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IEC222 - Signals and Systems
Problem Sheet - IV
IEC222 - Signals and Systems Problem Sheet - IV For each of the periodic signals shown in Fig, find the compact trigonometric Fourier series and sketch the amplitude and phase spectra. If either the sine or cosine terms are absent in the Fourier series, explain why. —207 —107 —T|T7 107 207 > (b) 4 | 8a 6m 4a Qn 0 2a 4a on 8ir t > (c) A_A A VW A A sz S\N LIN LN 2. If the two halves of one period of a periodic signal are identical in shape except that one is the negative of the other, the periodic signal is said to have a half-wave symmetry. If a periodic signal X(t) with a period 7p satisfies the half-wave symmetry condition, then x ('- 2) = —X(f) In this case, show that all the even-numbered harmonics vanish and that the odd-numbered harmonic coefficients are given by 4 plo/2 =— X(t) cos not dt To Jo Gn and 4 pTol2 Dy = aa X(t) sin naot dt To Jo Using these results, find the Fourier series for the periodic signals in Fig. pe me & & E (b) ti Figure below shows the exponential Fourier spectra of a periodic signal x(f). (a) By inspection of Figure. find the exponential Fourier series representing x(f). (b) By inspection of Figure. sketch the trigonometric Fourier spectra for x(1). (c) By inspection of the trigonometric Fourier spectra found in part (b), find the trigono- metric Fourier series for x(9). (d) Show that the series found in parts (a) and (c) are equivalent. Pil LLL, 7. For each of the periodic signals in Fig. . find exponential Fourier series and sketch the corresponding spectra. 1 —7 —5 3 -1 0 1 3 5 7 t‘— -|4 (a) 1 —207 —107 —T)7 107 207 1 (b) 4 | 89 —6T —47 -20 0 Pon 40 on 87 ti (c) — | A aal/4 | ZA [~- [7 «8 |A# i AAAAAAL [\ LN f\~ 13. Consider three continuous-time periodic signals whose Fourier series representa- tions are as follows: 100 7p \k x(t) = >(5) elk 30, k=0 100 x(t) = cos(karje!* 30", k=—100 100 kar\ _ x)= > jin Jer k=-100 “ Use Fourier series properties to help answer the following questions: (a) Which of the three signals is/are real valued? (b) Which of the three signals is/are even? 14. Suppose we are given the following information about a signal x(t): 1. x(7) is real and odd. 2. x(t) is periodic with period T = 2 and has Fourier coefficients a,. 3. ay = 0 for |k| > 1. 4. Ho |x@2dr = 1. Specify two different signals that satisfy these conditions. 15. Let x[n] be areal and odd periodic signal with period N = 7 and Fourier coefficients a,x. Given that a5 = fai = 2j,a17 = 3), determine the values of ag, a_|, a2, and a_3. 16. Suppose we are given the following information about a signal x[”]: 1. x[n] is a real and even signal. 2. x[n] has period NV = 10 and Fouricr coefficients a,. 3. ay, = 5. 4,1 5g |xtn]? = 50. 1. n=0 Show that x[”] = Acos(Bn + C), and specify numerical values for the constants A, B, and C. 17. Each of the two sequences x)[n] and x2[7] has a period N = 4, and the correspond- ing Fourier series coefficients are specified as x\[n] <= a,, xo[n] —— by, where 1 a = a3 = 5a = Say = 1 and by = hh = bp = by = 1. Using the multiplication property in Table 3.1, determine the Fourier series coeffi- cients c; for the signal g[n] = x;[n]x2[n]. 18. A discrete-time periodic signal x[n] is real valued and has a fundamental period N = S. The nonzero Fourier series coefficients for x[n] are a = 2,4, =a, = 2ei7 ay = a’ ,=e/ lst Express x[n] in the form x[n] = Ap + Ss Ax Sin(wyn + bx). k=1 19. Determine the Fourier series coefficients for each of the following discrete-time periodic signals. Plot the magnitude and phase of each set of coefficients a,. (a) Each x[n] depicted in Figure (a)—(c) (b) x[n] = sin(27n/3) cos(an/2) (c) x[#] periodic with period 4 and x[n] = 1 ~sin forO 
