Digital Signal Processing FIR Filter Design

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
• 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 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
4
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

5
Review of discrete-time systems 3/10

• Impulse response/convolution

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

15
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



16
FIR Filters 3/14

• retiming results in...
  

uk
uk-1
uk-3
uk-2
b1
b4
b3
b2
x
x
x
x
yk



17
FIR Filters 4/14

• retiming repeated application results in...
  
• Transposed direct-form realization

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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)

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








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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




21
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



22
FIR Filters 9/14

• Lattice form derivation (IV), end result
  is...

uk
k4
yk
ko
k1
k2
k3
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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)

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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

25
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
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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

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FIR Filters 14/14

• Linear-phase FIR filters
• efficient direct-form realization.
• example

bo
b4
b3
b2
b1
x
x
x
x
x
yk




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

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Filter Design by Optimization

• this is equivalent to
• i.e. convex Quadratic Optimization problem
• this is often supplemented with additional
  constraints...

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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

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PS Filter Specification
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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

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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
  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)

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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.

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Filter Specification - Step 1
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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.

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Window Method

• First stage of this method is to calculate the
  coefficients of the ideal filter.
• This is calculated as follows

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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
40
Window Method

• The third stage is to calculate the set of
  truncated or windowed impulse response
  coefficients, hn

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(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)
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Window Method
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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)

44
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