Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt variabl or variabls. Mathmatically, w dscrib a signal as a function of on or mor indpndnt variabls. Slid 3 CLASSIFICATIO OF SIGALS Continuous-Tim and Discrt-Tim Signals A signal x(t) is a continuous-tim signal if t is a continuous variabl. If t is a discrt variabl, that is, x(t) is dfind at discrt tims, thn x(t) is a discrt-tim signal. Illustrations of a continuous-tim signal x(t) and of a discrt-tim signal x[n] ar shown blow
Slid 4 Dtrministic and Random Signals: Dtrministic signals ar thos signals whos valus ar compltly spcifid for any givn tim. Random signals ar thos signals that tak random valus at any givn tim and must b charactrizd statistically. Evn andodd Signals: A signal x ( t ) or x[n] is rfrrd to as an vn signal if x ( - t ) = x ( t ) x [ - n ] = x [ n ] Slid 5 signal x ( t ) or x[n] is rfrrd to as an odd signal if x ( - t ) = - x ( t ) x [ - n ] = - x [ n ] Exampls of vn signals Exampls of odd signals Slid 6 Priodic and onpriodic Signals: A continuous-tim signal x ( t ) is said to b priodic with priod T if thr is a positiv nonzro valu of T for which x(t + T ) = x ( t ) all t
Slid 7 BASIC COTIUOUS-TIME SIGALS Th Unit Stp Function: Th unit stp function u(t), also known as th Havisid unit function, is dfind as Slid 8 Th Unit Impuls Function: Complx Exponntial Signals: Slid 9 BASIC DISCRETE-TIME SIGALS Unit impuls squnc n [n] n Unit stp squnc n n u[ n] [ k ] [ n k ] n k k Exponntial squncs n x[n] A
Slid SYSTEMS AD CLASSIFICATIO OF SYSTEMS A systm is a mathmatical modl of a physical procss that rlats th input (or xcitation) signal to th output (or rspons) signal. Lt x and y b th input and output signals, rspctivly, of a systm. Thn th systm is viwd as a transformation (or mapping) of x into y. This transformation is rprsntd by th mathmatical notation Y=Tx whr T is th oprator rprsnting som wll-dfind rul by which x is transformd into y. Slid Continuous Tim and Discrt-Tim Systms: If th input and output signals x and y ar continuous-tim signals, thn th systm is calld a continuous-tim systm. If th input and output signals ar discrt-tim signals or squncs, thn th systm is calld a discrt-tim systm. Slid Causal and oncausal Systms: Th output of a causal systm at th prsnt tim dpnds on only th prsnt and/or past valus of th input, not on its futur valus. Exampl for a causal systm is Y(t) = x(t-) Exampl for a on causal systm is Y(t) = x(t+)
3 5 7 9 3 5 7 9 -. -.44 -.64 -.8 3 -.98 -. 5 -. -.6 7 -.8 -.6 9 -. -. 3 -.98 -.8 33 -.64 -.44 35 -. 37..44.64.8.98...4.7.4...98.8.64.44. Slid 3 Linar Systms and onlinar Systms: A linar systm obys th principl of suprposition and it satisfis th following two conditions. (i) Additivity (ii) Homognity (or Scaling) whr y = T[x ] and y = T[x ] Slid 4 What is DSP? Convrting a continuously changing wavform (analog) into a sris of discrt lvls (digital) Slid 5 What is DSP? Th analog wavform is slicd into qual sgmnts and th wavform amplitud is masurd in th middl of ach sgmnt Th collction of masurmnts mak up th digital rprsntation of th wavform.5.5 -.5 - -.5 -
Slid 6 Convrting Analog into Digital Elctronically Th dvic that dos th convrsion is calld an Analog to Digital Convrtr (ADC) Thr is a dvic that convrts digital to analog that is calld a Digital to Analog Convrtr (DAC) Slid 7 Discrt Fourir Transform-DFT Rlationship btwn priodic squnc and finit-lngth squnc Priodic squnc can b sn as priodically copis of finitlngth squnc. Finit-lngth squnc can b sn as xtracting on priod from priodic squnc. Finit-duration Squnc Priodic Squnc Main priod Slid 8 Proprty of DFT Suppos : X( k) DFT[ x( n)], Y( k) DFT[ y( n)] () Linarity DFT[ ax( n) by( n)] ax ( k) by( k), a, b ar cofficint () Circular Shift Circular shift of x(n) can b dfind: x ( n) x(( n m)) R ( n) m 7//4 8
Slid 9 (3)Parsval s Thorm x( n) y ( n) X ( k) Y ( k) n k Whn x( n) y( n), x( n) X ( k) n k Consrvation of nrgy in tim domain and frquncy domain. nk Proof: x( n) y ( n) x( n) [ Y( k) W ] n n k nk x( n) [ Y ( k) W ] n k Y ( k) x( n) W X ( k) Y ( k) nk k n k Slid (4)Circular convolution Suppos F( k) X ( k) Y( k) thn : f ( n) IDFT[ F( k)] x( m) y(( n m)) R ( n) m m or : f ( n) y( m) x(( n m)) R ( n) Priodic convolution is convolution of two squncs with priod in on priod, so it is also a priodic squnc with priod. Circular convolution is acquird by xtracting on priod of priodic convolution, xprssd by. Circular convolution Slid DFT[ x( n)] X ( k) DFT[ y( n)] Y( k) IDFT[ X ( k) Y( k)] x( n) y( n) Circular convolution can b usd to comput two squnc s linar convolution.
