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Chapter 4

Chapter 4. Digital Processing of Continuous-Time Signals. §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,

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Chapter 4

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  1. Chapter 4 Digital Processing of Continuous-Time Signals

  2. §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

  3. §4.1 Introduction • Conversion of a continuous-time signal into digital form is carried out by an analog-to- digital (A/D) converter • The reverse operation of converting a digital signal into a continuous-time signal is performed by a digital-to-analog (D/A) converter

  4. §4.1 Introduction • Since the A/D conversion takes a finite amount of time, a sample-and-hold (S/H) circuitis used to ensure that the analog signal at the input of the A/D converter remains constant in amplitude until the conversion is complete to minimize the error in its representation

  5. §4.1 Introduction • To prevent aliasing, an analog anti-aliasing filteris employed before the S/H circuit • To smooth the output signal of the D/A converter, which has a staircase-like waveform, an analog reconstruction filteris used

  6. Anti- Aliasing filter S/H A/D D/A Reconstruction filter Digital processor §4.1 Introduction • Complete block-diagram • Since both the anti-aliasing filter and the reconstruction filter are analog lowpass filters, we review first the theory behind the design of such filters • Also, the most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype

  7. §4.2 Sampling of Continuous-Time Signals • As indicated earlier, discrete-time signals in many applications are generated by sampling continuous-time signals • We have seen earlier that identical discrete- time signals may result from the sampling of more than one distinct continuous-timefunction

  8. §4.2 Sampling of Continuous-Time Signals • In fact, there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal • However, under certain conditions, it is possible to relate a unique continuous-time signal to a given discrete-time signal

  9. §4.2 Sampling of Continuous-Time Signals • If these conditions hold, then it is possible to recover the original continuous-time signal from its sampled values • We next develop this correspondence and the associated conditions

  10. §4.2.1 Effect of Sampling in the Frequency Domain • Let ga(t) be a continuous-time signal that is sampled uniformly at t = nT, generating the sequence g[n] where g[n]=ga(nT), -∞ <n<∞ with Tbeing the sampling period • The reciprocal of Tis called the samplingfrequencyFT , i.e., FT=1/T

  11. §4.2.1 Effect of Sampling in the Frequency Domain • Now, the frequency-domain representation of ga(t) is given by its continuos-time Fourier transform(CTFT): • The frequency-domain representation of g[n] is given by its discrete-time Fourier transform (DTFT):

  12. §4.2.1 Effect of Sampling in the Frequency Domain • To establish the relation between Ga(jΩ) and G(ejω),wetreat the sampling operationmathematically as a multiplication of ga (t) by a periodic impulse trainp(t):

  13. §4.2.1 Effect of Sampling in the Frequency Domain • p(t) consists of a train of ideal impulses with a period Tas shown below • The multiplication operation yields an impulse train:

  14. §4.2.1 Effect of Sampling in the Frequency Domain • g p(t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nTweighted by the sampled value g a(nT) of g a(t) at the instant

  15. §4.2.1 Effect of Sampling in the Frequency Domain • There are two different forms of Gp(jΩ): • One form is given by the weighted sum of the CTFTs of δ(t-nT) : • To derive the second form, we make use of the Poisson’s formula: where ΩT=2π/T and Φ(jΩ) is the CTFT of φ(t)

  16. §4.2.1 Effect of Sampling in the Frequency Domain • For t=0 reduces to • Now, from the frequency shifting property of the CTFT, the CTFT of g a(t) e-jΨt is given by Ga(j(Ω+Ψ))

  17. §4.2.1 Effect of Sampling in the Frequency Domain • Substituting φ(t)=ga(t)e-jΨt in we arrive at • By replacing Ψ with Ω in the above equation we arrive at the alternative form of the CTFT G p(jΩ) of g p(t)

  18. §4.2.1 Effect of Sampling in the Frequency Domain • The alternative form of the CTFT of g p(t) is given by • Therefore, G p(jΩ) is a periodic function of Ω consisting of a sum of shifted and scaled replicas of G a(jΩ) , shifted by integer multiples of ΩTand scaled by 1/T

