Chapter 11ece6590/ch11.pdf · 11.2.3 Symbol-spaced versus fractionally-spaced equalizer A fractionally -spaced equalizer has the following advantages over its symbol-spaced counterpart:

$Page 1: Chapter 11ece6590/ch11.pdf · 11.2.3 Symbol-spaced versus fractionally-spaced equalizer A fractionally -spaced equalizer has the following advantages over its symbol-spaced counterpart:$
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:



I No sensitivity to timing phase.



I No sensitivity to timing phase.I Superior performance in most cases.



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];


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)


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)



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)




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





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




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

Chapter 11ece6590/ch11.pdf · 11.2.3 Symbol-spaced versus fractionally-spaced equalizer A fractionally -spaced equalizer has the following advantages over its symbol-spaced counterpart:

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