Discretetime Fourier transform The ztransform Example ROC Example ROC Example ROC 1 Unit circle 12 x x 13 Example ROC 1 Unit circle x 13 x ROC Property 1 The ROC of Xz consists ID: 653380
Download Presentation The PPT/PDF document "The z-transform Laplace transform" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
The z-transform
Laplace transform
Discrete-time Fourier transformSlide2
The z-transform
Example
ROCSlide3
Example
ROC
Example
ROC
1
Unit circle
1/2
x
x
1/3Slide4
Example
ROC
1
Unit circle
x
1/3
xSlide5
ROC
Property 1: The ROC
of X(z) consists
of
a ring in the z-plane centered about the origin
Property 2:
The ROC doesn’t contain any pole
Property 3: If
x[n] is finite duration, then the ROC is the entire z-plane, except possibly z=0 and z=. Slide6
Example
Property 4: If
x[n]
is
right-sided
, and if the
circle |z|
= r0 is in the ROC, then all values of s for which |z| >
r0 will also be in the ROC.
ROC is the entire
z-plane
ROC is the entire
z-plane, except z =0
ROC is the entire
z-plane, except z =
Slide7
Property 6: If x(t) is two sided, and if the
ring |z| = r0 is in the ROC, then the ROC will consist of a ring in the z-plane
that include the ring |z| = r0
.
Property 5: If
x[n] is left-sided, and if the circle |z| = r0 is in the ROC, then all values of s for which |z| <
r0 will also be in the ROC. Slide8
Example
b<1
ROC |z|>b
ROC |z|< 1/b
Property 7: If the
z-transform X(z)
of
x[n]
is rational, then its ROC is bounded by poles or extends to infinity. In addition, no poles are contained in the ROC.Slide9
Property 8: If the
z-transform X(z) of x[n] is rational, then if x[n] is right sided, then ROC is the region in the z-plane outside of the outmost pole, i.e. outside of the circle of radius equal to the largest magnitude of the poles of X(z). Furthermore, i
f x[n] is causal (i.e. if it is right-sided and equal to 0 for n<0), then the also include z=.
Property
9: If the z-transform X(z) of
x[n] is rational, then if x[n] is left-sided, then ROC is the region in the z-plane inside of the innermost non-zero pole, i.e. inside of the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z = 0 and extend inward to any possibly including z = 0. In particular, if
x[n] is anticausal (i.e. if it is left-sided and equal to 0 for n>0), then the ROC also include z=0. Slide10
Example
ROC ?
ROC
Unit circle
1
2
x
x
1/3
xSlide11
Example
ROC ?
ROC
1
Unit circle
x
1/3
2
x
xSlide12
Example
ROC ?
ROC
1
Unit circle
2
x
x
1/3
x