1. Analogue filtering UNIVERSITY of MANCHESTER School of Computer Science
11/16/2018
UNIVERSITY of MANCHESTER
School of Computer Science
Comp30291: Digital Media Processing
Section 2
Analogue filtering
4 Oct 2008
Comp30291 Section 2
Comp Oct'08

2. Analog system represented as ‘black box’
x(t)
y(t)
Inside we could have analogue components, or
Digital
processor
Analog
lowpass
filter 1
x(t)
Analog
lowpass
filter 2
ADC
y(t)
DAC Fs Fs
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3. Analog low-pass filters
Analog Lowpass Filter 1 is ‘antialiasing’ filter: removes any frequency components above Fs/2 before sampling process.
Analog Lowpass Filter 2 is ‘reconstruction’ filter: smoothes DAC output to remove all frequency components above Fs/2.
Digital processor controls ADC to sample at Fs Hz.
Also sends output sample to DAC at Fs samples per second.
DAC produces ‘staircase’ waveform: smoothed by ALpF2.
DAC output t
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4. Analogue filters Still needed in the world of DSP
Also, many digital filter designs are based on analog filters.
They are ‘linear’ & ‘time-invariant’ (LTI)
Analogue
filter
x(t)
y(t)
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5. Definition of ‘linearity’
System is LINEAR if
for any signal x(t),
if x(t)  y(t) then a.x(t )  a.y(t) for any constant a.
(2) for any signals x1(t) & x2(t),
if x1(t)  y1(t) & x2(t)  y2(t)
then x1(t) + x2(t)  y1(t) + y2(t)
(By x(t)  y(t) we mean that applying x(t) to the input produces the output signal y(t). )
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6. Alternative definition of ‘linearity’
System is linear if
for any signals x1(t) & x2(t), if x1(t)  y1(t) & x2(t)  y2(t)
then a1x1(t) + a2 x2(t)  a1y1(t) +a2 y2(t) for any a1 & a2
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7. Linearity (illustration)
system
If x1(t) y1(t) & x2(t) y2(t) then 3x1(t)+4x2(t)  3y1(t)+4y2(t) t
x2(t)
y2(t)
x1(t)
y1(t) +
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8. Definition of ‘time-invariance’
A time-invariant system must satisfy:
For any x(t), if x(t)  y(t) then x(t-)  y1(t-) for any 
Delaying input by  seconds delays output by  seconds
Not all systems have this property.
An LTI system is linear & time invariant.
An analogue filter is LTI.
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9. ‘System function’ for analogue LTI circuits
An analog LTI system has a system (or transfer) function
a0 + a1s + a2s aNsN
H(s) = 
b0 + b1s + b2s bMsM
Coeffs a0, a1, ...,aN, b0, ..., bM determine its behaviour.
Designer of analog lowpass filters must choose these carefully.
H(s) may be evaluated for complex values of s.
Setting s = j where  = 2f gives a complex function of f.
Modulus |H(j)| is gain at  radians/second (/2 Hz)
Argument of H(j) is phase-lead at  radians/s.
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10. Gain & phase response graphs
Gain: G() = |H(j)| Phase lead: () = Arg[H(j)|
-()
G(w)
Gain
Phase-lag 
f / (2)
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11. Effect of phase-response
It may be shown that:
when input x(t) = A cos(t),
output y(t) = A . G() . cos( t +() )
Output is sinusoid of same frequency as input.
‘Sine-wave in  sine-wave out’
Multiplied in amplitude by G() & ‘phase-shifted’ by ().
Example: If G() = 3 and () = /2 for all  what is the output?
Answer: y(t) = 3.A.cos(t + /2)
= 3.A.sin (t)
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12. Phase-shift expressed as a delay
Express y(t) = A . G() . cos( t +() )
as A . G() . cos ([t + ()/])
= A . G() . cos( [t - ()] ) where () = -()/
Cosine wave is delayed by -()/ seconds.
-()/ is ‘phase-delay’ in seconds
Easier to understand than ‘phase-shift’
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13. Linear phase
If -()/ is constant for all , all frequencies delayed by same time.
Then system is ‘linear phase’ - this is good.
Avoids changes in wave-shape due to ‘phase distortion’;
i.e different frequencies being delayed by differently.
Not all LTI systems are ‘linear phase’.
