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T he n ature of t he influence of pole a nd zero locations on t he frequency response
is similar t o t hat observed in continuous-time systems with a minor difference. In
place of t he i maginary axis of t he continuous-time systems, we have a u nit circle in
t he discrete-time case. T he n earer t he pole (or zero) is t o a p oint e jwT (on t he u nit
circle) representing some frequency w , t he more influence t hat pole (or zero) wields
on t he a mplitude response a t t hat frequency because t he l ength of t he vector joining
t hat pole (or zero) t o t he p oint eiwT is small. T he proximity of a pole (or zero) has
similar effect on t he p hase response. From Eq. (12.19a), i t is clear t hat t o enhance
t he a mplitude response a t a frequency w we s hould place a pole as close as possible
t o t he p oint e jwT (on t he u nit circle) representing t hat frequency w. Similarly, t o
s uppress t he a mplitude response a t a frequency w , we should place a zero as close
as possible t o t he p oint e jwT o n t he u nit circle. Placing repeated poles o r zeros will
further enhance their influence.
Total suppression of signal transmission a t a ny frequency c an be achieved by
placing a zero on t he u nit circle a t a p oint corresponding t o t hat frequency. This is
t he principle o f t he n otch (bandstop) filter.
Placing a pole o r a zero a t t he origin does not influence t he a mplitude response
because t he l ength o f t he vector connecting the origin t o any point o n t he u nit
circle is unity. However, a pole ( a zero) a t t he origin generates angle - wT (wT) i n 724 12 F requency Response a nd D igital Filters It roT .... - It LH (a) ! HI 12.2 Frequency Response F rom Pole-Zero Location 725 For a s table s ystem, all t he p oles m ust b e l ocated inside t he u nit circle. T he
zeros may lie anywhere. Also, for a physically realizable system, H[zJ m ust b e
a p roper f raction, t hat is, n 2: m . If, t o achieve a c ertain a mplitude r esponse,
we require m > n , we c an s till m ake t he s ystem r ealizable by placing a sufficient
number of poles a t t he o rigin. T his will n ot c hange t he a mplitude r esponse b ut i t
will increase t he t ime d elay o f t he response.
I n general, a pole a t a p oint h as t he o pposite effect of a zero a t t hat p oint.
P lacing a zero closer t o a p ole t ends t o c ancel t he effect of t hat pole o n t he frequency
response. Lowpass Filters
roT~ ( b) - It 0
I2 It roT~ LH (e) IHI
ideal It roT~ LH
(d) ideal 3 lt/2 0 LH
( e) F ig. 1 2.4 Various pole-zero configurations and the corresponding frequency response.
L H [ ejwTJ. T he p hase s pectrum - wT is a linear function of frequency a nd therefore represents a pure time-delay of T seconds (see Eq. (10.48) or Exercise EI2.2).
Therefore, a pole (a zero) a t t he origin causes a t ime d elay (time advance) of T
s econds in t he response. T here is no change in t he a mplitude response. A lowpass filter h as a m aximum g ain a t w = 0, which corresponds t o p oint
= 1 o n t he u nit circle. Clearly, placing a pole inside t he u nit circle n...
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