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Quantization of Filter Coefficients 275
Im{z}
0
z 3
z 2
z 1
Re{z}
z * 1
z *
z * 2
3
z 2
z 1
0 z * 1 0 0
z * 2
z 3
z * 3
(a) Direct-form Arrangement
FIGURE 6.29
(b) Cascade- or Parallel-form Arrangement
z-plane plots of tightly clustered poles of a digital filter
Finally, the total perturbation error △p l can be expressed as
△p l =
N∑
k=1
∂p l
∂a k
△a k (6.73)
This formula measures the movement of the lth pole, p l ,tochanges in
each of the coefficient {a k }; hence it is known as a sensitivity formula.
It shows that if the coefficients {a k } are such that if the poles p l and p i are
very close for some l, i, then (p l −p i )isvery small and as a result the filter
is very sensitive to the changes in filter coefficients. A similar result can
be obtained for the sensitivity of zeros to changes in the parameters {b k }.
To investigate this further in the light of various filter realizations,
consider the z-plane plot shown in Figure 6.29(a) where poles are tightly
clustered. This situation arises in wideband frequency selective filters such
as lowpass or highpass filters. Now if we were to realize this filter using the
direct form (either I or II), then the filter has all these tightly clustered
poles, which makes the direct-form realization very sensitive to coefficient
changes due to finite word length. Thus, the direct form realizations will
suffer severely from coefficient quantization effects.
On the other hand, if we were to use either the cascade or the parallel
forms, then we would realize the filter using 2nd-order sections containing
widely separated poles, as shown in Figure 6.29(b). Thus, each 2nd-order
section will have low sensitivity in that its pole locations will be perturbed
only slightly. Consequently, we expect that the overall system function
H(z) will be perturbed only slightly. Thus, the cascade or the parallel
forms, when realized properly, will have low sensitivity to the changes or
errors in filter coefficients.
□ EXAMPLE 6.24 Consider a digital resonator that is a 2nd-order IIR filter given by
1
H(z) =
(6.74)
1 − (2r cos θ) z −1 + r 2 z −2
Analyze its sensitivity to pole locations when a 3-bit sign-magnitude format is
used for the coefficient representation.
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