This equation shows that because of the 908 phase shift, x
0
n is orthogonal to
x
1
n and the off-diagonal terms in the R matrix are 0. The eigenvalues l
1
and l
2
of the R matrix are identical and equal to A
2
=2. Therefore the system has very fast
convergence since the eigenvalue spread equals 1. The time constant of the
adaptation is approximated as
t
mse
1
ml
2
mA
2
,
which is determined bythe power of the reference sinewave and the step size m.
8.5.5 Adaptive Channel Equalization
In digital communications, considerable effort has been devoted to the development of
data-transmission systems that utilize the available telephone channel bandwidth effi-
ciently. The transmission of high-speed data through a channel is limited by intersymbol
interference (ISI) caused bydistortion in the transmission channel. High-speed data
transmission through channels with severe distortion can be achieved in several ways,
such as (1) bydesigning the transmit and receive filters so that the combination of filters
and channel results in an acceptable error from the combination of ISI and noise; and
(2) bydesigning an equalizer in the receiver that counteracts the channel distortion. The
second method is the most commonlyused.
As illustrated in Figure 8.11, the received signal y(n) is different from the original
signal x(n) because it was distorted bythe overall channel transfer function C(z), which
includes the transmit filter, the transmission medium, and the receive filter. To recover
the original signal, x(n), we need to process y(n) using the equalizer W(z), which is the
inverse of the channel's transfer function C(z) to compensate for the channel distortion.
That is, we have to design the equalizer
Wz
1
Cz
,
8:5:11
i.e., CzWz 1 such that
^
xn xn.
In practice, the telephone channel is time varying and is unknown in the design stage
due to variations in the transmission medium. The transmit and receive filters that are
-
C(z)
W(z)
LMS
z
-
d(n)
+
x(n)
y(n)
x(n)
^
e(n)
Figure 8.11 Cascade of channel with an ideal adaptive channel equalizer
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