1. 15 Oct 2009Comp30291 Section 21 UNIVERSITY of MANCHESTER School of Computer Science Comp30291: Digital Media Processing 2009-10 Section 2 Analogue filtering

2. 15 Oct 2009Comp30291 Section 22 Analog system represented as ‘black box’ x(t)y(t) Inside we could have analogue components, or Analog lowpass filter 1 ADC Digital processor DAC Analog lowpass filter 2 x(t) y(t) Fs

3. 15 Oct 2009Comp30291 Section 23 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

4. 15 Oct 2009Comp30291 Section 24 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)

5. 15 Oct 2009Comp30291 Section 25 System is LINEAR if (1)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 x 1 (t) & x 2 (t), if x 1 (t)  y 1 (t) & x 2 (t)  y 2 (t) then x 1 (t) + x 2 (t)  y 1 (t) + y 2 (t) (By x(t)  y(t) we mean that applying x(t) to the input produces the output signal y(t). ) Definition of ‘linearity’

6. 15 Oct 2009Comp30291 Section 26 Alternative definition of ‘linearity’ System is linear if for any signals x 1 (t) & x 2 (t), if x 1 (t)  y 1 (t) & x 2 (t)  y 2 (t) then a 1 x 1 (t) + a 2 x 2 (t)  a 1 y 1 (t) +a 2 y 2 (t) for any a 1 & a 2

7. 15 Oct 2009Comp30291 Section 27 Linearity (illustration) Linear system If x 1 (t)  y 1 (t) & x 2 (t)  y 2 (t) then 3x 1 (t)+4x 2 (t)  3y 1 (t)+4y 2 (t) t x 2 (t) t y 2 (t) x 1 (t) t t y 1 (t) + +

8. 15 Oct 2009Comp30291 Section 28 Definition of ‘time-invariance’ A time-invariant system must satisfy: For any x(t), if x(t)  y(t) then x(t-  )  y 1 (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.

9. 15 Oct 2009Comp30291 Section 29 a 0 + a 1 s + a 2 s 2 +... + a N s N H(s) =  b 0 + b 1 s + b 2 s 2 +... + b M s M ‘System function’ for analogue LTI circuits An analog LTI system has a system (or transfer) function Coeffs a 0, a 1,...,a N, b 0,..., b M 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.

10. 15 Oct 2009Comp30291 Section 210 Gain & phase response graphs G(  Gain: G(  ) = |H(j  )| Phase lead:  (  ) = Arg[H(j  )| -()-() Gain Phase-lag  f / (2  )

11. 15 Oct 2009Comp30291 Section 211 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) Effect of phase-response

12. 15 Oct 2009Comp30291 Section 212 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’ Phase-shift expressed as a delay

13. 15 Oct 2009Comp30291 Section 213 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’. Linear phase

14. 15 Oct 2009Comp30291 Section 214 Linear phase response graph

15. 15 Oct 2009Comp30291 Section 215 Low-pass analog filters Would like ideal ‘brick-wall’ gain response & linear phase response as shown below:   (  ) G(  ) 1 0 CC  C = cut-off frequency

16. 15 Oct 2009Comp30291 Section 216 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) = 1 ( 0 dB gain at  =0) (ii) G(  C ) = 1/(  2) ( -3dB gain at  =  C )

17. 15 Oct 2009Comp30291 Section 217 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.

18. 15 Oct 2009Comp30291 Section 218 050100150200250300350400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 radians/second G(  ) n = 2 n=4 n=7 1 /  (2) LINEAR-LINEAR PLOT

19. 15 Oct 2009Comp30291 Section 219 Butterworth gain responses on dB scale Plot G(  ) in dB, i.e. 20 log 10 (G(  )), against . With  on linear or log scale. As 20 log 10 (1/  (2)) = -3, all curves are -3dB when  =  C

20. 15 Oct 2009Comp30291 Section 220 050100150200250300350400 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 dB radians/second -3dB dB-LINEAR PLOT n=2 n=4 n=7

21. 15 Oct 2009Comp30291 Section 221 10 0 1 2 3 -80 -70 -60 -50 -40 -30 -20 -10 0 dB radians/second dB-LOG PLOT n=2 n=4  3 dB

22. 15 Oct 2009Comp30291 Section 222 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’); MATLAB program to plot these graphs

23. 15 Oct 2009Comp30291 Section 223 ‘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.

24. 15 Oct 2009Comp30291 Section 224  G(  ) 1 CC Low-pass with  C = 1 Low-pass  G(  ) 1 1 radian/s Ideal Approximatn Filter types - low-pass

25. 15 Oct 2009Comp30291 Section 225   G(  ) 1 G(  ) 1 CC LL UU High-pass Band-pass Filter types - high-pass & band-pass

26. 15 Oct 2009Comp30291 Section 226  G(  ) 1 LL UU Filter types - band-stop

27. 15 Oct 2009Comp30291 Section 227   G(  ) 1 LL UU Narrow-band (  U < 2  L ) Broad-band (  U > 2  L ) Two types of band-pass gain-responses LL UU G(  ) 1

28. 15 Oct 2009Comp30291 Section 228   G(  ) 1 LL UU Narrow-band (  U < 2  L ) Broad-band (  U > 2  L ) Three types of ‘band-stop’ gain-responses LL UU G(  ) 1

29. 15 Oct 2009Comp30291 Section 229   G(  ) 1 NN Notch All-pass Third type of ‘band-stop’ gain-response Yet another type of gain-response G(  ) 1

30. 15 Oct 2009Comp30291 Section 230 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.

31. 15 Oct 2009Comp30291 Section 231 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. Other approximations

32. 15 Oct 2009Comp30291 Section 232 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;

33. 15 Oct 2009Comp30291 Section 233 Result obtained:-