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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 61 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
LECTURE 23 Optimum FIR Filters We will consider the case of Type I filters only optimum Type II Type III and Type IV FIR filters are considered in ECE 844 Digital Signal Processing First consider designing an optimum lowpass filter The filter will be optimum in the sense of minimizing the peak 39 approximation error over the passband and the stOpband min maxlEmH where Em weighted approximation error WlwllHdlejmlelelmll 7 I desired aCtual Leads to a mini max solution equiripple filter 0 s a s up pass band lf Wo 2 l K I 035 lt 03 s 713 stop band T467667 Introducticm to Digital Signal Processing 51 3 K52 lHltei gtl 151 143quot i62 4 1452 Note MATLAB s quotfirpm39r routine designs filters of this type Alternation Theorem states that optimum lowpass filter of order 2L has at least L2 alternations peaks of the approximation error over the passband and stopband Note For OWpass filter of order 2L the max possible no of alternations is L3 lt Called the extra ripple case Example L7 h A i 7 3 Va s C0 5 7 9 No of alterations 9 2 L 2 gt filter is optimum L ECE T4676639l Inn oducticm to Digital Signal Prcwcssing 7 102L3 zafilter is optimum extraripple case 39 EXAMPLE OPTIMUM FIR LOWPASS DIGITAL FILTER Hdejm 1 0 00 g 47 z 0 675300371 error weighting function we 1 39 0500347 corresponds to K 10 H 06 14 U 2 E n 7 E lt 0 o21 39 v 0 10 20 30 Sample number n a l 20 T J quot20 40 60 80 100 J 1 J i 0 0271 04m 06V 087239 1r Radian frequency w b 0015 0010 m 0005 quot2 5 0 A A A A g 7 V V V lt 4005 4 0 Figure 743 optimum type I FIR 1 l l J lowpass filter for cup 2 04m 0015 2 6 n 0277 047 0m 02M 71 05 06 K Wand M 2 a Impulse response b Log magnitude of the frequency response C c Approximation error unWeighted L213 Radian frequency w M Ml erhw vw 1 i C g 39 v L t L EXAMPLE OPTIMUM FER BANDPASS FELTER O O lt a g 03x HA6 2 1 03573 5 a S 06 0 077rgw57r error Weighting function 1 0 f a f 03m Wa 1 035 E a g 06m 02 077 E a 5 7r Ampliludu 04 i i r 0 20 4O 60 80 Sample number n a 20 I 1 O 20 o 390 40 f 60 80 J L J J 027r 047r 061r 08w 7r Radian frequency w b 0060 1 4 0040 g 0020 DAMw mmmr E JV Willi YWVW 0020 0040 r 0oeo 1 i r 0 027r 04rr 06rr 087139 7r Radian frequency w C L337 Figure 747 Optimum HR bandpass fiiterior M 74 a Impulse response b Log magnitude ofihe frequency response 0 Approximation error unweighted 36 7 41 qikrm x ovrj Design of Type I FIR Band Pass Diqital Filters Usinq Matiab39s quotfirpmquot Routine 12 I 39 i l I l I I I I 1 08 06 04 02 00quot Wu 03 011W 05 06 quot 7 quot quot 08 09 quotquot 1 1 1 f 0 3 4 5 6 1 a 0 0 1 10 0 wtx 1 1 1 b rpm 36faW1X hm freqz b 1 128 plot faWpiabsh T467667 Intmduction to Digital Sigma P1 c ccssin g 12 I I 1 1 T I l f 0 3 4 5 6 1 a O 0 1 1 0 O wtx 5 1 5 b rpm 36faWTX hm frqu 1 1 128 plot fawpiabsh T467667 InrmducLicm to Digital Signal Processing 12 r 391 f03 45 61 a O O 1 1 O 0 WTX 1 5 1 b rpm 3 6fawtx hw freqz b 1 128 plot fawpiabsh l0


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