Digital Signal Processing FIR Filter Design Marc Moonen Dept. E.E./ESAT, K.U.Leuven

FIR Filter Design Review of discrete-time systems FIR filters LTI systems, impulse response, transfer function, ... FIR filters Direct form, Lattice, Linear-phase filters FIR design by optimization Weighted least-squares design Minimax design FIR design using `windows’ Equiripple design Software (Matlab,…)

Review of discrete-time systems 1/10 Linear time-invariant (LTI) systems Linear : input u1[k] -> output y1[k] input u2[k] -> output y2[k] hence input a.u1[k]+b.u2[k]-> a.y1[k]+b.y2[k] Time-invariant (shift-invariant) input u[k] -> output y[k] hence input u[k-T] -> output y[k-T] u[k] y[k]

Review of discrete-time systems 2/10 Linear time-invariant (LTI) systems Causal systems: for all input u[k]=0, k<0 -> output y[k]=0, k<0 Impulse response : input 1,0,0,0,... -> output h[0],h[1],h[2],h[3],... input u[0],u[1],u[2],u[3] -> output y[0],y[1],y[2],y[3],... = `convolution’ u[k] y[k]

Review of discrete-time systems 3/10 Impulse response/convolution: u[0],u[1],u[2],u[3],0,0…. y[0],y[1],... h[0],h[1],h[2],0,0,... `Toeplitz’ matrix

Review of discrete-time systems 4/10 Z-Transform:

Review of discrete-time systems 5/10 Z-Transform : input-output relation `shorthand’ notation (for convolution operation/Toeplitz-vector product) stability bounded input u[k] -> bounded output y[k] --iff --iff poles of H(z) inside the unit circle (for causal,rational systems)

Review of discrete-time systems 6/10 Frequency response : given a system with impulse response h[k] given an input signal = complex sinusoid output signal : is `frequency response’ is H(z) evaluated on the unit circle

Review of discrete-time systems 7/10 Frequency response : for a real impulse response h[k] Magnitude response is even function Phase response is odd function example : Nyquist frequency

Review of discrete-time systems 8/10 `Popular’ frequency responses for filter design : low-pass (LP) high-pass (HP) band-pass (BP) band-stop multi-band

Review of discrete-time systems 9/10 Rational transfer functions (`IIR filters’): N poles (zeros of A(z)) , N zeros (zeros of B(z)) corresponds to difference equation

Review of discrete-time systems 10/10 `FIR filters’ (finite impulse response): `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

Review of discrete-time systems `FIR filter’ (finite impulse response) design this lecture +++ : phase control (linear phase) guaranteed stability design flexibility minor coefficient sensitivity/quantization/round-off problems,…. - - - : long filters, hence expensive

FIR Filters 1/14 `direct-form’ realization u[k] u[k-4] u[k-3] u[k-2] x bo + b4 b3 b2 b1 y[k]

FIR Filters 2/14 `retiming’ = select subgraph (shaded) remove delay element on all inbound arrows add delay element on all outbound arrows u[k] u[k-1] u[k-2] u[k-3] u[k-4] x bo + b1 b2 b3 b4 x x x x y[k] + + +

FIR Filters 3/14 `retiming’ : results in... u[k] u[k-1] u[k-2] u[k-3] x bo + b1 b2 b3 b4 x x x x y[k] + + +

FIR Filters 4/14 `retiming’ : repeated application results in... `Transposed direct-form’ realization u[k] x bo + y[k] b1 b2 b3 b4

FIR Filters 5/14 `Lattice form’ : derived from combined realization with reversed coefficient vector results in - same magnitude response - different phase response

FIR Filters 6/14 `Lattice form’ : derivation (I) u[k] u[k-1] u[k-2] bo b1 b2 b3 b4 x x x x x b4 b3 b2 b1 bo x x x x x y[k] + + + + y[k] + + + +

FIR Filters 7/14 `Lattice form’ : derivation (II), this is equivalent to... (=simple proof) u[k] u[k-1] u[k-2] u[k-3] u[k-4] b’o b’1 b’2 b’3 x x x x x y[k] b’3 b’2 b’1 b’o x x x x x + x + + + + ko + + + + x y[k] +

