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Digital Signal Processing FIR Filter Design

- Marc Moonen
- Dept. E.E./ESAT, K.U.Leuven

FIR Filter Design

- Review of discrete-time systems
- 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 u1k -gt output y1k
- input u2k -gt output y2k
- hence input a.u1kb.u2k-gt

a.y1kb.y2k - Time-invariant (shift-invariant)
- input uk -gt output yk
- hence input uk-T -gt output yk-T

uk

yk

Review of discrete-time systems 2/10

- Linear time-invariant (LTI) systems
- Causal systems
- for all input uk0, klt0 -gt output

yk0, klt0 - Impulse response
- input 1,0,0,0,... -gt output

h0,h1,h2,h3,... - input u0,u1,u2,u3 -gt output

y0,y1,y2,y3,... - convolution

Review of discrete-time systems 3/10

- Impulse response/convolution

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 uk -gt bounded output yk
- --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 hk
- 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 hk
- 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 z0 (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

FIR Filters 2/14

- retiming select subgraph (shaded)
- remove delay element

on all inbound arrows - add delay element on

all outbound arrows

uk

uk-4

uk-3

uk-2

uk-1

b4

b3

b2

b1

x

x

x

x

yk

FIR Filters 3/14

- retiming results in...

uk

uk-1

uk-3

uk-2

b1

b4

b3

b2

x

x

x

x

yk

FIR Filters 4/14

- retiming repeated application results in...

- Transposed direct-form realization

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)

uk

uk-1

uk-2

uk-3

uk-4

b1

b2

b3

bo

b4

x

x

x

x

x

b3

b2

b4

b1

bo

x

x

x

x

x

yk

FIR Filters 7/14

- Lattice form derivation (II), this is

equivalent to... -

(simple proof)

uk

uk-1

uk-2

uk-3

uk-4

b1

b2

b3

bo

0

x

x

x

x

x

b3

b2

yk

0

b1

bo

x

x

x

x

ko

FIR Filters 8/14

- Lattice form derivation (III), retiming leads

to... - repeat

procedure for shaded graph

uk

uk-3

uk-2

uk-2

bo

b1

b2

b3

x

x

x

x

b3

b2

yk

b1

bo

x

x

x

x

ko

FIR Filters 9/14

- Lattice form derivation (IV), end result

is...

uk

k4

yk

ko

k1

k2

k3

FIR Filters 10/14

- Lattice form
- kis are reflection coefficients
- procedure for computing kis from bis

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

kilt1 (i1,,N-1) - (ps procedure breaks down if ki1 is

encountered)

FIR Filters 11/14

- Linear-phase FIR filters
- Non-causal zero-phase filters
- example symmetric impulse response
- h-L,.h-1,h0,h1,...,hL
- hkh-k, k1..L
- frequency response is
- - real-valued (zero-phase) transfer

function - - causal implementation by introducing

(group) delay

FIR Filters 12/14

- Linear-phase FIR filters
- Causal linear-phase filters
- example symmetric impulse response N even
- h0,h1,.,hN
- N2L (even)
- hkhN-k, k0..L
- frequency response is
- - causal implementation of zero-phase

filter, by - introducing (group) delay

k

N

0

FIR Filters 13/14

- Linear-phase FIR filters
- Type-1 Type-2 Type-2

Type-4 - N2Leven N2L1odd N2Leven

N2L1odd - symmetric anti-symmetric symmetric

anti-symmetric - hkhN-k hk-hN-k

hkhN-k hk-hN-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

bo

b4

b3

b2

b1

x

x

x

x

x

yk

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 hdk to N1 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

wk 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

Window Method

- 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, hn

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(n0)eps

avoiding division by zero h

sin(n2pifc)./(npi) generate sequence of

ideal coefficients w 0.54

0.46cos(2pin/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 2104 -70 10) figure(2) ste

m(d) plot coefficient values xlabel(Coeffic

ient number) ylabel (Value) title(Truncated

Impulse Response) figure(3) freqz(d,1,512,4410

0) use freqz to plot magnitude and phase

response axis(0 2104 -70 10)

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 ais are such that the max

(maximum weighted approximation error) is

obtained at L2 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
- bfir1(n,Wn,type,window), windowed linear-phase

FIR design, n is filter order, Wn defines

band-edges, type is high,stop, - bfir2(n,f,m,window), windowed FIR design

based on inverse fourier transform with frequency

points f and corresponding magnitude response m - bremez(n,f,m), equiripple linear-phase FIR

design with Parks-McClellan (Remez exchange)

algorithm

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