JTU World JTU World DSP OTES PREPARED BY Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World DIGITAL SIGAL PROCESSIG A signal is dfind as any physical quantity that varis with tim, spac or anothr indpndnt variabl. A systm is dfind as a physical dvic that prforms an opration on a signal. Systm is charactrizd by th typ of opration that prforms on th signal. Such oprations ar rfrrd to as signal procssing. Advantags of DSP. A digital programmabl systm allows flxibility in rconfiguring th digital signal procssing oprations by changing th program. In analog rdsign of hardwar is rquird.. In digital accuracy dpnds on word lngth, floating Vs fixd point arithmtic tc. In analog dpnds on componnts. 3. Can b stord on disk.. It is vry difficult to prform prcis mathmatical oprations on signals in analog form but ths oprations can b routinly implmntd on a digital computr using softwar. 5. Chapr to implmnt. 6. Small siz. 7. Svral filtrs nd svral boards in analog, whras in digital sam DSP procssor is usd for many filtrs. Disadvantags of DSP. Whn analog signal is changing vry fast, it is difficult to convrt digital form.(byond KHz rang). w=/ Sampling rat. 3. Finit word lngth problms.. Whn th signal is wak, within a fw tnths of millivolts, w cannot amplify th signal aftr it is digitizd. 5. DSP hardwar is mor xpnsiv than gnral purpos microprocssors & micro controllrs. Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World 6. Ddicatd DSP can do bttr than gnral purpos DSP. Applications of DSP. Filtring.. Spch synthsis in which whit nois (all frquncy componnts prsnt to th sam lvl) is filtrd on a slctiv frquncy basis in ordr to gt an audio signal. 3. Spch comprssion and xpansion for us in radio voic communication.. Spch rcognition. 5. Signal analysis. 6. Imag procssing: filtring, dg ffcts, nhancmnt. 7. PCM usd in tlphon communication. 8. High spd MODEM data communication using puls modulation systms such as FSK, QAM tc. MODEM transmits high spd (-9 bits pr scond) ovr a band limitd (3- KHz) analog tlphon wir lin. 9. Wav form gnration. Classification of Signals I. Basd on Variabls:. f(t)=5t : singl variabl. f(x,y)=x+3y : two variabls 3. S = A Sin(wt) : ral valud signal. S = A jwt : A Cos(wt)+j A Sin(wt) : Complx valud signal 5. S (t)= S( t) S( t) S3( t) : Multichannl signal Ex: du to arth quak, ground acclration rcordr 6. I(x,y,t)= Ir( x, y, t) Ig( x, y, t) Ib( x, y, t) II. Basd on Rprsntation: multidimnsional 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World III. Basd on duration.. right sidd: x(n)= for n<. lft sidd :x(n)= for n> 3. causal : x(n)= for n<. Anti causal : x(n)= for n 5. on causal : x(n)= for n > IV. Basd on th Shap.. (n)= n = n=. u (n) = n = n< Arbitrary squnc can b rprsntd as a sum of scald, dlayd impulss. Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World P (n) = a -3 (n+3) +a (u-) +a (u-) +a 7 (u-7) Or k x(n) = x( k) ( n k) u(n) = (k) n k = (n) + (n-)+ (n-).. k = ( n k) 3.Discrt puls signals. Rct (n/) = n = ls whr. 5.Tri (n/) = - n / n = ls whr.. Sinc (n/)= Sa(n /) = Sin(n /) / (n /), Sinc()= Sinc (n/) = at n=k, k=, Sinc (n) = (n) for =; (Sin (n ) / n == (n)) 6.Exponntial Squnc x (n) = A n If A & ar ral numbrs, thn th squnc is ral. If < < and A is +v, thn squnc valus ar +v and dcrass with incrasing n. For -< <, th squnc valus altrnat in sign but again dcrass in magnitud with incrasing n. If >, thn th squncs grows in magnitud as n incrass. 7.Sinusoidal Squnc x(n) = A Cos(w o n+ ) for all n 8.Complx xponntial squnc 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World If = jwo A = A j x(n) = A j n jwon = A n Cos(w o n+ ) + j A n Sin(w o n+ ) If >, th squnc oscillats with xponntially growing nvlop. If <, th squnc oscillats with xponntially dcrasing nvlop. So whn discussing complx xponntial signals of th form x(n)= A jwon or ral sinusoidal signals of th form x(n)= A Cos(w o n+ ), w nd only considr frquncis in a frquncy intrnal of lngth such as < Wo < or Wo<. VI. VII. V. Dtrministic (x (t) = t x (t) = A Sin(wt)) & on-dtrministic Signals. (Ex: Thrmal nois.) Priodic & non priodic basd on rptition. Powr & Enrgy Signals Enrgy signal: E = finit, P= Signal with finit nrgy is calld nrgy signal. Enrgy signal hav zro signal powr, sinc avraging finit nrgy ovr infinit tim. All tim limitd signals of finit amplitud ar nrgy signals. as t VIII. Ex: on sidd or two sidd dcaying. Dampd xponntials, dampd sinusoidal. x(t) is an nrgy signal if it is finit valud and x (t) dcays to zro fastn than t. Powr signal: E =, P, P Ex: All priodic wavforms ithr nrgy nor powr: E=, P= Ex: / t t E=, P=, Ex: t n Basd on Symmtry. Evn x(n)=x (n)+x o (n). Odd x(-n)=x (-n)+x o (-n) 3. Hiddn x(-n)=x (n)-x o (n). Half-wav symmtry. x (n)= [x(n)+x(-n)] 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x o (n)= [x(n)-x(-n)] Signal Classification by duration & Ara. a. Finit duration: tim limitd. b. Smi-infinit xtnt: right sidd, if thy ar zro for t < whr = finit c. Lft sidd: zro for t > Picwis continuous: possss diffrnt xprssions ovr diffrnt intrvals. Continuous: dfind by singl xprssions for all tim. x(t) = sin(t) Priodic: x p (t) = x p (t nt) For priodic signals P = T dt x( t) T X rms = P For non priodic P = Lt T x( t) dt To To Xavg = Lt x( t) dt x(t) = A cos( f o t + ) P=.5 A j( fo t + ) x(t) = A P=A 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World E= A b E = A b E = 3 A b Q. - t dt = Q. Ex = A.5T + (-A).5T =.5 A T 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Px =.5 A Q. Ey = [ 3 A.5T] = 3 A T Py = 3 A x(t) = A jwt is priodic Px = T x( t) T dt = A x(t -6 ): comprssd by and shiftd right by 3 OR shiftd by 6 and comprssd by. x(-t): fold x(t) & shift right by OR shift right and fold. x(.5t +.5) Advanc by.5 & strtchd by OR strtchd by & advanc by. y (t) = x [- ; = /3 ( t ) t ] = x[ 3 3 ] x( t + ) ; 5 + =-; - + = => = -/3 3 Ara of symmtric signals ovr symmtric limits (-, ) Odd symmtry: x (t) dt = Evn symmtry: x (t) dt = X (t) +Y (t): vn symmtry. x (t) dt 9 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X (t) Y (t): vn symmtry. Xo (t) +Yo (t): odd symmtry. Xo (t) Xo (t): vn symmtry. X (t) +Yo (t): no symmtry. X (t) Yo (t): odd symmtry. X (n)= [x(n)+x(-n)] Xo (n) = [x (n)-x (-n)] Ara of half-wav symmtry signal always zro. Half wav symmtry applicabl only for priodic signal. F = GCD ( f,f ) T = LCM (T, T) Y(t) = x (t) + x (t) P y = P x +P x Y(t)rms = Py U() =.5 is calld as Havisid unit stp. X(t) = Sin(t) Sin( t) =.5 cos (- )t.5 cos (+ ) t W =- W =+ almost priodic OR non priodic. P x =.5[.5 +.5 ] =.5 W Ara of any sinc or Sinc quals ara of triangl ABC inscribd within th main lob. Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Evn though th sinc function is squar intgrabl ( an nrgy signal), it is not absolutly intgrabl( bcaus it dos not dcay to zro fastr than t ) (t) = t = t= ( ) d = An impuls is a tall narrow spik with finit ara and infinit nrgy. Th ara of impuls A (t) quals A and is calld its strngth. How vr its hight at t= is. -t (t) = (t) = (t) -t u(t) [ [t- ]] = ( t ) I = cos( t) (t ) dt = cos( t).5 ( t.5) dt =.5 cos( t) at t=-.5 = -.5 x(t) = x(t) k (t-kt s ) = k x(kt s ) (t-kt s) Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x(t) is not priodic. Th doublt (t) = t = undfind t= '( t) dt (-t) = - (t) thn Odd function. [ [t- ]] = ( t ) Diffrntiating on both sids [ [t- ]] = '( t ) With =- (-t) = - (t) d dt Or d dt [ x( t) ( t )] = x (t) (t- ) + x (t) (t- ) = x ( ) (t- ) + x (t) (t- )----------- d [ x( t) ( t )] = [ x( ) ( t )] = x ( ) (t- ) ----------- dt = x ( ) (t- ) + x (t) (t- ) = x ( ) (t- ) Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x (t) (t- ) = x ( ) (t- ) - x ( ) (t- ) x (t) (t- ) dt = = - x ( ) = - x ( ) x ( ) (t- ) dt - x ( ) (t- ) dt Highr drivativs of (t) oby n (t) = (-) n n (t) ar altrnatly odd and vn, and possss zro ara. All ar liminating forms of th sam squnc that gnrat impulss, providd thir ordinary drivativs xits. on ar absolutly intgrabl. Th impuls is uniqu in bing th only absolutly intgrabl function from among all its drivativs and intgrals (stp, ramp tc) What dos th signal x(t) = -t (t) dscrib? x(t) = (t) (-) (t) = (t) + (t) I = [( t 3) (t )] 8cos( t) '( t.5)] dt d =.5 (t-3) t - 8 [cos t] t. 5 dt = 3.37 Answr. Opration on Signals:. Shifting. x(n) shift right or dlay = x(n-m) x(n) shift lft or advanc = x(n+m). Tim rvrsal or fold. x(-n+) is x(-n) dlayd by two sampls. x(-n-) is x(-n) advancd by two sampls. Or x(n) is right shift x(n-), thn fold x(-n-) x(n) fold x(-n) shift lft x(-(n+)) = x(-n-) Ex: x(n) =, 3,, 5, 6, 7. Find. y(n)=x(n-3). x(n+) 3. x(-n). x(-n+) 5. x(-n-). y(n)= x(n-3) = {,,3,,5,6,7} shift x(n) right 3 units. 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World. x(n+) = {,3,,5, 6,7} shift x(n) lft units. 3. x(-n) = { 7,6,5,,3,} fold x(n) about n=.. x(-n+) = { 7,6, 5,,3,} fold x(n), dlay by. 5. x(-n-) = { 7,6,5,,3,} fold x(n), advancd by. 3. a. Dcimation. Suppos x(n) corrsponds to an analog signal x(t) sampld at intrvals Ts. Th signal y(n) = x(n) thn corrsponds to th comprssd signal x(t) sampld at Ts and contains only altrnat sampls of x(n)( corrsponding to x(), x(), x() ). W can also obtain dirctly from x(t) (not in comprssd vrsion). If w sampl it at intrvals Ts (or at a sampling rat Fs = Ts ). This mans a two fold rduction in th sampling rat. Dcimation by a factor is quivalnt to sampling x(t) at intrvals Ts and implis an -fold rduction in th sampling rat. b. Intrpolation. y(n) = x(n/) corrsponds to x(t) sampld at Ts/ and has twic th lngth of x(n) with on nw sampl btwn adjacnt sampls of x(n). Th nw sampl valu as for Zro intrpolation. Th nw sampl constant = prvious valu for stp intrpolation. Th nw sampl avrag of adjacnt sampls for linar intrpolation. Intrpolation by a factor of is quivalnt to sampling x(t) at intrvals Ts/ and implis an -fold incras in both th sampling rat and th signal lngth. Ex: Dcimation Stp intrpolation {,, 6,, 8} {, 6, 8} {,, 6, 6, 8, 8} n n n n/ Stp intrpolation Dcimation {,, 6,, 8} {,,,,6, 6,,,8, 8} {,, 6,, 8} Ex: x(n) = {,, 5, -} n n/ n n Sinc Dcimation is indd th invrs of intrpolation, but th convrs is not ncssarily tru. First Intrpolation & Dcimation. Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x(n/3) = {,,,,,,5,,,-,,} Zro intrpolation. = {,,,,,,5,5,5,-,-,-} Stp intrpolation. = {, 3, 3 5,, 3,,5,3,,-, - 3,- 3 } Linar intrpolation.. Fractional Dlays. M ( n M ) It rquirs intrpolation (), shift (M) and Dcimation (n): x (n - ) = x ( ) n x(n) = {,, 6, 8}, find y(n)=x(n-.5) = x ( ) g(n) = x (n/) = {,,,, 6, 6, 8,8} for stp intrpolation. h(n) =g(n-) = x( n ) = {,,,, 6, 6,8,8} n y(n) = h(n) = x(n-.5) = x( ) = {,, 6, 8} OR g(n) = x(n/) = {,3,,5, 6,7,8,} linar intrpolation. h(n) = g(n-) = {,3,, 5, 6, 7,8,} g (n) = h(n)={3,5,7,} Classification of Systms. a. Static systms or mmory lss systm. (on Linar / Stabl) Ex. y(n) = a x (n) = n x(n) + b x 3 (n) = [x(n)] = a(n-) x(n) y(n) = [x(n), n] If its o/p at vry valu of n dpnds only on th input x(n) at th sam valu of n Do not includ dlay lmnts. Similarly to combinational circuits. b. Dynamic systms or mmory. If its o/p at vry valu of n dpnds on th o/p till (n-) and i/p at th sam valu of n or prvious valu of n. Ex. y(n) = x(n) + 3 x(n-) 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World = x(n) - x(n-) + 5 y(n-) Similar to squntial circuit.. Idal dlay systm. (Stabl, linar, mmory lss if nd=) Ex. y (n) = x(n-nd) nd is fixd = +v intgr. 3. Moving avrag systm. (LTIV,Stabl) y(n) = / (m +m +) m k m x( n k) This systm computs th n th sampl of th o/p squnc as th avrag of (m +m +) sampls of input squnc around th n th sampl. If M=; M=5 y(7) = /6 [ x (7 k) ] 5 k = /6 [x(7) + x(6) + x(5) + x() + x(3) + x()] y(8) = /6 [x(8) + x(7) + x(6) + x(5) + x() + x(3)] So to comput y (8), both dottd lins would mov on sampl to right.. Accumulator. ( Linar, Unstabl ) y(n) = x ( k) n k n = x ( k) + x(n) k = y(n-) + x(n) 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Ex: x(n) = {,3,,,,,,3,,.