10/5/2012 S. THAI SUBHA CHAPTER-V
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1 /5/ /5/ S. THAI SUBHA CHAPTER-V FIR is finit impuls rspons. FIR systm s n impuls rspons tt is ro outsi of sm finit tim intrvl. FIR systm s finit mmory of lngt M smpls. /5/ S. THAI SUBHA CHAPTER-V
2 /5/ IIR Filtr All t infinit smpls of impuls rspons r onsir. T impuls rspons nnot b irtly onvrt to igitl filtr trnsfr funtion. T sign involvs nlog filtr sign n tn trnsforming nlog filtr to igitl filtr. T spifitions inlu t sir rtristis for mgnitu rspons only. IIR filtr n bom unstbl (if t pols of t IIR filtr r outsi t unit irl) FIR Filtr Only N smpls of impuls rspons r onsir. T impuls rspons n b irtly onvrt to igitl filtr trnsfr funtion. T igitl filtr n b irtly sign to iv t sir spifitions. T spifitions inlu t sir rtristis for bot mgnitu n ps rspons. FIR filtr is lwys stbl /5/ S. THAI SUBHA CHAPTER-V 3 T FIR filtr n b rtri by its systm funtion H() N n (n) n T frquny rspons is givn by, H( jω ) H( ω) N n (n) jωn /5/ S. THAI SUBHA CHAPTER-V
3 /5/ Symmtri onition: If t unit impuls for FIR filtr s t proprty tt, (n) (N--n), tn its ps is linr. /5/ S. THAI SUBHA CHAPTER-V 5 Gibb s pnomnon /5/ S. THAI SUBHA CHAPTER-V 6 3
4 /5/ Fturs of goo winow for FIR filtrs:. Si lob lvl soul b smll.. Bron mil stion. 3. Attnution soul b mor.. Smootr mgnitu rspons. 5. T tr off btwn min lob wits n si lob lvl n b just. 6. Smootr ns. 7. If osin trm is us tn si lobs r ru furtr. /5/ S. THAI SUBHA CHAPTER-V 7 Winowing Bs on Trunt Fourir sris /5/ S. THAI SUBHA CHAPTER-V 8
5 /5/ Winowing Rtngulr: w(n) Hnning: (n) n N n w (n).5 os N Hmming: (n) w H.5.6 os n N /5/ S. THAI SUBHA CHAPTER-V 9 Prour for signing FIR filtrs using winows:. For t sir frquny rspons H ( jω ), fin t impuls rspons (n) using,. Multiply t infinit impuls rspons wit osn winow squn w(n) of lngt N to obtin filtr offiints (n), i.., (n)w(n), (n), jω H ( ) 3. Fin t trnsfr funtion of t rlibl filtr. (n) for ω n N - otrwis /5/ S. THAI SUBHA CHAPTER-V 5
6 /5/ (N + (N) H() () n ) (n)( n + n ) /5/ S. THAI SUBHA CHAPTER-V Ex 3.(NK): Dsign LPF using rtngulr winow by tking 9 smpls of w(n) n wit utoff frquny of. rins/s. Solution: ω. rins/s n N 9 For LPF t sir frquny rspons is, jωα ;- ω ω ω H ( ω) ; otrwis Tk invrs Fourir trnsform of H (ω) to obtin (n), /5/ S. THAI SUBHA CHAPTER-V 6
7 /5/ (n) ω ω j (n) H ( ω) jω(nα) ω (nα) jωn jω(nα) ω j(n α) j ω jω (nα) sin ω (n α) ω ω (n α) jωα jωn ω ω ω /5/ S. THAI SUBHA CHAPTER-V 3 sin ω (n α) α Wn n α, will bom intrmint (n ) form. For n α; using L Hospitl rul, sin Aθ Lim A θ θ sin ω(n α) ω (n) Lim (n α) n α T impuls rspons of FIR filtr is obtin by multiplying (n) by winow squn, (n) (n) W R (n) /5/ S. THAI SUBHA CHAPTER-V 7
