Chapter 4

1. Chapter 4 Digital Processing of Continuous-Time Signals

2. Outline • Sampling of Continuous-Time Signals • Recovery of Analog Signals • Implication of the Sampling Process • Sampling of Bandpass Signals • Analog Lowpass Filter Design

3. 4.1 Introduction • Digital processing of a continuous-time signal involves the following basic steps: (1) Conversion of the continuous-time signal into a discrete-time signal; (2) Processing of the discrete-time signal; (3) Conversion of the processed discrete-time signal back into a continuous-time signal.

4. Some concepts: 1. ADC: Analog-to-Digital Converter 2. DAC: Digital-to-Analog Converter 3. S/H : Sample-and-Hold (circuit). 4. Anti-aliasing filter. 5. Reconstruction or interpolation filter.

5. (a) (b) t t t t t t 0 0 0 0 0 0 T T 4.2 Sampling of Continuous-Time Signals Ideal sampling models

6. 4.2.1 Effect of Sampling in the Frequency Domain • When τ« T, p(t)~δT(t); so we firstly discuss ideal sampling: The output of ideal sampling is: (1)In time domain:

7. (2)In frequency domain: If given: Then:

8. … … 0 0 … … 0 The frequency procedure of sampling • Gp(jΩ) is a periodic function of Ω consisting of a sum of shifted and scaled replicas of Ga(jΩ) , shifted by integer multiples of ΩT and scaled by 1/T.

9. Assume ga(t) is a band-limited signal with a CTFT Ga(j) as shown below: • The spectrum P(j) of p(t) having a sampling period T=2/T is indicated below:

10. Two possible spectra of Gp(j) are shown below:

11. It is evident from the top figure on the previous slide that if T>2 m , there is no overlap between the shifted replicas of Ga(j) generating Gp(j); • On the other hand, as indicated by the figure on the bottom, if T<2 m , there is an overlap of the spectra of the shifted replicas of Ga(j) generating Gp(j).

12. Sampling theorem • Let ga(t) be a band-limited signal with CTFT Ga(j)=0 for | |> m , Then ga(t) is uniquely determined by its samples ga(nT) , -n if T  2 m , whereT=2/T. • The condition T  2 m is often referred to as the Nyquist condition. • The frequency T/2 is usually referred to as the folding frequency.

13. The highest frequency m contained in ga(t) is usually called the Nyquist frequency since it determines the minimum sampling frequency T =2m that must be used to fully recover ga(t) from its sampled version. • The frequency 2m is called the Nyquist rate.

14. Oversampling - The sampling frequency is higher than the Nyquist rate; • Undersampling - The sampling frequency is lower than the Nyquist rate; • Critical sampling - The sampling frequency is equal to the Nyquist rate; • Note: A pure sinusoid may not be recoverable from its critically sampled version.

15. Examples • In digital telephony, a 3.4 kHz signal bandwidth is acceptable for telephone conversation; • Here, a sampling rate of 8 kHz, which is greater than twice the signal bandwidth, is used. • In high-quality analog music signal processing, a bandwidth of 20 kHz has been determined to preserve the fidelity; • Hence, in compact disc (CD) music systems, a sampling rate of 44.1 kHz, which is slightly higher than twice the signal bandwidth, is used.

16. Example4.1 Consider the three continuous-time sinusoidal signals: • Their corresponding CTFTs are:

17. These three transforms are plotted below

18. These continuous-time signals sampled at a rate of T = 0.1 sec, i.e., with a sampling frequency T =20 rad/sec • The sampling process generates the continuous-time impulse trains, g1p(t), g2p(t) , and g3p(t) • Their CTFTs are given by:

19. Plots of the 3 CTFTs are shown below

20. And • These figures also indicate by dotted lines the frequency response of an ideal lowpass filter with a cutoff at c=T/2=10 and a gain T=0.1. • In fact, the 3 discrete-time sinusoidal signals are:

21. The Relationship BetweenG(ejw) and Gp(jΩ) • We now derive the relation between the DTFT of g[n] and the CTFT of gp(t) • To compare: with And make use of g[n]=ga(nT), -∞<n<∞

22. Observation: We have G(ejω)=Gp(jΩ)|Ω=ω/T Or, equivalently, Gp(jΩ)=G(ejω)|ω=ΩT From the above observation and

23. We arrive at the desired result:

24. The relation derived on the previous slide can be alternately expressed as From Or from It follows that G(ejω) is obtained from Gp(jΩ) by applying the mapping Ω=ω/T ★

25. Now, the CTFT Gp(j W) is a periodic function of W with a period WT = 2p/T • Because of the mapping, the DTFT G(ej) is a periodic function of w with a period 2p

26. Supplement: The practical sampling • In practical condition, p(t) is a periodic rectangle impulse with τ-width . If the τ and T are fixed, then when k varies, the magnitude of Ck varies with:

27. 0 0 Supplement: The practical sampling • So we can see, in practical sampling, use Ck to substitute Ak, but Ck is varying.