FIR Filters 8/14 `Lattice form’ : derivation (III), retiming leads to... …repeat procedure for shaded graph u[k] u[k-2] u[k-2] u[k-3] b’o b’1 b’2 b’3 x x x x y[k] b’3 b’2 b’1 b’o x x x x + x + + + ko + + + x y[k] +

FIR Filters 9/14 `Lattice form’ : derivation (IV), end result is... u[k] k4 x y[k] + + + + x x x x ko k1 k2 k3 x x x x y[k] + + + +

FIR Filters 10/14 `Lattice form’ : ki’s are `reflection coefficients’ procedure for computing ki’s from bi’s corresponds to `Schur-Cohn’ stability test (control theory): all zeros of B(z) are stable (i.e. lie inside unit circle) iff all reflection coefficients statisfy |ki|<1 (i=1,…,N-1) (ps: procedure breaks down if |ki|=1 is encountered)

FIR Filters 11/14 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 - real-valued (=zero-phase) transfer function - causal implementation by introducing (group) delay k L

FIR Filters 12/14 Linear-phase FIR filters Causal linear-phase filters : 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 - causal implementation of zero-phase filter, by introducing (group) delay k N

FIR Filters 13/14 Linear-phase FIR filters Type-1 Type-2 Type-2 Type-4 N=2L=even N=2L+1=odd N=2L=even N=2L+1=odd symmetric anti-symmetric 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 HP

FIR Filters 14/14 Linear-phase FIR filters efficient direct-form realization. example: u[k] + + + + + bo b1 b2 b3 b4 x x x x x y[k] + + + +

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,…) e.g.: LP optimization criterion is where is a weighting function

Filter Design by Optimization …this is equivalent to i.e. convex `Quadratic Optimization’ problem this is often supplemented with additional constraints...

Filter Design by Optimization Example: Low-pass (LP) design optimization criterion is an additional constraint may be imposed to control the pass-band ripple … … as well as the stop-band ripple

PS: Filter Specification

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

Filter Design using `Windows’ Example : Low-pass filter design (continued) it can be shown that this corresponds to the solution of weighted least-squares optimization with the given Hd, and weighting function for all freqs. 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, which cannot be reduced by increasing the filter order N.

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)

Design Procedure To fully design and implement a filter five steps are required: (1) Filter specification. (2) Coefficient calculation. (3) Structure selection. (4) Simulation (optional). (5) Implementation.

Filter Specification - Step 1

Coefficient Calculation - Step 2 There are several different methods available, the most popular are: Window method. Frequency sampling. Parks-McClellan. We will just consider the window method.

Window Method First stage of this method is to calculate the coefficients of the ideal filter. This is calculated as follows: ( ) ï î í ì = ¹ × ò - n for 2 sin 1 c j d f e H h w p

Window Method Using the Hamming Window: Second stage of this method is to select a window function based on the passband or attenuation specifications, then determine the filter length based on the required width of the transition band. Using the Hamming Window:

Window Method The third stage is to calculate the set of truncated or windowed impulse response coefficients, h[n]: for Where: for

Window Method Matlab code for calculating coefficients: close all; clear all; fc = 8000/44100; % cut-off frequency N = 133; % number of taps n = -((N-1)/2):((N-1)/2); n = n+(n==0)*eps; % avoiding division by zero [h] = sin(n*2*pi*fc)./(n*pi); % generate sequence of ideal coefficients [w] = 0.54 + 0.46*cos(2*pi*n/N); % generate window function d = h.*w; % window the ideal coefficients [g,f] = freqz(d,1,512,44100); % transform into frequency domain for plotting figure(1) plot(f,20*log10(abs(g))); % plot transfer function axis([0 2*10^4 -70 10]); figure(2); stem(d); % plot coefficient values xlabel('Coefficient number'); ylabel ('Value'); title('Truncated Impulse Response'); figure(3) freqz(d,1,512,44100); % use freqz to plot magnitude and phase response

Window Method

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 ai’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)

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