AN2072 Freescale Semiconductor / Motorola, AN2072 Datasheet - Page 3

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

The ideal, infinite-length feed-forward filter is a noise whitening filter that results in an overall response with
minimum phase. Choosing an infinite-length filter eliminates the delay optimization, because the overall response,
consisting of transmit, channel, receiver, sampler, and FF filters is simply a filter with minimum phase. The first
tap in the overall response is the main tap. Spectral factorization is a very important concept used in deriving DFE
taps. A finite version of the infinite tap DFE is derived in [1]. The Cholesky factorization theory is a useful tool in
deriving the finite-length equivalent to the infinite-length DFE. Terms used in DFE development are described as
follows:
Freescale Semiconductor
1.
2.
3.
4.
x
k
A filter response F(D), is canonical if it is:
— causal (f
— monic (f
— minimum phase
If F(D) is canonical then F
— anti-causal
— monic
— maximum-phase
An n
are positive definite
Matrices with constant entries along a diagonal are called Toeplitz matrices. Equation 2 expresses a
Toeplitz matrix with the number three as the constant entry along the diagonal.
Channel
×
n symmetric matrix, A, is positive definite if
k
0
=0, for k<0)
=1)
Decision Feedback Equalizer for StarCore™-Based DSPs, Rev. 1
X 0
n
k
≠
. The definition for a positive definite matrix is expressed in Equation 1.
T
y
k
=
⇒
*
(D
X
Figure 2. DFE Block Diagram
3 4 0
2 3 4
1 2 3
T
-*
AX
Feed Forward
) is:
Filter
w
=
,
-k
i
T
=
∑
n
is Toeplitz
1
i
∑
=
n
1
a
ij
DFE
x
z
i
k
x
all diagonal elements and submatrices of A
j
>
0
Feedback
Decision
Filter
δ
k
−
b
Decision Feedback Equalization Theory
k
x
k
+N
f
-1-
Equation 1
Equation 2
∆
3