Complex z plane 丌<w<丌冷 unit circle Z-plane Unit circle 4=e gRe 2021/1/31 Zhongguo Liu_Biomedical Engineering_ shandong Univ
7 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Complex z plane − w unit circle
Condition for convergence of the z-transform X()=∑xX(z)=∑小] <0 z=re"→x(ve)=∑(p"km Xx(c)s∑网小 7"<O Convergence of the z-transform for a given sequence depends only on r=z 8 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
8 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Condition for convergence of the z-transform ( ) jw n n X re x n r − =− ( ) = − = n 0 n X z x n z ( ) 0 n n X z x n z − = = jw z = re ( ) ( ) =− − − = n j w n jwn X re x n r e ◆Convergence of the z-transform for a given sequence depends only on r z =
Region of convergence(Roc) l Fbr any given sequence, the set of values of z for which the z-transform converges is called the region Of Convergence(ROC) z-plane X()=∑” n=0 if some value of Z, say Z=Z1 is in the rocr Re then all values of z on the circle defined by z=z, 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
9 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. z1 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). 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 0 n X z x n z
Region of convergence(Roc) SIn w n n= coS( w on),hp[n 0o<n< oo Neither of them is absolutely summable neither of them multiplied byr-n(oo<n<oo) would be absolutely summable for any value of r Thus, neither them has a z-transform that converges absolutely 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
10 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Region of convergence (ROC) sin , lp w nc h n n x n w n [ ] cos , = ( 0 ) = ➢Neither of them is absolutely summable, neither of them multiplied by r-n (-∞<n<∞) would be absolutely summable for any value of r. Thus, neither them has a z-transform that converges absolutely. -∞<n<∞
Region of convergence(Roc) cos(wn)→∑[6(m-1+27)+6(m++27) k=-0 sin w n W<w H 0,w<w≤x >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 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
11 2021/1/31 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. it is not strictly correct to think of the Fourier transform as being the z-transform evaluated 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 =