} y(n) = {,3,5,6,6,7,9,, } O/p at th n th sampl dpnds on th i/p s till n th sampl x(n) = n u(n) ; givn y(-)=. i.. initially rlaxd. y(n) = x ( k) + x( k) k k n n = y(-) + x( k) = + n = k 5. Linar Systms. n k n( n ) If y (n) & y (n) ar th rsponss of a systm whn x (n) & x (n) ar th rspctiv inputs, thn th systm is linar if and only if [ x( n) x( n)] = [ x( n)] + [ x( n)] = y (n) + y (n) (Additiv proprty) [ ax( n)] = a [ x( n)] = a y(n) (Scaling or Homognity) Th two proprtis can b combind into principl of suprposition statd as [ ax( n) bx( n)] = a [ x( n)] Othrwis non linar systm. 6. Tim invariant systm. + b [ x( n)] Is on for which a tim shift or dlay of input squnc causs a corrsponding shift in th o/p squnc. y(n-k) = [ x( n k)] TIV 7. Causality. TV A systm is causal if for vry choic of n o th o/p squnc valu at indx n= n o dpnds only on th input squnc valus for n n o. y(n) = x(n) + x(n-) causal. y(n) = x(n) + x(n+) + x(n-) non causal. 8. Stability. 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World For vry boundd input x(n) B x < for all n, thr xists a fixd +v finit valu By such that y(n) B y <. PROPERTIES OF LTI SYSTEM.. x(n) = x ( k) ( n k) k y(n) = [ x ( k) ( n k) ] for linar k k x ( k) [ (n-k)] for tim invariant k x ( k) h( n k) = x(n) * h(n) Thrfor o/p of any LTI systm is convolution of i/p and impuls rspons. y(n o ) = h ( k) x( no k) k = h ( k) x( no k) k + k h( k) x( no k) = h(-) x(n +) + h(-) x(n +).+h() x(n ) + h() x(n -) +. y(n) is causal squnc if h(n) = n< y(n) is anti causal squnc if h(n) = n y(n) is non causal squnc if h(n) = n > Thrfor causal systm y(n) = k h( k) x( n k) If i/p is also causal y(n) = h( k) x( n k) n k. Convolution opration is commutativ. x(n) * h(n) = h(n) * x(n) 3. Convolution opration is distributiv ovr additiv. x(n) * [h (n) + h (n)] = x(n) * h (n) + x(n) * h (n). Convolution proprty is associativ. x(n) * h (n) * h (n) = [x(n) * h (n)] * h (n) 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World 5 y(n) = h * w(n) = h(n)*h(n)*x(n) = h3(n)*x(n) 6 h (n) = h (n) + h (n) 7 LTI systms ar stabl if and only if impuls rspons is absolutly summabl. y (n) = h ( k) x( n k) k k Sinc x (n) is boundd x(n) h (k) x( n k) b x < y(n) B x k h(k) S= k h (k) is ncssary & sufficint condition for stability. 8 (n) * x(n) = x(n) 9 Convolution yilds th zro stat rspons of an LTI systm. Th rspons of LTI systm to priodic signals is also priodic with idntical priod. y(n) = h (n) * x(n) 9 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World = h ( k) x( n k) k y (n+) = h ( k) x( n k ) k put n-k = m = h ( n m) x( m ) m = h ( n m) x( m) m m=k = h ( n k) x( k) = y(n) (Ans) k Q. y (n)-. y(n-) =x (n). Find causal impuls rspons? h(n)= n<. h(n) =. h(n-) + (n) h() =. h(-) + () = h() =. h() =. h() =. h(n) =. n for n Q. y(n)-. y(n-) = x(n). find th anti-causal impuls rspons? h(n)= for n h(n-) =.5 [h(n)- (n)] h(-) =.5 [h()- () ] = -.5 h(-) = -.5... h(n) = -.5 n valid for n - Q. x(n)={,,3} y(n)={3,} Obtain diffrnc quation from i/p & o/p information y(n) + y(n-) + 3 y(n-) = 3 x(n) + x(n-) (Ans) Q. x(n) = {,,}, y(n)= x(n)-.5x(n-). Find th diffrnc quation of th invrs systm. Sktch th ralization of ach systm and find th output of ach systm. Solution: Th original systm is y(n)=x(n)-.5 x(n-) Th invrs systm is x(n)= y(n)-.5 y(n-) y (n) = x (n).5 x(n-) Y (z) = X (z) [-.5Z - ] Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Y ( z) =-.5 Z - Systm X ( z) Invrs Systm y (n).5 y(n-) =x(n) Y (z) [-.5 Z - ] = X (z) Y ( z) [-.5 Z - ] - X ( z) g (n) = (n) - (n-) + (n-) - (n-) = (n) + (n-) - (n-) y (n) =.5 y(n-) + (n) + (n-) (n-) y () =.5y(-) + () = y() = y() =.5 y() - () = y(n) = {, } sam as i/p. on Rcursiv filtrs Rcursiv filtrs y(n) = k a k x(n-k) for causal systm = k a k x(n-k) For causal i/p squnc y(n) = k a k x(n-k) k b k y(n-k) Prsnt rspons is a function of th prsnt and past valus of th xcitation as wll as th past valus of rspons. It givs IIR o/p but not Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World y(n) = k a k x(n-k) Prsnt rspons dpnds only on prsnt i/p & prvious i/ps but not futur i/ps. It givs FIR o/p. always. y(n) y(n-) = x(n) x(n-3) Q. y(n) = 3 [x (n+) + x (n) + x (n-)] Find th givn systm is stabl or not? Lt x(n) = (n) h(n) = 3 [ (n+) + (n) + (n-)] h() = 3 h(-) = 3 h() = 3 S= h (n) < thrfor Stabl. Q. y(n) = a y(n-) + x(n) givn y(-) = Lt x(n) = (n) h(n) = y(n) = a y(n-) + (n) h() = a y(-) + () = = y() h() = a y() + () = a h() = a y() + () = a....... h(n) = a n u(n) stabl if a<. y(n-) = [ y(n) x(n)] a y(n) = a [ y(n+) x(n+)] Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World y(-) = a [ y() x()]= y(-) = Q. y(n) = n y(n-) + x(n) for n = othrwis. Find whthr givn systm is tim variant or not? Lt x(n) = (n) h () = y(-) + () = h() = ½ y() + () = ½ h() = /6 h(3) = / if x(n) = (n-) y(n) = h(n-) h(n-) = y(n) = n h(n-) + (n-) n= h(-) = y() = x + = n= h() = y() = ½ x + ()= n= h() = y() = /3 x + = /3 h() = / h (n, ) h(n,) TV Q. y (n) = n x(n) Tim varying Q. y (n) = [x (n+) + x (n) + x (n-)] Linar 3 Q. y (n) = x (n-) + x(n-) TIV Q. y (n) = 7 x (n-) non linar Q. y (n) = x (n) non linar Q. y (n) = n x (n+) linar Q. y (n) = x (n ) linar Q. y (n) = x(n) non linar Q. y (n) = x(n) x (n) non linar, TIV 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World (If th roots of charactristics quation ar a magnitud lss than unity. It is a ncssary & sufficint condition) on rcursiv systm, or FIR filtr ar always stabl. Q. y (n) + y (n) = x(n) x(n-) non linar, TIV Q. y (n) - y (n-) = x(n) x (n) non linar, TIV Q. y (n) + y (n) y (n) = x (n) non linar, TIV Q. y (n+) y (n) = x (n+) is causal Q. y (n) - y (n-) = x (n) causal Q. y (n) - y (n-) = x (n+) non causal Q. y (n+) y (n) = x (n+) non causal Q. y (n-) = 3 x (n-) is static or Instantanous. Q. y (n) = 3 x (n-) dynamic Q. y (n+) + y (n+3) = x (n+) causal & dynamic Q. y (n) = x ( n ) If = causal, static < causal, dynamic > non causal, dynamic TV Q. y (n) = (n+) x (n) is causal & static but TV. Q. y (n) = x (-n) TV Solution of linar constant-co-fficint diffrnc quation Q. y(n)-3 y (n-) y(n-) = dtrmin zro-input rspons of th systm; Givn y(-) = & y(-) =5 Lt solution to th homognous quation b y h (n) = n n - 3 n- - n- = n- [ - 3 - ] = = -, y h (n) = C n + C n y() = 3y(-) + y(-) = 5 = C (-) n + C n Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World C + C =5 y () = 3y () + y (-) = 65 -C +C = 65 Solv: C = - & C =6 y(n) = (-) n+ + n+ (Ans) If it contain multipl roots y h (n) = C n n + C n + C 3 n n n or [C + nc + n C 3.] Q. Dtrmin th particular solution of y(n) + a y(n-) =x(n) x(n) = u(n) Lt y p (n) = k u(n) k u(n) + a k u(n-) =u(n) To dtrmin th valu of k, w must valuat this quation for any n k + a k = k = a y p (n) = a u(n) Ans x(n). A. Am n 3. An m. A Cosw o n or A Sinw o n y p (n) K Km n K o n m + K n m- +. K m K Cosw o n + K Sinw o n Q. y(n) = 6 5 y(n-) - 6 y(n-) + x(n) x(n) = n n Lt y p (n) = K n K n u(n) = 6 5 K n- u(n-) - 6 K n- u(n-) + n u(n) For n K = 6 5 (K) - 6 K + Solv for K=8/5 y p (n) = 5 8 n Ans Q. y(n) 3 y(n-) - y(n-) = x(n) + x(n-) Find th h(n) for rcursiv systm. 