AN2072 Freescale Semiconductor / Motorola, AN2072 Datasheet - Page 4

• Manufacturer Part Number: AN2072
• Description: AN2072, Decision Feedback Equalizer for StarCore-Based DSPs
• Manufacturer: Freescale Semiconductor / Motorola

Derivation of DFE Filter Coefficients
2
The name MMSE-DFE implies that the DFE coefficients are derived under MMSE criteria. Consider the
development, where b
square-sense, by making the error orthogonal to the received sequence. Equation 3 represents the received signal
as, y(t), the channel-input data symbols as, x
white Gaussian noise and T is the symbol duration.
If l oversampling is used, then the sampled channel-output is expressed in Equation 4. For our implementation, l is
set to 1, and channel memory is expressed as v in the equations.
By combining N
A more compact representation of Equation 6 is expressed in Equation 7.
The equalizer output error is expressed in Equation 8.
4
Derivation of DFE Filter Coefficients
y
k
f
=
successive l-tuples of samples y
y k
⎛
⎝
k
+
and w
y kT
(
(
-------------- - T
°
°
l 1
Decision Feedback Equalizer for StarCore™-Based DSPs, Rev. 1
–
l
y
y
)
k
k
k
)
+
+
are the feedback and feed forward filter coefficients derived in minimum-mean-
y
⎞
⎠
N f 1
N f 2
°
k
≡
–
–
y
k
y l 1
+
y
Nf 1
–
=
0 k
°
°
,
n
,
–
h
k
0 h
° … … … … … …
0 … 0 h
k
0
,
y t ( )
=
k,
h
y
k
k
1
0
and the channel-impulse response as, h(t);
n k
=
h
=
⎛
⎝
… h
=
1
Hx
k
m
∑
m
+
:
n kT
… h
ν
∑
0
ν
(
x m
=
(
-------------- - T
k
°
°
l 1
h
+
0
–
0 … 0
l
ν
1
)
⋅
h
Nf 1
m
…h
)
0 …
h t mT
–
(
⋅
⎞
⎠
ν
–
x
,
≡
k m
⋅
k ν
–
n
–
l 1
)
n 0 k
x k
x k
–
+
+
°
°
+
,
x
n t ( )
+
+
h
n
k ν
,
n
m
k
°
N f 1
N f 2
–
k
k
+
=
–
–
Nf 1
h m
–
+
⎛
⎝
,
n
n k
h mT
+
k
k
(
(
-------------- - T
°
°
+
+
l 1
n
°
–
N f 1
N f 2
l
k
)
–
–
)
⎞
⎠
≡
where n(t) is additive-
h l 1
Freescale Semiconductor
h
–
0 m
°
°
,
,
m
Equation 3
Equation 4
Equation 5
Equation 6
Equation 7