Introduction to Digital Signal and System Analysis - Tutorsindia
Introduction to Digital Signal and System Analysis - Tutorsindia
Introduction to Digital Signal and System Analysis - Tutorsindia
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<strong>Introduction</strong> <strong>to</strong> <strong>Digital</strong> <strong>Signal</strong> <strong>and</strong> <strong>System</strong> <strong>Analysis</strong><br />
Z Domain <strong>Analysis</strong><br />
Definition or property <strong>Signal</strong> z-transform<br />
z-transform definition x[n]<br />
X ( z)<br />
= x[<br />
n]<br />
z<br />
Inverse z-transform<br />
x[<br />
n]<br />
1<br />
2π j<br />
<br />
X (z)<br />
n−1<br />
= X ( z)<br />
z dz<br />
Linearity ax n]<br />
+ bx [ ]<br />
aX z)<br />
+ bX ( )<br />
1[ 2<br />
n<br />
Time-shifting property x n n ] u[<br />
n − ]<br />
Convolution [ n]*<br />
y[<br />
n]<br />
−n0<br />
[ −<br />
0<br />
n0<br />
X ( z)<br />
z<br />
∞<br />
n=<br />
0<br />
1( 2<br />
z<br />
x X ( z) Y ( z)<br />
−n<br />
5.5 Z-plane, poles <strong>and</strong> zeros<br />
For the z-transform of a digital signal or a transfer function of an LTI system, generally it can be expressed as fac<strong>to</strong>rised<br />
form for both the numera<strong>to</strong>r <strong>and</strong> denomina<strong>to</strong>r:<br />
N ( z)<br />
K(<br />
z − z1)(<br />
z − z2<br />
)( z − z3)...<br />
X ( z)<br />
= =<br />
D(<br />
z)<br />
( z − p )( z − p )( z − p )... (5.4)<br />
1<br />
2<br />
3<br />
where z 1<br />
, z2,<br />
z3<br />
... are called the zeros as which make X(z) =0; <strong>and</strong> p<br />
1<br />
, p2,<br />
p<br />
3 , ... are the poles as which make X(z) ® ¥.<br />
Zeros <strong>and</strong> poles are either a real number or complex conjugate pairs.<br />
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