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by: Eloy Ferry


Eloy Ferry
GPA 3.84

John Gowdy

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John Gowdy
Class Notes
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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 54 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 quot Lecture 13 Continue with Butterworth analog filter General form for a filter with n poles and no zeros Hrs M M fie so Skw Butterworth polynomial of order n n In order to have 1 at 20 choose MH SK 0 k1 Example n3 1E 55 2 s1z ce3 szz ce 3QCe3Qe 3 3 l7 39 3 Sk ch 3 c quotace 3 Qc k1 1 Examgle n 4 57 711 39 7 5n A 1 3129693 szz cee 32969 8 s4ch 8 4 15 quot i215 j5quot j H Sk ch 8 ch 8 ch 8 che 8 93 k1 Note Table 31a in text gives Bns for n1 gt5 for case of QC 1 normalized Butterworth filters T467667 Intrtxiuction to Digital Si gnai Processing Desion of Low Pass Butterworth Analoq Filter Design Specs in gassband 0 2 20 log1oHjQ 2 k1 for pl 3 21 In stogband 20og1oiHjQ s k2th 2 22 20IogmH0 21 O k1 k2 For Butterworth filter the passband requirement becomes 20 iogmiHGQ 1Olog1olHj 212 10og10 l l1 2K1 f39orl l lt Q1 Q 1 QC 39 1 k1 use quotquot sign for case where 381 390910 mg 7n 75 condition is minimally satisfied QC BCE T467667 Introduction to Digital Signal Processing k1 1 10 Zn QC 2n k1 2n 39 i I 1EL 1010amp 10quot 1 0 20 20 Similarly the minimum stogband requirement for Butterworth filters at Q 22 is 2n k 92 1075g 1 6 Now divide G by k1 912110754 23 k2 1073 1 W if calculated n is not an 10 1 integer round to next hh t Zloga lg er In eger ECE T467667 Introduction to Digital Signal Processing 39 Now solve for no method 1 use 9 passband condition H mm 4T Note If n was quotrounded up then Passband condition will be me exactly Stopband condition will be exceeded Method 2 use 69 stopband condition Get QC 92 5 2n 1010 1 if rounded up then 962 lllgt oquot Stopband Condition is met exactly Passband condition is exceeded ECE T467667 Immiduction to Digital Signal Prmessing Method 3 Use QC 9 i39g ca HCgt 203 Both stopband and passband conditions are slightly exceeded if n was rounded up Note If n was 391 rounded up ther i39Qc1 2 202 2 163 Desidn Example k1 2db k2 10db iHGQi Odb 2db 10db 20 30 2 gt 31 I 1010 1 0910 T 1010 1 d n 393 337 39 L39roun u 4 210939 30 Q 20 213868 c1 2 2 8 1010 AT stopband requirement at 92 30 will be exceeded ECE T467667 Introduction to Digital Signal Processing Method 1 for findinq H5 Express H4s in terms of pole locations Since n is even quotstarting anglequot z in JP 2n 8 0 Angle spacing l M 5 plane n14 118 W Poles of H4s In general for complex conjugate poles at s reiiquot denominater term in Hs is s requot3 XS re 39 82 2rs cos 9 r2 524 Z H4S 39 57c 77 2 2 2C cosm s 92c 132 20C cos gs 92c 20 213868 ECE T 467667 Inmxiuction 10 Digital Signal Prawcssing Method 2 for findind HgLSl Transform a normalized 4th order Low Pass Filter from a table to get the desired Hs 44091 0 db k1 301db QC 1 o IHZUQI 39 0 db 301de do 213868 9 T467667 Intrrxiuction to Digital Signal Processing Consider the mapping sew U 2 U U s 39Q Hn S Hns Hn 7 Check when S2 u Hj 2 Q mphEL HnJQL1 301 db for Butterworth filters 99 u Note All frequencies of HnjQ are mapped by the same multiplier in forming H 39 eg HnjQ H300 09 29 Qu9 Show this H309 ECE T467667 Introducticm to Digital Signal Processing Examgle S S quot 5 k1 k1 k2 N 9 2 M Q 1 If nu Qr QrQu Now apply to previous example 1 Find H4s from Table 31 bnormaized 4th order Butterworth filter 1 TSZL 76536s 1st 1 847766 3 H4s 2 Let HsH4 sH4s se 21 3868 HS 209210 x 106 sz163666s 457394st 3951765 45739 ECE T467667 Introduction to Digital Signal Processing Other transformations of normalized lowpass filters H09 H GQ Odb k1 k1 I M k2 I K 1 Q 2 ou 2 9 e 1 Q Q Q ConSIder the mapping H sHs H so H jg2 HL u u S I Q 1 39 Then H pi H u H H Danni nnu 1 J n1 Since is even fn 091924 2 91 k1db 7 I When 20 gii H GQH H j r and lH m inn91 kzdb U Qr ECE T467667 Introduction to Digital Signal Processin I Design steps for designing highpass analog filters by transforming normalized filters 0 Given the performance parameters k1 u and k2 2r which describe the desired high pass filter k1 k2 U o Qr 1 Find Q 2 Design a low pass filter with performance parameters khl and k2 2 ie the appropriate normalized lowpass filteruse tables when possible k1 K2 f r l 1 Q ECE T467667 Introduction to Digital Signal Processing For Butterworth filters 39 If k1301 db 261 solve for n find normalized filter39in Table HLps Hns I 39 irom table If k1 301 db 39 solve forn and QC use LP gtLP mapping of n th order normalized filter in Table to get necessary Low Pass Filter HLPS HnS 54 9c from table slt 9 S 3 Let HHg sHLPIs 4 Simplify result of step 3


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