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World W know that y h (n) = C (-) n + C n for n= y p (n) = whn x(n) = (n) y() - 3y(-) - y(-) = () + (-) y() = y() = 3 y() + = 5 C + C = -C + C =5 Solving C = 5 ; C = 5 6 h(n) = [ 5 (-) n + 5 6 n ] u(n) Ans h(n) 3 h(n-) - h(n-) = (n) + (n-) h() = h() =3 h() + = 5 plot for h(n) in both th mthods ar sam. OR Q. y(n).5 y(n-) = 5 cos.5n n with y(-) = y h (n) = n n.5 n- = n- [ -.5] = =.5 y h (n) = C (.5) n y p (n) = K cos.5n + K sin.5n y p (n-) = K cos.5(n-) + K sin.5(n-) = - K sin.5n - K cos.5n y p (n) -.5 y p (n-) = 5 cos.5 n K +.5 K = 5 = (K +.5 K ) cos.5 n -(.5 K K ) sin.5n.5 K K = Solving w gt: K = & K = y p (n) = cos.5 n + sin.5n 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Th final rspons y (n) = C (.5) n + cos.5 n + sin.5n with y(-) = = C- i.. C=3 y (n) = 3 (.5) n + cos.5 n + sin.5n for n Concpt of frquncy in continuous-tim and discrt-tim. ) x a (t) = A Cos ( t) x (nts) = A Cos ( nts) = A Cos (wn) w = Ts = rad / sc w = rad / Sampl F = cycls / sc f = cycls / Sampl ) A Discrt- tim sinusoid is priodic only of its f is a Rational numbr. x (n+) = x (n) Cos f (n+) = Cos f n K f = K => f = Ex: A Cos ( 6 ) n w = 6 = f f = = Sampls/Cycl ; Fs= Sampling Frquncy; Ts = Sampling Priod Q. Cos (.5n) is not priodic 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Q. x (n) = 5 Sin (n) f = => f = Q. x (n) = 5 Cos (6 n) f = 6 => f = 3 Q. x (n) = 5 Cos 6n 35 6 3 f = => f = 35 35 on-priodic = for K=3 Priodic for =35 & K=3 Priodic Q. x (n) = Sin (. n) f =. => f =. for = & K= Priodic Q. x (n) = Cos (3 n) for = Priodic f o = GCD (f, f ) & T = LCM (T, T ) ------- For Analog/digital signal [Complx xponntial and sinusoidal squncs ar not ncssarily priodic in n with priod ( Wo ) and dpnding on Wo, may not b priodic at all] = fundamntal priod of a priodic sinusoidal. w = or - 3. Th highst rat of oscillations in a discrt tim sinusoid is obtaind whn 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Discrt-tim sinusoidal signals with frquncis that ar sparatd by an intgral multipl of ar Idntical.. - Fs Fs F - Fs F Fs - Ts Ts - Ts Thrfor - w 5. Incrasing th frquncy of a discrt- tim sinusoid dos not ncssarily dcras th priod of th signal. n x (n) = Cos ( ) =8 3 n x (n) = Cos ( ) =6 3/8 > / 8 f = 3 /8 => f = 6 3 6. If analog signal frquncy = F = sampls/sc = Hz thn digital frquncy f = Ts W = T s f = F T s => f = F = ; f = / 9 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World F = 8 ; T = 8 ; f = 8 =8 7. Discrt-tim sinusoids ar always priodic in frquncy. Q. Th signal x (t) = Cos ( t) + Sin (6 t) is sampld at 75Hz. What is th common priod of th sampld signal x (n), and how many full priods of x (t) dos it tak to obtain on priod of x(n)? F = Hz f = 75 F = 3Hz K f = 5 3 75 5 K Th common priod is thus =LCM (, ) = LCM (5, 5) = 5 Th fundamntal frquncy F o of x (t) is GCD (, 3) = Hz And fundamntal priod T = Sinc =5 sampl ---------- sc 75. s Fo 5 5 sampl -----------? =>.S 75 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World So it taks two full priods of x (t) to obtain on priod of x (n) or GCD (K, K ) = GCD (, ) = Frquncy Domain Rprsntation of discrt-tim signals and systms For LTI systms w know that a rprsntation of th input squnc as a wightd sum of dlayd impulss lads to a rprsntation of th output as a wightd sum of dlayd rsponss. Lt x (n) = jwn y (n) = h (n) * x (n) = h ( k) x( n k) h( k) k = jwn h ( k) -jwk k k jw (n-k) Lt H ( jw ) = h ( k) -jwk is th frquncy domain rprsntation of th systm. k y (n) = H ( jw ) jwn jwn = ign function of th systm. H ( jw ) = ign valu Q. Find th frquncy rspons of st ordr systm y (n) = x (n) + a y (n-) (a<) Lt x (n) = jwn y p (n) = C jwn C jwn = jwn jw (n-) + a C C jwn [-a -jw ] = jwn C = [ a jw ] Thrfor H ( jw ) = [ a jw ] = a(cos w j sin w) H ( jw ) = a cos w a asinw H( jw ) Tan ( ) acosw 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Q. Frquncy rspons of nd ordr systm y(n) = x(n) - y( n ) jwn x (n) = y p ( n) c jwn c jwn = jwn c jwn jw( n ) - c jw (+ ) = jwn c = jw 6Cosw c 5 Cosw Sinw c tan Cosw 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World UIT - II Continuous Tim o t = o nt s = w o n Priodic f (t) = k c k on priodic C k = T Ts T = Ts T jkot f ( t) t = n Ts : dt = Ts on-priodic f(t) = F( w) j t d jkot x( n) on-priodic F(w) = f ( t) jt dt dt nts jk T Priodic x p (n) = DTFS n Discrt Tim k c Priodic C k = x p k= to - ( n) on Priodic x(n) = X ( w) jwn dw Priodic X(w) = X(w) = FT of DTS n jk n k j nk x( n) jwn 33 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Enrgy and Powr E = n * x ( n) x( n) x ( n) n * jwn ( X ( w) dw x n) = n Thrfor: E = P = n = jwn = X w * ( ) x( n) dw n X = X ( w) dw x( n) * ( w) X ( w) dw X ( w) dw -------- Parsval s Thorm Lt x( n) for non priodic signal n = = n x( n) for priodic Signal j nk * * x( n) x ( n) x( n) Ck n n k * = Ck k n x( n) j nk Thrfor P = Ck k E = k Ck Ex: Unit stp 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World P = Lt n u ( n) = Lt E = Powr Signal Ex: x (n) = A jwon P = Lt n A jwon Lt = A [...] = Lt A ( ) A it is Powr Signal and E = Ex: x (n) = n u(n) nithr nrgy nor powr signal Ex: x (n) = 3 (.5) n n Ex: E = P = n x ( n) 9(.5) n x (n) = 6 Cos 3 n x n ( n) n [36 9 J.5 not: [ n n ] whos priod is = x (n) = { 6,, 6, } 36] 8W n j Ex: x (n) = 6 whos priod is = P = 3 n x( n) [36 36 36 36] 36Watts 35 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World DISCRETE COVOLUTIO It is a mthod of finding zro input rspons of linar Tim Invariant systm. Ex: x(n) = u(n) h(n) = u(n) y(n) = k u ( k) u( n k) u(k) = k< u(n-k) = k>n n k n u( k) u( n k) = k = (n+) u(n) = r(n+) Q. x(n) = a n u(n) and h(n) = a n u(n) a< find y(n) 36 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World n y(n) = k a k a n-k = a n (n+) u(n) Q. x(n) = u(n) and h(n) = n u(n) < find y(n) y(n) = k k u(k) u(n-k) = n k k = (- n+ ) / (- ) Th convolution of th lft sidd signals is also lft sidd and th convolution of two right sidd also right sidd. Q. x(n) = rct ( h(n) = rct ( y(n) n n ) = n = ls whr ) = x(n) * h(n) = [u (n+) u (n--)] * [u (n+) u (n--)] = u (n+) * [u (n+) u (n--)] u (n--)* [u (n+) u (n--)] = u (n+) * u (n+) u (n+) * u (n--)] + u (n--) * u (n--) = r(n++) r(n) + r(n--) = (+) Tri ( n ) Tri ( n ) = - n for n = lswhr. 