1. Digital Signal Processing II Chapter 3: FIR & IIR Filter Design
Marc Moonen
Dept. E.E./ESAT, K.U.Leuven

2. PART-I : Filter Design/Realization
Step-1 : define filter specs
(pass-band, stop-band, optimization criterion,…)
Step-2 : derive optimal transfer function
FIR or IIR design
Step-3 : filter realization (block scheme/flow graph)
direct form realizations, lattice realizations,…
Step-4 : filter implementation (software/hardware)
finite word-length issues, …
question: implemented filter = designed filter ?
Chapter-3
Chapter-4
Chapter-5
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3. Chapter-3 : FIR & IIR Filter Design
FIR filters
Linear-phase FIR filters
FIR design by optimization
Weighted least-squares design, Minimax design
FIR design in practice
`Windows’, Equiripple design, Software (Matlab,…)
IIR filters
Poles and Zeros
IIR design by optimization
IIR design in practice
Analog IIR design : Butterworth/Chebyshev/elliptic
Analog->digital : impulse invariant, bilinear transform,…
Software (Matlab)
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4. FIR Filters FIR filter = finite impulse response filter
Also known as `moving average filters’ (MA)
N poles at the origin z=0 (hence guaranteed stability)
N zeros (zeros of B(z)), `all zero’ filters
corresponds to difference equation
impulse response
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5. Linear Phase FIR Filters
Non-causal zero-phase filters :
example: symmetric impulse response
h[-L],….h[-1], h[0] ,h[1],...,h[L]
h[k]=h[-k], k=1..L
frequency response is
i.e. real-valued (=zero-phase) transfer function k L
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6. Linear Phase FIR Filters
Causal linear-phase filters = non-causal zero-phase + delay
example: symmetric impulse response & N even
h[0],h[1],….,h[N]
N=2L (even)
h[k]=h[N-k], k=0..L
frequency response is
= i.e. causal implementation of zero-phase filter, by
introducing (group) delay k N
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7. Linear Phase FIR Filters
Type Type Type Type-4
N=2L=even N=2L+1=odd N=2L=even N=2L+1=odd
symmetric symmetric anti-symmetric anti-symmetric
h[k]=h[N-k] h[k]=h[N-k] h[k]=-h[N-k] h[k]=-h[N-k]
zero at zero at zero at
LP/HP/BP LP/BP BP HP
PS: `modulating’ Type-2 with 1,-1,1,-1,.. gives Type-4 (LP->HP)
PS: `modulating’ Type-4 with 1,-1,1,-1,.. gives Type-2 (HP->LP)
PS: `modulating’ Type-1 with 1,-1,1,-1,.. gives Type-1 (LP<->HP)
PS: `modulating’ Type-3 with 1,-1,1,-1,.. gives Type-3 (BP<->BP)
PS: IIR filters can NEVER have linear-phase property ! (proof see literature)
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8. Filter Specification Ex: Low-pass
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9. FIR Filter Design by Optimization
(I) Weighted Least Squares Design :
select one of the basic forms that yield linear phase
e.g. Type-1
specify desired frequency response (LP,HP,BP,…)
optimization criterion is
where is a weighting function
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10. FIR Filter Design by Optimization
…this is equivalent to
= `Quadratic Optimization’ problem
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11. FIR Filter Design by Optimization
Example: Low-pass design
optimization function is
i.e.
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12. FIR Filter Design by Optimization
a simpler problem is obtained by replacing the F(..) by…
where the wi’s are a set of (n) selected sample frequencies
This leads to an equivalent (`discretized’) quadratic optimization function:
+++ : simple
: unpredictable behavior in between sample freqs.
Compare to p.10
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13. FIR Filter Design by Optimization
This is often supplemented with additional constraints, e.g. for pass-band and stop-band ripple control :
The resulting optimization problem is :
minimize : (=quadratic function)
subject to (=pass-band constraints)
(=stop-band constraints)
= `Quadratic Programming’ problem
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14. FIR Filter Design by Optimization
(II) `Minimax’ Design :
select one of the basic forms that yield linear phase
e.g. Type-1
specify desired frequency response (LP,HP,BP,…)
optimization criterion is
where is a weighting function
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15. FIR Filter Design by Optimization
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this is equivalent to
the constraint is equivalent to a so-called `semi-definiteness’ constraint
where D>=0 denotes that the matrix is positive semi-definite
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16. Filter Design by Optimization
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a realistic way to implement these constraints, is to impose the constraints (only) on a set of sample frequencies :
i.e. a `Semi-Definite Programming’ (SDP) problem, for which efficient interior-point algorithms and software are available.
