LL(O=∑ snT rini e (e) n=-00 x()=∑h21etz@e n=-00 Laplace transform ( continuous time signal relation z-transform discrete-time signal e o+jQ)T OT 2T 三7e SpOT relation between s0: O=QT s and z 数字频率和模拟频率的关系
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 @ 数字频率和模拟频率的关系
ST (+j92)T。o,oj2T re J X(z)=∑ lInz O=97,9=0/T7 n=-00 JO -1 O=0.r=1 DTFT X(e)= x(ne on Discrete Time n=-00 Fourier Transform 2 s plane 3r/Th 0>0. r>1 Zplane x丌/T 0<0 < 丌/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, 1 r ( ) [ ] n n X z x n z − =− = = T Go
O=9T=2n//=2mff"@f/ .0→>2xf/2 O:0→>兀 → 0→2丌 Q→>29 2丌-)4兀 3丌 s plane plane mlz 丌 2 T 0/Re[= 3丌
14 2 2 T f f f s s = = = z plane Re[ ]z Im[ ]z 0r 0 2 / 2 0 2 2 s s s s s f f → → = → : 0 0 2 2 4 →→→ : s f f f @ js plane 0 2 2s s f T = Ts −3Ts 3Ts −
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) X(=∑42=x(e-)=∑km n=-00 Absolute Summability|Y(re)≤ <0 n=-00 X(2)∑x[小] <0 the roc consists of all values of z such that the inequality in the above holds 15 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
15 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 − =− ( ) 0 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(2)s∑x[l]”<∞ 2=7 K Convergence of the z-transform for a given sequence depends only on r=z Z-plane if some value of z, say Z=Z1 is in the roc then all values of z on the 9 circle defined by z= z1 Will also be in the roc Z uniform convergence if roc includes unit circle, then Fourier transform and all its derivatives with respect to w must be continuous functions of w
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. ( ) 0 , 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 uniform convergence