37 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Q. x(n) = {,-,3} h(n) = {,,,3} Graphically Fold-shift-multiply-sum y(n) = 3 6 - - - - -3 3 3 6 6 9 y(n) = {,3,5,,3,9} Q. x(n) = {,,3} h(n) = {,5,,} 5 8 6 5 3 6 5 y(n) = { 8,,,3,,} ot that convolution starts at n=-3 38 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Q) h(n): 5 x(n): 3 8 6 5 6 5 y(n): 8 3 Q. Convolution by sliding stp mthod: h(n) =, 5,, ; x(n)=,, 3 i) 5 ii) 5 3 3 y() = 8 y() = + = iii) 5 iv) 5 3 3 6 5 y() = 5 6 y(3) = 3 v) 5 Vi) 5 3 3 y() = y(5) = If w insrt zros btwn adjacnt sampls of ach signal to b convolvd, thir convolution corrsponding to th original convolution squnc with zros insrtd btwn its adjacnt sampls. 39 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Q. h(n) =, 5,, ; x(n)=,, 3 X(z) = z 3 +5z + ; X(z) = z +z+3 Thir product Y(z) = 8z 5 +z +z 3 +3z +z+ y(n) = 8,,,3,, h(n) =,, 5,,,, ; x(n) =,,,, 3 H(z) = z 6 +5z + ; X(z) = z +z +3 Y(z) = 8z +z 6 +3z +z + y(n) = { 8,,,,,,3,,,,} Q. Comput th linar convolution of h(n)={,,} and x(n) = {, -,,,, -,, 3, } using ovrlap-add and ovrlap-sav mthod. h (n): x (n): - - 3 x (n): - x (n): - x 3 (n): 3 y (n) = (h (n) * x (n)) 3 y (n) = - y 3 (n) = 5 8 5 y(n) = { 6 8 5 } OVER LAP and SAVE mthod h (n): ( =3) x (n): - ( 3 + -) = 5 x (n): - 3 x 3 (n): 3 y (n) = 6 5 y (n) = 7 3 y 3 (n) = 5 8 5 y(n) = { 6 8 5 } Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Q. Dtrmin th spctra of th signals a. x(n) = Cos n w o = f o = Discrt Fourir Sris is not rational numbr Signal is not priodic. Its spctra contnt consists of th singl frquncy b. x (n) = Cos n 3 f o = 6 =6 aftr xpansion x(n)={,.5,-.5,-,-.5,.5} C k = 6 5 n j nk 6 x( n) k= to 5 C k = 5 j k j k j k j j k x x 3 3 3 3 () () x() x(3) x() x(5 ) 6 6 For k= Co = x() x() x() x(3) x() x(5) Similarly = K= C =.5, C = = C 3 = C, C 5 =.5 k Or x (n) = Cos 3 j n 6 n + j n 6 5 = k C j kn 6 k Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World = C o +C j n 6 +C j n 6 + C 3 6 j n 6 +C 8 j n 6 +C 5 j n 6 By comparison C = Sinc j n 6 = 56 j n 6 = n j 6 C 5 c. x (n) = {,,,} C k = 3 n x( n) nk j k=,,, 3 k j = c ; j c C o & C = c & C = ; c & C undfind c ; c j 3 c 3 & C 3 = Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World PROPERTIES OF DFS. Linarity ~ x ( n) C k DFS ~ DFSx ( n) C k a~ x ( ~ n) bx ( n) ac k bc k DFS. Tim Shifting ~ mk j x( n m) C DFS k 3. Symmtry ~ x * ( n) C * DFS k C k = n ~ x ( n) nk j ~ x ( n) C DFS * * k R~ x n) DFS ( ~ x ( n) ~ x ~ x ( n ) ( n) k C k j nk * DFS C k C k Ck j Im ~ x( n) DFS * ~ x ( n) ~ x ( n) * DFS C k C k Cko ~ n If x ( ) is ral thn ~ x ( n) ~ x ( n) ~ x * ( n) * 3 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World ~ x o ( n) ~ x ( n) ~ x * ( n) ~ x k DFS ( n) Ck C * R Ck ~ x k DFS o ( n) Ck C * j Im Ck Priodic Convolution m ~ x ( m x ( n m) ) ~ DFS k k If x(n) is ral C k R[ C Im[ C C * k k ] R[ Ck ] k ] Im[ C k ] C C C k C k C k C k PROPERTIES OF FT (DTFT). Linarity y (n) = a x (n) + b x (n) Y ( jw ) = a X ( jw ) + b X ( jw ). Priodicity H ( j( w ) ) = H ( jw ) 3. For Complx Squnc Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World h (n) = h R (n) + j h I (n) H ( jw ) = [ h n- R (n) j h I (n) ][Cos(wn)- jsin(wn)] n- [ h R (n) Cos(wn) h I (n)sin(wn) = H R ( jw ) n- [ h I (n) Cos(wn) h R (n)sin(wn) = H I ( jw ) H ( jw jw jw ) = H ( ) jh ( ) H ( R I jw jw jw * jw = H ( ) H ( ) H( ) H ( ) jw R ) tan H H. For Ral Valud Squnc I R ( ( I jw jw ) ) H ( jw ) = n h ( n) jwn From (a) & (b) = ( n) Cos( wn) j n h h( n) Sin( wn) jw jw = ( ) ( ) R I n H jh -------------------- (a) jw H( ) = h ( n) n H R ( jwn = ( n) Cos( wn) j n h h( n) Sin( wn) n jw jw = H ( ) jh ( ) -------------------- (b) jw R ) H R ( jw ) I 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World H I ( jw ) H I ( jw ) Ral part is vn function of w Imaginary part is odd function of w jw * jw H( ) H ( ) jw jw * jw * -jw -jw -jw => H ( ) H ( )H ( ) H ( )H( ) H( ) Magnitud rspons is an vn function of frquncy H ( -jw ) tan - H I( HR ( -jw -jw 5. FT of a dlayd Squnc ) ) tan - H I( HR ( jw jw ) H ( ) Phas rspons is odd function. jw ) FT [h (n-k)] = n Put n-k = m = m h ( n k) h( m) jwn jw( mk ) jwk = 6. Tim Rvrsal x (n) X (w) x (-n) X (-w) m h ) jwm ( m = H ( jw ) jwk F T [x (-n)] = n Put n = m x ( n) jwn jwm x( m) X ( w) 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World 7. Frquncy Shifting x(n) jw o n X (w-w o ) F T [x (n) jw o n ] = n x (n) jw o n -jwn = n x (n) j( ww o ) n = X (w-w o ) 8. a. Convolution x (n) * x (n) X (w) X (w) n [x (n) * x (n) ] -jwn = n Put n-k = m k [ x (k) x (n-k) ] -jwn = n x (k) m -jw (m+k) [x (m)] b. = n x (k) -jwk = X (w) X (w) [X (w) * X (w)] x (n) x (n) 9. Parsvals Thorm m [x (m)] -jwm n x (n) x * (n) = n x (n) j. F T of Evn Symmtric Squnc [X (w) X * (w)] dw dx ( w) dw H ( jw ) = n h (n) -jwn 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World = n Lt n = -m h (n) -jwn + h () + n h (n) -jwn = m h (-m) jwm + h () + n Lt h (-m) = h (m) for vn h (n) -jwn frquncy Thrfor = h () + n ; H ( jw ) > ; H ( jw ) <. F T of Odd Symmtric Squnc For odd squnc h () = h (n) Cos (wn) is a ral valud function of H ( jw ) = n h (n) [ -jwn - jwn ] = -j n of frq. and a odd function of w i., H ( jw ) = - H ( jw ) h (n) Sin (wn) H I ( jw ) is a imaginary