Signal Processing and Linear Systems-B.P.Lathi copy

# We now show t hat for a n asymptotically stable ltid

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Unformatted text preview: Response From Pole-Zero Location 723 lines connecting Z1&gt; Z2, . .. , Zn t o t he p oint eiwT . Similarly, let d 1, d2, . .. , d n b e t he l engths a nd 01, O2 , . .. , On be t he angles, respectively, o f t he lines connecting '11&gt; '12, . .. , 'In t o eiwT . T hen 1m z - plane H [e i wT _ _ (qei&lt;l&gt;. )(T2ei4&gt;2) . .. (rnei&lt;l&gt;n) - H[zllz=e;wT - bn ( dleiO. )(d2ei02) . .. ( dne iOn ) = b Tl T2 . .. Tn ei[(&lt;I&gt;. +&lt;1&gt;2+&quot;·+&lt;I&gt;n)-(O. +02+&quot;'+On)] n d l d 2··· dn (12.17) (12.18) T herefore IH[ejwTlI = b T1T2'&quot; Tn n d 1d2'&quot; d n n =b (a) p roduct of t he d istances of zeros t o eiwT p roduct of distances o f poles t o eiwT (12.19a) a nd L H[e iwT ] = (&lt;1&gt;1 F ig. 1 2.3 ( a) vector r epresentation o f c omplex n umbers (b) vector representation o f f actors o f H[z]. 1 2.2 + &lt;1&gt;2 + ... &lt;l&gt;n) - (0 1 + O2 + ... + On) = s um of zero angles t o e jwT - sum o f pole angles t o eiwT (b) Frequency Response From Pole-Zero l ocation T he frequency response (amplitude and phase response) of a system are determined by pole-zero locations o f t he t ransfer function H [z]. J ust as in continuoustime systems, it is possible t o d etermine quickly t he a mplitude a nd t he phase response a nd t o obtain physical insight into the filter characteristics of a discrete-time system by using a graphical technique. Consider t he t ransfer function (12.16) We c an c ompute H[z] graphically using t he concepts discussed in Sec. 7.3. T he d irected line segment from Zi t o Z in t he complex plane (Fig. 12.3a) represents the complex number z - Zi. T he l ength of this segment is Iz - zil a nd i ts angle with the horizontal axis is L (z - Zi). I n filtering applications, t he i nputs are often t he s ampled continuous-time sinusoids. Earlier, we showed t hat a s ampled continuous-time sinusoid cos w t a ppears as a discrete-time sinusoid cos nk (0, = w T). T he a ppropriate function for computing t he frequency response in such a situation, therefore, is H [e jwT ] (0, = w T). To compute t he frequency response H [e iwT ] we e valuate H[z] a t z = e jwT . B ut for JwT z= e , Izi = 1 a nd L z = w T so t hat z = eiwT r epresents a point on the u nit circle a t a n angle w T w ith the horizontal. We now connect all zeros ( Zl, Z2, . .. , z n) a nd all poles (&quot;n, '12, . .. , 'In) t o t he p oint eiwT , as indicated in Fig. 12.3b. Let T l, T2, . .. , Tn b e t he l engths and 'Pl, &lt;1&gt;2, . .. , &lt;l&gt;n be t he angles, respectively, of t he s traight (12.19b) In this manner, we c an compute t he frequency response H [e jwT ] for any value of w by selecting t he p oint on t he u nit circle a t a n angle w T corresponding t o t hat value of w. T his point is eiwT . I n s ummary, t o c ompute t he frequency response H [e jwT ], we c onnect all poles a nd zeros t o t his point a nd d etermine IH[ejwTlI a nd L H[e jwT ] using t he above equations. We r epeat t his procedure for all values of w T from 0 t o 7r t o o btain t he frequency response. Controlling Gain by P lacement o f Poles and Z...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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