8 /5/ Rtngulr winow squn is givn by, ; for n to N - W R (n) ; otrwis (n); n to N - (n) Hr, ω. rins/s n N 9; sin ω (n α) (n) ; n α (n α) N α Wn n ; sin(.x( )) ().793 ( ) /5/ S. THAI SUBHA CHAPTER-V 5 Wn n ; sin(.x( 3)) ().7 ( 3) n ; sin(.x( )) ().75 ( ) n 3; sin(.x( )) (3).967 ( ) ω (n) ; n α Wn n (n α);. ().38 /5/ S. THAI SUBHA CHAPTER-V 6 8
9 /5/ Wn n 5; n 6; n 7; n 8; sin(.x()) (5).967 () sin(.x()) (6).75 () sin(.x(3)) (7).7 (3) sin(.x()) (8).793 () T impuls rspons is stisfying t symmtry onition (N--n) (n) /5/ S. THAI SUBHA CHAPTER-V 7 T mgnitu funtion of FIR filtr wn t impuls rspons is symmtri & N is o is givn blow, N N N ( ω) + n os n ( ) ( ( )) ω H n T trnsfr funtion of t filtr is givn by, H() N n 8 n 3 n (n) (n) (n) n n n + () + 8 n 5 (n) /5/ S. THAI SUBHA CHAPTER-V 8 n 9
10 /5/ H() Y() X() 3 n () (n) n (8n) [ + ] [ + ] + () [ + ] + () [ + ] + (3) + () 3 5 [ + ] + () /5/ S. THAI SUBHA CHAPTER-V 9 X(Z) Z Z - - X(Z) Z - X(Z) Z Z - Z -3 X(Z) Z - Z - X(Z) Z - Z Z -7 X(Z) - Z Z -8 X(Z) - Z -6 X(Z) Z -5 X(Z) () () () (3) Z - () /5/ S. THAI SUBHA CHAPTER-V Y(Z)
11 /5/ Ex 3.3(NK): Dsign HPF using mming winow by tking 9 smpls of w(n) n wit utoff frquny of. rins/s. Solution: ω. rins/s n N 9 For HPF t sir frquny rspons is, jωα ; - ω ω & ω ω H ( ω) ; otrwis Tk invrs Fourir trnsform of H (ω) to obtin (n), /5/ S. THAI SUBHA CHAPTER-V (n) ω jω(nα) j(n α) H ( ω) jω(nα) jω (nα) jωn ω + ω ω ω jω(nα) + j(n α) j(n jω(nα) α) j + j(n α) jωα ω ω (nα) jωn ω jω (nα) /5/ S. THAI SUBHA CHAPTER-V
12 /5/ (n) (n) (n α) (n α) j(nα) j j(nα) jω (nα) j jω (nα) [ sin(n α) sin(n α) ω ]; for n α Wn n α, (n) will bom intrmint form. For n α; using L Hospitl rul, (n) Lim n α [ ω ] sin(n α) sin(n α) ω (n α) (n α) sin Aθ θ Lim /5/ S. THAI SUBHA CHAPTER-V 3 θ A Wn n α, ω (n) T winow squn for mming winow is givn by, n (n).5.6 os ; for n to N - ( ) WH N (n) (n)wh (n); n to N - (n) sin(n α) sin(n α) ω (n α) ω [ ] n [.5.6 os( )]; N for n α n [.5.6 os( )]; for n α N /5/ S. THAI SUBHA CHAPTER-V
13 /5/ Hr, ω. rins/s n N 9; N α Sin bot n n α r intgrs, sin(n α) ω (n) (n α) ω n [.5.6 os( )]; n [.5.6 os( )]; for n α N N sin( )(.) Wn n ; () [.5.6 os( ) ]. 63 ( ) sin( 3)(.) Wn n ; ().5.6 os. ( 3) 8 for n α /5/ S. THAI SUBHA CHAPTER-V 5 n ; ().58 n 3; (3).567. Wn n ; () n 5; (5).567 n 6; (6).58 n 7; (7). n 8; (8).63 [.5.6 os( ) ]. 68 /5/ S. THAI SUBHA CHAPTER-V 6 3
14 /5/ Dsign low pss filtr wit H (ω) jω ; ω H ( ω) ; ω using Hnn winow, wit N 5. Solution: (n) jωn H ( ω) jω (n) j(n ) j j(n ) ω jnω ω j(n) j(n) [ ] [ sin(n ) ] H ( ω) j(n) ω ω /5/ S. THAI SUBHA CHAPTER-V 7 (n) sin(n ) (n ) ; n () ; n () sin () sin (3) () () () /5/ S. THAI SUBHA CHAPTER-V 8
15 /5/ From t Hnning winow funtion, W (n) W () n n ( os ) ( os ) N Now, W () W (); W () (n) (n)w (n) () () W (3) n () () (3) /5/ S. THAI SUBHA CHAPTER-V 9 Dsign n il bn pss filtr wit frquny rspons, Fin t vlus of (n) for N 7. From t sir frquny rspons, w n fin tt t givn rspons is symmtri N o /5/ S. THAI SUBHA CHAPTER-V 3 5