valud function H ( jw ) = H I ( jw ) for H I ( jw ) > = - H I ( jw ) for H I ( jw ) < jw H ( ) For w ovr which H I ( jw ) > = for w ovr which H I ( jw ) <. x() = X ( w) dw Cntral Co-ordinats 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X () = n 3. Modulation x ( n) Cos (w o n) x (n) X ( w w ) X ( w w ) FOURIER TRASFORM OF DISCRETE TIME SIGALS X (w) = n x (n) -jwn F T xists if n x(n) Th FT of h (n) is calld as Transfr function Ex: h (n) = 3 for - n Sol: = othrwis H ( jw ) = n 3 jwn jw jw = 3 3 = Cos( w) w 3-3 9 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Ex: h (n) = a n u (n) H ( jw ) = n = n ( a n jwn jw n a ) = jw Q. x(n) = n n u(n) < a d n n u(n) j jw dw = jw ( jw ) Hint: n u(n) n Q. x(n) = n n Or n x(n) = n [ u(n) u(n-)] X(w) = Q. x(n) = jwn = n ( jw n ) = jw = n u(n) n- u(n-) Using Shifting Proprty ( jw jw = jw n ) Ans [ jw jw ] two sidd dcaying xponntial x(n) = n u(n) + -n u(-n) - (n) using folding proprty = jw jw = Cosw Q. x (n) = u (n) Sinc u (n) is not absolutly summabl 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World w know that u (t) ( w) Similarly X (w) = jw jw + (w) 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World DFT (Frquncy Domain Sampling) Th Fourir sris dscribs priodic signals by discrt spctra, whr as th DTFT dscribs discrt signals by priodic spctra. Ths rsults ar a consqunc of th fact that sampling on domain inducs priodic xtnsion in th othr. As a rsult, signals that ar both discrt and priodic in on domain ar also priodic and discrt in th othr. This is th basis for th formulation of th DFT. x ( n) jwn Considr apriodic discrt tim signal x (n) with FT X(w) = n Sinc X (w) is priodic with priod, sampl X(w) priodically with quidistanc sampls with spacing w. X k j Kn n x( n) K =,,..- Th summation can b subdividd into an infinit no. of summations, whr ach sum contains k X... n x( n) j Kn n j Kn x( n) + n x( n) j Kn... 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World = l Put n = n-l l nl x( n) j Kn = l = n n l x( n l ) x( n l ) j K ( nl ) j Kn X(k) = n x p (n) j Kn W know that x p (n) = k C k j Kn n= to - C k = n x p (n) j Kn k= to - Thrfor C k = X(k) k= to - IDFT ------------ x p (n) = k X(w). X(k) j Kn n = to - This provids th rconstruction of priodic signal x p (n) from th sampls of spctrum Th spctrum of apriodic discrt tim signal with finit duration L<, can b xactly k rcovrd from its sampls at frquncy W k = Prov: x(n) = x p (n) n -. 53 Downloadd From JTU World (http://jtu World)
JTU World JTU World Using IDFT x (n) = k X(k) j Kn X (w) = n [ k X (k) j Kn ] -jwn = k X (k) [ n If w dfin p(w) jn( w K ) = n ] -jwn = Thrfor: X (w) = k k At w = P () = k And P (w- jw jw = k X (k) P(w- ) = for all othr valus w Sin w Sin ) jw X (w) = k X(k) = k k X( ) Ex: x(n) = a n u(n) <a< k Th spctrum of this signal is sampld at frquncy W k =. k=,..-, dtrmin rconstructd spctra for a =.8 and = 5 & 5. X (w) = jw a X (w k ) = k=,, - a j k 5 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x p (n) = l x( n L) l a a n n l = a a l a nl n - Aliasing ffcts ar ngligibl for =5 If w dfin aliasd finit duration squnc x(n) xˆ ( n) x p ( n) n - = othrwis Xˆ ( w) n xˆ( n) jwn = n x p ( n) jwn = n n a a jwn = a n ( a jw ) n Xˆ ( w) a a a jw jw 55 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Xˆ K a = a jk a a k j K j K = X Although Xˆ ( w) X ( w), th k sampls at W k = ar idntical. Ex: X (w) = Apply IDFT a jw & X (k) = a j k x (n) = j k a nk k j using Taylor sris xpansion = a k r nk kr j j r ( nr) jk = r a r k = xcpt r = n+m x (n) = m a nm n a = m n a a = ( a ) m Th rsult is not qual to x (n), although it approachs x (m) as bcoms. Ex: x (n) = {,,, 3} find X (k) =? 3 X (k) = x( n) k j n 56 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World 3 X () = x ( n) = +++3 = 6 n 3 X () = n x( n) j n X () = - X (3) = --j DFT as a linar transformation = -+j Lt W j X (k) = n nk x( n) W k = to - x (n) = k X ( k) nk W n =, - x() x() Lt x = x( ) W = W W W W W W ( ) X () X () X = X ( ) W W W ( ) ( ) ( )( ) Th point DFT may b xprssd in matrix form as 57 Downloadd From JTU World (http://(http:// )JTU World
58 DFT IDFT X = W x x = W X * X W x. K K W W * W W. K K W W Ex: x (n) = {,,, 3} DFT W = 9 6 3 6 3 W W W W W W W W W = 3 3 W W W W W W W W W = j j j j X = x W = j j j j 3 = j j 6 IDFT x = X W * = j j j j j j 6 = 3 Ans Q. x (n) = { },.5 h (n) = {.5, } JTU World JTU World Downloadd From JTU World (http://(http:// )JTU World )
JTU World JTU World Find y (n) = x (n) h (n) using frquncy domain. Sinc y (n) is priodic with priod. Find -point DFT of ach squnc. X () =.5 H () =.5 X () =.5 H () = -.5 Y (K) = X (K) H (K) Y () =.5 Y () = -.5 Using IDFT y () = ; y () =.5 ~ y ( n) h ~ ( n) ~ x ( n) = k ~ y () = k k h ~ ( k) ~ x( n k) ~ ~ x ( k) h( n k) ~ ~ x ( k) h ( k) ~ ~ ~ ~ () = x() h() x h( ) = *.5 +.5 * = ~ y () = k ~ ~ x ( k) h ( k) 59 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World ~ ~ ~ ~ () h = x() h() x () = * +.5 *.5 =.5 ~ y () = ~ ( n ) k ~ ~ x ( k) h ( k) ~ ~ ~ ~ () h = x() h() x () = *.5 +.5 * = y = {,.5,,.5..} Q. Find Linar Convolution of sam problm using DFT Sol. Th linar convolution will produc a 3-sampl squnc. To avoid tim aliasing w convrt th -sampl input squnc into 3 sampl squnc by padding with zro. For 3- point DFT X () =.5 H () =.5 X () = +.5 j 3 H () =.5+ 3 j X () = +.5 Y (K) = H (K) X (K) Y () =.5 j 3 H () =.5+ 3 j Y () =.5 +.5 j 3 +.5 3 j j 3 Y () =.5 +.5 +.5 Comput IDFT 8 3 j y(n) = 3 k Y( k) kn j 3 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World y() =.5 y() =.5 y() =.5 y(n) = {.5,.5,.5} Ans PROPERTIS OF DFT. Linarity If h(n) = a h (n) + b h (n) H (k) = a H (k) + b H (k). Priodicity H(k) = H (k+) ~ h ( n) h( n m) 3. m. y(n) = x(n-n ) Y (k) = X (k) 5. y (n) = h (n) * x (n) Y (k) = H (k) X (k) 6. y (n) = h(n) x(n) kn j Y (k) = H ( k) X ( k) 7. For ral valud squnc H H R I ( k) n ( k) n h( n) Cos h( n) Sin kn a. Complx conjugat symmtry h (n) H(k) = H*(-k) h (-n) H(-k) = H*(k) = H(-k) kn 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World i. Producs symmtric ral frquncy componnts and anti symmtric imaginary frquncy componnts