FIR Digital Filter Design.

Transcription

1 FIR Digital Filter Deig. Ulike IIR, FIR Caual Digital Filter are iheretly table. (All ole at the origi i uit irle) Ulike IIR, exat liear hae deig i oible with FIR Filter (rovided imule reoe i either ymmetry or ati ymmetry). For ame frequey reoe eifiatio, the order of FIR filter are muh higher tha the IIR filter.

2 Bai Deig Aroah FIR filter deig doe ot have ay oetio with aalog filter a for the ae of IIR digital filter. Deig i baed uo diret aroximatio of the eified magitude reoe with ofte added requiremet of liear hae reoe.

3 FIR Deig baed o Widow Futio DTFT air equatio : Frequey Reoeor Setrumi - give by theaalyi equatio : - DTFT The imule reoeamlei IDTFT H h d d ( e j [] ) h d H [ ] e d ( e j give by theytheiequatio : j ) e j d, - -.

4 Ideal Lowa Filter H ( LP e j ) 1,, 0,. Stobad 1 Pabad Stobad (utoff frequey)

5 Takig IDTFT of Frequey Reoe of a Ideal Lowa Filter. -, ) i( ) ( ). ( 2 1 ] [ h. 0,, 1, ) ( LP j e j e d e d e e H e H j j j j j LP j LP Imule reoe i doubly ifiite. It i ot aual ad otai ifiite delay. Therefore urealizable

6 Aroximatig the ifiite reoe to fiite reoe by multiliatio of retagular widow i time domai. w h ret LP [ ] 1, 0,. M, -M i( ) [ ], 0, otherwie. M. ^ hlp[ ] 0 M i( ( M ), ( M ) 0, otherwie. 0 N. (order of filter N 2M) Fiite but ot aual. Shift M amle to right for aual.

7 Blok Diagram of FIR Digital Filter (1) FromCovolutio um: - y[] (2) Liear otat oeffiiet differeeequatio : - y[] 2M k 0 2M k 0 h[k]x[- k] b k x[ k] x[] D D D D D b0 h[0] b1 h[1] b2 h[2] b3 h[3] b2 M h[2m ] + y[]

8 Ideal Higha Filter H HP ( e j ) DTFT h HP [ ] 1, for 0 i( ) -, for 0 1 Pabad Stobad Pabad (utoff frequey)

9 Ideal Bada Filter H BP ( e j ) DTFT h BP i( 2) i( 1) [ ] - i( 2) 2 f 2-2f1 2 f 2, for i( 2 f 1 1 ) 0, for 0 1 Pabad Stobad Pabad Stobad (utoff frequey)

10 H BS ( e j ) DTFT Ideal Badto Filter h BS 2 1 [ ] 1, i( 1) i( - i( 1) 2 f 1-2f 2 f 1 2 for 0, ), for 0 2 i( 2), for 2 f Stobad Pabad Stobad Pabad (utoff frequey)

11 Effet of Multilyig the Ifiite Imule Reoe with A Retagular Widow. By truatig the Ideal Ifiite Imule Reoe, the FIR filter are realizable but they uffer from oillatory behavior i their reetive magitude reoe. Thi effet i kow a Gibb heomeo. Arie from the fat that multiliatio i time domai i equivalet to ovolutio i frequey domai.

12 Gibb Pheomeo for a Lowa FIR Filter imlemeted uig retagular widow H ( e t j ) Stobad Stobad Pabad 2 Traitio Bad

13 Gibb Pheomeo a be redued by uig other widow futio. Tyeof Widow Retagular Ha (Haig) Hammig Blakma Mai Lobe Width ML 4/(2M 1) 8/(2M 1) 8/(2M 1) 12/(2M 1) Relative Sidebad Level A 13.3dB 31.5dB 42.7dB 58.1dB l Miimum Stobad Atteuatio 20.9dB 43.9dB 54.5dB 75.3dB Traitio Badwidth 0.92 /M 3.11 /M 3.32 /M 5.56 /M Order of Filter N=2M, No of amle/equee legth=n+1=2m+1 Table of Proertie of Fixed Widow Futio.

