Region of convergence(Roc) cos(wn)→∑[6(m-1+27)+6(m++27) k=-0 sin w n Hilen) w< w n 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 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
18 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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
Zero and pole The z-transform is most useful when the infinite sum can be expressed in closed form usually a ratio of polynomials in Z(or z-i) P(z Zero: The value of z for which X(=)=0 ◆Poe: The value of z for which X(z)=∞ 19 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
19 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Zero and pole ( ) ( ) Q(z) P z X z = ◆Zero: The value of z for which X(z) = 0 ◆The z-transform is most useful when the infinite sum can be expressed in closed form, usually a ratio of polynomials in z (or z-1 ). ◆Pole: The value of z for which X(z) =
Example 3.1: Right-sided exponential sequence Determine the z-transform including the roc in Z-plane and a sketch of the pole-zero-plot for sequence xn]=a"un DtEt >1? Solution X()=2∑a"=∑( az a2 0 n=0 0 az RoC: az <1or >la zero: 2=0, pole: z=a 20 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
20 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 3.1: Right-sided exponential sequence ( ) ( ) 1 0 0 n n n n n X z a z az − − = = = = zero z pole z a : 0 : = = ， xn a un n = ◆Determine the z-transform, including the ROC in z-plane and a sketch of the pole-zero-plot, for sequence: Solution: ROC: 1 az o1 r z a − ( ) 1 1 1 1 az az − − − = − a 1 DTFT a 1 ? z z a = − 0
J z-plane Unit circle rIn=a uln Re 2 o: zeros az X: poles or z> Gray region Roc 21 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
21 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. : zeros : poles Gray region: ROC xn a un n = ( ) z X z z a = − for z a 1 1 1 az − = −
Ex 3.2 Left-sided exponential sequence Determine the z-transform, including the roc pole-zero-plot, for sequence xn]=au-n Solution aul-n-11z n=- n=1 n- a z< ROC. 2<a zero: z=0 pole: z=a 22 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
22 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 3.2 Left-sided exponential sequence ( ) 1 1 n n n n n n X z a u n z a z − − − =− =− = − − − = − xn= −a u− n −1 n z a , ( ) 1 1 1 n n n n n a z a z − − = = = − = − ◆Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: ROC: z z a = − zero z pole z a : 0 : = = ( ) 1 1 1 1 a z a z a z − − − − = − − 0 -1 a z 1