Chapter 10 The z-transform Example 10.5 6l小<2→∑lk"=1k z≥0 n=-0 Example 10.6 xn]=a"un-uln-ND z-a 多 storder pole N 2-a re J N。j2k丌 = e Zeros Ok=j2K/N z>0 11
11 Chapter 10 The Z-Transform Example 10.6 xn a un un N n = − − ( ) z (z a) z a X z N N N − − = −1 ( ) N j k N j re a e 2 = = = ω j k N r a k 2 / Zeros: (N-1)st order pole a z 0 Example 10.5 ⎯→ =1 − + =− n n n n z Z z 0
Chapter 10 The z-transform Property 4: If xn] is right sided, x[n=0, n<N 彐6ROC>2>6∈ROC If xn] is right sided, z> max Furthermore if N, <0 z=oo ROC If[n] is not causal, (N, <O) X(x)=∑[lzn+∑ x[n]z n=N positive powers ofz 12
12 Property 4: If is right sided, 1 xn x n = 0 , n N Chapter 10 The Z-Transform r0 ROC z r0 ROC max If is right sided, xn z r Furthermore, if N1 0 z = ROC ( 0) If is not causal, xn N1 ( ) n n n n N X z x n z x n z − + = − − = = + 0 1 1 positive powers of z
Chapter 10 The z-transform Example x a un N n-n z n=N n=m+N +oo ∑(az 2=0 z z ①N≥0 ②N<0z> a and z≠
13 Chapter 10 The Z-Transform Example xn a un N n = − ( ) n n n N X z a z − + = = ( ) m N m n m N az + − + = = + = 1 0 ( ) ( ) 1 1 1 − − − = az az X z N ① N 0 z a ② N 0 z a and z
Chapter 10 The z-transform Property 5: If xn is left sided, x[n]=0,n> N 彐6ROC><6∈ROC If xin] is left sided, z min Furthermore. if N,>0 z=o ROC Ifx[n] is not anticausal, (N,>O) X(z)=∑lz"+∑l]zn n=1 negative powers of z 14
14 Property 5: If is left sided, 1 xn x n = 0 , n N Chapter 10 The Z-Transform r0 ROC z r0 ROC min If is left sided, xn z r Furthermore, if N1 0 z = 0 ROC ( 0) If is not anticausal, xn N1 ( ) n N n n n X z x n z x n z − = − =− = + 1 1 0 negative powers of z
Chapter 10 The z-transform Example xn]=aul-n-NI x()=∑a""=∑(a2 n=-0 m=N m=k+N+∞ ∑(a2z k=0 X(z) -4Z ①N≥0(N≤0)→k<a ②N<0(N>0)→0<2<a 15
15 Chapter 10 The Z-Transform Example xn a u n N n = − − ( ) n n N n X z a z − − =− = ( ) m m N n m a z −1 + = =− = ( ) ( ) a z a z X z N 1 1 1 − − − = ① N 0 (− N 0) ( ) k N k m k N a z + − + = = + = 1 0 z a ② N 0 (− N 0) 0 z a