14 Widow Futio Widow'legth are all amegive by- M M. Retagular : - w[] 1 Ha/Haig : - 2 w[] 0.5[1 o( )]. 2M 1 Hammig : - 2 w[] o( ). 2M 1 Blakma : w[] o( ) 0.08o( ). 2M 1 2M 1

15 Ste i Deig of Lowa FIR Digital Filter Uig the Widow Futio. (1) Normalizig to agular edgebadfrequey ( & ) if egive i Hz. (2) ChooeMiimum StobadAtteuatio a (3) Refer totable of (4) Traitio Badwidth. (5) betwee the rile i abador atteuatio bad. Chooe the right widow through. DetermieM by referig to theolum of i the Table of Sequeelegth of (6) Calulate the3dbutoff frequey. 2 2 (7) Determie oeffiiet of the FIR filter through tabulatio of h D 2F F [ ] T h l, mi(, ). 2F F T Proertieof Proertie. Order of h[]i N 1. [ ],w[]& h[ ]. fixed w[ ] h D widow futioa i reviou lide. filter N [ ]. themiimum Traitio Bad 2M.

16 Examle i Deig of Lowa FIR Digital Filter Uig the Widow Futio. Deig a Liear PhaeLowaFIR Digital Pabadedge frequey Stobadedge frequey Samlig Frequey Peak abadrile 2KHz. 300Hz, 500Hz. Miimum tobadatteuatio 0.2dB, 40dB, Filter with thefollowig eifiatio : -

17 Examle i Deig of Lowa FIR Digital Filter Uig the Widow Futio. (1) Normalizig to agular edgebad frequey ( & ) if egive i Hz. (2) ChooeMiimum StobadAtteuatio a betwee the rile i abador atteuatio bad. mi(, ). 2F F T 20log / 20 (1 ), 110 mi(, ) 0.01, 40dB / 20 (3) Refer totable of Proertieof fixed widow futioa i reviou 2 lide. Chooe the right widow uig theolum of For 40dB ad iorder tohave (4) Traitio Badwidth, 0.3, log - / 20. ( ) , 2F F theleat order of T themiimum 10 Miimum Stobad Atteuatio / 20 1/ filter, weeed to hooeha/ Haig Widow.

18 Examle i Deig of Lowa FIR Digital Filter Uig the Widow Futio. (5) DetermieM (Order of i the Table of 3.11 Traitio Badwidth, M 3.11 M 15.55, N 2M Chooe ext highet iteger, theorder of filter N i.e. M 16, N 32, N Atually we have Thi i tye1ie N 1 (6) Calulate the3db utoff frequey 0.3 twootio either hooetye1(odd legth h[])or tye2 (eve legth h[]) Proertie. filter/2) by referig to theolum of 33, for tye2, N , Legth of imule reoe. 2 Traitio Bad N 1 33.

19 h Examle i Deig of Lowa FIR Digital Filter Uig the Widow Futio. (7) Filter i lowa,therefore: - h D i( ) [ ], [ ], 0. D Ha Widow i w[ ] h D 0.4 Tomakeit aualadd M 16 to idie h[16] h[17] h[18] [] w[ ] h[15] h[14] [1 o( )], 2M 1 h[] h [] w[ ] D - M M. with M 16

20 Aroximatig the ifiite reoe to fiite reoe by multiliatio of widow w[] i time domai. h[ ] i( ) w[ ], 0, otherwie. 0 Imulereoe/Tye1Filter legth Note that exet for Ret.& Hammig, (refer tofigure10.6 Page 533of N. (order of Mitra) h[ ] h[ M ] filter N 2M N 1 33 firt & lat amlemut be for Shift M amle to right for aual ) auality. 0. h[16]

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