16 /5/ For symmtry rspons, /5/ S. THAI SUBHA CHAPTER-V 3 For n, For n -3, -, -,,, 3: /5/ S. THAI SUBHA CHAPTER-V 3 6
17 /5/ Dsign ig pss filtr wit H (ω) ; ω H ( ω) jω ; ω using Rtngulr winow givn by, ; n W(n) ; otrwis /5/ S. THAI SUBHA CHAPTER-V 33 Homwork (RB son E.). Exmpl.. Exmpl. 3. Exmpl.(NK). Exmpl 5.6 (Us bot trnsformtions) 5. Exmpl Exris Exmpl Exmpl 3.(NK) 9. Exmpl 3.5(NK). Exmpl 7.5 Lst t:.. /5/ S. THAI SUBHA CHAPTER-V 3 7
18 /5/ /5/ S. THAI SUBHA CHAPTER-V 35 T funmntl oprtion in igitl filtrs r multiplition n ition. Wn ts oprtions r prform in igitl systm t input t n output t (prout & sum) v to b rprsnt in finit wor lngt, wi pns on t si(lngt of t rgistr) us to stor t t. In igitl omputtion t I/P & O/P t (sum & prout) r qunti by Rouning or Truntion to onvrt tm to finit wor si. /5/ S. THAI SUBHA CHAPTER-V 36 8
19 /5/ Tis rts rror (nois) in t input or rts osilltions (limit yls) in t output. Ts ffts u to finit prision rprsnttion of numbrs in igitl systm r ll finit wor lngt ffts. /5/ S. THAI SUBHA CHAPTER-V 37 /5/ S. THAI SUBHA CHAPTER-V 38 9
20 /5/ /5/ S. THAI SUBHA CHAPTER-V 39 T following r som of t finit wor lngt ffts in igitl filtrs:. Errors u to quntition of input t by A/D onvrtr.. Errors u to quntition of filtr o-ffiint. 3. Errors u to rouning t prouts in multiplition.. Errors u to ovrflow in ition. 5. Limit yls /5/ S. THAI SUBHA CHAPTER-V
21 /5/ Typs:. Input quntition rror Errors u to rouning of I/P t. Prout quntition rror Errors u to rouning t prout in multiplition 3. Coffiint quntition rror Errors u to quntition of filtr offiints Limit yls: Du to prout quntition & ovrflow in ition /5/ S. THAI SUBHA CHAPTER-V Input quntition rror: In DSP, t ontinuous tim I/P signls r onvrt into igitl using b-bit ADC. T rprsnttion of ontinuous signl mplitu by fix igit prous n rror, wi is known s Input quntition rror. /5/ S. THAI SUBHA CHAPTER-V
22 /5/ Prout quntition rror Ariss t t output of multiplir. Multiplition of b bit t wit b bit offiint rsults prout ving b bits. Sin b bit rgistr is us, t multiplir output must b roun or trunt to b bits, wi prous n rror. /5/ S. THAI SUBHA CHAPTER-V 3 Coffiint quntition rror In igitl omputtion t filtr offiints r rprsnt in binry & r stor in rgistrs. If b bit rgistr is us, t filtr offiints must b roun or trunt to b bits, wi prou n rror. Du to quntition of offiints, t frquny rspons of t filtr my iffr ppribly from t sir rspons & somtims t filtr my tully fil to mt t sir spifitions. If t pols of sir filtr r los to t unit irl, tn tos of t filtr wit qunti offiints my li just outsi t unit irl, ling to unstbility. /5/ S. THAI SUBHA CHAPTER-V
