BIEN425 – Lecture 14
• By the end of the lecture, you should be able to:
– Design and implement IIR filters using frequency transform and
bilinear transform
– Compare the advantages and disadvantages of IIR filter design
strategies (zero-pole versus freq-transform and bilinear
transform)
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In general
This is an alternative way of representing Method 2 from the last lecture.
This time, we don’t even need to do partial fraction expansion, the variable
in s-domain is simply changed into z-domain.
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Prove:
s
2( z  1)
T ( z  1)
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Bilinear transformation
• Simply going between s-plane and z-plane
• This is very fun…. A circle becomes a rectangle and a
line becomes a arc.
• Lecture14.m
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Frequency warping
• Given bilinear
transformation and
s = j2pF
• Let’s look at how freq
in analog filters (F) can
be translated to freq in
digital filters (f)
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Learning through example
• Building a digital lowpass filter from Chebyshev-I given
our digital specs: (Butterworth example 8.8)
– f0 = 2.5Hz, f1 = 7.5Hz
– dp = 0.1, ds = 0.1 (Could have given Ap and As instead)
• Recall the following procedure:
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• Step 1a) pre-wrap frequencies to analog specs
• Step 1b) compute r, d, minimum order
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• Step 1c) Find poles
• Step 1d) Write H(s)
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• Step 2) Determine fs =20hz
• Step 3) Re-write into H(z)
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Another example
• Given H ( s ) 
s  0.2
( s  0.1) 2  3
• Find the resonance frequency of this filter
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Analog frequency transformation
• Design digital HP,BP,BS filters
• Always start off with a normalized lowpass filter
Normalized
Lowpass
Filter
Analog
Frequency
Transformation
(Analog)
(Analog)
Bilinear
Transformation
(Digital)
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Digital frequency transformation
Normalized
Lowpass
Filter
(Analog)
Bilinear
Transformation
(Digital)
Digital
Frequency
Transformation
(Digital)
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Beware of potential problems
• Double check your filter response after design
– Stability?
– Actual frequency response
– Impulse response (check for limit cycles or deadband effects:
oscillations even when input has gone to zero)
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