for N + 1 evenejπn(n+1)/(N+1), for N + 1 odd.
Comparisons (symbol-spaced)
Table: Performance comparison of the cyclic equalizers for direct andindirect setting.
Average MSE of Cyclic EqualizerChannel MMSE Direct Indirect
c1 0.000138 0.00380 0.000240c2 0.000085 0.00241 0.000151c3 0.000644 0.00547 0.000750c4 0.000311 0.01832 0.000676
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