Lecture 19: Filter design

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Lecture 19Filter design

• Svetoslav NikolovØrstedDTU, Building 349

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Today

• Causality
• Linear phase filters
• Filter design using window functions
• Filter design using windowed sinc function
• Design IIR using SPtool.

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Filter causality

• Frequency response H(?) cannot be 0, except at a
  finite set of points
• H(?) cannot be constant in any finite range of
  ?
• Transition from stop- to pass-band cannot be
  infinitely sharp
• Real and imaginary parts are related via Hilbert
  transform.

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Characteristics of practical filters
Fdatool demo!
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Linear phase filters
Input signal
Non-linear phase
Demo non-linear phase
Linear phase
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Symmetric and Antisymmetric FIR filters
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Shapes of window functions
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Rectangular window
61 coefficients
31 coefficients
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Hamming window
Hamming
Rectangular
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Windowed sinc rectangular window
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Using other windows
Rectangular window
Hamming window
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Example

• Design a 15 coefficient causal law-pass filter
  with a bandwidth of B Hz and sampling rate of 4B
  Hz. Calculate the amplitude of the actual
  transfer function.
• Apply a Hamming filter on the designed filter and
  determine the resulting frequency response.

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Example result 1
i 014 b sinc((i-7)/2)/2 freqz(b,1)
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Example result 2
i 014 b sinc((i-7)/2)/2 b
b.hamming(15)' freqz(b,1)
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Design of optimum linear phase filter
f 0 400 600 900 1100 2500/2500 m 0 0 1 1
0 0 b remez(31, f, m) freqz(b,1,400, 5000)
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Recursive filters

• Chebyshev (???????, Chebychev, Tcsheysheff,
  Tchebichef) filters are based on a mathematical
  strategy for achieving a faster roll-off by
  allowing a ripple in frequency response.
• Butterworth filter has a ripple of 0 .
• Filters with ripple in pass-band are type 1, and
  with ripple in the stop-band are type 2 (seldom
  used).
• Elliptic filters have ripples both in pass- and
  stop-band

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Designing recursive filters in Matlab

• Use Matlab to design an 8th order elliptic
  lowpass filter with a cutoff freq. of 300 Hz, 0.5
  dB ripple in pass band and a minimum stop-band
  attenuation of 50 dB for use with a signal
  sampled at 4 kHz.
• Evaluate the response using 53, 20 and 12 bit
  precision.
• Implement it as a cascade of second order
  sections.

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Example with elliptic filter
b, a ellip(8, .5, 50, 300/2000) z, p, k
ellip(8, .5, 50, 300/2000) zplane(b,a) b12
round(b211)/211 a12 round(b211)/211 zpl
ane(b12,a12)
53-bit precision (double)
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12-bit coefficients
b, a ellip(8, .5, 50, 300/2000) z, p, k
ellip(8, .5, 50, 300/2000) zplane(b,a) b12
round(b211)/211 a12 round(b211)/211 zpl
ane(b12,a12)
12-bit precision
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The transfer function
53 bit
subplot(3,1,1) h,f freqz(b,a, 512, 4000) m
20log10(abs(h)) plot(f,m) axis(0 2000 -80
20) ylabel ('H(f) dB')
20 bit
12 bit
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Using 4 2nd-order sections
m zp2sos(z,p,k) m12 round (m211)/211 fig
ure subplot(2,2,1) zplane(m12(1,13),
m12(1,46))
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Summary

• Symmetric FIR filters have a linear phase.
• A simple method to design them is to use
  windowing functions in combination with a
  truncated sinc.
• Recursive filters have better performance, but
  the phase is not linear and are subject to
  stability issues with integer coefficients.