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17. FIR Filter Design by Optimization
Conclusion:
(I) weighted least squares design
(II) minimax design
provide general `framework’, procedures to translate filter design problems into standard optimization problems
In practice (and in textbooks):
emphasis on specific (ad-hoc) procedures :
- filter design based on `windows’
- equiripple design
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18. FIR Filter Design using `Windows’
Example : Low-pass filter design
ideal low-pass filter is
hence ideal time-domain impulse response is
truncate hd[k] to N+1 samples :
add (group) delay to turn into causal filter
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19. FIR Filter Design using `Windows’
Example : Low-pass filter design (continued)
PS : it can be shown that the filter obtained by such time-domain truncation is also obtained by using a weighted least-squares design procedure with the given Hd, and weighting function
truncation corresponds to applying a `rectangular window’ :
+++: simple procedure (also for HP,BP,…)
- - - : truncation in time-domain results in `Gibbs effect’ in frequency domain, i.e. large ripple in pass-band and stop-band (at band edge discontinuity), which cannot be reduced by increasing the filter order N.
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20. FIR Filter Design using `Windows’
Remedy : apply windows other than rectangular window:
time-domain multiplication with a window function w[k] corresponds to frequency domain convolution with W(z) :
candidate windows : Han, Hamming, Blackman, Kaiser,…. (see textbooks, see DSP-I)
window choice/design = trade-off between side-lobe levels (define peak pass-/stop-band ripple) and width main-lobe (defines transition bandwidth)
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21. FIR Equiripple Design Starting point is minimax criterion, e.g.
Based on theory of Chebyshev approximation and the `alternation theorem’, which (roughly) states that the optimal d’s are such that the `max’ (maximum weighted approximation error) is obtained at L+2 extremal frequencies…
…that hence will exhibit the same maximum ripple (`equiripple’)
Iterative procedure for computing extremal frequencies, etc. (Remez exchange algorithm, Parks-McClellan algorithm)
Very flexible, etc., available in many software packages
Details omitted here (see textbooks)
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22. FIR Filter Design Software
FIR Filter design abundantly available in commercial software
Matlab:
b=fir1(n,Wn,type,window), windowed linear-phase FIR design, n is filter order, Wn defines band-edges, type is `high’,`stop’,…
b=fir2(n,f,m,window), windowed FIR design based on inverse Fourier transform with frequency points f and corresponding magnitude response m
b=remez(n,f,m), equiripple linear-phase FIR design with Parks-McClellan (Remez exchange) algorithm
See exercise sessions
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23. IIR filters Rational transfer function :
N poles (zeros of A(z)) , N zeros (zeros of B(z))
infinitely long impulse response
stable iff poles lie inside the unit circle
corresponds to difference equation
= also known as `ARMA’ (autoregressive-moving average)
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24. IIR Filter Design +++ - - -
low-order filters can produce sharp frequency response
low computational cost
- - -
design more difficult
stability should be checked/guaranteed
phase response not easily controlled
(e.g. no linear-phase IIR filters)
coefficient sensitivity, quantization noise, etc. can be a problem (see Chapter-5)
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25. IIR filters Frequency response versus pole-zero location :
(cfr. frequency response is z-transform evaluated on the unit circle)
Example-1 :
Low-pass filter
poles at
DC (z=1)
Nyquist freq (z=-1)
pole Re Im
pole near unit-circle introduces `peak’ in frequency response
hence pass-band can be set by pole placement
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26. hence stop-band can be emphasized by zero placement
IIR filters
Frequency response versus pole-zero location :
Example-2 :
Low-pass filter
poles at
zeros at DC
Nyquist freq
zero
pole
zero near (or on) unit-circle introduces `dip’ (or transmision zero) in freq. response
hence stop-band can be emphasized by zero placement
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27. IIR Filter Design by Optimization
(I) Weighted Least Squares Design :
IIR filter transfer function is
specify desired frequency response (LP,HP,BP,…)
optimization criterion is
where is a weighting function
stability constraint :
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28. IIR Filter Design by Optimization
(II) `Minimax’ Design :
IIR filter transfer function is
specify desired frequency response (LP,HP,BP,…)
optimization criterion is
where is a weighting function
stability constraint :
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29. IIR Filter Design by Optimization
These optimization problems are significantly more
complex than those for the FIR design case… :
Problem-1: presence of denominator polynomial leads to non-linear (non-quadratic) optimization
A possible procedure (`Steiglitz-McBride’) consists in iteratively (k=1,2,…) minimizing
…which leads to iterative Quadratic Programming, etc..
