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# INTRO TO DIG SIG PRO E C E 467

Clemson

GPA 3.84

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This 15 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 21 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 19 quotDirect Methodsquot for Designing Digital Filters Do not involve the design of an intermediate analog filter Minimum MeanSquare Error Method produces an HR digital filter produces a filter which is the best meansquare fit to some desired frequency response curve at M points uses computer iteration Hdej loon quot 1 T 39 c 0 C01 T D3T CD5 T D7 T 039 T 011T n mts 032 04 a 08 6 mm 912 Assumed form of Hz k 1 ajz 1 biz 2 Hz 2 AH i1 1 012 1 djz 2 Let 5 4k1 element parameter vectorA1a1b1c1d1 dk Squared magnitude error functions Esp iliremrridreiw 1 Design goal Find 5 such that E s ECG for all 5 15 ECE T467667 Introducticm to Digital Signal Processing To find 5 set aaE eLo n12 4k1 Get 4k 1 nonlinear n equations in 4k1 unknowns Use iterative method on computer to find F Fletcher Powell method Example from Digital Siqnal Processinq 1st ed by Oppenheim and Schafer 39 The following example 14 illustrates the use of the above procedure An idmoii m lt r was39 sp d ed39 ith cutoff frequency 011r as depicted in Fig 527 That is 39 r I 39 1 r w 0 091quot 002 009 05 w 01 1 quot quotl quot0 w 011 01212 019 O r a In 027 03a 09m 1r H l 5 as 55 av3d VaVAe o 39J tr 1 1 1 139 r3939 1 1 39 on 021 03 04 067 05 07109 09 1 u 39 Fig 511 Fixed ulna of lb tnqueneyngpenu for triumph of the nu of Sulgllu39l design procedure 39 I IJHI Exam II at or respond bullied by mlnimlndoo of H uranium Afur5ni llamp 1 g 39 Note The minimum meansquared error design method sometimes gives poles outside the unit circle To get a stableI causal filter having T467667 Introduction to Digital Signal Processing the same frequency response replace the pole at z Lei witha gt1 L pole at z gejquot hr lt1 Let Hz filter with a pole at z rejewhere rgt1 Now let H1Z H Zero to cancel the pole 1 outside the unit circle z r New pole inside the unit quotav 2 circle 2 H2ejm cosm jsmco rcose Jrsme 1 1 coswjsmco cose JFsm9 2 2 H2ejm12 cosm rcose smmrsrn9 1 2 1 2 cosm Fcosej sinw Fsin9 134671667 Inntoducticm to Digital Sigma Processin 2 cos2 0 2r cos moose r392 0052 9 sin2 0 2r sinoasin6 r2 sin2 e H2e39w 2 t 1 2 39 1 cos2 w coswoosequot c052 6 sun2 o smcosm9 731n2 6 r 39 r r r 2 H2eim2 1r 2rcoscocoseo smmsme 1 1 Ecoscooos e sinm in 6 r2 r 1r2 2rcos03 e 1J coscn G r J 1 2cosoo 6 r2 r r 1l2 cosa e r 2 r So the new cascaded filter H1z has frequency response H1 91 31 Hej H2 9 Hejml r Same shape as He only differs by a constant scaie factor T4671667 Introduction to Digital Signal Processing Example 1 12 j O8 39 06 02 03 3903 075 06 07 68 09 102001391001100 139 02 3 4 5 67 8 1 bra yulwajk10 m i huh I r0qbn128h P1 mwp1lablh ECE T467667 Introduction to Digital Signal Processing Examgle 2 12 39 I I l l 08 04 0 m 00001110 039000139110001 l t E0 05 1 15 2 2s39393 3 5 4 45 5 55 6 65 7 75 8 11 ha in yulmlkllhtmir 39 i than roqziba128a plotltnwpiobl hH FIR Dioital Filter Desicin hn has finite duration 0 no poles only zeros Advantages of FIR filters vs IIR filters 1 Can have exactly linear phase 2 Less sensitive to roundoff errors due to MPYs ECE T467667 Introduction to Digital Signal Processing Disadvantages of FlRfilters 1 FIR filters typically require more computation per output term than R filters 2 FIR filters are generally more difficult to design They often involve a computer algorithm Recall linear phase Heim lHelw1 laa lf input lies within passband of a flat39passbandfilter with linear phase then yn xn or no timedistortion Linear phase lt gt symmetry condition on hn Example Nonzero range of hn O s n s N t Symmetry condition hn hNln For N539 hn h4n Osns4 h0h4 h1h3 h2h2 hn 