AN2072 Freescale Semiconductor / Motorola, AN2072 Datasheet - Page 6

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

Derivation of DFE Filter Coefficients
The middle term in the right-hand side of Equation 18, is defined as a Cholesky factorization, where LDL’ is the
Lower-Diagonal-Upper.
Where L is a lower-triangular monic matrix, D is a diagonal matrix shown in Equation 19.
Because L is a monic matrix, its columns constitute a basis for the (N
[1] that the optimal setting for b is column L which corresponds to the highest value of “d” in the diagonal matrix.
When the feedback coefficients are located, the solution expressed in Equation 20, gives the optimal setting for w,
the feed forward coefficient.
The first coefficient of b is always unity, therefore a simple back substitution method can be used to solve for the
feed forward coefficients. This process for derivation of DFE filter coefficients, eliminates the need for the matrix
inversion. It has been seen that the:
6
I
Nf
2. The matrix inversion lemma:
+
ν
•
•
–
H?asteriskframe? HH?asteriskframe?
H?asteriskframe? HH?asteriskframe?
number of taps in the feedback filter N
optimum delay ∆ is between ν and N
kframe?R nn
⎛
⎝
⎛
⎝
Decision Feedback Equalizer for StarCore™-Based DSPs, Rev. 1
1 –
H
=
⎛
⎝
R xx R xy R
–
ame?
+
+
---------- - I
SNR
---------- - I
SNR
1
1
D
1 –
L
opt
yy
N
=
=
f
R
f
Nf
b
+ ν
⎞
⎠
yx
1 –
=
= ν,
d 0 …
l
+
0 …
0 …d
0
⎞
⎠
H
ν
1 –
…l
b?asteriskfram
⎞
⎠
=
1 –
=
N
⎛
⎝
N
H
f
HH?asteriskframe?
N
---------- - I
SNR
f
+
+
f
1
0
0
ν 1
> N
ν 1
I
–
ν
–
0
Nf
+
b
1
+
=
f
ν
+ν) dimensional vector space. It is shown in
+
N
0
H?asteriskframe?H
0 I ν
+
---------- - I
SNR
+
1
1
⎛
⎜
⎜
⎝
N
⎛
⎝
f
HH?asteriskframe?
⎞
⎠
1 –
H?asteriskframe?H
=
Freescale Semiconductor
L
+
Equation 17
Equation 18
Equation 19
Equation 20
---------- - I
SNR
1
Nf
+
ν