about th DFT ii. Only frquncy componnts from to nd to b computd in ordr to dfin th output compltly. b. Ral Componnt is vn function H R (k) = H R (-k) c. Imaginary componnt odd function H I (k) = -H I (-k) d. Magnitud function is vn function H( k) H( k). Phas function is odd function H ( k) H ( k) f. If h(n) = h(-n) H (k) is purly ral g. If h(n) = -h(-n) H (k) is purly imaginary 8. For a complx valud squnc x * (n) X * (-k) = X * (-k) DFT [x(n)] = X(k) = n X * (-k) = n X * (k) = n x * x( n) ( n) nk W nk W * nk x ( n) W = X * (-k) DFT [x * (n)] = n x * ( n) Similarly DFT [x * (-n)] = X * (k) nk W = X * (-k) provd 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World 9. Cntral Co-ordinats x () = k X ( k) x ( ) = k ( ) k X ( k) =vn X () = n. Parsval s Rlation x ( n) X ( ) = n ( ) n x( n) =vn n x( n) k X ( k) Proof: LHS n x( n) x * ( n) * nk = x( n) X ( k) W m k = * nk X ( k) x( n) W k n = k. Tim Rvrsal of a squnc * X ( k) X (k) = k X ( k) x(( n)) x( n) X (( k)) X ( k) Rvrsing th -point sq in tim is quivalnt to rvrsing th DFT valus. jk n DFT x( n) x( n) Lt m=-n n = n x( m) jk ( m) m= to = to - = m x( m) jk m 63 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World = m x( m). Circular Tim Shift of a squnc j m ( k ) = X(-k) x( n l) X ( k) j k l DFT x( n l) n x( n l) jk n l = n x ( n l ) jk n + n jk x n ( n l ) l l = n x( n l) jk n + n jk n l x( n l) Put +n-l = m = m l jk ( ml ) x( m) to --L is shiftd to to --L Thrfor to - = ( to --L) to ( -L to -) + l jk ( ml) x( m) m Thrfor m x( m) k j ( ml) = m x( m) k j m j k l = X(k) j k l RHS 3. Circular Frquncy Shift 6 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World l j n x ( n) X ( k l) DFT l j n x( n) = n x( n) l j n k j n = n x( n) n j ( kl) X ( k l) RHS =. x(n) X(k) {x(n), x(n), x(n).x(n)} M X ( m k ) (m-fold rplication) n x( ) { X ( k), X ( k),... X ( k)} (M- fold rplication) m, 3,, 8, -j,, j Zro intrpolatd by M {, 3,,,, 3,,,, 3,, } {,,, -j6,,,,,, j6,, } 5. Duality x(n) X(k) X(n) x(-k) K x(n) = x(-k) = j X ( ) n j ( ) X ( ) k j k = X ( ) 65 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x(-k) = X ( ) j k = n X ( n) (k) 6. R[x(n)] x p k j n X p (k) X ) j Im[x(n)] op (k x op (n) (n) R[X(k)] j Im[X(k)] = DFT [ X(n) ] LHS provd X p = X (( k)) X (( k)) * x p (n) Evn part of priodic squnc = x ( n) x(( n)) x op (n) Odd part of priodic squnc = x ( n) x(( n)) Proof: X(k) = n x( n) nk W X(-k) = n x ( n) W X (( k)) nk X * (k) = n x * ( n) nk W X * (-k) = n * * x ( n) W X (( k)) nk 66 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World 7. n X (( k)) * X (( k)) n x = * ( n) x ( n) * Lt y(n) = x ( n) x( n) x( n) x = DFT of [R[x (n)]] LHS k Y(k) = X ( ) ( ) * k X k X = X ( ) ( ) * l X k l l Y() = X ( ) ( ) * l X l l Using cntral co-ordinat thorm Y() = n x * ( n) x ( n) * ( n) W * ( k) X ( k) nk Thrfor n QUESTIOS x * ( n) x ( n) = k X * ( k) X ( k) Q. (i) {,,,.} (impuls) {,,..} (constant) (ii) {,,, } (constant) ) {,,,.} (impuls) (iii) n jk (iv) Cos nk o k jk n ( ( k k ) ( k ( ) (Impuls pair) o k o jk ) jk 67 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Or Cos ( nf ) = Cos (wn) Sol. x(n) = jnk o jnk o ) = jnk o jn( k W know that (k) x( n) jnko o ) X ( K ( k k ) ( ( ) x(n) o k k o I. Invrs DFT of a constant is a unit sampl. II. DFT of a constant is a unit sampl. Q. Find point IDFT of X(k) = 3 k= = k 9 Sol. X(k) = + (k) = + (k) 5 x(n) = 5 + (n) Ans Ko) 3 Q. Suppos that w ar givn a program to find th DFT of a complx-valud squnc x(n). How can this program b usd to find th invrs DFT of X(k)? X(k) = n x( n) W nk 68 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x(n) = k X ( k) W nk x*(n) = k X * ( k) W nk. Conjugat th DFT cofficints X(k) to produc th squnc X*(k).. Us th program to fing DFT of a squnc X*(k). 3. Conjugat th rsult obtaind in stp and divid by. Q. x p (n) = {,, 3,, 5,,, } (i) f p (n) = x p (n-) = {,,,, 3,, 5, } (ii) g p (n) = x p (n+) = { 3,, 5,,,,, } (iii) h p (n) = x p (-n) = {,,,, 5,, 3, } 5 Q. x(n) = {,,,,,,, } n = to 7 Find DFT. X(k) = n x( n) jk n 8 = + jk k = to 7 X() = + = X() = + X() = + j j =.77 - j.77 = - j X(3) = + j3 =.93 - j.77 X() = - = By conjugat symmtry X(k) = X*(-k) = X*(8-k) X(5) = X*(3) =.93 + j.77 69 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X(6) = X*() = +j X(7) = X*() =.77 + j.77 X(k) = {,.77 - j.77,.93 - j.77, -j,, +j,.93 + j.77,.77 + j.77 } 6 Q. x(n) = {,,, } = X(k) = {, -j,, j} (i) y(n) = x(n-) = {,,, } Y(k) = X(k) jk ( no) (ii) X(k-) = {j,, -j, } =, j,, -j IDFT x(n) j ln = x(n) j n (iii) g(n) = x(-n) =,,, = {, j, -, } G(k) = X(-k) = X*(k) = {, j,, -j} (iv) p(n) = x*(n) = {,,, } P(k) = X*(-k) = {, j,, -j}* = {, -j,, j} (v) h(n) = x(n) x(n) = {,,, } H(k) = ( k) X ( k) (vi) c(n) = x(n) x(n) X = = {,,, } {,,, } = {,,6,} C(k) = X(k)X(k) = {6, -,, -} (vii) s(n) = x(n) x(n) = {,, 6,,,, } [, -j6,, j6] = {6, -j,, j} S(k) = X(k) X(k) = {6, -.35- j.8, -.8 + j.5,. + j.3,. - j.3, -.8 - j.5, -.35 + j.8} (viii) x( n) 6 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X ( k) [6 ] 6 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World III UIT: FFT X(k) = n x nk ( n) W K = n nk { R[x(n)] + j Im[x(n)] } { R( W ) + j Im( W ) } nk = n R[x(n)] R( W nk ) - n Im[x(n)] Im( W ) + nk j{ n nk Im[x(n)] R( W ) + Im( W )R[x(n)]} nk Dirct valuation of X(k) rquirs additions. ral multiplications complx multiplications and (-) complx { (-) + } = (-) ral additions Th dirct valuation of DFT is basically infficint bcaus it dos not us th symmtry & priodicity proprtis DITFFT: K W K W & nk W nk W X(k) = n x (n) (vn) nk W (odd) nk = x ( n) W + n + n K W x(n ) n x (n) k W ( n) nk o W = n x ( n) nk W / + W K n x ( n) nk o W / = X(k) + K W Xo(k) 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Although k= to -, ach of th sums ar computd only for k= to / -, sinc X(k) & Xo(k) ar priodic in k with priod / For K / K W = - K W X(k) for K / X(k) = X (k-/) - K W X o (k-/) = 8 x(n) = x (n) ; x(n+) = x o (n) x () = x() x o () = x() x () = x() x o () = x(3) x () = x() x o () = x(5) x (3) = x(6) x o (3) = x(7) X(k) = X(k) + W k Xo( ) k = to 3 8 k = X(k-) - W k Xo( k ) k = to 7 8 