Digital Signal Processing II Chapter 3: FIR & IIR Filter Design Marc Moonen Dept. E.E./ESAT, K.U.Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/scd/

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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) Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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) Version 2010-2011 Chapter-3: FIR/IIR Filter Design

Filter Specification Ex: Low-pass Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

FIR Filter Design by Optimization …this is equivalent to = `Quadratic Optimization’ problem Version 2010-2011 Chapter-3: FIR/IIR Filter Design

FIR Filter Design by Optimization Example: Low-pass design optimization function is i.e. Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

FIR Filter Design by Optimization Skip this slide 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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

Filter Design by Optimization Skip this slide 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. Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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. Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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) Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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) Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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) Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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) Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 : Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 : Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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) Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

Analog IIR Filter Design Skip this slide 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: Version 2010-2011 Chapter-3: FIR/IIR Filter Design

Analog IIR Filter Design Skip this slide 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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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... Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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. Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design

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 Version 2010-2011 Chapter-3: FIR/IIR Filter Design