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

4fnwc d isppoles o f t he t ransfer f unction a res o

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Unformatted text preview: ; . . , n (7.51) tWe can show 2 that at higher frequencies (in the stopband), the Chebyshev filter gain is smaller than the comparable Butterworth filter gain by about 6 (n - 1) dB. = C:<nn(s) = sn + an _ls n - IKn ... + alB + aD + (7.52) T he c onstant K n is selected t o have p roper dc gain, as shown in Eq. (7.45). As a result n o dd K n = { aD aD aD v I + f 2 = IO f / 2o n e ven (7.53) T he design procedure is considerably simplified by ready-made t ables o f t he p olynomial C~ (s) i n Eq. (7.52) or t he p ole locations of 1 t(s). T able 7.4 lists t he coefficients aD, a i, a2, . .. , a n-I o f t he p olynomial C~(s) in Eq. (7.52) for r = 0.5, 1 ,2, a nd 3 dB ripples corresponding t o t he values of f = 0.3493, 0.5088, 0.7648, a nd 0.9976, respectively. T able 7.5 lists t he poles of various Chebyshev filters for t he s ame values of r ( and f). T ables listing more extensive values of r ( or f) c an b e found in t he l iterature. We c an also use MATLAB functions for t his p urpose. o C omputer E xample C 7.7 Using MATLAB, find poles, zeros, and the gain factor of a normalized 3rd-order Chebyshev filter with f = 2 dB. [ z,p,kj=cheblap(3,2) 0 518 7 F requency R esponse a nd A nalog F ilters T able 7 .5: T able 7 .4: C~(s) = s n 519 C hebyshev F ilters 7.6 C hebyshev F ilter P ole L ocations C hebyshev F ilter C oefficients o f t he D enominator P olynomial + a n_lS n - l + a n_2Sn-2 + ... + a lS + aO 1 '=3 1'=2 1 '=1 l ' = 0 .5 n - 1.3076 -1.0024 ± jO.8951 - 0.4019 ± jO.8133 - 0.3224 ± jO.7772 ± jO.9660 -0.3689 - 0.1845 ± jO.9231 - 0.2986 - 0.1493 ± jO.9038 - 0.1395 ± jO.9834 - 0.3369 ± jO.4073 - 0.1049 ± jO.9580 - 0.2532 ± jO.3968 - 0.0852 - 0.2056 ± jO.9465 ± jO.3920 - 0.3623 - 0.1120 ± j l.0116 - 0.2931 ± jO.6252 - 0.2895 - 0.0895 ± jO.9901 - 0.2342 ± jO.6119 -0.2183 - 0.0675 ± jO.9735 - 0.1766 ± j O.6016 - 0.1775 - 0.0549 ± jO.9659 - 0.1436 ± jO.5970 6 - 0.0777 ± j l.0085 - 0.2121 ± jO.7382 - 0.2898 ± jO.2702 - 0.0622 ± jO.9934 - 0.1699 ± jO.7272 - 0.2321 ± jO.2662 - 0.0470 ± jO.9817 - 0.1283 ± jO.7187 - 0.1753 ± jO.2630 - 0.0382 ± jO.9764 - 0.1044 ± jO.7148 - 0.1427 ± jO.2616 7 - 0.2562 - 0.0570 ± j l.0064 - 0.1597 ± jO.8071 - 0.2308 ± jO.4479 -0.2054 - 0.0457 ± jO.9953 - 0.1281 ± jO.7982 - 0.1851 ± jO.4429 -0.1553 - 0.0346 ± jO.9866 - 0.0969 ± jO.7912 - 0.1400 ± jO.4391 - 0.1265 - 0.0281 ± jO.9827 - 0.0789 ± jO.7881 - 0.1140 ± jO.4373 8 - 0.0436 ± j l.0050 - 0.1242 ± jO.8520 - 0.1859 ± jO.5693 - 0.2193 ± jO.1999 - 0.0350 ± jO.9965 - 0.0997 ± jO.8447 - 0.1492 ± jO.5644 - 0.1760 ± jO.1982 - 0.0265 ± - 0.0754 ± - 0.1129 ± - 0.1332 ± jO.9898 jO.8391 jO.5607 jO.1969 - 0.0216 ± - 0.0614 ± - 0.0920 ± - 0.1085 ± 9 - 0.1984 - 0.0345 ± j l.0040 - 0.0992 ± jO.8829 - 0.1520 ± jO.6553 - 0.1864 ± jO.3487 - 0.1593 - 0.0277 ± - 0.0797 ± - 0.1221 ± - 0.1497 ± jO.9972 jO.8769 jO.6509 jO.3463 - 0.1206 - 0.0209 ± - 0.0603 ± - 0.0924 ± - 0.1134 ± jO.9919 jO.8723 jO.6474 jO.3445 - 0.0983 - 0.0171 - 0.0491 - 0.0753 - 0.0923 - 0.0279 ± j l.0033 - 0.0810 ± jO.9051 - 0.1261 ± jO.7183 - 0.1589 ± jO.4612 - 0.1761 ± jO.1589 - 0.0224 ± - 0.1013 ± - 0.0650 ± - 0.1277 ± - 0.1415 ±...
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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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