Introduction to Digital Signal Processing(DSP)

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Pearl Shepherd
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1 Forth Clss Commictio II Electricl Dept Nd Nsih Itrodctio to Digitl Sigl ProcessigDSP Recet developmets i digitl compters ope the wy to this sject The geerl lock digrm of DSP system is show elow: Bd limited filter ADC Compter DAC Smoothig filter xt f s xt s yt s yt The ipt sigl xt is loge sigl speech, video, This sigl is first d limited sig LPF hvig ct-off freqecy f mx The d limited sigl is the coverted ito digitl xt s sig smpler with smplig freqecy f s f mx d qtier The discrete sigl xt s or simply writte x is etered to digitl compter with sitle iterfce crd sod crd, video crd, Iside this compter progrm high or low level d rel time or off time is writte to perform y sort of sigl lysis to x sch s lier mplifictio, log, expoetil, covoltio, correltio, d filterig The reslt of the digitl processig is yt s this sigl is the coverted to loge form sig DAC, de-smpler, d filly smoothig filter to remove the stir cse shpe of y Geerl cocepts i DSP: Lierity : A DSP system is clled lier if sperpositio theory pplies For exmple if y=x, d x=x +x the the system is lier sice: y=x=x +x = x +x = y +y where y d y re the otpts de to x d x

2 Forth Clss Commictio II Electricl Dept Nd Nsih x y x y If y= y +y the system is lier If y # y +y the system is ot lier Cslity: A DSP system is sid to e csl if the preset vle of the otpt is ot the fctio of ftre vle of the ipt x y memory-less csl x y - memory csl x y + csl Stility: A DSP system is sid to e stle if the otpt is oded for oded ipt For exmple, if y=x-5x-, x <G where G is fiite, the y <G-5G hece if x is oded y G the y is lso oded ie the system is stle 4 Time Vrit d Time Ivrit: A system is sid to e time vrit if its chrcteristics depeds o the time idex For exmple, y=x is time vrit system A system is sid to e time ivrit if its chrcteristics does't chge with time idex For exmple, y=e -x- is time ivrit system

3 Forth Clss Commictio II Ipt/Otpt reltios of the lier systems: Electricl Dept Nd Nsih Aloge cotios systems: If ht is the implse respose of the system the Hw is the trsfer fctio which is the Forier trsform of ht yt=xt ht where is the covoltio t x h t d h x t y t d t xt Xw ht Hw yt Yw Also Yw=XwHw d Yw = Xw Hw Or G y w=g x w Hw where G y w d G x w re spectrl desities of x d y respectively Also otpt power y t H w Gx w dw Discrete digitl systems: Here h is the implse respose of the system, d H is its trsfer fctio: H=Y/X xt ht yt Ad Xw Hw Yw y=x h or y x k h k h k x k which is clled the discrete covoltio k k DSP systems re clssified ccordig to their resposes h ito: i FIR Fiite Implse Respose: where h hs fiite mer of elemets sch s : h={,,4,,,} where the crsor idictes the positio where =

4 Forth Clss Commictio II Electricl Dept Nd Nsih ii IIR Ifiite Implse Respose: where h hs ifiite mer of elemets sch s : h=/ where is the it step fctio: = for =,,,, else where 4 5 h Discrete covoltio Methods: Grphic Method: This iclde the sic covoltio steps :- reversig i time sig k s time idex shiftig y smples c Mltiplictio of the correspodig smples d Addig 4

5 Forth Clss Commictio II Electricl Dept Nd Nsih Ex: fid the vle of y sig grphicl method h={,-,}, x={,,-,} y x k h k k y x k h k 6 k y x k h k 5 k y x k h k k Ad so o, shiftig of h left d right til the over lppig etwee xk d h-k disppers givig 's t the otpt y h x h- - - h-- h

6 Forth Clss Commictio II Electricl Dept Nd Nsih Tlr Method: this is very simple method sed for FIR systems with fiite mer of smples x A rectglr tle with N rows mer of elemets i h d N colms mer elemets of x, or vis vers, is rrged The the cross mltiplictios re crried ot The sm of the mltiplictios digolly will give the vle of y Ex: Repet previos exmple sig tlr method h={,-,}, x={,,-,} y-= y-=- sm sm sm sm y-= y=6 y=-5 y=6 sm sm The y={,-,,6,-5,6} Note tht N=N +N -= mer of elemets i y = +4-=6 Ad tht O d O re positios of the crsors i h d x from the left, the O= O +O -=+-=4 which is the positio of y Add-overlp Method: This is modified method from the tlr method, whe either h d x hs lrge mer of elemets, the this c e divided ito s-segmets of smller legth 6

