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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 10 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 25 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 Fall 2001 LECTURE 16 Elliptic Analoq Filters Ripple in both passband erg stopband o Optimum filter in the sense of narrowest transition region for a specified amount of passband ripple stopband attenuation and no of poles Hnj 22 1 Q 1 where Rn is a quotChebyshev rational fn 162 R 2 C Inherent parameters eQcn For a quotnormalizedquot elliptic filter QC 1our design methods will be based on tables of normalized elliptic filters 1 if QC 1then 19192 1gt91Q 2 Q Forelliptlcfilters r 9 62 Note Qgt1 1 c As 2 decreases toward a value of 1 the width of the transition region decreases ECE 1 46766397 Introduction to Digital Signal Processing For normalized elliptic filters 92 o 1 ofgtg22gt or and Q1 1 X22 1 22 is Example Design of normalized elliptio filterusing Table Design specs G1 12 db passband ripple G2 30 db stopband attenuation Q lt 121 relates to steepness of transition region lHUQi Odb ldb Q 239J 2r11 2 21 w i 909 quotSOdb M 92 909 III Q G 1 db Use table 36 for 1 39 2 Find minimum n which gives org 121 From Table 36 p141 Qr 480880 1 9232239 1 32446 112912 Target value of 121 mean gtChoose n5 EDGE 14676671 Introduction to Digital Signal Processing Since S2 112912 for n 5 the transition width will be narrower than required by problem specs From Table 36 Ho 1 Am r A02 r Aquot quotquot 2 2 r39quot 118807 5 214490 s 118132 H55 So 811 301 312 s 51 1761152 48077453 648774132 08808039072 802 Actual Q value 9 1 r 94105 compare with 909 1 1 Qr Actual 22 value 22 1l r 210626 compare with 11 Deslon of General non normalized elliptic analoq filters I Performance requirements G1 G2 91 02 Odb G1 GZ W 9391 2392 9 oc 2192 also called no in text MAP Desired mapping Hnorm Odb 1 G1 62 W 1 A r 911 9 co 49192 1 ECE 391 467667 Introduction to Digital Signal Processing Design Stags 1 Find n 9 for H in I I r norm 39 Recall LP gtLP mapping of Table 32 5 l n 52 n For LPgtLP mapping use QU QC xl 192 Use backward design equation for LP gtLP transformation to find 9102 and then 9r W Q139T 9239 1 Qc39 2 Find minimum n for39Hnorms in Table 36 for specified G1 52 values which satisfies the 2r requirement 3 Let HsHnoms S S LP 9LP mapping 9c EOE 14676672 Introduction to Digitai Sigma Processing Examgie Design of nonnormalized elliptic anaiog filter Design specs G1 2 db 21 10000 32 40 db 22 14400 QC 19192 12000 1 Find 2 for Hnoms 9 Q 1440010000 144 2 Find n using Table 36 for G12 G2 40 p 156 n 9 2 576107 I 3 213923 1 4 140842 W target valueof Qr 144 From Table 36 for n4 Ho 39 A01 A02 0100001 32 1725202 32 157676 Hrtornigs quot 32 4672903 212344 s2 127954s 677934 LVJ W YJ v1 I B11 801 B12 302 3 HS Hnorm 3 s s s 5 73 F 39 12000 0132 1 04429 gtlt109Xs2 227053 x108 s2 5607483 30577536st 1535448s 97622496 ECE 1 467 667 Introduction to Digital Signal Processing Example Design a high pass elliptic analog filter lion 0 db etc k 1 k12 db k A k220 db 2 39150 nu 350 Q We backward design equation for High Pass gr 9y Q2333 Q 150 I39 We need to design this low pass filter lepQQ 0 4 Hm2091 k1 K2 9039 1 2333 1526 From Table 36 for k1 2 db and k2 20 db gig 100103s392 360961 53 5373233 454391 and H2s ECE 1 467667 introduction to Digital Sigma Processing slt v 1526 rim mg s Then HHPS HLP SLF E i S 5 Combine the above mappings HHPS H2s 229357798 se 5 2 100103 2293577982 36096132 2293577982 537326229357798s 454891 52 100103 5260x104 3609652 5260x104 123240345532 220 5260x104 3609652 s2 27092251156x105 7941s2 1458x1o4 32 27O92251156x105 EOE 391 46766397 Introduction to Digital Signal Processing Summarize Analoq Filters Butterworth Chebyshev and Elliptic filters all have the general form 1 me Wquot 1 62 aim QC n For Butterworth e 1 and Fn 8 E Q Q 9 s1 For Butterworth and Chebyshev QXQC Properties of FEM 1 ForlQlSQxOs For Elliptic OX 91 1 16 2 3 H1912 s 1 for oi s ox for all three filter types 2 Let K large finite value Then there is some stuch that gt K for all pl 2 oy C 1 gt S W ltlt1 for a gt y EOE 1 467687 Introduction to Digital Signal Processing Chapter 4DiciLtal Filter Design Using the Bilinear Transformation to transform analog filters to digital filters Based on approximating an integral Consider 1St order analog filter Hltsgt ac 128 Ysa1s a0 b0Xs at g T 3030 boxt For tnT a1y nT aoynT b0xnT Fundamental Theorem of Integral Calculus t yit J Y 39rd c Vito 0 For tnT and to n1T yinTi lyrm m n 1gtn n 1T 1 I Approximate using trapezmdal rule 5 y nT y n 1To T EOE 1 467 667 2 Introduction to Digital Signal Processing y39r r n1T 39I39 nT quot r ynTgiy nTy n 1iryn 1T From first order system equation your 99 xnT39 Eeyore a1 a1 Rewriting with n replaced by n12 I be 30 y n 01 3 Xn Ur g Yn 1T 1 391 Now use these two expressions in bo ynT yn1iris xnT fyni Z fxn i i gf wm UU 31 xnTxn1ir f ynTyn iir Yz i 71 39 91916 2quotquot XzF2 t19 1 2 1 a1 a1 Hz b Hs 1 4 s fie 211 1O


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