AN2072 Freescale Semiconductor / Motorola, AN2072 Datasheet - Page 5

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

The w
For a decision delay of ∆, the corresponding MMSE is expressed in Equation 10.
Equation 11 shows where b is the vector of the coefficients for the feedback FIR filter and x
the data symbols in the feedback path.
Applying the orthogonal principle by making the error orthogonal to the output we get Equation 12.
In other words, the optimum error sequence is uncorrelated with the observed data. This simplifies to Equation 13,
which gives the relation between the DFE feedback and feed forward filter coefficients
The FIR MMSE-DFE autocorrelation matrix is expressed in Equation 14.
Where R
correlation matrix,
The mean-square error is expressed in Equation 16.
Simplifying Equation 16 by using the matrix inversion lemma
Freescale Semiconductor
1. (.)
R
* 1
1 –
yy
*
nn
feed forward filter taps are expressed in Equation 9.
R
is the complex conjugate transpose
k
yx⎠
2
= N
⎫
⎬
⎭
⎞
=
=
0
E e k
I
S
Nf
{
x
where S
0 I ν
⋅
and where N
e k ?asteriskframe?
y
+
=
E Y ˜ k
1
Decision Feedback Equalizer for StarCore™-Based DSPs, Rev. 1
⎧
⎨
⎩
E x k k
x
[
I
Nf
is the signal power
+
,
Nf 1 k
+
ν
0
–
–
is the noise power, and where I is an identity matrix, the input-output cross-
–
ν
?
y?asteriskframe? k
H?asteriskframe? HH?asteriskframe?
,
=
} E x k ∆
eriskframe?
=
Y?asteriskframe?
[
w?asteriskframe?
˜
⎧
⎨
⎩
(
–
ame?R
b
⋅
x k k ν
Y
=
–
,
k
as expressed in Equation 15.
⎛
⎝
?asteriskframe?
wY
+
xy
–
1 b 0 …b Nb
Nf 1 k
k
–
k
=
(
+
–
+
–
Nf 1
w?asterisk
bx k ∆
Nf 1 k
w?asteriskframe?y k
,
–
–
–
]
)
2
w?asteriskframe?
,
=
results in Equation 17.
–
S x 0
}
⎫
⎬
⎭
1
⎞
⎟
⎠
) x k ∆
[
(
=
(
ν
+
S
–
+
---------- - I
SNR
x
HH
1
1
) Nf 1
–
(
wY
?asteriskfram
Nf
–
–
Derivation of DFE Filter Coefficients
k
+
+
ν
) … I
⎞
⎠
bx k ∆
1 –
(
ν
H
–
+
1
–
I
)
ν
]
1
k
0
+
)?asteriskfram
–
1
∆ − 1
is the vector of
Equation 10
Equation 11
Equation 12
Equation 13
Equation 14
Equation 15
Equation 16
Equation 8
Equation 9
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