0. Chapter 7: FILTER DESIGN TECHNIQUES
Department of Computer Eng.
Sharif University of Technology
Discrete-time signal processing
Chapter 7: FILTER DESIGN TECHNIQUES
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, © Prentice Hall Inc.

1. Filter Design Techniques
DSP
Filter:
Any discrete-time system that modifies certain frequencies
Frequency-selective filters pass only certain frequency components and totally reject all others
Filter Design Steps:
Specification
Problem or application specific
Approximation of specification with a discrete-time system
Our focus is to go from spec to discrete-time system
Implementation
Realization of discrete-time systems depends on target technology
Desired filter is generally implemented with digital computation
It is used to filter a sampled and digitized continuous-time signal
Chapter 7: Filter Design Techniques 1

2. Filter Design Techniques
DSP
We have already studied the use of discrete-time systems to implement a continuous-time system
If our specifications are given in continuous time we can use:
It can be noted that the specifications for filters are typically given in frequency domain
D/C
xc(t)
yr(t)
C/D
H(ej)
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3. Filter Specifications
DSP
Specifications
Passband
Stopband
Parameters
Specs in dB
Ideal passband gain =20log(1) = 0 dB
Max passband gain = 20log(1.01) = 0.086dB
Max stopband gain = 20log(0.001) = -60 dB
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4. Discrete Signal Processing
7.1 Discrete-Time IIR Filter Design from Continuous-Time Filters
Chapter 7: Filter Design Techniques 4

5. 7.1.1 Filter Design by Impulse Invariance
DSP
Remember impulse invariance
Mapping a continuous-time impulse response to discrete-time
Mapping a continuous-time frequency response to discrete-time
If the continuous-time filter is bandlimited to
In this design procedure:
Start with discrete-time spec in terms of  (Td has no role in the process)
Go to continuous-time =  /T and design continuous-time filter
Use impulse invariance to map it back to discrete-time = T
Works best for practical filters due to possible aliasing
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6. Impulse Invariance of System Functions
DSP
Develop impulse invariance relation between system functions
Partial fraction expansion of transfer function of continuous-time filter:
Corresponding impulse response:
Impulse response of discrete-time filter:
System function:
Pole s=sk in s-plane transforms into a pole at in the z-plane
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7. Chapter 7: Filter Design Techniques
Example
DSP
Impulse invariance applied to Butterworth
Since sampling rate Td cancels out we can assume Td=1
Map spec to continuous time
Butterworth filter is monotonic so spec will be satisfied if
Determine N and c to satisfy these conditions
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8. Chapter 7: Filter Design Techniques
Example Cont’d
DSP
Satisfy both constrains
Solve these equations to get
N must be an integer so we round it up to meet the spec
Poles of transfer function
The transfer function
Mapping to z-domain
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9. Chapter 7: Filter Design Techniques
Example Cont’d
DSP
Chapter 7: Filter Design Techniques 9

10. Filter Design by Bilinear Transformation
DSP
Avoids the aliasing problem of impulse invariance
Maps the entire s-plane onto the unit-circle in the z-plane
Nonlinear transformation
Frequency response subject to warping
Bilinear transformation
Transformed system function
Again Td cancels out so we can ignore it
We can solve the transformation for z as
Maps the left-half s-plane into the inside of the unit-circle in z
Stable in one domain would stay in the other
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11. Bilinear Transformation
DSP
On the unit circle the transform becomes
To derive the relation between  and 
Which yields
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12. Bilinear Transformation
DSP
Chapter 7: Filter Design Techniques 12

13. Chapter 7: Filter Design Techniques
Example
DSP
Bilinear transform applied to Butterworth
Apply bilinear transformation to specifications
We can assume Td=1 and apply the specifications to
To get
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14. Chapter 7: Filter Design Techniques
Example Cont’d
DSP
Solve N and c
The resulting transfer function has the following poles
Resulting in
Applying the bilinear transform yields
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15. Chapter 7: Filter Design Techniques
Example Cont’d
DSP
Chapter 7: Filter Design Techniques 15