  19. §4.2.1 Effect of Sampling in the Frequency Domain • The term on the RHS of the previous equation for k=0 is the basebandportion of G p(jΩ), and each of the remaining terms are the frequency translated portions of G p(jΩ) • The frequency range is called the basebandor Nyquist band

  20. Ga(jΩ) Ω Ωm -Ωm 0 P(jΩ) 1 … … Ω ΩT -ΩT 0 2ΩT 3ΩT §4.2.1 Effect of Sampling in the Frequency Domain • Assume ga(t) is a band-limited signal with a CTFT Ga(jΩ) as shown below • The spectrum P(jΩ) of p(t) having a sampling period T=2π/ ΩT is indicated below

  21. §4.2.1 Effect of Sampling in the Frequency Domain • Two possible spectra of G p(jΩ) are shown below

  22. §4.2.1 Effect of Sampling in the Frequency Domain • It is evident from the top figure on the previous slide that if ΩT>2Ωm,there is no overlap between the shifted replicas of Ga(jΩ) generating Gp(jΩ) • On the other hand, as indicated by the figure on the bottom, if ΩT<2Ωm ,there is anoverlap of the spectra of the shifted replicas of Ga(jΩ) generating Gp(jΩ)

  23. If ΩT>2Ωm , ga(t) can be recovered exactly from gp(t) by passing it through an ideal lowpass filterHr(jΩ) witha gain T and a cutoff frequency Ωc greater than Ωm and less than ΩT -Ωm as shown below gp(t) Hr(jΩ) ga(t) p(t) §4.2.1 Effect of Sampling in the Frequency Domain

  24. §4.2.1 Effect of Sampling in the Frequency Domain • The spectra of the filter and pertinent signals are shown below

  25. §4.2.1 Effect of Sampling in the Frequency Domain • On the other hand, if ΩT<2Ωm ,due to the overlap of the shifted replicas of Ga(jΩ) , the spectrum Ga(jΩ) cannot be separated byfiltering to recover Ga(jΩ) because ofthedistortion caused by a part of the replicas immediately outside the baseband folded back or aliasedinto the baseband

  26. §4.2.1 Effect of Sampling in the Frequency Domain Sampling theorem– Let ga(t) be a band- limited signal with CTFTGa(jΩ) =0 for|Ω|>Ωm • Then ga(t) is uniquely determined by its samples ga(nT),-∞≤n≤∞ if ΩT≥2Ωm where ΩT=2π/T

  27. §4.2.1 Effect of Sampling in the Frequency Domain • The condition ΩT≥2Ωm is often referred to as the Nyquist condition • The frequency ΩT/2 is usually referred to asthe folding frequency

  28. §4.2.1 Effect of Sampling in the Frequency Domain • Given {ga(nT) }, we can recover exactly ga(t)by generating an impulse train and then passing it through an ideal lowpass filter Hr(jΩ) with a gain T and a cutoff frequency Ωcsatisfying m< c<(T - m)

  29. §4.2.1 Effect of Sampling in the Frequency Domain • The highest frequency Ωm contained in ga(t)is usually called the Nyquist frequency since it determines the minimum sampling frequency ΩT=2Ωm that must be used to fully recover ga(t) from its sampled version • The frequency 2Ωm is called the Nyquist rate

  30. §4.2.1 Effect of Sampling in the Frequency Domain • 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

  31. §4.2.1 Effect of Sampling in the Frequency Domain • 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

  32. §4.2.1 Effect of Sampling in the Frequency Domain • 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

  33. §4.2.1 Effect of Sampling in the Frequency Domain • Example - Consider the three continuous- time sinusoidal signals: ga(t)=cos(6πt) ga(t)=cos(14πt) ga(t)=cos(26πt) • Their corresponding CTFTs are: G1(jΩ)=π[δ(Ω-6π)+δ(Ω+6π)] G2(jΩ)=π[δ(Ω-14π)+δ(Ω+14π)] G3(jΩ)=π[δ(Ω-26π)+δ(Ω+26π)]