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14. Linear phase response graph
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15. Low-pass analog filters
Would like ideal ‘brick-wall’ gain response
& linear phase response as shown below: 
() G() 1 C
C = cut-off frequency
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16. Butterworth low-pass gain response
Cannot realise ideal ‘brick-wall’ gain response nor linear phase.
Can realise Butterworth approximation of order n:
Properties
(i) G(0) = ( 0 dB gain at =0)
(ii) G(C) = 1/(2) ( -3dB gain at  = C)
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17. Examples of Butterwth low-pass gain responses
Let C = 100 radians/second.
G(C) is always 1/(2)
Shape gets closer to ideal ‘brick-wall’ response as n increases.
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18. n=7 n = 2 n=4 LINEAR-LINEAR PLOT G() 1 / (2) radians/second 1 0.9
0.8
1 / (2)
0.7
0.6
0.5
0.4
0.3
0.2
n=7
n=4
n = 2
0.1 50
100
150
200
250
300
350
400
radians/second
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19. Butterworth gain responses on dB scale
Plot G() in dB, i.e. 20 log10(G()), against .
With  on linear or log scale.
As 20 log10(1/(2)) = -3, all curves are -3dB when  = C
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20. dB-LINEAR PLOT dB -10 n=2 -20 -30 n=4 -3dB -40 -50 -60 n=7 -70 -80 -90dB
-10
n=2
-20
-30
n=4
-3dB
-40
-50
-60
n=7
-70
-80
-90 50
100
150
200
250
300
350
400
radians/second
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21. dB-LOG PLOT dB 3 dB n=2 n=4 radians/second -10 -20 -30 -40 -50 -60dB
-10
3 dB
-20
-30
n=2
-40
-50
-60
n=4
-70
-80 1 2 3 10 10 10 10
radians/second
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Comp30291 Section 2

22. MATLAB program to plot these graphs
clear all;
for w = 1 : 400
G2(w) = 1/sqrt(1+(w/100)^4);
G4(w) = 1/sqrt(1+(w/100)^8) ;
G7(w) = 1/sqrt(1+ (w/100)^14);
end;
plot([1:400],G2,'r',[1:400],G4,'b',[1:400],G7,'k'); grid on;
DG2=20*log10(G2); DG4=20*log10(G4); DG7=20*log10(G7);
plot([1:400],DG2,'r',[1:400],DG4,'b',[1:400],DG7,'k'); grid on;
semilogx([1:990], DG2,'r', [1:990], DG4, 'b’);
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23. ‘Cut-off’ rate Best seen on a dB-Log plot
Cut-off rate is 20n dB per decade
or 6n dB per octave
at frequencies  much greater than C.
Decade is a multiplication of frequency by 10.
Octave is a multiplication of frequency by 2.
So for n=4, gain drops by 80 dB if frequency is multiplied by 10
or by 24 dB if frequency is doubled.
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24. Filter types - low-pass
G() 1
1 radian/s
Ideal
Approximatn
Low-pass with C = 1 
G() C
Low-pass
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25. Filter types - high-pass & band-pass1
High-pass  C
G()
Band-pass  L U
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26. Filter types - band-stop
G() 1  L U
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27. Two types of band-pass gain-responses1
Narrow-band
(U < 2 L )  L U
G() 1
Broad-band
(U > 2 L )  L U
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28. Three types of ‘band-stop’ gain-responses1
Narrow-band
(U < 2 L )  L U
G() 1
Broad-band
(U > 2 L )  L U
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29. Third type of ‘band-stop’ gain-response1
Notch  N
Yet another type of gain-response
G() 1
All-pass 
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30. Approximatns for high-pass, band-pass etc
Fortunately these can be derived from the formula for a Butterworth LOW-PASS gain response.
MATLAB does all the calculations.
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31. Other approximations
A filter which uses a Butterworth gain-response approximation of order n is an ‘nth order Butterworth type filter’.
In addition to Butterworth we have other approximations
Chebychev (types 1 & 2)
Elliptical
Bessel, etc.
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32. Problems 1. An analog filter has: H(s) = 1 / (1 + s)
Give its gain & phase responses & its phase delay at  = 1.
2. Use MATLAB to plot gain response of Butterwth type analog low-pass filter of order 4 with C = 100 radians/second.
Solution to (2):-
for w = 1 : 400
G(w) = 1/sqrt(1+(w/100)^8) ;
end;
plot(G); grid on;
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33. Result obtained:-
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