X() = X() + X() = X() + X() = X() + X(3) = X(3) + W Xo() ; X() = X() - W Xo() 8 W Xo() ; X(5) = X() - W Xo() 8 W Xo() ; X(6) = X() - W Xo() 8 3 W Xo(3) ; X(7) = X(3) - W Xo(3) 3 8 8 8 8 8 X() & X() having sam i/ps with opposit signs 73 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World This pt DFT can b xprssd as combination of pt DFT. X(k) = X(k) + W k Xo ( k) k = to - ( to ) ( k ) = X(k- )- W Xo ( k ) k = to - ( to 3 ) Xo(k) = Xo(k) + W k Xoo( k) k = to - For =8 ( k ) = Xo(k- ) - W Xoo( k ) k = to - 7 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X() = X() + X() = X() + X() = X() - W8 Xo() ; W8 Xo() ; W8 Xo() ; x () = x () = x() x () = x () = x() xo () = x () = x() X(3) = X() - W8 Xo() ; xo (3) = x (3) = x(6) Whr X(k) is th point DFT of vn no. of x (n) & Xo(k) is th point DFT of odd no. of x (n) Similarly, th squnc x o (n) can b dividd in to vn & odd numbrd squncs as x o () = x o () = x() x o () = x o () = x(5) x oo () = x o () = x(3) x oo () = x o (3) = x(7) X o () = Xo() + Xo() = Xo() + Xo() = Xo() - Xo(3) = Xo() - W8 Xoo() ; W8 Xoo() ; W8 Xoo() ; W8 Xoo() ; Xo(k) is th -pt DFT of vn-numbrd of x o (n) Xoo(k) is th -pt DFT of odd-numbrd of x o (n) X() = x () + x () = x () + x () = x() + x() X() = x () - x () = x () - x () = x() - x() 75 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X() = x () + x () = x () + x () = x() + x() X() = x () - x () = x () - x () = x() - x() 76 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World o. of o. of Complx Spd o. of points Multiplications Improvmnt Factor: Stags Dirct FFT Log Log 6 3 8 6 5.33 6 56 3 8 5 3 8.8 6 6 96 9.33 For =8 o of stags givn by= Log = Log 8 = 3. o. of i/p sts = ( Log - ) = Total o. of Complx additions using DITFFT is Log 77 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World = 8 * 3 = Each stag no. of buttrflis in th stag= m-q whr q = stag no. and = m Each buttrfly oprats on on pair of sampls and involvs two complx additions and on complx multiplication. o. of buttrflis in ach stag / DITFFT: ( diffrnt rprsntation) (u can follow any on) ( both rprsntations ar corrct) X(k) = x (n) n nk W + x(n ) n (n) k W nk = x ( n) W + n k W n x ( n) nk o W / pt DFT X(k) + K W Xo(k) k= to / - = to 3 ( K ) X(k- ) - W Xo(k- ) k = / to - = to 7 K pt DFT X(k) = X(k) + W Xo(k) k = to /- = to = X(k-/) - ( k ) W Xo(k-/) k = / to / - = to 3 Xo(k) = Xo(k) + K W Xoo(k) k = to /- = to = Xo(k-/) - ( k ) W Xoo(k-/) k = / to / - = to 3 W 8 W =8 X() = X() + W8 Xo() ; X() = X() + W8 Xo() ; X() = X() + W8 Xo() ; 3 X(3) = X(3) + W8 Xo(3) ; X() = X() - W8 Xo() X(5) = X() - W8 Xo() X(6) = X() - W8 Xo() 3 X(7) = X(3) - W8 Xo(3) X () = X() + W8 Xo() ; X () = X() - X() = X() + W8 Xo() ; X(3) = X() - W8 Xo() W8 Xo() 78 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Xo() = Xo() + W8 Xoo() ; Xo() = Xo() - Xo() = Xo() + W8 Xoo() ; Xo(3) = Xo() - W8 Xoo() W8 Xoo() X(k) = x (n) nk W = n x (n) nk W =x() + x() W k 8 X() = x()+x() X() = x()-x() x() x() x() x() x() x() x() x() x() 79 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World x(6) x() x(5) x(3) x(7) x(6) x(3) x() x() x(3) x(5) x(5) x(6) x(7) x(7) Othr way of rprsntation 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World DIFFFT: X(k) = n x( n) nk W + n / x( n') n' k W put n = n+/ = n x ( n) nk W + n x( n / ) ( n / ) W k nk = x ( n) W + n k W n x( n / ) nk W = n W [ x ( n) nk + (-) k x(n+ )] X(k) = n W / [ x ( n) nk + x(n+ )] X(k+) = n {[ x( n) - x(n+ Lt f(n) = x(n) + x(n+/) n nk )] W } W / g(n) = { x(n) x(n+/) } n W 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World =8 f() = x() + x() f() = x() + x(5) f() = x() + x(6) f(3) = x(3) + x(7) g() = [x() - x()] g() = [x() - x(5)] g() = [x() - x(6)] g(3) = [x(3) - x(7)] W 8 W 8 W 8 3 W 8 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X(k) = n [ f ( n) nk + f(n+ )] W / X(k+) = n [{ f ( n) n - f(n+ )} nk W / ] W / X(k+) = n [ g( n) + g(n+ nk )] W / X(k+3) = n [{ g( n) nk - g(n+ )} nk W / ] W / X(k) = f() + f() + [ f() + f(3) ] W k 8 X(k+) = f() f() + { [ f() f(3) ] W 8 } W 8 k X() = f() + f() + f() + f(3) 83 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World X() = f() + f() [ f() + f(3) ] X() = f() - f() + [ f() - f(3)] X(6) = f() - f() - [ f() - f(3)] W 8 W 8 Find th IDFT using DIFFFT X(k) = {, -j.,, -j.,, +j.,, +j. } Out put 8x*(n) is in bit rvrsal ordr x(n) = {,,,,,,,} 8 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Th diffrnc quation UIT-IV DIGITAL FILTER STRUCTURE P y(n) = k F M ak x(n-k) + k bk y(n-k) H(z) = P k M a F k k b z k k z k or = A Z If b k = non rcursiv or all zro filtr. Dirct Form I F p F k ( C M k k ( d ) Z k ) Z. Easily implmntd using computr program.. Dos not mak most fficint us of mmory = M+p+ F dlay lmnts. Dirct form-ii 85 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World Smallr no. of dlay lmnts = Max of (M, p) + F Disadvantags of D-I & D-II. Thy lack hardwar flxibility, in that, filtrs of diffrnt ordrs, having diffrnt no. of multiplirs and dlay lmnts.. Snsitivity of co-fficint to quantization ffcts that occur whn using finit-prcision arithmtic. Cascad Combination of scond-ordr sction (CSOS) y(n) = x(n) + a x(n-) + a x(n-) + b y(n-) + b y(n-) az az H(z) = b Z b Z Ex: 86 Downloadd From JTU World (http://(http:// )JTU World
JTU World JTU World H(z) = = z 3 5 5 Z Z Z Z z 3 Z Z Z Z = Z z 3 5 Z Z 5Z = z 3 Z Z Z Z Z 3 Z Z Ex: H(z) = Z = Z = Z Z Z Z Z Z 8.65.5Z Z.5 Z Z.5 Z Z = Z 8 Z Z.65.5Z Z Z Z 3 Z Z Z Z 3 Z 8 87 Downloadd From JTU World (http://(http:// )JTU World
88 Paralll Combination of Scond Ordr Sction (PSOS) Ex: H(z) = 5 5 3 Z Z Z Z z = 5 5 3 3 Z Z Z Z Z Z 3 5 3 5 Z Z Z Z Z 3 7 3 Z 3 6 3 Z Z Z - + - 3 7 Z Z 3 7 6 7 7 Z Z 3 7 5 Z H(z) = Z 5 3 7 3 Z Z Z Z JTU World JTU World Downloadd From JTU World (http://(http:// )JTU World )
JTU World JTU World Ex: H(z) = Z Z Z Z Z Z Z Z Z Z 8 Z 8 A = 8/3 B = C = -35/3 obtain PSOS A B = Z Z C Z 8 Jury Stability Critrion ( z) H(z) = D( z) D(z) = i b i Z i = b o Z +b Z - + b Z - +.. b - Z + b ROWS 3 5 6 COEFFICIETS b o b b. b b -. bo C o C. C - C - C -. Co d o d. d - d - d -3. do 89 Downloadd From JTU World (http://(http:// )JTU World