Digital Signal Processing Reference
In-Depth Information
input samples in matrix form:
x(
0
)
x(
−
1
)
···
x(
−
n
pre
)
0
···
0
.
x(
1
)
x(
0
)
···
x(
−
n
pre
+
1
)
−
n
pre
)
0
.
.
.
.
0
.
x(n
post
)
post
−
1
)
x(
−
n
pre
+
1
)
x(
−
n
pre
)
x
=
.
0
x(n
post
)
x(
−
n
pre
+
2
)
−
n
pre
+
1
)
.
.
.
.
.
.
.
.
0
···
x(n
post
)
(
post
−
1
)
···
x(
0
)
x(
−
1
)
0
···
0
x(n
post
)
···
x(
1
)
x(
0
)
(12-24)
The columns of
x
represent the taps of the equalizer and the rows represent con-
secutive time steps with an interval between steps that is equal to the tap spacing
of the equalizer. The matrix is square with
n
pre
1 rows and columns. In
this form,
x
shows the propagation of the input samples through the equalizing
filter. For example,
x
(0) appears in the first row and the first column, which
indicates that it is at the first tap at the first time sample. It also appears in the
second row and second column, which corresponds to the second tap and second
time sample. This is exactly what we expect for a discrete linear equalizer.
In matrix form, equation (12-21) is written
+
n
post
+
y
=
xc
(12-25)
where
y
is the vector containing the output from the equalizer and
c
is the
vector of equalizer tap coefficients. The number of elements in both
y
and
c
is
n
pre
1. We have essentially expressed the discrete convolution of the
data stream and equalizer as a matrix multiplication, which we now apply to our
problem.
By transmitting a lone pulse, we define the expected output results from the
equalizer according to Nyquist's first method for the elimination of ISI [Nyquist,
1928; Couch, 1987]:
+
n
post
+
0
for
k
=
0
y
target
=
(12-26)
1
for
k
=
0
In (12-26) the expected value is 1 for the cursor sample and zero for all others.
The equalizer coefficients that give the zero forcing solution are then
x
−
1
y
target
c
ZFS
=
(12-27)
The coefficients calculated are not constrained by power or by the maximum
signal swing that can be achieved. Application of such constraints requires that
the coefficients be adjusted using equation (12-22), as we illustrate next.
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