equaliser

Structure and Design of
MMSE Channel Equalizers
Cenk Toker [e-mail] www.ee.hacettepe.edu.tr/~toker/equalizers.pdf

Intersymbol Interference (ISI) vk xk

H

+

yk received signal-no noise

N H −1

y k = ∑ h n x k −n v k n=0 transmitted data

ISI channel noisy received signal

MMSE FIR Equalizers v[k] xk

h

+ δ -m

Equalizer: b=1 y[k]

w b xk

●

Partial response equalizer: b: fixed (for example, b=1+z-1)
● Channel shortening equalizer: b: design parameter
●

^ ε[k] z[k]
+
z[k] v[k] +

h

y[k]

w

-

+

ε[k]

δ -m

xk

v[k]

+

h δ -m

y[k]

w b -

+

ε[k]

MMSE FIR
Decision Feedback Equalizer (DFE) xk v[k] h +

y[k]

w

Q

+

+

b δ -m
● Assumption: Past decisions are correct xk v[k]

+

h δ -m

y[k]

b δ -m

+

w

-

-

+

ε[k]

ε[k]

MMSE FIR Equalizers v[k] xk

h

y[k]

+ δ -m

n w −1

[

z k = ∑ w * y k −n= w * w * ⋯ w * −1

n
0
1 n n=0

w

]

n h−1

nb −1

m=0

[

zk = w

*
0

w

*
1

n=0

⋯ w

* nw −1

]

[

h0
0
0
0

h1 h0 ⋱
⋯

⋯ h n −1 0 h1 ⋯ hn −1
⋱ ⋱
⋯
0 h0 h1 h

h

H

z k = ∑ b n−m x k −n

=w H y k

w−1

*

n=0



y k = ∑ w  ∑ hn x k −n−m v k −m 
*
m

b

[ ] yk y k −1
⋮

y k −n

n w −1

w

^ ε[k] z[k]
- + z[k] ][ ] [ ]

⋯ 0 xk vk
⋯ 0 x k −1 v  k −1
⋯
⋮
⋱ 0
⋯ hn −1 x k −n n −2 v k −n h H

z k =w H x k w v k


h

w

w−1



[

*

*

*

= 0 ⋯ 0 b0 b1 ⋯ bn −1 0 ⋯ 0 b ]

[ ] xk x k−1
⋮

x k−n n −2 w [ ]

0m×1
= b
0

H

xk

H

z k =b x k

h

MMSE FIR Equalizers v[k] xk

+

h δ y[k]

w

- +

ε[k]

z[k]

b

-m

^ z[k] k =z k − z k


=[ b H −w H H ] x k −w H v k

●

Error:

●

Mean Square Error (MSE):

*

J =E {k k }
H
H
H
H

=[ b −w H