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Some Preliminaries 389
8.1.2 PROPERTIES OF |H a (jΩ)| 2
Analog filter specifications (8.1), which are given in terms of the
magnitude-squared response, contain no phase information. Now to evaluate
the s-domain system function H a (s), consider
H a (jΩ) = H a (s)| s=jΩ
Then we have
|H a (jΩ)| 2 = H a (jΩ)H ∗ a(jΩ) = H a (jΩ)H a (−jΩ) = H a (s)H a (−s)| s=jΩ
or
H a (s)H a (−s) =|H a (jΩ)| 2∣ ∣
∣Ω=s/j (8.5)
Therefore the poles and zeros of the magnitude-squared function are distributed
in a mirror-image symmetry with respect to the jΩ axis. Also for
real filters, poles and zeros occur in complex conjugate pairs (or mirrorimage
symmetry with respect to the real axis). A typical pole-zero pattern
of H a (s)H a (−s) isshown in Figure 8.2. From this pattern we can
construct H a (s), which is the system function of our analog filter. We
want H a (s) torepresent a causal and stable filter. Then all poles of H a (s)
must lie within the left half-plane. Thus we assign all left-half poles of
H a (s)H a (−s) toH a (s). However, zeros of H a (s) can lie anywhere in the
s-plane. Therefore they are not uniquely determined unless they all are
on the jΩ axis. We will choose the zeros of H a (s)H a (−s) lying left to or
on the jΩ axis as the zeros of Ha(s). The resulting filter is then called a
minimum-phase filter.
jΩ
s-plane
σ
FIGURE 8.2
Typical pole-zero pattern of H a(s)H a(−s)
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