6.003 Homework #13 Solutions

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1 6.003 Homework #3 Soluions Problems. Transformaion Consider he following ransformaion from x() o y(): x() w () w () w 3 () + y() p() cos() where p() = δ( k). Deermine an expression for y() when x() = sin(/)/(). k= X(j) W (j) / W (j) / W 3 (j) / Y (j) / y() = δ()

2 6.003 Homework #3 Soluions / Fall 0. Muliplied Sampling The Fourier ransform of a signal x a () is given below. X a (j) This signal passes hrough he following sysem x a () x b () uniform x c [n] x d [n] sample-oimpulse x e () K H(j) sampler x f () cos (7) where x c [n] = x b (nt ) and x e () = x d [n]δ( nt ) and n= H(j) = { T if < T 0 oherwise. a. Skech he Fourier ransform of x f () for he case when K = and T =. X f (j) Use your skech o deermine an expression for X f (j) for he following inervals: 0 < < /: 0 / < < : /

3 6.003 Homework #3 Soluions / Fall 0 3 < < 3/: 0 3/ < < : 0 b. Is i possible o adjus T and K so ha x f () = x a ()? If yes, specify a value T and he corresponding value of K (here may be muliple soluions, you need only specify one of hem). If no, wrie none. T = 7, 4 7, 6 7, 8 7, 0 7, 7, or K = c. Is i possible o adjus T and K so ha he Fourier ransform of x f () is equal o he following, and is zero ouside he indicaed range? 3 X f (j) If yes, specify all possible pairs of T and K ha work in he able below. If here are more rows in he able han are needed, leave he remaining enries blank. If no, ener none. T K

4 6.003 Homework #3 Soluions / Fall Paerns The ime waveforms for six signals are shown in he lef panels below. The righ panels show he magniudes of he Fourier ransforms of x () o x 6 (), however, he order has been shuffled. For each panel on he lef, find he corresponding panel on he righ. All of he ime funcions are ploed on he same ime scale. Similarly, all of he frequency funcions are ploed on he same frequency scale. x () X A (j) x () X B (j) x 3 () X C (j) x 4 () X D (j) x 5 () X E (j) x 6 () X F (j)

5 6.003 Homework #3 Soluions / Fall 0 5 C A 3 E 4 B 5 D 6 F

6 6.003 Homework #3 Soluions / Fall Inpus and Oupus A causal, sable LTI sysem wih frequency response H(j) has inpu x() and oupu y(). The problem is o deermine which of he following inpus can or canno give rise o he oupu y() = sin( 00 ). For each par of he problem, deermine if he saemen is True (T) or False (F) and give an explanaion. Par a. x () is a periodic impulse rain of period 0.05 s. x () (T or F) x () can generae he response y() = sin( 00 ). True. Since x() has a period of 0.05 seconds, he fundamenal frequency is 0 Hz (= 0 radians/second). The Fourier ransform of x() is an impulse rain wih impulses a ineger muliples of 0 Hz. Thus, he fifh harmonic occurs a 00 Hz. Thus, H(j) could be a narrowband filer cenered a 00 Hz wih a phase shif of / a 00 Hz. Tha would produce he desired oupu. However, his filer is no causal. An alernaive is o consruc a filer wih zeros a all he unwaned frequencies. Par b. x () is a periodic funcion of period 0. s. Each period consiss of five cycles of a sinewave of he form sin( 00 ). x () (T or F) x () can generae he response y() = sin( 00 ). True. Since he period of x () is 0.s, he fundamenal frequency is /0. Hz. Thus, x () will conain impulses a he frequencies f = k/0. Hz and he h harmonic will appear a he frequency f = 00 Hz. The only remaining issue is wheher he area of his impulse is non-zero. The pulse of sinusoid has a specrum which is a sinc funcion cenered on 00 Hz. Hence, is value a 00 Hz is non-zero and so he ampliude of he impulse a 00 Hz is also non-zero. The filer can be chosen as indicaed in par a. Par c. x 3 () is a periodic pulse rain of period 0.0 s. Each pulse has duraion s. x 3 ()

