Download Z Transform-Digital Signal Processing-Lecture Slides and more Slides Digital Signal Processing in PDF only on Docsity!
The Z-Transform
Quote of the Day
Such is the advantage of a well-constructed
language that its simplified notation often
becomes the source of profound theories.
Laplace
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck,
The z-Transform
- Counterpart of the Laplace transform for discrete-time signals
- Generalization of the Fourier Transform
- Fourier Transform does not exist for all signals
- The z-Transform is often time more convenient to use
- Definition:
- Compare to DTFT definition:
- z is a complex variable that can be represented as z=r ej
- Substituting z=ej^ will reduce the z-transform to DTFT
n
n Xz xn z
j n
n
j X e xn e
Convergence of the z-Transform
- DTFT does not always converge
- Infinite sum not always finite if x[n] no absolute summable
- Example: x[n] = anu[n] for |a|>1 does not have a DTFT
- Complex variable z can be written as r ej^ so the z-transform
- DTFT of x[n] multiplied with exponential sequence r -n
- For certain choices of r the sum maybe made finite
n
n j n
n
j j n Xre xn re xn r e
j n
n
j X e xn e
^
n
Region of Convergence
- The set of values of z for which the z-transform converges
- Each value of r represents a circle of radius r
- The region of convergence is made of circles
- Example: z-transform converges for
values of 0.5<r<
- ROC is shown on the left
- In this example the ROC includes
the unit circle, so DTFT exists
- Not all sequence have a z-transform
- Example:
- Does not converge for any r
- No ROC, No z-transform
- But DTFT exists?!
- Sequence has finite energy
- DTFT converges in the mean-
squared sense
Re
Im
x n cos (^) on
Same Example Alternative Way
- For the term with infinite exponential to vanish we need
- Determines the ROC (same as the previous approach)
- In the ROC the sum converges to
n 0
1 n
n
n n n xn aun X z aunz az
2
1
N 1 2
n N
N N 1 n
1
n 0
1
1 0 1 1 n
1 az
az az az
az 1 a z
1
n 0
1
1 n
1 az
X z az
|z|>
Two-Sided Exponential Sequence Example
(^) u- (^) n- 1 2
un - 3
xn
n n
1 1
1
0 1
n 0
n 1
z 3
z 3
z 3
z 3
z 3
1 1
0 1 1 1
n
n 1
z 2
z 2
z 2
z 2
z 2
z 3
z 1 3
ROC :
1
z 2
z 1 2
ROC :
1
z 3
z
2 z z
z 2
z 3
X z 1 1
Re
Im
2
1
oo
12
1
(^3) x x
1
Properties of The ROC of Z-Transform
- The ROC is a ring or disk centered at the origin
- DTFT exists if and only if the ROC includes the unit circle
- The ROC cannot contain any poles
- The ROC for finite-length sequence is the entire z-plane
- except possibly z=0 and z=
- The ROC for a right-handed sequence extends outward from
the outermost pole possibly including z=
- The ROC for a left-handed sequence extends inward from the
innermost pole possibly including z=
- The ROC of a two-sided sequence is a ring bounded by poles
- The ROC must be a connected region
- A z-transform does not uniquely determine a sequence without
specifying the ROC
Stability, Causality, and the ROC
- Consider a system with impulse response h[n]
- The z-transform H(z) and the pole-zero plot shown below
- Without any other information h[n] is not uniquely determined
- If system stable ROC must include unit-circle: ½<|z|<
- If system is causal must be right sided: |z|>
