文档介绍：Outline of chapter 4
The z-Transform
Z变换
The Bilateral z-Transform
Difinition
Comments:
Complex variable z
Ragion of convergence
Values on the unit (radius) circle = DTFT
Properties of ROC
Properties of z-transform
Properties of ROC
Bounded by a circle
For right-sided-seq.: outside of a circle
For left-sided-seq.: inside of a circle;
For two-sided-seq.: an open ring;
For finite-duration-seq.: entire z plane;
Pole(s) must on (and outside of) the boundary
One contiguous region;
Properties of z-transform
Linearty:
Shifting: (in time and frequency domain)
Folding n ↔ inverting plex conjugation:
Differential in z ↔ times n
Convolution ↔ Multiplication
Proved in -56 on chinese. ref.
Examples
calculate z-trans for positive-sided seq.
calculate z-trans for negative-sided seq..
calculate z-trans for two-sided seq.
Multiply two z-transforms by conv
Multiply two z-transforms by conv_m
calculate z-trans for a seq. with many math. operations;
Inversion of z-transform
Method 1: Using partial fraction expansion:
Inver. Z-trans. With MATLAB
1. [R,p,C] = residuez(b,a)
R,p – column vectors
C – row vector
h=R(1)*p(1).^n+ …+R(i)*p(i).^n
[b,a]=residue(R,p,C)
causual finite duration seq.
2. Impulse resp: h=filter(b,a,impseq(0,0,N))
3. Long devision
h=deconv([b,zeros(1,N)],a)
Examples for partial fraction
Z-1 for rational polynomial function(3 possibilities for different ROCs)
Z-1 for partial fraction using MATLAB
Method 2: impulse ;
Z-1 using imp. Resp. with double roots
Z-1 using imp. Resp. with ROC check
Method 3: long division
LTI Systems in the z- domain
Systems function definition:
Y(z)= H(z)*X(z)
LTI Systems with difference Equation
Transfer function expression
Zero/pole expression
Frequency response expression