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606 Chapter 11 APPLICATIONS IN ADAPTIVE FILTERING
decision rule:
Re [â(n)] > 0 and Im [â(n)] > 0 −→ 1+j
Re [â(n)] > 0 and Im [â(n)] < 0 −→
1 − j
Re [â(n)] < 0 and Im [â(n)] > 0 −→ −1+j
Re [â(n)] < 0 and Im [â(n)] < 0 −→ −1 − j
The effectiveness of the equalizer in suppressing the ISI introduced by
the channel filter may be seen by plotting the following relevant sequences
in a two-dimensional (real–imaginary) display. The data generator output
{a(n)} should consist of four points with values ±1 ± j. The effect
of channel distortion and additive noise may be viewed by displaying
the sequence {x(n)} at the input to the equalizer. The effectiveness of
the adaptive equalizer may be assessed by plotting its output {â(n)} after
convergence of its coefficients. The short-time average squared error
ASE(n) may also be used to monitor the convergence characteristics of
the LMS algorithm. Note that a delay must be introduced into the output
of the data generator to compensate for the delays that the signal encounters
due to the channel filter and the adaptive equalizer. For example, this
delay may be set to the largest integer closest to (N + K)/2. Finally, an
error counter may be used to count the number of symbol errors in the
received data sequence, and the ratio for the number of errors to the total
number of symbols (error rate) may be displayed. The error rate may be
varied by changing the level of the ISI and the level of the additive noise.
It is suggested that simulations be performed for the following three
channel conditions:
a. No ISI: c(0) = 1, c(n) =0,1≤ n ≤ K − 1
b. Mild ISI: c(0) = 1, c(1) = 0.2, c(2) = −0.2, c(n) =0,3≤ n ≤ K − 1
c. Strong ISI: c(0) = 1, c(1) = 0.5, c(2) = 0.5, c(n) =0,3≤ n ≤ K − 1
The measured error rate may be plotted as a function of the signalto-noise
ratio (SNR) at the input to the equalizer, where SNR is defined
as P s /P n , where P s is the signal power, given as P s = s 2 , and P n is the
noise power of the sequence at the output of the noise generator.
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