28. 4.2.2 Recovery of the Analog Signal By filtering in frequency domain or interpolation in time domain, the continuous signal can be recovered from The Sampled Discrete signal. • Here, the filter is called reconstruction filter or interpolation filter or smoothing filter.

29. The spectra of the filter and pertinent signals are shown below:

30. If T>2 m , ga(t) can be recovered exactly from gp(t) by passing it through an ideal lowpass filter Hr(j) with a gain T and a cutoff frequency c greater than m and less than T - m; • On the other hand, if T< 2 m , due to the overlap of the shifted replicas of Ga(j) , the spectrum Gp(j) cannot be separated by filtering to recover Ga(j). The distortion caused by a part of the replicas outside the baseband folded back or aliased into the baseband.

31. We now derive the expression for the output of the ideal lowpass reconstruction filter Hr(j) as a function of the samples g[n]. ^ • The impulse response hr(t) of the ideal lowpass reconstruction filter is obtained by taking the inverse DTFT of Hr(j)

32. Thus, the impulse response is given by • The input to the lowpass filter is the impulse traingp(t):

33. Therefore, the output of the ideal lowpass filter is given by: ^ ^ ^ * Applying hr(t)=sin(ct)/(Tt/2), and assuming: c= T/2= /T , we get:

34. ^ is called Poisson sum formula or interpolation formula.

35. 4.3 Sampling of Bandpass Signals • The conditions developed earlier for the unique representation of a continuous-time signal by the discrete-time signal obtained by uniform sampling assumed that the continuous-time signal is bandlimited in the frequency range from DC to m ---lowpass signal. • There are continuous-time signals which are bandlimited to a higher frequency range L || H with L >0 --- bandpass signals.

36. To prevent aliasing a bandpass signal can of course be sampled at a rate greater than twice the highest frequency, i.e. by ensuring T2 H • However, due to the bandpass spectrum of the continuous-time signal, the spectrum of the discrete-time signal obtained by sampling will have spectral gaps with no signal components present in these gaps. • Moreover, if H is very large, the sampling rate also has to be very large which may not be practical in some situations.

37. Let = H - L define the bandwidth of the bandpass signal. • Assume:H= M() • We choose the sampling frequency T to satisfy the condition: T= 2() = 2H/M • T is smaller than 2H , the Nyquist rate undersampling

38. As before, Gp(j) consists of a sum of Ga(j) and replicas of Gp(j) shifted by integer multiples of twice the bandwidth DW and scaled by 1/T; • The amount of shift for each value of k ensures that there will be no overlap between all shifted replicas. no aliasing

39. 0 0 • Figure below illustrate the idea:

40. Discuss: (1)If ga(t) is bandlimited with its frequency band in (fL, fH) . (2)If using sampling rate: Where, n is a maximum integer which meets fs>=2(fH-fL)=2B (n = 0,1,2……) Then, the sampled signal ga(nTs) can be used to represent original signal ga(t).

41. Note: • Before band-sampling, the signal in only one frequency-band can be permitted. • For lowest sampling rate, that is fs=2B, the signal center frequency f0 should be:

42. The band-sampling result is moving the signal in the frequency band of (nB, (n+1)B)(n=0,1,2,…) to the frequency band of (0, B). • When n is odd number, the relationship between them is reversal; while n is even, the relationship is opposite.

43. As can be seen, ga(t) can be recovered from gp(t) by passing it through an ideal bandpass filter with a passband given by L || H and a gain of T. • Note: Any of the replicas in the lower frequency bands can be retained by passing through bandpass filters with passbands L- k()  || H- k() , 1k  M-1providing a translation to lower frequency ranges.

44. 4.4 Analog Lowpass Filter Design • Filter specifications • Several kinds of analog filters (1) Butterworth approximation (2) Chebyshev approximation:Type1 and Type2 (3) Elliptic approximation • Analog filter design using MATLAB • Design of analog Highpass, Bandpass and Bandstop filters

45. 4.4.1 Filter Specifications

46. Analog Lowpass Filter Specifications • In the passband, defined by 0  p , we require 1-p  |Ha(j)|  1+ p , ||  p i.e., |Ha(j)| approximates unity within an error of p • In the stopband, defined by s   , we require |Ha(j)|  s s    i.e., |Ha(j)| approximates zero within an error of s

47. dB dB Analog Lowpass Filter Specifications • p - passband edge frequency • s - stopband edge frequency • p - peak ripple value in the passband • s - peak ripple value in the stopband • Peak passband ripple • Minimum stopband attenuation

48. Analog Lowpass Filter Specifications • Magnitude specifications may alternately be given in a normalized form as indicated below

49. In this situation minimum stopband attenuation : Transtion ratio or selectivity parameter Discrimination parameter Obviously, the smaller the ε is, the better the Ha(jΩ) is.

50. Attenuation function or Loss function