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System Identification or System Modeling 599
We will apply the LMS algorithm to several practical applications
involving adaptive filtering.
11.2 SYSTEM IDENTIFICATION OR SYSTEM MODELING
To formulate the problem, let us refer to Figure 11.2. We have an unknown
linear system that we wish to identify. The unknown system may
be an all-zero (FIR) system or a pole-zero (IIR) system. The unknown
system will be approximated (modeled) by an FIR filter of length N. Both
the unknown system and the FIR model are connected in parallel and are
excited by the same input sequence {x(n)}. If{y(n)} denotes the output
of the model and {d(n)} denotes the output of the unknown system, the
error sequence is {e(n) =d(n) − y(n)}. Ifweminimize the sum of squared
errors, we obtain the same set of linear equations as in (11.7). Therefore,
the LMS algorithm given by (11.8) may be used to adapt the coefficients of
the FIR model so that its output approximates the output of the unknown
system.
11.2.1 PROJECT 11.1: SYSTEM IDENTIFICATION
There are three basic modules that are needed to perform this project.
1. A noise signal generator that generates a sequence of random numbers
with zero mean value. For example, we may generate a sequence of
uniformly distributed random numbers over the interval [−a, a]. Such
a sequence of uniformly distributed numbers has an average value of
zero and a variance of a 2 /3. This signal sequence, call it {x(n)}, will be
used as the input to the unknown system and the adaptive FIR model.
In this case the input signal {x(n)} has power P x = a 2 /3. In MATLAB
this can be implemented using the rand function.
FIGURE 11.2 Block diagram of system identification or system modeling
problem
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