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# INTRO TO DIG SIG PRO E C E 467

Clemson

GPA 3.84

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This 12 page Class Notes was uploaded by Eloy Ferry on Saturday September 26, 2015. The Class Notes belongs to E C E 467 at Clemson University taught by John Gowdy in Fall. Since its upload, it has received 45 views. For similar materials see /class/214305/e-c-e-467-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.

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Date Created: 09/26/15

ECE 467667 Introduction to Digital Signal Processing LECTURE 15 Chebvshev Analoq Filters Type I 140912 1 1 1 62 QC 2 1 1111le Q 1 62 Tn2 820 where Tnx Chebyshev polynomial of order n T0x1 T1x x Tnx 2X 39 Tn1X Tn2xn 2 2 See table 33p135 cpencos 1x xs1 Also Tnm coshncosh 1 x gt1 For all n Tnx oscillates between 1 and 1 for xlt1 and Tnx gets large as x gets large I ECE quot14676567 Introduction to Digital Signal Processing 00 TnX 391 l v 00 1 1 652 Tn2 9c oscillates between 1 andquot 391 2 for 2lt QC 10 16 39 Therefore Consider 26 1 Normalized filter 2 1 Hm 1 e2 Tn282 HnjQHnj 2 Hnj 2Ilg 19 69 Therefore 1 1 62 Tn2 7 Hn u S using sj 2 Poles of Hn s are left hand plane poles of Denote these 1 62 Hi poles assk Gk ij k122n 39 ECE 1 467667 Introduaion to Digital Signal ProceSslng 39 2 2 quotCan be shownquot 932 952 1 equation of an ellipse a where 1 1 611 1162 quotl 1J162 quot 2 e 2 e 1 39 1 2 n 2 n and b1e 1 e 1amp2 n23 case k l 2 n Poles in lefthand plane k n n1 2n Poles in right half plane 39 ECE 391 467667 Introduction to Digital Signal Processing 1 Use n lefthand plane poles for Hns LHP 39 LHP poles of Hns can also be expressed as s asin T r i jbcos l 2n n 2n n k01 g 1 for n even k0i iorn dd Li 2 T gives single real pole at s a 40 for n odd Let Vns 3 sk where skLHP poles of Hns Hn s for Chebyshev filter Then Vn s 2 5n 39 bn1squot 391 bmsn 2 b1s b0 When n is 9clj HnjQ has a maximum at 820 lHn ECE 14676672 InLI oduction to Digital Signal Processing When n is even Hnj 2 has a Eelative minimuLn atsz01 Y n4 9 K K At 30 H s z n Vns 30 b0 K Fornggg wa1nt 1gtsetKbo o 1 b0 For n even want gt set K 39 b0 62 62 EOE 391 467667 Introduction to Digital Signal I3 rocessing Table 34 gives Vns for several values of e and n for 5201 p140 Table 35 gives goles of Hns for values at e and n for 901 p142 Desiqn of Normalized Low Pass Chebvshev FiltersAnalog Typical DTOblem SPeCi Cationi in terms of Filter Parameters Min PM 0 db 1 1 k1 db 39 1 27 39 2 16 kadb u g A JI 9 1 Q x Q c 2 1 C er gr 1 Find 6 1 Set 20lo k 910 6 1 2 Find n l I 1 a Trial and Error evaluate Hn m I 1 62 This g QC Qc1 99 for increasing values of n until lHJjQ2Qr s kzdb I 14676671 Introduction to Digital Signal Processing quot QB b Use the formula n logg 92 1 r Assumes 8201 logQlr 922 1 u A2 A 12 where g 2 1 and A 10 20 E 3 Use Table 34if n s 10 to find Hns Design example Problem specification k1 2db k2 20 db H19 o 13 2 1 ea szb QC Qr DO I 391 467667 hm39oclucLion Lo Digital Signal Processing Find 6 based only on k1 1 39 2 20og 1 6 2 1 1 log 162 36 12 100 16 16 1 e2 10 gte 0 176478 51 quot In genera e 1390 101 39 I3ij 14676671 Introduction L0 Digital Signal Processing quot3 2 Find n Iogg 1192 1i log 2 92 1 next higher integer 2 5 k where gA 21 and A1O 5 I 20 For this example A 10 4 10 1 2 E 9110 1 21301 76478 and using these values of A and g with S2 13 we can solve for n using to get n43L5 3 Use Table 34 With n5 ripplequot2 db to get k5 39 k 0817225 5a7064606844L494543333n693477032u4593491sn0817225 le1 391 467667 Introduction to Digital Signal Processing Another design example non normalized Chebyshev lowpass filter Method 1 Convert problem to the design of an appropriate normalized filter design this filter 39 i l 2 Map the filter of step 1 to get the desired nonnormalized filter Desired Filter Normalized Filter Mtg M152 0 db 0 db 2 db 2 db 20 db a 20 db E S2 1 52 S2 52 40 Q39 52 r C r A Q 1 a Backward design equation for LP gtLP mappingTable 32 or 9i 31agt needfilterwith QC 19 13 9 4o U D Design this filter was designed in previous example Call 1 Hmrmsgt 2 Hdesir9ds Hnorm s LP gtLP mappingTable 32 39 56 fo 8366x10 5 s 873212s2 539693 152344Xs2 1412923 628984 10 39 ECE quot1 467667 Introduction to Digital Signal Processing Examgle Design a highgass Chebyshev analog filter whose gain in the passband gt 60 rad see is between 0 db and ldb and whose gain in the stopband 0 s Q s 30 radsec is lt 20 db HM Odb 12db 20db 30 60 Q on on Find 52 for LP counterpart Odb 1I2 db N 1 9r Q gr 39 Qu g9 S2 30 11 ECE 14676367 Introduction to Digital Signal Processing Now design the LE filter shown above 20 k A210 01o 20 10 M 451 K 6210 X04210 1 1220 2 g A 2 1 28484 E log Lgr Qr 2 n Wig 9 111143069 4 From Table of normalized QC 1 filter H s K4 39 4 s4 1197385633 1716866232 102545523 3790506 b0 3790506 quot where k 357 8469 4 WW WW Map to thedesired 51 lter HHP S 2 H43 5639 8 12

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