Advanced Digital Signal Processing

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Advanced Digital Signal Processing
• Lecture 7
• FIR Digital Filters
• Alexander Kolesnikov
• 13.10.205

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Z-transform
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Z-transform X(z)
Let x(n) is a discrete set of points
Inverse Z-transform
where z is complex variable, C is closed contour
in the region of convergence includes all
singular points (poles) of X(z).
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Properties of Z-transform
1) Linearity
2) Time-shifting
3) Convolution
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Examples
1)
2)
3)
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Fourier- and Z- transforms
Discrete-time Fourier-transform
Z-transform
Fourier-transform equals its z-transform at zei?
zei? defines unit circle.
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Filters
Filter
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Types of filters
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Parameters of filters
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Parameters of filter
Rp? passband ripples Rs ? stopband atenuation ?p
? passband frequency ?s ? stopband frequency
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Types of digital filters

• Finite impulse responce (FIR) digital filters
• Infnite impulse responce (IIR) digital filters

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FIR Digital Filters
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Basic equation
Output signal y(n) is convolution of input signal
x(n) with profile function h(n)
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Block scheme

z-1delay element
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Impulse response
Input signal
Output signal
h(n)
n
1 2
M
FIR finite impulse responce
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Example 1 Average of two samples
1/2
0 1
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Example 2 Difference of two samples
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?1
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Transfer function

• Z-transform of impulse responce function

• Convolution in time-domain

• Multiplication in z-domain

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Transfer function block scheme
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Example 1 Average
Equation
Transfer function
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Example 1
Amplitude-frequency responce Phase-frequency
responce
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Example 1
Lowpass (smooth) linear-phase filter
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Example 2 Differentiator
Equation
Transfer function
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Example 2 Differentiator
Highpass (sharp) linear-phase filter
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Frequency reponce
Amplitude-frequency response
Phase-frequency response
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FIR filter design Frequency sampling

• Specify the desired frequency responce H0(?)
• Estimate the number of filter coefficients M
• Calculate ideal impulse responce function h0(n)
  as inverse Fourier
• transform of frequency responce H0(?) taken in
  K points
• H0(0), H0(??), , H0((K ? 1)???).
• Calculate windowing function w(n).
• Multiplay the ideal impulse responce h0(n) to M
  values h(n) to windowing function w(n)
  h0(n)h0(n)w(n)
• Perform Fourier transform of h(n) to calculate
  real frequency
• responce H (?).

The simplest window function is rectangular
coefficient truncation
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FIR filter design Frequency sampling

• Red points samples of desired
• frequency response
• Black line the actual frequency
• responce of truncated impulse
• responce.

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Frequency sampling Windowing
w kaiser(200,2.5) wvtool(w)
Use another windows Hamming, Hanning, Kaiser,
Blackman, etc.
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FIR filter design Frequency sampling
FS1000 Sampling frequency FBFS/2
Bending frequency (Nyquist freq in MtaLB) N73
Filter length beta5.65 Kaser window
riple param fc1125/FB Left cut-off
freq fc2275/FB Right cut-off freq FCfc1
fc2 Get windowed filter coeffs hnfir1(N-1,FC
,kaiser(N,beta)) Calc freq responce H,ffreq
z(hn,1,512,FS) mag10log10(abs(H)) plot(f,mag)
grid on xlabel('Frequency, Hz') ylabel('Magnitu
de Responce, Db')
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The filter coefficients
M36
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Frequency responce
Passband 125-275 Hz
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Signal filtration 50 Hz
x 10sin(2? f 1t)
f150 Hz
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Signal filtration 200Hz
x 10sin(2? f1t)
f1200 Hz
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Signal filtration 350 Hz
x 10sin(2? f 1t)
f1350 Hz
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Linear-phase filter design optimal method
The Parks-McClellan method is based on Remez
exchange algorithm and Chebyshev approximation
theory to design filters with an optimal fit
between the desired and actual frequency
responses. See remez() and firpm() functions.
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Approximation of frequency responce
FS10000 Sampling freq N41
Filter length Desired magnitude responce M0
0 1 1 0 0 Band edge relative freqs
(2f/fS) F0, 0.1, 0.2, 0.3, 0.4, 1 Calc
filter coeffs bremez(N-1,F,M) or
firpm(N-1,F,M) Calc freq responce
H,ffreqz(b,1,512,FS) mag10log10(abs(H)) pl
ot(f,mag)grid on xlabel('Frequency,
Hz') ylabel('Magnitude Responce, Db')
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The filter coeffficients

M20
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Amplitude- and phase-frequency responces
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More about filters

• Linear-phase filters
• Filter stability
• Filter casuality
• Noise and quantization effects in digital
  filters.

• MatLAB
• Unit frequency is the Nyquist frequency
• The Nyquist frequency is half the sampling
  frequency.

http//www.dsptutor.freeuk.com/analyser/SA102.html
http//www.dsptutor.freeuk.com/filter/FIRDigitalFi
lterDemo.html