23 /5/ Quntition stp si: Lt us ssum sinusoil signl vrying btwn + & - ving ynmi rng. If ADC us to onvrt t sinusoil signl mploys b+ bits inluing sign bit, t numbr lvls vilbl for quntiing x(n) is b+. Tus t intrvl btwn sussiv lvls rprsnts t stp si. Quntition stp si is givn by, b q b+ /5/ S. THAI SUBHA CHAPTER-V 5 If 8-bit rgistr is vilbl, tn stp si vris wit rspt to rng of t signl. For t rng btwn V to 5V, 5 5 stp si 9.53mV 8 56 For t rng btwn -5V to 5V, stp si 39.65mV 8 56 /5/ S. THAI SUBHA CHAPTER-V 6 3
24 /5/. Truntion Pross of isring ll bits lss signifint tn LSB tt is rtin.. Rouning Roun to t losst of t originl numbr. /5/ S. THAI SUBHA CHAPTER-V 7 Wt is rouning fft? Rouning is t pross of ruing si of binry numbr to finit si of b bits su tt t roun b-bit numbr is losst to t originl unqunti numbr. T rouning pross onsists of truntion n ition. In rouning of numbr to b-bits, first t unqunti numbr is trunt to b-bits by rtining t most signifint b-bits. Tn ro or on is to t lst signifint bit of t trunt numbr pning on t bit tt is nxt to t lst signifint bit tt is rtin. For Exmpl:. roun to four bits is itr. or. (Hr ing on is ll rouning up). /5/ S. THAI SUBHA CHAPTER-V 8
25 /5/ Error u to rouning: T quntition rror is fix point numbr u to rouning is fin s In fix point rprsnttion t rng of rror m by rouning numbr to b bits is Error u to Truntion: N N t In fix point rprsnttion t rng of rror m by trunting numbr to b bits is t b < /5/ S. THAI SUBHA CHAPTER-V 9 Quntition of input t: Quntition rror, (n) x q (n) x(n) x x(n) q (n) Mn of t rror signl Vrin of t rror signl, q σ Sty stt nois powr u to For t rng of V, σ input quntition Qq b b+ /5/ S. THAI SUBHA CHAPTER-V 5 (n) (n) ; y(n) 5
26 /5/ Quntition of filtr offiints: Consir son orr IIR filtr wit fin t fft on quntition on pol lotions of t givn systm funtion in irt form-i & in s form. Tk b 3 bits. Solution: Dirt Form-I: H() () (.5. )(.5 H ) /5/ S. THAI SUBHA CHAPTER-V 5 (.95) (..) (-.95) (..) Aftr Truntion, (.) (-.875) (.5) (. ) Aftr Truntion, (.) (.5) So,. H() /5/ S. THAI SUBHA CHAPTER-V 5 6
27 /5/ Cs Form: () (.5. )(.5 H ) (-.5) (..) (-.5) (..) Aftr Truntion, (.) (-.5) Aftr Truntion, (.) (-.375) () (.5. )(.375 H ) /5/ S. THAI SUBHA CHAPTER-V 53 Sty stt output nois powr: ε(n) (n) (n) k (k)(n k) Sty stt vrin, n x(n) (n) x q (n) (n) y(n) σ (n) σ ε n (n) Using Prsvl s torm, t sty stt output nois vrin u to t quntition rror is givn by, σ σε (n) σ (n) H()H( ) j n (n) wr t los ontour of intgrtion is roun t unit irl in wi s only t pols tt lis in t unit irl r vlut using rsiu torm. /5/ S. THAI SUBHA CHAPTER-V 5 (n) ε(n) 7
28 /5/ Exmpl 7.5 (NK): Consir t trnsfr funtion H() H ()H () wr () n H () H (i). Fin out output rounoff nois powr. (ii). Assum,.5 n.6 n fin out output rounoff nois powr. x(n) y(n) (n) Z - (n) Z - /5/ S. THAI SUBHA CHAPTER-V 55 H() ( H() ( )( ) [ s sn by (n)] [ s sn by (n)] Totl sty stt nois vrin is givn by σ σ σ σ j [ σ j ) H()H( ) ( )( sum of rsius of H()H( t pols, ) ( )(, σ σ + σ /5/ S. THAI SUBHA CHAPTER-V 56 - ) -, ) 8
