Adaptive system identification is a technique that uses an adaptive filter for the model
W(z). This section presents the application of adaptive estimation techniques for direct
system modeling. This technique has been widely applied in echo cancellation, which
will be introduced in Sections 9.4 and 9.5. A further application for system modeling is
to estimate various transfer functions in active noise control systems [8].
Adaptive system identification is a very important procedure that is used frequently
in the fields of control systems, communications, and signal processing. The modeling
of a single-input/single-output dynamic system (or plant) is shown in Figure 8.6, where
x(n), which is usuallywhite noise, is applied simultaneouslyto the adaptive filter and the
unknown system. The output of the unknown system then becomes the desired signal,
d(n), for the adaptive filter. If the input signal x(n) provides sufficient spectral excita-
tion, the adaptive filter output y(n) will approximate d(n) in an optimum sense after
convergence.
Identification could mean that a set of data is collected from the system, and that a
separate procedure is used to construct a model. Such a procedure is usuallycalled off-
line (or batch) identification. In manypractical applications, however, the model is
sometimes needed on-line during the operation of the system. That is, it is necessary to
identifythe model at the same time that the data set is collected. The model is updated at
each time instant that a new data set becomes available. The updating is performed with
a recursive adaptive algorithm such as the LMS algorithm.
As shown in Figure 8.6, it is desired to learn the structure of the unknown system
from knowledge of its input x(n) and output d(n). If the unknown time-invariant system
P(z) can be modeled using an FIR filter of order L, the estimation error is given as
en dn À yn
LÀ1
l0
Â
pl À w
l
n
Ã
xn À l,
8:5:1
where p(l) is the impulse response of the unknown plant.
Bychoosing each w
l
n close to each p(l), the error will be made small. For white-
noise input, the converse also holds: minimizing e(n) will force the w
l
n to approach
p(l), thus identifying the system
w
l
n % pl, l 0, 1, . . . , L À 1:
8:5:2
The basic concept is that the adaptive filter adjusts itself, intending to cause its output to
match that of the unknown system. When the difference between the physical system
response d(n) and adaptive model response y(n) has been minimized, the adaptive model
approximates P(z). In actual applications, there will be additive noise present at the
adaptive filter input and so the filter structure will not exactlymatch that of the unknown
system. When the plant is time varying, the adaptive algorithm has the task of keeping the
modeling error small bycontinuallytracking time variations of the plant dynamics.
8.5.2 Adaptive Linear Prediction
Linear prediction is a classic signal processing technique that provides an estimate of the
value of an input process at a future time where no measured data is yet available. The
APPLICATIONS
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