7 Forth Clss Commictio II Ex: fid the discrete covoltio etwee: h={,-,}, x={,,-,,4,-,,} Electricl Dept Nd Nsih Here x is divided ito three segmets, first two segmets of legth, d lst segmet of legth Hece previos tlr method will e repeted times y ={,,-,5,-} y ={,,,9,-} y ={,,-,6} The, dd y, y, d y, with y shifted to the left y elemets legth of ech segmet d y shifted to the left y 6 elemets: y y 9 - y - 6 y y={,,-,8,-,,9,,-,6} with the crsor for y t O=+4-=5 7

8 Forth Clss Electricl Dept Commictio II Nd Nsih 8 4 Mtrix Method: Here, Y=Ah, where y=otpt, h=h d the A mtrix hs N=N +N - rows d N colms mer of elemets of h The st row i A is the st elemet i x from the left d the remiig elemets re 's the d row i A is the d elemet i x, the the st elemet i x d the remiig elemets re 's d so o til the lst elemet i x is etered t the N th row N is the mer of elemets i x After tht 's re pplied isted of the elemets of x til the lst row t N=N +N - Ex: fid the vle of y sig mtrix method h={,-,}, x={,,-,} h A the Y The y={,-,,6,-5,6} where O=+-= 5 the Z-trsform Method: This is geerl method sed either or oth of h d x hs ifiite elemets The procedre here is to tke the -trsform of x d h, the mltiply to fid Y from which y is fod sig iverse -trsform

9 Forth Clss Electricl Dept Commictio II Nd Nsih 9 Ex: fid the vle of y where: h={,-,}, x=/ Note, sice x hs ifiite mer of elemets, the we mst se -trsform method Now, tkig the -trsform of oth h d x: h H 5 5 x X 5 5 X The Y=HX=-+ - / Ad y= Ex: Fid sig the -trsform where d re costts d,the: Z Z Hece: = d,the: Z Z Hece: = c d Z Z

10 Forth Clss Commictio II Discrete Decovoltio: Electricl Dept Nd Nsih To fid h if oth x d y re give, we se the -trsform method sice it is vlid for oth FIR d IIR DSP system H=Z - H=Z - Y/X Ex: Fid h if x={,-,} d y={,,-,7,-6} X= divisio d Y= the we se the log Hece H= or h={,5,-} d there is o remider, the the system is FIR Ex: Fid h if x= d y=-5 X The H= d Y By tkig the iverse -trsform we get : h=5

11 Forth Clss Commictio II Electricl Dept Nd Nsih DSP System Implemettios: FIR systems: Here h hs fiite mer of elemets: h={h, h, h, hm} with m+ elemets H= h+h - + h -, hm -m d if: H=Y/X, the: y= hx+ hx-+ hx- hmx-m ie, y is otied from x y the weighted sm weighted y h of the delyed smples of x these delyed smples of x re otied sig tpped dely lie with m-tps d with T s time dely per tp x Z - Z - Z Z - h h hm Z - T s Ʃ y Note tht the FIR system is lwys stle, where the system is ope loop withot feed ck d tht is why it is clled o-recrsive

12 Forth Clss Commictio II Electricl Dept Nd Nsih IIR systems: For IIR system hvig m-eros d r-poles, the: Y H X m m r r Where we c lwys set the first term t the deomitor to ity y dividig with sitle costt Fro which : y= x+ x-+ x-+ + m x-m- y-- y-- - r y-r Note tht y depeds o preset d previos vles of x d previos vles of y Hece: IIR system cotis feedck from the otpt to the ipt possiility of istility if some poles of H lies otside the it circle to implemet the IIR system, the tpped dely lies re reqired, oe with m-tps for the x ipt d the other with r-tps for the feedck from the y otpt x Ʃ y Z - Z - Z - Z - Z - Z - Z - Z - Z - Z - Z - m r Z -

13 Forth Clss Commictio II Ex: implemet the DSP system: H 4 Electricl Dept Nd Nsih This is IIR system First we divide y to set the first term t the domitor to ity, the: H This eeds tpped dely lie with tps for x d tpped dely lie with tps for y x Ʃ y Z - 5 Z - Z Z - 5 Z -

The z Transform. The Discrete LTI System Response to a Complex Exponential

The z Transform. The Discrete LTI System Response to a Complex Exponential The Trsform The trsform geerlies the Discrete-time Forier Trsform for the etire complex ple. For the complex vrible is sed the ottio: jω x+ j y r e ; x, y Ω rg r x + y {} The Discrete LTI System Respose