29 /5/ 9 [ ] + σ σ σ ) )( )( )( ( ) ( ) )( )( )( ( ) (, t pols ) H()H( rsius of sum of r ro., t ) H()H( lis outsi t unit irl,so t rsius of, pols, tn r lss n If /5/ 57 S. THAI SUBHA CHAPTER-V Similrly, + σ σ σ σ ) ( ()H H j σ σ σ σ σ ) )( ( ) ( ) ( ) ( j /5/ 58 S. THAI SUBHA CHAPTER-V
30 /5/ σ σ + + T sty stt nois powr for.5,.6 is givn by, σ σ [ 5.35] /5/ S. THAI SUBHA CHAPTER-V 59 Limit yl osilltions: For n IIR filtr, implmnt wit infinit prision ritmti, t output soul ppro ro in t sty stt if t input is ro & it soul ppro onstnt vlu, if t input is onstnt. Howvr, wit implmnttion using finit lngt rgistr n output n our vn wit ro input if tr is non-ro initil onition on on of t rgistrs. T output my b fix vlu (or) it my osillt btwn finit positiv & ngtiv vlus. /5/ S. THAI SUBHA CHAPTER-V 6 3
31 /5/ Tis fft is rfrr to s (ro-input) limit yl osilltions & is u to t non-linr ntur of t ritmti quntition. T mplitu of t output uring limit yl r onfin to rng of vlus ll t bn of t filtr. In t s of FIR filtrs, tr r no limit yl osilltions, if t filtr is rli in irt form or s form sin, ts struturs v no fbk. /5/ S. THAI SUBHA CHAPTER-V 6 Consir first orr IIR filtr wit iffrn qution, y(n) α y(n-) + x(n) D bn for t first orr filtr is givn by, b y(n ) α /5/ S. THAI SUBHA CHAPTER-V 6 3
32 /5/ Exmpl 7.8 (RB): Explin t rtristis of limit yl osilltion wit rspt to t systm srib by t iffrn qution, y(n).95 y(n-) + x(n) Dtrmin t bn of t filtr. Solution: Lt us ssum -bit sign mgnitu rprsnttion (xluing sign bit) Lt t input, x(n).875 for n, otrwis /5/ S. THAI SUBHA CHAPTER-V 63 Bus of finit lngt rgistr t prout.95 y(n- ) in t iffrn qution must b roun to - bits bfor ing to x(n). T output y(n) wit rouning is givn by, y(n) x(n) + Q[.95y(n-)] wr Q[.] stns for quntition. For n, y() x() + Q[.95y(-)].875 [y(-) ] For n, y() Q[.95y()] + x() Q[.95(.875)] + Q[.835] (.835) (. ) /5/ S. THAI SUBHA CHAPTER-V 6 3
33 /5/ Aftr rouning to -bits, w gt, Q[.835] (.) (.85) y().85 For n, y() Q[.95y()] + x() Q[.95(.85)] Q[.77875] ( ) (. ) Aftr rouning to -bits, w gt, (. ) (.) (.75) y().75 /5/ S. THAI SUBHA CHAPTER-V 65 For n 3, y(3) Q[.95(.75)] Q[.75] (.75) (. ) (.) (.6875) y(3).6875 Similrly, y().65 y(5).65 y(6).65 From t bov lultions it n b obsrv tt for n 5, t output rmins onstnt t.65 using limit yl bviour. /5/ S. THAI SUBHA CHAPTER-V 66 33
34 /5/ T bn is givn by b D bn α For, b, /5/ S. THAI SUBHA CHAPTER-V 67 3
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CS 461, Lecture 17. Today s Outline. Example Run
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Types of Transfer Functions. Types of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr x[n] X( LTI h[n] H( y[n] Y( y [ n] h[ k] x[ n k] k Y ( H ( X ( Th tim-domain classification of an LTI digital transfr function is basd on th lngth of its impuls rspons h[n]:
Introduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by: