Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Z Domain Analysis
For poles, those close to the unit-circle can produce sharp peaks in the frequency response. Therefore, high gain can be
achieved by placing the poles close to the unit circle. Equal number of zeros and poles are normally placed to ensure
no system delay or causing non-causality, i.e. the impulse response begins at n=0. Consider the first order system with
a zero the origin
The first-order system
Its frequency response
z
H ( 1
z)
= z −α
exp( jW)
H1(
W)
=
exp( jW)
− α
To achieve low-pass, adopting 0 < 1
smallest, and the maximum gain value is
< α , i.e. the pole is on the positive axis. The denominator 1 −α
becomes the
G max
=
exp(0) 1
=
exp(0) −α 1−α
and the minimum gain
is
G
min
=
exp( jp
) 1
=
exp( jp
) −α
1+
α
To achieve a high-pass, adopting 1 < < 0
biggest, and the maximum gain at peak value is
− α , i.e. the pole is on negative axis, the denominator + α
1 becomes the
G
max
=
exp( jp
) 1
=
exp( jp
) −α
1+
α
,
and the minimum gain is
G min
=
exp(0) 1
=
exp(0) −α 1−α
When α is close to 1, the peak gain gets high, the bandwidth gets more narrow, and the impulse response decays more
slowly. The following figures illustrate the above low and high pass filters.
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