16. Chapter 7: Filter Design Techniques
DSP
7.2 Design of FIR Filters By Windowing
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17. Filter Design by Windowing
DSP
Simplest way of designing FIR filters
Method is all discrete-time no continuous-time involved
Start with ideal frequency response
Choose ideal frequency response as desired response
Most ideal impulse responses are of infinite length
The easiest way to obtain a causal FIR filter from ideal is
More generally
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18. Windowing in Frequency Domain
DSP
Windowed frequency response
The windowed version is smeared version of desired response
If w[n]=1 for all n, then W(ej) is pulse train with 2 period
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19. Chapter 7: Filter Design Techniques
Properties of Windows
DSP
Prefer windows that concentrate around DC in frequency
Less smearing, closer approximation
Prefer window that has minimal span in time
Less coefficient in designed filter, computationally efficient
So we want concentration in time and in frequency
Contradictory requirements
Example: Rectangular window
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20. Chapter 7: Filter Design Techniques
Rectangular Window
DSP
Narrowest main lobe
4/(M+1)
Sharpest transitions at discontinuities in frequency
Large side lobes
-13 dB
Large oscillation around discontinuities
Simplest window possible
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21. Bartlett (Triangular) Window
DSP
Medium main lobe
8/M
Side lobes
-25 dB
Hamming window performs better
Simple equation
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22. Chapter 7: Filter Design Techniques
Hanning Window
DSP
Medium main lobe
8/M
Side lobes
-31 dB
Hamming window performs better
Same complexity as Hamming
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23. Chapter 7: Filter Design Techniques
Hamming Window
DSP
Medium main lobe
8/M
Good side lobes
-41 dB
Simpler than Blackman
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24. Chapter 7: Filter Design Techniques
Blackman Window
DSP
Large main lobe
12/M
Very good side lobes
-57 dB
Complex equation
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26. Incorporation of Generalized Linear Phase
DSP
Windows are designed with linear phase in mind
Symmetric around M/2
So their Fourier transform are of the form
Will keep symmetry properties of the desired impulse response
Assume symmetric desired response
With symmetric window
Periodic convolution of real functions
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27. Linear-Phase Lowpass filter
DSP
Desired frequency response
Corresponding impulse response
Desired response is even symmetric, use symmetric window
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28. Kaiser Window Filter Design Method
DSP
Parameterized equation forming a set of windows
Parameter to change main-lobe width and side-lobe area trade-off
I0(.) represents zeroth-order modified Bessel function of 1st kind
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29. Determining Kaiser Window Parameters
DSP
Given filter specifications Kaiser developed empirical equations
Given the peak approximation error  or in dB as A=-20log10 
and transition band width
The shape parameter  should be
The filter order M is determined approximately by
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30. Example: Kaiser Window Design of a Lowpass Filter
DSP
Specifications
Window design methods assume
Determine cut-off frequency
Due to the symmetry we can choose it to be
Compute
And Kaiser window parameters
Then the impulse response is given as
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31. Chapter 7: Filter Design Techniques
Example Cont’d
DSP
Approximation Error
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32. General Frequency Selective Filters
DSP
A general multiband impulse response can be written as
Window methods can be applied to multiband filters
Example multiband frequency response
Special cases of
Bandpass
Highpass
Bandstop
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33. Chapter 7: Filter Design Techniques
DSP
7.4 Optimum Approximation of FIR Filters
Chapter 7: Filter Design Techniques 33

34. Chapter 7: Filter Design Techniques
Optimum Filter Design
DSP
Filter design by windows is simple but not optimal
Like to design filters with minimal length
Optimality Criterion
Window design with rectangular filter is optimal in one sense
Minimizes the mean-squared approximation error to desired response
But causes large error around discontinuities
Alternative criteria can give better results
Minimax: Minimize maximum error
Frequency-weighted error
Most popular method: Parks-McClellan Algorithm
Reformulates filter design problem as function approximation
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35. Function Approximation
DSP
Consider the design of type I FIR filter
Assume zero-phase for simplicity
Can delay by sufficient amount to make causal
Assume L=M/2 an integer
After delaying the resulting impulse response
Example approximation mask
Low-pass filter
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36. Polynomial Approximation
DSP
Using Chebyshev polynomials
Express the following as a sum of powers
Can also be represented as
Parks and McClellan fix p, s, and L
Convert filter design to an approximation problem
The approximation error is defined as
W() is the weighting function
Hd(ej) is the desired frequency response
Both defined only over the passpand and stopband
Transition bands are unconstrained
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37. Lowpass Filter Approximation
DSP
The weighting function for lowpass filter is
This choice will force the error to = 2 in both bands
Criterion used is minmax
F is the set of frequencies the approximations is made over
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38. Chapter 7: Filter Design Techniques
Alternation Theorem
DSP
Fp denote the closed subset
consisting of the disjoint union of closed subsets of the real axis x
The following is an rth order polynomial
Dp(x) denotes given desired function that is continuous on Fp
Wp(x) is a positive function that is continuous on Fp
The weighted error is given as
The maximum error is defined as
A necessary and sufficient condition that P(x) be the unique rth order polynomial that minimizes is that Ep(x) exhibit at least (r+2) alternations
There must be at least (r+2) values xi in Fp such that x1<x2<…<xr+2
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39. Chapter 7: Filter Design Techniques
Example
DSP
Examine polynomials P(x)
that approximate
Fifth order polynomials shown
Which satisfy the theorem?
Not alternations
Not alternations
Chapter 7: Filter Design Techniques 39

40. Optimal Type I Lowpass Filters
DSP
In this case the P(x) polynomial is the cosine polynomial
The desired lowpass filter frequency response (x=cos)
The weighting function is given as
The approximation error is given as
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41. Typical Example Lowpass Filter Approximation
DSP
7th order approximation
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42. Properties of Type I Lowpass Filters
DSP
Maximum possible number of alternations of the error is L+3
Alternations will always occur at p and s
All points with zero slope inside the passpand and all points with zero slope inside the stopband will correspond to alternations
The filter will be equiripple except possibly at 0 and 
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43. Flowchart of Parks-McClellan Algorithm
DSP
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44. Butterworth Lowpass Filters
DSP
Passband is designed to be maximally flat
The magnitude-squared function is of the form
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45. Chapter 7: Filter Design Techniques
Chebyshev Filters
DSP
Equiripple in the passband and monotonic in the stopband
Or equiripple in the stopband and monotonic in the passband
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