2A1H Time-Frequency Analysis II

Transcription

1 2AH Time-Frequency Analysis II Bugs/queries to HT 209 For any corrections see the course page DW Murray at dwm/courses/2tf. (a) A signal g(t) with period T is constructed by repeating the f (t) of Figure. Find the coefficients C n of the complex Fourier series describing g(t). (b) Use the coefficients to deduce the Fourier transform G(ω) of g(t) directly from the standard formula for the FT of a complex Fourier series. (c) Now rewrite the periodic function g(t) as a convolution between f (t) and a train of δ-functions. (d) Determine G(ω) using the time-convolution property of Fourier transforms, and confirm that it agrees with that derived in part (b). f(t) T/ 2 a/ 2 0 a/ 2 T/ 2 Figure t 2. The signal x(t) whose spectrum is shown in Figure 2 is sampled using a train of δ-functions p(t) before being passed through an ideal reconstruction filter. (a) What is the Nyquist limit? For sampling frequencies above and below the Nyquist limit, sketch and explain the appearance of the frequency spectra (i) after sampling and (ii) after reconstruction. (b) When is an anti-aliasing filter useful? Where is it placed in the processing chain? (c) Music is recorded on CDs at a sampling rate of 44. khz per channel, and players usually have reconstruction filters with a nearly flat frequency response over 0-20 khz, and rejection of greater than 70 db above 22 khz. Suggest why these various frequencies are chosen. x(t) y(t) x ω m ω r(t) m p(t) Figure 2 Reconstruction Filter

2 HT 209/DW Murray 2AH/p2 3. A signal x(t) = cos(ω 0 t) is sampled at ω s with p(t) = δ(t kt s ) and then reconstructed from the sampled signal using an ideal low-pass filter with cut-off frequency of ω s /2. (a) Show that for any signal x(t) with spectrum X(ω) the sampled signal has spectrum Y (ω) = ω s 2π and, for this particular signal, + k= X(ω kω s ), Y (ω) = ω s 2 + k= [δ(ω kω s ω 0 ) + δ(ω kω s + ω 0 )]. (b) Frequency ω s is kept fixed, but the cosine signal frequency ω 0 is varied. Determine the reconstructed signals x r (t) when (i) ω 0 = ω s /5 (ii) ω 0 = 3ω s /5 (iii) ω 0 = ω s Illustrate your answers with sketches showing the frequency spectra. (Use the template shown in Fig. 3. A printable sheet is on the course page.) (c) In each of these three cases, what would be the reconstructed signal were an anti-aliasing LPF with cut-off frequency ω s /2 used before sampling? Reconstruction filter ω s 0 ω s 2ω s ω s 0 ω s 2ω s ω s 0 ω s 2ω s Figure 3

3 HT 209/DW Murray 2AH/p3 4. (a) Explain what is meant by a finite energy signal and contrast its properties with those of a finite power signal. (b) For finite energy signals Parseval s theorem states that the total energy in the time domain is equal to the total energy in the frequency domain. Prove the theorem. (c) By using a Fourier transform pair from HLT, verify the theorem by calculating, in both the time and frequency domains, the energy dissipated in a 47Ω resistor subjected to a time-varying voltage v(t) = 2u(t) exp( 2t) where u(t) is the unit step function. (See Useful Integral # at the end.) 5. (a) By finding the energy in the time domain, confirm that f (t) = u(t)ae at, where a is a positive constant, is a finite energy signal, (b) For any finite energy signal f (t) derive the Wiener-Khinchin theorem that the Fourier transform of the autocorrelation R f f (τ) is equal to the energy spectral density E(ω). (c) By integration in the frequency domain, show that the total energy in f (t) is E = A 2 /2a and identical with that found in part (a). (d) Let f (t) be the input to a filter with impulse response h(t) = Be bt, b > 0, and let y(t) be the output of the filter. Compute the total energy of y(t). (See Useful Integral #2.)

4 HT 209/DW Murray 2AH/p4 6. (a) Explain the terms time-average, ensemble-average, stationary random function and ergodic random function. (b) A zero-mean stationary random process x(t) has autocorrelation R xx (τ) = Ae τ. Determine its average power and its power spectral density S xx (ω). (c) Determine the integral S xx (ω)dω. Is the value what you expected? 2π (d) Signal x(t) is input to a system with transfer function H(s) = /( + T s). The output is y(t). Determine the expected value and variance of y(t). Show that the variance is also the average power of y(t). 7. (a) Write down the power spectral density S xx (ω) of an ideal white noise signal with unit variance (σ 2 = ). Determine the signal s autocorrelation R xx (τ). (b) An ideal white noise signal x(t) is band-limited to frequencies ω < ω 0 by an ideal low pass filter with transfer function H(ω) = for ω < ω 0, and H(ω) = 0 elsewhere. Determine the autocorrelation of the filter s output. (c) Ideal white noise is now limited by the st order low pass filter shown in Figure 4 which has a half power (break) point at angular frequency ω 0. Derive expressions for the power spectral density S yy and autocorrelation R yy of the output. (d) Sketch the autocorrelations of parts (a), (b) and (c), and comment on your results. x(t) R C y(t) Figure 4

5 HT 209/DW Murray 2AH/p5 8. Each pulse in a continuous binary train has a fixed duration, T, but takes the value, 0, or - with probabilities /4, /2 and /4 respectively, independently of the previous pulse. (An example is sketched in Figure 5.) (a) If x i is the random variable representing the height of pulse i, calculate E[x i ], E[x 2 i ], and E[x i x j ] for i j. (b) Show that the auto-correlation of the pulse train is R xx (τ) = (/2) ( τ /T ) for τ T, and zero elsewhere. (c) Hence find the power spectral density S xx (ω) of the pulse train. (d) Use the autocorrelation to find the mean power, and confirm that this agrees with the result from part (a). 0 t Figure 5 9. In communications, a cosine carrier is amplitude modulated by a signal f (t) to give an output for broadcast of v(t) = [ + mf (t)]v c cos ω c t. Here, m is the modulation depth, v c is the unmodulated carrier s amplitude, and ω c is the carrier s angular frequency. Suppose that f (t) is a square wave of amplitude, the modulation depth is m = 0.5, and the signal s angular frequency ω s is much smaller than that of the carrier. (a) By considering the components making up v(t) in turn, sketch the general appearance of v(t). (b) Use the square wave s Fourier Series in HLT to determine the spectrum of v(t). Sketch the spectrum for w 0. (c) The square wave has a frequency (not angular frequency) of khz. Assuming that components in the spectrum with amplitude less than 5% of the maximum component can be neglected, explain how you would find the bandwidth in Hz required for broadcast.

6 HT 209/DW Murray 2AH/p6 0. Be prepared to discuss the following with your tutor: (a) Explain how a sampled signal from can be reconstructed in the time domain, rather than using an analogue reconstruction filter. (b) What is over-sampling? Describe its effect on the frequency spectrum of a sampled signal. What benefit might it have for reconstruction in the time domain? (c) Describe a practical use for under-sampling.. A tan substitution. Useful Integrals ( ) 2π ω 2 + a 2 dω = 2a 2. When one tan is not enough: two tan substitutions... ( ) ( ) 2π ω 2 + a 2 ω 2 + b 2 dω = 2π b 2 a 2 = = b 2 a 2 2ab(a + b) [ 2a 2b [ ω 2 + a 2 ω 2 + b ] 2 ] dω Some Answers and Hints See the course page at dwm/courses/2tf