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Unformatted text preview: omplex c onjugate
p oles t o o btain z7 - 1 [ z
H = - - - z
z7 - 1 [z z (2zcos !If - 2 cos !If)]
+ -'--:--.,..-'----,,::---':-'z2 - (2 cos ~z + 1)
1 .802z(z - 1) ] ~ , :;-:::-I - z2 - 1.247z + 1 ,
Hl!z] H;[zl We c an realize t his filter by placing t he c omb filter HI[z] i n cascade w ith H2[Z], which
c onsists o f a f irst-order a nd a s econd-order filter in parallel. • P ole-Zero Cancellation in Frequency Sampling Filters I n t he method of frequency sampling filters, we make a n intriguing observation
t hat t he required nonrecursive (FIR) filter is realized by a cascade of H r[z] a nd 777 12.9 Summary H2[Z]. However, H2[Z] is recursive (IIR). This strange fact should alert us t o t he
possibility of something interesting going on in this filter. For a nonrecursive filter
there can be no poles other t han t hose a t t he origin [see Eq. (12.69a)]. I n t he
frequency sampling filter [Eq. (12.100)]' in contrast, H2[Z] has No poles a t eirno
( r = 0, 1, 2, " ', n ). All these poles lie on t he u nit circle a t e qually spaced points.
These poles simply cannot be in a nonrecursive filter. They must somehow get
canceled along t he way somewhere. This is precisely w hat h appens. T he zeros of
H dz] a re exactly where t he poles of H2[Z] a re because
'0 z No - 1 = (z - e J 211" NO . 211" '2 ) (z - eJNiJ)(z - e J 211" N o) . . . 211' · (z _ eJnNO) T hus, t he poles o f H2[Z] a re canceled by t he zeros o f H dz], r endering t he final filter
Pole-zero cancellation in this filter is a p otential cause for mischief because
such a perfect cancellation assumes exact realization o f b oth H I [z] a nd H 2 [z]. Such
a realization requires t he use o f infinite precision arithmetic, which is a p ractical
impossibility because o f q uantization effects. Imperfect cancellation o f poles a nd
zeros means there will still be poles on t he u nit circle, a nd t he filter will n ot have a
finite impulse response. More serious, however, is t he fact t hat t he r esulting system
will be marginally stable. Such a system provides no damping o f t he round-off noise
t hat is i ntroduced in t he c omputations. In fact, such noise t ends t o increase with
time, a nd m ay render t he filter useless.
We c an partially mitigate this problem by moving b oth t he poles (of H2[Z]) a nd
zeros (of Hl[Z]) t o a circle of radius r = 1 - €, where € is a small positive number
--+ O. T his artifice will make t he overall system asymptotically stable.
Spectral Sampling w ith Windowing T he frequency sampling method can be modified t o t ake advantage o f windowing. We first design a frequency sampling filter using a value No' t hat is much
higher t han t he design value No. T he r esult is a filter t hat m atches with t he desired
frequency response a t a very large number (No') o f points. T hen we use a suitable
No-point window t o t runcate t he N o'-point impulse response. This procedure yields
t he final design of a desired order. 1 2.9 Summary T he response of...
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