  34. §4.2.1 Effect of Sampling in the Frequency Domain • These three transforms are plotted below

  35. §4.2.1 Effect of Sampling in the Frequency Domain • These continuous-time signals sampled at a rate of T = 0.1 sec, i.e., with a sampling frequency ΩT=20πrad/sec • The sampling process generates the continuous-time impulse trains,g1p(t), g2p(t),andg3p(t) • Their corresponding CTFTs are given by

  36. §4.2.1 Effect of Sampling in the Frequency Domain • Plots of the 3 CTFTs are shown below

  37. §4.2.1 Effect of Sampling in the Frequency Domain • These figures also indicate by dotted lines the frequency response of an ideal lowpass filter with a cutoff at Ωc=ΩT/2=10π and a gain T=0.1 • The CTFTs of the lowpass filter output are also shown in these three figures • In the case of g1(t), the sampling rate satisfies the Nyquist condition, hence no aliasing

  38. §4.2.1 Effect of Sampling in the Frequency Domain • Moreover, the reconstructed output is precisely the original continuous-time signal • In the other two cases, the sampling rate does not satisfy the Nyquist condition, resulting in aliasing and the filter outputs are all equal tocos(6πt)

  39. §4.2.1 Effect of Sampling in the Frequency Domain • Note: In the figure below, the impulse appearing at Ω = 6π in the positive frequency passband of the filter results from the aliasing of the impulse in G2(jΩ) at Ω = −14π • Likewise, the impulse appearing atΩ = 6πin the positive frequency passband of the filter results from the aliasing of the impulseinG3(jΩ)atΩ = 26π

  40. §4.2.1 Effect of Sampling in the Frequency Domain • We now derive the relation between the DTFT of g[n] and the CTFT of gp[t] • To this end we compare with and make use of g[n]=ga[nT] ,-∞<n< ∞

  41. §4.2.1 Effect of Sampling in the Frequency Domain • Observation: We have or, equivalently, • From the above observation and

  42. §4.2.1 Effect of Sampling in the Frequency Domain we arrive at the desired result given by

  43. §4.2.1 Effect of Sampling in the Frequency Domain • 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

  44. §4.2.1 Effect of Sampling in the Frequency Domain • Now, the CTFT Gp(jΩ)is a periodicfunction of Ω with a period ΩT=2π/T • Because of the mapping, the DTFT G(ejω) is a periodic function of ω with a period 2π

  45. §4.2.2 Recovery of the Analog Signal • We now derive the expression for the output ĝa(t) of the ideal lowpass reconstruction filter Hr(jΩ) as a function of the samples g[n] • The impulse response hr(t) of the lowpass reconstruction filter is obtained by taking the inverse DTFT of Hr(jΩ) :

  46. §4.2.2 Recovery of the Analog Signal • Thus, the impulse response is given by • The input to the lowpass filter is the impulse train gp(t) :

  47. ^ * §4.2.2 Recovery of the Analog Signal • Therefore, the output ĝa(t) of the ideal lowpass filter is given by: • Substituting hr(t) =sin(Ωct)/(ΩTt/2) in the above and assuming for simplicity c= T/2= /T, we get

  48. §4.2.2 Recovery of the Analog Signal • The ideal bandlimited interpolation process is illustrated below

  49. §4.2.2 Recovery of the Analog Signal • It can be shown that when Ωc= ΩT/ 2 in hr(0)=1 andhr(nT)=0 for n≠0 • As a result, from we observe for all integer values of r in the range-∞<r<∞

  50. §4.2.2 Recovery of the Analog Signal • The relation holds whether or not the condition of the sampling theorem is satisfied • However, ĝa(rT)=ga(rT) for all values of t only if the sampling frequency ΩTsatisfiesthe condition of the sampling theorem

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