Unformatted text preview: cursive are easily understood in terms of a specific example. Consider a t hird-order system with t he t ransfer function (12.23)
T he i nput J[k] a nd t he c orresponding o utput y[k] of this system are related by t he
difference equation. y[k] + a2y[k - 1] + aIy[k - 2] + aoy[k - 3] = (b) b3I[k] + b2I[k - 1] + b d[k - 2] + boI[k - 3] (12.24a) or o .lL y[k] It 2 o = :-a2y[k - 1] - aIY~k - 2] - aoy[k - 31 o utput t erms + ?3I[k] + b2f[k - 1] + b d[k - 2] + boI[k - lOOOlt 500!t 31 (12.24b) i nput t erms ¥tY ~lay lIfi!; hIlS (c) F ig. 12.6 Designing a notch (bandstop) filter in Example 12.3.
Figure 12.6b shows IH[ejwTll for values of a = 0.3,0.6, and 0.95. Figure 12.6c shows
a realization of this filter. •
f:,. E xercise E 12.3
U sing t he g raphical argument, show t hat a filter w ith t ransfer f unction H[z] = z - 0.9
z a cts a s a h ighpass filter. Make a r ough s ketch o f t he a mplitude r esponse. 'V Here y[k], t he o utput a t i nstant k, is determined b y t he i nput values I[k], I[k - 1],
I[k - 2], a nd I[k - 3] as well as by t he p ast o utput values y[k - 1], y[k - 2], a nd
y[k - 3]. T he o utput is therefore determined iteratively or recursively from t he i ts
p ast values. To compute t he p resent o utput of this third-order system, we use t he
p ast t hree values o f t he o utput. In general, for a n n th-order system we use t he p ast
n values of t he o utput. S uch a s ystem is called a r ecursive s ystem.
An interesting feature of a recursive system is t hat once a n o utput exists, i t
t ends t o p ropagate itself forever because of its recursive nature. This is also seen
from t he c anonical realization of H[z] in Fig. 12.7a. Once a n i nput (any input) is
applied, t he feedback connections loop t he o utput continuously back into t he system,
a nd t he o utput continues forever. This propagation of t he o utput in perpetuity
occurs because o f t he nonzero values of coefficients ao, a lo a2, . .. , a n-I· T hese
coefficients [appearing in t he d enominator of H[z] in Eq. (12.23)] are t he r ecursive
c oefficients. T he coefficients bo, b l, b2, . .. , bn ( appearing in t he n umerator of
H [z]) a re t he n onrecursive c oefficients. I f a n i nput 8[k] is applied a t t he i nput
of a recursive filter, t he response h[k] will continue forever up to k = 0 0. For this
reason, recursive filters are also known as i nfinite i mpulse r esponse ( IIR) filters. 730 12 Frequency Response a nd Digital Filters 1 731 12.4 Filter Design Criteria system, t he i nput will pass t hrough t he s ystem a nd will be completely o ut o f t he
s ystem by k = 4. T here a re no feedback connections t o p erpetuate t he o utput.
Therefore, h[k] will be nonzero only for k = 0, 1, 2, a nd 3. F or a n n th-order
nonrecursive filter, h[k] is zero for k > n . Therefore, t he d uration o f h[k] is finite for
a nonrecursive filter. For this reason, nonrecursive filters are also known as f inite
i mpulse r esponse ( FIR) filters. Nonrecursive filters are a special case of recursive
filters, in which all t he recursive coefficients a o, a l, a 2, . .. , a n-l a re zero.
A n n th-or...
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