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

Recall t hat a pole a nd a zero in t he vicinity tend

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Unformatted text preview: n . Therefore, t he 3-dB (or half power) bandwidth is 1 r ad/s for all n . 3. For large n , t he a mplitude response approaches t he ideal characteristic. To determine t he corresponding transfer function h (s), recall t hat h (-jw) is t he complex conjugate of h (jw). T herefore h (jw)h(-jw) 1 = Ih(jw)1 2 = - - 2 1 +w n t Butterworth filter e xhibits m aximally flat characteristic also a t w = 0 0. e j ll'(2k-l) for integral values of k, a nd j = e j ll'/2 t o o btain k i nteger This equation yields t he poles of h (s ) h( - s) as k = 1, 2, 3, . .. , 2 n (7.33) Observe t hat all poles have a u nit m agnitude; t hat is, they are located on a u nit circle in t he s -plane separated by angle 11' In, a s illustrated in Fig. 7.21 for o dd a nd even n . Since h (s) is s table a nd causal, its poles must lie in t he LHP. T he poles of h (-s) a re t he m irror images of t he poles of h (s) a bout t he vertical axis. Hence, t he poles of h (s) are those in t he L HP a nd t he poles of h ( - s) a re those in t he R HP in Fig. 7.21. T he poles corresponding t o H (s) a re obtained by setting k = 1, 2, 3, . .. , n in Eq. (7.33); t hat is sk = e ~(2k+n-l) = cos .!!:...(2k 2n +n - 1) +j sin .!!:...(2k + n - 1) 2n k = 1, 2, 3, . .. , n (7.34) a nd h (s) is given by (7.35) 7 .5 B utterworth F ilters C oefficients o f B utterworth P olynomial B n(8) = 8n T able 7 .1: a, n 5 09 aa a2 a4 2.00000000 3.41421356 5.23606798 7.46410162 10.09783468 13.13707118 16.58171874 20.43172909 T able 7 .2: 1 &lt; -u 8 bO ol :\ ,~J ; . I =!...... I=! ~ C &gt;:: 'g&quot;&quot; ·····x..' ';;~ &quot; \&gt;1.... / .. , '. ~ t- ., '0 E &lt;a ..c: &quot; ...., .. ~ ~ .3 -= ill &quot;I ~~ o .~ ........... ~...// ::I ~ ~ ... ::l . I &gt;:: 0. :3 &gt;&lt;+;:;&quot; ~ ~. 0 e K~&lt; .... X/ ~ ~ .~ &lt;il 4.49395921 13.13707118 31.16343748 64.88239627 t-: 5.12583090 16.58171874 42.80206107 5.75877048 20.43172909 6.39245322 '2 ' -' . . , .s 0.. N N 1/ ~ &gt;:: ..... r::- ~ ~o~ ... 'fii ;;t .~ ~- g .s &quot; a&quot;,).\$ u::z:: ;;t .=: g;: '&quot; ;.~ ~ + tog;: '&quot; o + &gt; ::-0&gt; to &quot; , 0 '&quot; . I ~_ ~ o i .M &quot; ,,00C'&gt; M ____ O&gt; ~ &quot;&quot; r--:'&quot;'o ~ M .~ &amp;~ -11 'o &quot; &quot;&quot; J :il&gt;::tolC'&gt; ' &lt;-, SooM . 00 o J :j ? 6 _ en I N g;: o + a ?&gt;:: ~t-~ ~ 00 'bb ----'&quot; @ -;:;- + ., j!'; .o... q ~ ., &quot;if + :l ~ N &lt;0 + &quot;., &quot; 1&quot;~ .... 1....+, '.... . -~ + 0) ..... ,..,1;; ,.., '&quot; M ~i . -&lt; &quot;+ &quot; 1j CD '&quot; 1l &quot;&quot; ~ 0 &lt;:l s @-u 7 ~ 5~ ~ se ' -._ ;g ~ o + &quot; ~ M + t 'l :l ~ &lt;0 N .., + ;;t -u + g ~ a1 &gt;:: to o ..., Cf.l &quot;CQ II b 0 0. 0 ~ 'I ... Q) ~ ~ I&quot;'-&quot;&quot; '0i .&quot;'&quot;'_ _j fg ' ;;' ~ ,.., eq&quot; ~ 'bE-&lt; &quot;'~ .~ ~ .z 1/ :::; &lt;:l ~ .s -U ~ u o .0. .. a u ~ &quot;';;' ~ ;.: &quot; ~ :- 1j . ..., ~ c:; &gt;:: ..., &lt;:l ..... .~ ..: S ... ~&lt;:l ~0 ) ' &quot; 0 ....,&quot;&quot; ~.§ ] ~ EQ) t~ El Q) &gt; i8~ '&quot;' '&quot; C'&gt; .~ .., &quot; r': 00 &...
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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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