0 1 L2 3 4 n ECE T467667 Introduction to Digital Signal Prooessing 39 For N6 hnh5n hn 0 011 2 3 14 5 n Windowino method of FIR Filter Desion 1 Start with desired frequency response Hdej 39 a 1 1 iw icon 2 Obtain hdn t j Hde e do mm is usually of infinite duration gt 1LFIR 3 Truncate mm to get hn for FIR filter Let hn hdnwn finite duration window tar I 39 ln frequency domain Helm39H2ej 39Wejm 511 EHd 910iw eim 9d9 ECE T467667 Introduction to Digital Signal Processing Example Design a FIR approximation to an ideal lowpass filter 1 HA9 27 7c DC 0 0C 1139 2n 0 2 hn 5quot an n i 0 hO 9i 1m 1t hn lt etc etc gt 3Truncate and shift to make causal Assume that N desired length of hn N odd sin 030 n on nn 0L g na TC N 1 nioc Shift hn by 2quot a h1n wN 1 n OSnSN l Then h1nh1nW where Mn 0 other T467667 Introduction to Digital Signal Processin 1 OSnsN1 Consider a rectangular window wn 39 39 r 0 other n M sun We eju 2 D sin 3 75 a 2n 0 N H elmWequot V l L quota quot r I coc no 7 21 0 Principal window properties Width of main lobe affects width of filter39s transition region Size of quotside lobesquot affects filter39s stopband attenuation 1o T467667 Innmiuction to Digital Signal Processing 9 Window Type Shage Width Max Side 5 91 Lobe m in Io e Rectangular 13 db 47cN 0 N1 Bartlett 39 27db BitN O N1 39 Hanning A 0 N91 STEN 39 32 db 21m 1 cos N 1 W n w han 2 Hamming 39 BicN 43 db 0 W quot N1 Wham n 54 46 cos MW Blackman 39 I A 121cN 58 db O N139 WBn 42 5005N2 rl1 39 47m 08 cos a N4 1139 ECE T 467667 Introduction to Digital Signal Processing 39 Wall39l Eiln Ee ilx 0 at a l Modified Oorder Bessel function of 1St kind Kaiser wk n o wa trades off main lobe width vs side lode magnitude 0 Kaiser window has max energy in main lobe for a given peak side lobe amplitude Design Table FIR lowpass filters using windowing method Window Type Transition Min Stopband Width Attenuation Rectangular 41rN 21 db Bartlett STEN 25db Hanning 87tN 44db Hamming BTCN 53db Blackman 121tN 74db k 39 J lnd epencfjent of N Design Steps Given 031002k1k2 as design specs 1 Choose the window typecall it wx n that satisfies stopband attenuation condition k2 and which has narrowest transition 2 For this window type choose N large enough to satisfy the transition width requirement wt 032 31 use gdd N 12 ECE T467i 667 lutmtlucticm to Digital Signal Processing N 1 3 Set 0t 2 2 and me 01 on IS an integer if N is odd Candidate filter hnw wxn n 2 on 1tn on w 1 noa 7t 4Evaluate lHej at cot and 02 Eli quotk1 at w quot is not met adjust wc and go back to step 4 6 Check Can N be reduced and still permit the filter to satisfy the kg at 61quot conditionGo back to step 4 to check using smaller N Examgle 013712 k1 3 db 002 457t k2 50 db 1Wlndow candidates which satisfy stopband conditionk2 Hammingtransition width 81rN 4 Choose Blackman transition width 12739CN 2wtm2 co1157r Choose N so that s 151 gt N 2 533 gt use N55 odd 3 Let aN 1 27 and we m13n 2 Candidate filter sin 37tn 27 21th 54 46008 lt 54 dn 27 Mn n27 54 O h an wHamn hn 3n 27 13 ECE T467667 Introductionto Digital Signal Processing 4Evaluate Helm at co 01 37r Note that He 3 1 6 db L cl 0 20 log ll r U f L lI H quotk1 at 01quot requirement QL satisfied 4 Increase no slightlytry 01223370 then return to step 4 5 Try a lower value of N which still permits the to be metFind that N29 works mm 20 mamaquot 704r OJ nut cw o 03 0A539 kg at 12 requirement 1 Using DC 2337 N 29 b 14 ECE T467667 Introduction to Digital Signal Processing Designed filter 39 W54I46CO5 O S n S 28 and n 14 hn xn14 28 33 n 14 To implement this filter 28 yn 2 hkxn k 29 MPYS 28 ADD39s k0 B using the symmetry of hn yn h0xn xn N 1 h1xn 1 xn n 2 M i i h x n 2 2 one quotunpairedquot term J and since N29 yn h01xn xn 28 hiXxn 1 xn 27 h13xn 13 xn 15 h14xn 14 15 MPYS 28 ADDS 15

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