x()=x()∑6(t-n7)=∑x(n7)6(-n7) L[x:()=∑x7)em=X(e) X(2)=∑难n]zn let: zae 1= Laplace transform (> continuous time signal [-relation z-transform discrete-time signal QT z三已 (o+jQ2)T' JO r=ooT relation between s0: O=QT F s and z 数字频率(单位rad)和模拟频率(rad/s)的关系 12
12 sT T ( ) j T j T j z e e e e re + = = = = ( ) [ ] n n X z x n z − =− = so： T r e = relation between = T s z and Laplace transform continuous time signal z-transform discrete-time signal relation [ ] ( ) ( ) ( ) snT sT s n x t x nT e X e − =− L = = sT let： z e @ 数字频率(单位rad)和模拟频率(rad/s)的关系 s c c ( ) ( ) ( ) ( ) ( ) n n x t x t t nT x nT t nT =− =− = − = −
2=7 (a+j92)T 2T e e Jo X(2)=∑xnz",O=927,9=0 n=-0 O O 0=0,F=e07 z三 DTFT X(e)=∑x(n)e Jon Discrete Time 1=-0 Fourier transform s plane 0>0, r>l Pla ane S=0+ 丌 J O<0 \F< -T /T Go
13 = T, DTFT : Discrete Time Fourier Transform sT j T T j T j ( ) z e e e e re + = = = = ( ) ( ) j j n n X e x n e − =− = S plane Z plane - 3 / T j /T − /T 1 | , j j r z re e = = = 0, 1 r r =1 0 r 1 = 0, ( ) [ ] , n n X z x n z − =− = = T Go j z re = s = + j 1 T r e = =
Region of convergence (Roc) lFor any given sequence, the set of values of Z for which the z-transform converges is called the region of Convergence (roc) X()=∑=”=XVe")=∑(py")km 1三-00 1=-00 Absolute X e x nlr<oo Summability Ax(x)≤∑l] <OO n=-0 the roc consists of all values of z such that the inequality in the above holds 15
15 Region of convergence (ROC) ◆For any given sequence, the set of values of z for which the z-transform converges is called the Region Of Convergence (ROC). ( ) n n X z x n z − =− = ( ) jw n n X re x n r − =− ( ) - n n X z x n z − = jw z re = Absolute Summability ( ) ( ) =− − − = n j w n jwn X re x n r e the ROC consists of all values of z such that the inequality in the above holds
Region of convergence(ROc) Ax(z)≤∑l]"<∞, 9 z三7 1=-00 K Convergence of the z-transform for a given sequence depends only on r= la Z-plane if some value of Z, say z=zu is in the roc then all values of z on the Re circle defined by 1z=1Zl will also be in the roc 21 致收敛u uniform convergence ifroc includes unit circle. then fourier transform and all its derivatives with respect to w must be continuous functions of w 16
z1 if some value of z, say, z =z1 , is in the ROC, then all values of z on the circle defined by |z|=|z1 | will also be in the ROC. if ROC includes unit circle, then Fourier transform and all its derivatives with respect to w must be continuous functions of w. ( ) , - n n X z x n z − = r z = ◆Convergence of the z-transform for a given sequence depends only on . Region of convergence (ROC) jw z = re 一致收敛uniform convergence 16
Region of convergence (Roc) coS( Won n)→∑[6(m-1+27)+7(m+1+27) Sin w n W<w h e / C 0,w<w≤丌 The fourier transforms are not continuous, infinitely differentiable functions, so they cannot result from evaluating a z-transform on the unit circle. it is not strictly correct to think of the Fourier transform as being the z transform evaluated on the unit circle
18 Region of convergence (ROC) ➢The Fourier transforms are not continuous, infinitely differentiable functions, so they cannot result from evaluating a z-transform on the unit circle. cos(w n0 ) ( ) ( ) 0 0 2 2 k w w k w w k =− − + + + + ( ) = , w w , w w H e c j w c l p 0 1 sin lp w nc h n n = it is not strictly correct to think of the Fourier transform as being the ztransform evaluated on the unit circle