ECE651 Digital Signal Processing I

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ECE651 Digital Signal Processing I Digital
IIR Filter Design
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• Introduction
• Some Preliminaries on Analog Filters
• Digital IIR Filter Design (s z)
• Impulse Invariance Transformation
• Bilinear Transformation
• Frequency Band Transformations
• Analog Domain (s s )
• Digital Domain (z z)

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Introduction
Analog filter Infinitely long impulse response
S Z (complex-valued mapping)
Digital IIR filter Infinitely long impulse
response
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Introduction
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Introduction

• Advantages
• Analog filter design tables available
• Filter transformation (s z) tables
  available
• Frequency band transformation (s s / z z)
  available
• Disadvantages
• No control over the phase characteristics of
  the IIR filter
• Magnitude only design

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Introduction

• Other Design Approaches
• Simultaneously approximate both the magnitude
  and the phase response
• Require advanced optimization tools
• Not covered in the class

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Preliminaries On Analog Filters
Analog lowpass filter specifications
passband ripple parameter
A stopband attenuation parameter
passband cutoff frequency (rad/sec)
stopband cutoff frequency (rad/sec)
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Preliminaries On Analog Filters
Analog lowpass filter specifications
passband ripple in dB
stopband attenuation in dB
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Preliminaries On Analog Filters
Analog lowpass filter system function

• Poles and zeros of magnitude-squared function
  are distributed in a mirror-image symmetry with
  respect to the imaginary axis
• For real filters, poles and zeros occur in
  complex conjugate pairs ( mirror symmetry with
  respect to real axis)

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Preliminaries On Analog Filters
Analog lowpass filter system function

• Pick up poles On LHP
• Pick up zeros on LHP or
• Imaginary axis

Causal
Stable
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Preliminaries On Analog Filters

• Prototype analog filters
• Butterworth
• Chebyshev (Type I and II)
• Elliptic

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Preliminaries On Analog Filters
Butterworth lowpass filters (Magnitude-Squared
Response)
The Cutoff frequency (rand/sec)
N The order of the filter
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Preliminaries On Analog Filters
Butterworth lowpass filters (System Function)
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Preliminaries On Analog Filters
Butterworth lowpass filters (Design equations)
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Digital IIR Filter Design

• S - Z transformation
• Complex-valued mappings
• Derived by preserving different aspects of analog
  filters and digital filters

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Digital IIR Filter Design

• Impulse Invariance transformation
• Preserve the shape of impulse response

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Digital IIR Filter Design

• Impulse Invariance transformation (Design
  Procedure)
• (MATLAB
  function impinvar)
• Choose T and determine the analog frequencies
• Design an analog filter using
  specifications
• Partial fraction expansion
• Transform analog poles into digital poles
  to obtain

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Digital IIR Filter Design

• Impulse Invariance transformation (Aliasing)

gtgt f00.015T0.1 gtgt zexp(j2pifT) gtgt
zH(1-0.8966./z)./(1-1.5595./z0.6065./z./z) gtgt
sj2pif gtgt sH(1s)./(s.25s6) gtgt
plot(f,abs(zH),f,abs(sH)/T)legend('Digitital','An
alog') gtgttitle('Magnitude Response of Analog and
Digital IIR Filters')
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Digital IIR Filter Design

• Impulse Invariance transformation

• Advantages
• Stable design
• Analog frequency and digital frequency are
  linearly related
• Disadvantage
• Aliasing
• Useful only when the analog filter is
  band-limited (LPF and BPF)

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Digital IIR Filter Design

• Bilinear transformation

• Preserve the system function representation

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Digital IIR Filter Design

• Bilinear transformation (Design Procedure)
• (MATLAB
  function bilinear)
• Choose T (1)and determine the analog frequencies
• Design an analog filter using
  specifications
• Bilinear transformation

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Digital IIR Filter Design

• Bilinear transformation
• Advantages
• Stable design
• No aliasing
• No restriction on the type of filters that can be
  transformed

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Frequency DomainTransformations

• Analog Domain

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Frequency DomainTransformations

• Digital Domain