Digital Signal Processing @VRG_ET_YCCEDigital Signal Processing Digital Signal ProcessingMr. Vikas R. GuptaLecturerDepartment of Electronics & TelecommunicationEngineeringYCCE, Nagpur.Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Mr. Vikas R. GuptaLecturerDepartment of Electronics & TelecommunicationEngineeringYCCE, Nagpur.Digital Signal Processing @VRG_ET_YCCEUnit Unit -- 33Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Digital Signal Processing @VRG_ET_YCCESyllabus Syllabus Unit 3: The Z-transform:Definition,Region of convergence for the Z-transform,Z- transform properties,Inverse Z-transform using contour integration,Complex convolution theorem,Parsevals theorem,Unilateral Z-transform,Stability interpretation using Jurys array.Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur. Unit 3: The Z-transform:Definition,Region of convergence for the Z-transform,Z- transform properties,Inverse Z-transform using contour integration,Complex convolution theorem,Parsevals theorem,Unilateral Z-transform,Stability interpretation using Jurys array.Digital Signal Processing @VRG_ET_YCCEZ-Transform It is a mathematical tool for the design, analysis and monitoringof systems. The z-transform is the discrete-time counter-part of the Laplacetransformanda generalizationof the Fourier transformof asampled signal. LikeLaplacetransformthez-transformallowsinsight intothetransient behavior, the steady state behavior, and the stability ofdiscrete-time systems. A working knowledge of the z-transformis essential to the study of digital filters and systems.Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur. It is a mathematical tool for the design, analysis and monitoringof systems. The z-transform is the discrete-time counter-part of the Laplacetransformanda generalizationof the Fourier transformof asampled signal. LikeLaplacetransformthez-transformallowsinsight intothetransient behavior, the steady state behavior, and the stability ofdiscrete-time systems. A working knowledge of the z-transformis essential to the study of digital filters and systems.Digital Signal Processing @VRG_ET_YCCEWhy we require Z-Transform? Characterizes the signals &systems, in mostgeneral ways possible. To find stability by pole zero analysisMr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur. Characterizes the signals &systems, in mostgeneral ways possible. To find stability by pole zero analysisDigital Signal Processing @VRG_ET_YCCEThe direct Z transform The Z-transform of a discrete time signal x(n) is defined as thepower series: Where: z is the complex variable This relation is sometimes called the direct z-transform because ittransformsthetimedomainsignal x(n)intoitscomplexplanerepresentation X(z).( ) ( )nX z x n zMr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur. The Z-transform of a discrete time signal x(n) is defined as thepower series: Where: z is the complex variable This relation is sometimes called the direct z-transform because ittransformsthetimedomainsignal x(n)intoitscomplexplanerepresentation X(z).( ) ( )nX z x n zDigital Signal Processing @VRG_ET_YCCERegion of convergence [ROC] Since the z- transform is an infinite power series, it existsonly for those values of z for which this series converges. The region of convergence (ROC) of X(z) is the set of allvalues of z for which X(z) attains a finite value.Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur. Since the z- transform is an infinite power series, it existsonly for those values of z for which this series converges. The region of convergence (ROC) of X(z) is the set of allvalues of z for which X(z) attains a finite value.Digital Signal Processing @VRG_ET_YCCEThe direct Z transform Determine the Z-transformof the following finite durationsignals.1234567( ) { , 2, 5, 7, 0,1}( ) {1, 2, , 7, 0,1}( ) {, 0,1, 2, 5, 7, 0,1}( ) {2, 4, , 7, 0,1}( ) ( )( ) ( ), 0( )150( ), 05x nx nx nx nx n nx n n k kx n n k k===== c= c >= c + >Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.1234567( ) { , 2, 5, 7, 0,1}( ) {1, 2, , 7, 0,1}( ) {, 0,1, 2, 5, 7, 0,1}( ) {2, 4, , 7, 0,1}( ) ( )( ) ( ), 0( )150( ), 05x nx nx nx nx n nx n n k kx n n k k===== c= c >= c + >Digital Signal Processing @VRG_ET_YCCEThe direct Z transform from the definition of Z-transform, we have:1 2 3 512 1 322 3 4 5 732 1 1 34567( ) 1 2 5 7( ) 2 5 7( ) 2 5 7( ) 2 4 5 7( ) 1( )( )kkXZ z z z zX Z z z z zX Z z z z z zX Z z z z zX ZX Z zX Z z = + + + += + ++ += + + + += + ++ +===Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.1 2 3 512 1 322 3 4 5 732 1 1 34567( ) 1 2 5 7( ) 2 5 7( ) 2 5 7( ) 2 4 5 7( ) 1( )( )kkXZ z z z zX Z z z z zX Z z z z z zX Z z z z zX ZX Z zX Z z = + + + += + ++ += + + + += + ++ +===Digital Signal Processing @VRG_ET_YCCEThe direct Z transform From this example it is clear that the ROC of a finite-durationsignal istheentirez-plane, except thepoint z=0and/orz=.These points are excluded. Since , becomes unbounded for z = &becomes unbounded for z = 0.