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1、The Z-TransformContent and Figures are from Discrete-Time Signal Processing,2e by Oppenheim,Shafer,and Buck,1999-2000 Prentice Hall Inc.Copyright(C)2005 Gner Arslan351M Digital Signal Processing2The z-TransformCounterpart of the Laplace transform for discrete-time signalsGeneralization of the Fourie
2、r Transform Fourier Transform does not exist for all signalsThe z-Transform is often time more convenient to useDefinition:Compare to DTFT definition:z is a complex variable that can be represented as z=r ejSubstituting z=ej will reduce the z-transform to DTFT nnz nxzX njnje nxeXCopyright(C)2005 Gne
3、r Arslan351M Digital Signal Processing3The z-Transform;nnxExample1 Example2;1nnxExample3;)5.0(nunxnExample4;22nunxCopyright(C)2005 Gner Arslan351M Digital Signal Processing4The z-TransformSystem function from the difference equation representationNkMllklnxbknyany10)()()()()()()()()()(0010nNNbbMMNkMl
4、llkkazzzzbzAzBzXzYzHzXzbzYzazYMAfter factorization,we obtain)()()(110kNklMlMNpzzzzbzHZ-transformCopyright(C)2005 Gner Arslan351M Digital Signal Processing5The z-Transform Matlab exampleExample 2.7:convolution and filterExample 2.8:stability of LTICopyright(C)2005 Gner Arslan351M Digital Signal Proce
5、ssing6The z-transform and the DTFTThe z-transform is a function of the complex z variableConvenient to describe on the complex z-planeIf we plot z=ej for=0 to 2 we get the unit circleReImUnit Circler=102 02 jeXCopyright(C)2005 Gner Arslan351M Digital Signal Processing7Convergence of the z-TransformD
6、TFT does not always converge Infinite sum not always finite if xn no absolute summable Example:xn=anun for|a|1 does not have a DTFTComplex variable z can be written as r ej so the z-transformDTFT of xn multiplied with exponential sequence r-n For certain choices of r the sum maybe made finite nnjnnn
7、jje r nxer nxreX njnje nxeX nn-r nxCopyright(C)2005 Gner Arslan351M Digital Signal Processing8Region of ConvergenceThe set of values of z for which the z-transform convergesEach value of r represents a circle of radius r The region of convergence is made of circlesExample:z-transform converges for v
8、alues of 0.5r2Copyright(C)2005 Gner Arslan351M Digital Signal Processing11Two-Sided Exponential Sequence Example 1-n-u21-nu31nxnn111010nn1z3111z311z31z31z31110111nn1z2111z211z21z21z21z31 1 z31:ROC1z211 z21:ROC1 21z31z121zz2z2111z3111zX11ReIm21oo121xx31Copyright(C)2005 Gner Arslan351M Digital Signal
9、Processing12Finite Length Sequence otherwise01Nn0anxn azazz1az1az1azzazXNN1N1N11N0nn11N0nnnCopyright(C)2005 Gner Arslan351M Digital Signal Processing13Properties of The ROC of Z-TransformThe ROC is a ring or disk centered at the originDTFT exists if and only if the ROC includes the unit circleThe RO
10、C cannot contain any polesThe ROC for finite-length sequence is the entire z-plane except possibly z=0 and z=The ROC for a right-handed sequence extends outward from the outermost pole possibly including z=The ROC for a left-handed sequence extends inward from the innermost pole possibly including z
11、=0The ROC of a two-sided sequence is a ring bounded by poles The ROC must be a connected regionCopyright(C)2005 Gner Arslan351M Digital Signal Processing14EndCopyright(C)2005 Gner Arslan351M Digital Signal Processing15Stability,Causality,and the ROCConsider a system with impulse response hnThe z-transform H(z)and the pole-zero plot shown belowWithout any other information hn is not uniquely determined|z|2 or|z|or|z|2If system stable ROC must include unit-circle:|z|2
数字信号处理与DSP器件：The Z-Transform
