EEM478-DSPHARDWARE. WEEK12:FIR & IIR Filter Design

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Christine Sherman
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1 EEM478-DSPHARDWARE WEEK12:FIR & IIR Filter Design

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) accuracy issues, question: implemented filter = designed filter?

3 Lecture-2 : 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 Weighted least-squares design, Minimax design IIR design in practice Analog IIR design : Butterworth/Chebyshev/elliptic Analog->digital : impulse invariant, bilinear transform, Software (Matlab)

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

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 L k i.e. real-valued (=zero-phase) transfer function

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 0 N k = i.e. causal implementation of zero-phase filter, by introducing (group) delay

7 Linear Phase FIR Filters Type-1 Type-2 Type-3 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)

8 Filter Specification Ex: Low-pass

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

10 FIR Filter Design by Optimization this is equivalent to = `Quadratic Optimization problem

11 FIR Filter Design by Optimization Example: Low-pass design optimization function is i.e.

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 problem: +++ : simple : unpredictable behavior in between sample freqs. Compare to p.27

13 FIR Filter Design by Optimization then all this is often supplemented with additional constraints Example: Low-pass (LP) design (continued) pass-band ripple control : stop-band ripple control :

14 FIR Filter Design by Optimization Example: Low-pass (LP) design (continued) a realistic way to implement these constraints, is to impose the constraints (only) on a set of sample frequencies, e.g. in the pass-band and in the stop-band The resulting optimization problem is : minimize : subject to (pass-band constraints) (stop-band constraints) = `Quadratic Programming problem (see HG94)

15 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

16 FIR Filter Design by Optimization 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

17 Filter Design by Optimization 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.

18 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

19 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

20 FIR Filter Design using `Windows Example : Low-pass filter design (continued) ps : it can be shown that time-domain truncation corresponds to solving a weighted least-squares optimization problem with the given Hd, and weighting function truncation corresponds to applying a `rectangular window : +++: simple procedure (also for HP,BP, ) : truncation in the time-domain results in `Gibbs effect in the 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.

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) :

21 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,. window choice/design = trade-off between side-lobe levels (define peak pass-/stop-band ripple) and width main-lobe (defines transition bandwidth)

22 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)

23 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

24 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)

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 pole pole poles at Nyquist freq (z=-1) Im DC (z=1) Re

26 IIR filters Frequency response versus pole-zero location : Example-2 : Low-pass filter pole pole poles at Nyquist freq zeros at zero DC

27 IIR filters Frequency response versus pole-zero location : Example-3 : Band-pass filter pole poles at zeros at zero zero DC

28 IIR filters Frequency response versus pole-zero location : * pole near unit-circle introduces `peak in frequency response hence pass-band can be set by pole placement * zero near unit-circle introduces `dip in frequency response zero on unit-circle introduces `transmission zero in frequency response hence stop-band can be emphasized by zero placement

29 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

30 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 :

31 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 :

32 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 rational approximation, and hence to highly non-linear optimization (instead of - for instance- quadratic optimization in the FIR design case) A possible procedure (`Steiglitz-McBride ) consists in iteratively (k=1,2, ) minimizing which leads to iterative Quadratic Programming, etc.. (similar for minimax)

33 IIR Filter Design by Optimization 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

34 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)

35 Analog IIR Filter Design Commonly used analog filters : Lowpass Butterworth filters all-pole filters characterized by magnitude response. (N=filter order) Poles of H(s)H(-s) are equally spaced points on a circle of radius in s-plane poles of H(-s) poles of H(s) N=4

36 Analog IIR Filter Design Commonly used analog filters : Lowpass Butterworth filters monotonic in pass-band & stop-band N `maximum flat response : (2N-1) derivatives are zero at and

37 Analog IIR Filter Design 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:

38 Analog IIR Filter Design 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

39 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, bandstop, 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...

40 Analog -> Digital Principle : 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.

41 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) : s-plane jw z-plane j 1 -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)

42 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 s-plane jw z-plane j 1 -aliasing (!) if continuous-time response has significant frequency content above the Nyquist frequency

43 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. s-plane jw z-plane j -for low-frequencies, this is an approximation of -for high frequencies : significant frequency compression (`warping ) (sometimes pre-compensated by `pre-warping ) 1

44 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...

DIGITAL FILTERS. !! Finite Impulse Response (FIR) !! Infinite Impulse Response (IIR) !! Background. !! Matlab functions AGC DSP AGC DSP

DIGITAL FILTERS. !! Finite Impulse Response (FIR) !! Infinite Impulse Response (IIR) !! Background. !! Matlab functions AGC DSP AGC DSP DIGITAL FILTERS!! Finite Impulse Response (FIR)!! Infinite Impulse Response (IIR)!! Background!! Matlab functions 1!! Only the magnitude approximation problem!! Four basic types of ideal filters with magnitude