( 0)kz k >Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur. From this example it is clear that the ROC of a finite-durationsignal istheentirez-plane, except thepoint z=0and/orz=.These points are excluded. Since , becomes unbounded for z = &becomes unbounded for z = 0.( 0)kz k >( 0)kz k>Digital Signal Processing @VRG_ET_YCCEThe direct Z transformQ) Determine the z-transform of the signalSolution: the signal x(n) consist ofThe z-transform of x(n) is the infinite power series,This is a infinite geometric series.1( ) ( )2nx n u n (=( 2 3 41 1 1 1 1( ) {1, , , , ,......., .....}2 2 2 2 2nx n( ( ( (=( ( ( ( Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Q) Determine the z-transform of the signalSolution: the signal x(n) consist ofThe z-transform of x(n) is the infinite power series,This is a infinite geometric series.10 021 21 1( ) [ ]2 21 1 11 ............ ......2 2 2n nnnnX z z zz z z (= =(( (= + + + + +( ( Digital Signal Processing @VRG_ET_YCCEThe direct Z transformRecall that: , if A1/22 3 4 511 ........1A A A A AA+ + + + + + =1112 zMr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Recall that: , if A1/211( )112X zz=Z=1/2Digital Signal Processing @VRG_ET_YCCEThe direct Z transformLet us express the complex variable z in polar form asWhere, r =z and = Zz.Then X(z) can be expressed asin the ROC of X(z), |X(z)| < . Butjz re =( ) ( )n j nnX z x n r e == Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Let us express the complex variable z in polar form asWhere, r =z and = Zz.Then X(z) can be expressed asin the ROC of X(z), |X(z)| < . But( ) ( )n j nnX z x n r e == ( ) ( ) ) ( ( )n j n nn nn j nnx n r X z x n n e e r r x = == s = = Digital Signal Processing @VRG_ET_YCCEThe direct Z transformThus | X(z) |is finite if the sequence x(n)r^-n is absolutely summable.To find the ROC of X(z) is equivalent to determining the values of rfor which the sequence is absolutely summable.If X(z) converges in some region of the complex plane, bothsummations in the above equation must be finite in that region.10( )( ) ( )nnn nx nX z x n rr= =s + ( )nx n rMr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Thus | X(z) |is finite if the sequence x(n)r^-n is absolutely summable.To find the ROC of X(z) is equivalent to determining the values of rfor which the sequence is absolutely summable.If X(z) converges in some region of the complex plane, bothsummations in the above equation must be finite in that region.10( )( ) ( )nnn nx nX z x n rr= =s + 1 0( )( )nnn nx nx n rr = =s + Digital Signal Processing @VRG_ET_YCCEThe direct Z transform If thefirst sumconverges theremust exist values of r smallenoughsuchthat the productsequencex(-n)r^n, 1 n < , isabsolutely summable. ROC for the first sum1( )nnx n r=Im(z)Z-planeMr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.r1Im(z)Re(z)Z-planeDigital Signal Processing @VRG_ET_YCCEThe direct Z transform ROC for the second sumIm(z)Z-plane0( )nnx nr=Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.r2Re(z)Z-planeDigital Signal Processing @VRG_ET_YCCEThe direct Z transformSince the convergence of X(z) requires that both sums be finite, it follows thatthe ROC of X(z) is generally specified as the annular region in the z-plane, r2< r < r1,which is the common region where both sums are finite.Region of convergence for |X(z)|r2 < r < r1Im(z)Z-planeMr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Since the convergence of X(z) requires that both sums be finite, it follows thatthe ROC of X(z) is generally specified as the annular region in the z-plane, r2< r < r1,which is the common region where both sums are finite.Region of convergence for |X(z)|r2 < r < r1r2Re(z)Z-planer1Digital Signal Processing @VRG_ET_YCCEThe direct Z transform Determine the Z-transform of the signal., 0( ) ( )0, 0nnnx n u nn >= = `< )Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Digital Signal Processing @VRG_ET_YCCEThe direct Z transform, 0( ) ( )0, 0nnnx n u nn >= = `< ), 10 0( )nn nn nX z z z = == = z >Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.11 if z 111 z Or equivalently,This power series converges toDigital Signal Processing @VRG_ET_YCCEThe direct Z transform Thus we have thez-transform pair11( ) ( )1; ( : )nx ROC z n u n X zz = > =If we set =1 thenMr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.1( ) ( ) ; ) : 11(1X zzC z n O x n u R= > =If we set =1 thenDigital Signal Processing @VRG_ET_YCCEThe direct Z transform Determine the Z-transform of the signal.0, 0( ) ( 1), 1nnnx n u nn > = = ` < )Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.Digital Signal Processing @VRG_ET_YCCEThe direct Z transform From the definition of z-transform, 111( ( ) )lln nnX z z z == = = Where l =-n. Now by using formula2 3 2..... (1 ....)1 AA A A A A AA+ + + = + + + + =Mr. Vikas R. Gupta, Lecturer,Department of Electronics & Telecommunication Engineering,YCCE, Nagpur.2 3 2..... (1 ....)1 AA A A A A AA+ + + = + + + + =11 11( )1 1zX zz z = = 1_ _ 1 ___ tha z provided t o z r