7 6.003 Homework #3 Soluions / Fall 0 7 (T or F) x 3 () can generae he response y() = sin( 00 ). True. x 3 () can be represened by a uniform impulse rain of period 0.0 s convolved wih a recangular pulse of duraion s. Thus, he Fourier ransform of x 3 () is a uniform impulse rain, whose period in frequency is /0.0 = 50 Hz, muliplied by a sinc funcion. Thus, here are clearly impulses a he frequencies ±00, he only issue is wheher he sinc funcion has a zero a 00 Hz. Since he duraion of he recangular pulse is s, he firs zero of he sinc funcion is a 50 Hz. The filer can be chosen as indicaed in par a. Par d. x 4 () is a periodic sinc pulse rain of period 0. s. Each sinc pulse has he formula sin( ) x 4 () (T or F) x 4 () can generae he response y() = sin( 00 ). False. x 4 () is a uniform impulse rain of period 0. s convolved wih a sinc funcion. Hence, he Fourier ransform of x 4 () is he produc of an impulse rain in frequency wih an ideal lowpass filer. Since he period of x 4 () is 0. s, he fundamenal frequencey is 0 Hz and here will be an impulse a 00 Hz in he Fourier ransform of he impulse rain. Bu, he oal widh of he ideal lowpass filer is /0.006 = Hz so ha he passband is from o Hz. Thus, here is no componen a 00 Hz a he inpu o he filer H(j) and x 4 () canno generae he oupu y(). Par e. x 5 () is a periodic riangular wave of period 0.0 s. x 5 () (T or F) x 5 () can generae he response y() = sin( 00 ). False. x 5 () is a riangular wave which has half-wave symmery and herefore is even harmonics are zero. Since he fundamenal frequency is 50 Hz, he componen a 00 Hz is he second harmonic and is magniude mus be zero. An alernaive approach is o recognize ha he riangular wave is he convoluion of a periodic impulse rain in ime wih a riangular pulse. The riangular pulse has duraion 0.0 s and can be generaed by convolving a square pulse of duraion 0.0 s wih iself. Thus, he Fourier ransform is he squared sinc funcion. Bu he sinc funcion has a zero a 00 Hz. Hence, he componen a 00 Hz a he inpu o he filer has a magniude of zero, and x 5 () canno give rise o y().

8 6.003 Homework #3 Soluions / Fall DT Radio Demodulaion Commercial AM radio saions broadcas radio frequencies wihin a limied range: (f c 5 khz) < < (f c + 5 khz), where f c = c /() = n 0 khz and n is an ineger beween 54 and 60. The sysem shown below is inended o decode one of he AM radio signals using DT signal processing mehods. Assume ha all of he filers are ideal. Par a. Deermine he cener frequency f c for he AM saion ha his receiver will deec. = Ω T = f = = 8 07 =.5 MHz Par b. Which of he following saemen(s) is/are correc? b. Increasing he cuoff frequency r of LPF by a facor of.5 will cause aliasing. b. Decreasing he cuoff frequency r of LPF by a facor of will have no effec on he oupu y r (). b3. Halving he sampling inerval T would have no effec on he oupu y r (). b4. Doubling he sampling inerval T would have no effec on he oupu y r (). b Par c. Which of he following saemen(s) is/are correc? c. Increasing he cuoff frequency Ω d of LPF will change y r () by adding signals from unwaned radio saions. c. Increasing he cuoff frequency Ω d of LPF will change y r () because aliasing will occur.

9 6.003 Homework #3 Soluions / Fall 0 9 c3. Doubling he cuoff frequency Ω d of LPF will have no effec on y r (). c4. Halving he cuoff frequency Ω d of LPF will have no effec on y r (). c3