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

2. 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,…)

3. 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]

4. 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]

5. 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

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

7. 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)

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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]

15. 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] + + +

16. 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] + + +

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

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

19. 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] + + + +

20. 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] +

21. 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] +

22. 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] + + + +

23. 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)

24. 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

25. 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

26. 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

27. 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] + + + +

28. 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

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

30. 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

31. PS: Filter Specification

32. 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

33. 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.

34. 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)

35. 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.

36. Filter Specification - Step 1

37. 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.

38. 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 = × ò - n
for 2
sin 1 c j d f e H h w p

39. 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:

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

41. 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] = *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^ ]);
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

42. Window Method

43. 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)

44. 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