Slid Conjugat symmtric proprtis a)dft of conjugat squnc Suppos x ( n) is complx conjugat squnc of x( n), thn: DFT[ x ( n)] X ( k), k nk n( k ) n n Proof : DFT[ x ( n)] x ( n) W [ x( n) W ], k QW n j n j n DFT[ x ( n)] [ x( n) W W ] [ x( n) W ] n n( k n( k ) ) n n X ( k), k Attntion:X(k) has only k valid valus:k - but: whn k, X (( )) X (), not X ( ), so : DFT[ x ( n)] X (( k)) R ( k) in a strict way. Slid 3 b) DFT of squnc s ral and imaginary part Suppos x( n) ' s ral and imaginary part ar x ( n) and jx ( n), thn : x( n) x ( n) jx ( n) r i xr( n) [ x( n) x ( n)], jxi( n) [ x( n) x ( n)] x ( n) and jx ( n)'s DFT ar X ( k) and X ( k), thn: r i o X ( k) DFT[ xr( n)] DFT[ x( n) x ( n)] [ X ( k) X ( k)] X o( k) DFT[ jxi( n)] DFT[ x( n) x ( n)] [ X ( k) X ( k)] X ( k) X ( k) X ( k) o r i Slid 4 Analysis of X ( k) and X ( k)'s symmtric o Q X ( k) [ X ( k) X ( k)] X ( k) [ X ( k) X ( k)] [ X ( k) X ( k)] thn : X ( k) X ( k) X (k) is vn componnts of X(k), X (k) is conjugat symmtric; that is ral part is qual, imaginary part is opposit. thn : Xo( k) X o( k) X o(k) is odd componnts of X(k), X o(k) is conjugat asymmtric; that is ral part is opposit, imaginary part is qual.
Slid 5 Discrt Fourir Transform (DFT) Th DFT provids uniformly spacd sampls of th Discrt-Tim Fourir Transform (DTFT) DFT dfinition: IDFT dfinition: X[ k] n x[ n] x[ n] nk j DFT Rquirs complx multiplis and (-) complx additions n j X[ k] nk Slid 6 Fastr DFT computation? Tak advantag of th symmtry and priodicity of th complx xponntial (lt W = -j/ ) k[ n] kn kn * symmtry: W W ( W ) kn k[ n ] [ k ] n priodicity: W W W ot that two lngth / DFTs tak lss computation than on lngth DFT: (/) < Algorithms that xploit computational savings ar collctivly calld Fast Fourir Transforms Slid 7 Dcimation-in-Tim Algorithm Considr xprssing DFT with vn and odd input sampls: X[ k] n n vn r r x[ n] W nk x[ n] W nk rk x[r]( W ) W x[r] W rk / n odd W x[ n] W k k r r nk x[r ]( W ) x[r ] W rk / rk
Slid 8 DIT Algorithm (cont.) Rsult is th sum of two / lngth DFTs X[ k] G [ k] k W H[ k] of / DFT vn sampls / DFT oddsampls Thn rpat dcomposition of / to /4 DFTs, tc. x[,,4,6] x[,3,5,7] / DFT / DFT...7 W of X[ 7] Slid 9 BIT REVERSAL PERMUTATIO Bfor th in-plac implmntation of th DIT FFT algorithm can b don, it is ncssarily to first shuffl th squnc x(n) according to this prmutation. Slid 3 8-point DFT Diagram x[,4,,6,,5,3,7] X[ 7] W W W W W W W W W W W 3 W ECE4 Spring 3 FFT Intro R. C. Mahr 3
Slid 3 Thr stags in th computation of an = 8-point DFT Slid 3 DECIMATIO I FREQUECY (DIF) Flow graph for 8-point dcimation in frquncy