(similar for minimax)
Skip this part
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30. IIR Filter Design by Optimization
These optimization problems are significantly more
complex than those for the FIR design case… :
Problem-2: stability constraint
(zeros of a high-order polynomial are related to the polynomial’s
coefficients in a highly non-linear manner)
Solutions based on alternative stability constraints, that e.g. are affine functions of the filter coefficients, etc…
Topic of ongoing research, details omitted here
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31. IIR Filter Design by Optimization
Conclusion:
(I) weighted least squares design
(II) minimax design
provide general `framework’, procedures to translate filter design problems into ``standard’’ optimization problems
In practice (and in textbooks):
emphasis on specific (ad-hoc) procedures :
- IIR filter design based analog filter design (s-domain
design) and analog->digital conversion
- IIR filter design by modeling = direct z-domain design
(Pade approximation, Prony, etc., not addressed here)
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32. Analog IIR Filter Design
Commonly used analog filters :
Lowpass Butterworth filters
- all-pole filters (H) characterized by a specific
magnitude response (G): (N=filter order)
- poles of G(s)=H(s)H(-s) are equally spaced on circle of radius
- Then H(jw) is found to be monotonic in pass-band & stop-band, with
`maximum flat response’, i.e. (2N-1) derivatives are zero at
poles of H(s)
N=4
poles of H(-s) N
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33. Analog IIR Filter Design
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Commonly used analog filters :
Lowpass Chebyshev filters (type-I)
all-pole filters characterized by magnitude response
(N=filter order)
is related to passband ripple
are Chebyshev polynomials:
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34. Analog IIR Filter Design
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Commonly used analog filters :
Lowpass Chebyshev filters (type-I)
All-pole filters, poles of H(s)H(-s) are on ellipse in s-plane
Equiripple in the pass-band
Monotone in the stop-band
Lowpass Chebyshev filters (type-II)
Pole-zero filters based on Chebyshev polynomials
Monotone in the pass-band
Equiripple in the stop-band
Lowpass Elliptic (Cauer) filters
Pole-zero filters based on Jacobian elliptic functions
Equiripple in the pass-band and stop-band
(hence) yield smallest-order for given set of specs
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35. Analog IIR Filter Design
Frequency Transformations :
Principle : prototype low-pass filter (e.g. cut-off frequency = 1 rad/sec) is transformed to properly scaled low-pass, high-pass, band-pass, band-stop,… filter
example: replacing s by moves cut-off frequency to
example: replacing s by turns LP into HP, with cut-off frequency
example: replacing s by turns LP into BP
etc...
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36. Analog -> Digital Principle : Conversion methods:
design analog filter (LP/HP/BP/…), and then convert it to a digital filter.
Conversion methods:
- convert differential equation into difference equation
- convert continuous-time impulse response into discrete-time
impulse response
- convert transfer function H(s) into transfer function H(z)
Requirement: the left-half plane of the s-plane should map into the inside of the unit circle in the z-plane, so that a stable analog filter is converted into a stable digital filter.
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37. Skip this slide Analog -> Digital Conversion methods:
(I) convert differential equation into difference equation :
-in a difference equation, a derivative dy/dt is replaced by a `backward
difference’ (y(kT)-y(kT-T))/T=(y[k]-y[k-1])/T, where T=sampling interval.
-similarly, a second derivative, etc…
-eventually (details omitted), this corresponds to replacing s by (1-1/z)/T in Ha(s) (=analog transfer function) :
-stable analog filters are mapped into stable digital filters, but pole location for digital filter confined to only a small region (o.k. only for LP or BP) jw
s-plane
z-plane j 1
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38. Skip this slide Analog -> Digital Conversion methods:
(II) convert continuous-time impulse response into discrete-time impulse response :
-given continuous-time impulse response ha(t), discrete-time impulse
response is where T=sampling interval.
-eventually (details omitted) this corresponds to a (many-to-one) mapping
-aliasing (!) if continuous-time response has significant frequency
content above the Nyquist frequency
s-plane jw
z-plane j 1
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39. Analog -> Digital Conversion methods:
(III) convert continuous-time system transfer function into discrete-time system transfer function : Bilinear Transform
-mapping that transforms (whole!) jw-axis of the s-plane into unit circle in the z-plane only once, i.e. that avoids aliasing of the frequency components.
-for low-frequencies, this is an approximation of
-for high frequencies : significant frequency compression (`warping’)
(sometimes pre-compensated by `pre-warping’)
s-plane jw
z-plane j 1
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40. IIR Filter Design Software
IIR filter design considerably more complicated than FIR design (stability, phase response, etc..)
(Fortunately) IIR Filter design abundantly available in commercial software
Matlab:
[b,a]=butter/cheby1/cheby2/ellip(n,…,Wn),
IIR LP/HP/BP/BS design based on analog prototypes, pre-warping,
bilinear transform, …
immediately gives H(z) (oef!)
analog prototypes, transforms, … can also be called individually
filter order estimation tool
etc...
See exercise sessions
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