110 7. ADAPTIVE FILTERING
for( i = N-1; i > 0; i--) {
y[i] = y[i-1];
}
y[0] = bSum - aSum;
return y[0];
}
Compared to FIR filters, IIE filters have advantages and disadvantages. IIR filters allow meeting
a desired frequency response characteristic via a lower number of coefficients than an equiva-
lent FIR filter, resulting in a lower computation time. FIR filters provide linear phase response
whereas IIR filters do not. Unlike FIR filters which have no poles and are thus guaranteed to
be stable, special care must be taken with IIR filters. Stability of IIR filters is heavily dependent
on quantization. Recall the finite word length effect discussed in previous chapters. e amount
of shift in the positions of poles and zeros can be related to the amount of quantization error in
the coefficients. For an N th order IIR filter, the sensitivity of the ith pole p
i
with respect to the
kth coefficient a
k
can be derived to be [1],
@p
i
@A
k
D
p
N k
i
N
X
lD1
l¤i
.
p
i
p
l
/
: (7.2)
is means that the change in the position of a pole is influenced by the positions of all the other
poles. at is the reason an N th order IIR filter is normally implemented by having a number
of second-order IIR filters in series in order to decouple this dependency of poles and having
real value coefficients.
7.2 ADAPTIVE FILTERING
Adaptive filtering is used in many applications such as noise cancellation and system identifica-
tion. e most common method of adaptive filtering is to modify the coefficients of an FIR filter
according to an error signal to adapt to a desired signal. In system identification, the behavior of
an unknown system is modeled by comparing the output d Œn of the unknown system with the
output yŒn of a known system, in our case an adaptive FIR filter. e difference in the output
of the unknown system and the adaptive FIR filter is considered to be the error term eŒn, which
is used to update the coefficients of the adaptive filter. is process is illustrated in Figure 7.1.
e least mean square (LMS) algorithm is widely used to update each coefficient of the
FIR filter according to the following equation:
h
n
Œk D h
n1
Œk C ı eŒn xŒn k; (7.3)
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