SIGNALS AND SYSTEMS PROBLEM SHEET 2 ECE, Exercises of Signals and Systems

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IEC222 - Signals and Systems

Problem Sheet - II

IEC222 - Signals and Systems Problem Sheet - II Consider the conunuous-time signal x(t) = 6(t + 2) — 6(f — 2). Calculate the value of £., for the signal y(t) = [ x(n)dr. Let x[n] be a signal with x[m] = 0 for n < —2 and vn > 4. For each signal given below, determine the values of n for which it is guaranteed to be zero. (a) x[n — 3] (b) x[n + 4] (c) x[~n] (d) x[-n + 2] {e) x[-n — 2] Let x(r) be a signal with x(t) = 0 for r < 3. For each signal given below, determine the values of ¢ for which it is guaranteed to be zero. (a) x(1 - 1) (b) x11 — 1) + x2 -1) {e) x(1 — pa(2- 1) (d) x(32) (e) x(t/3) Let x[] and y[#] be given in Figs. (a) and (b), respectively. Carefully sketch the following signals: (a) x[2”] (b) x[ (c) yf (d) y[2 — 2n] (e) x[ — 2] + y[n + 2] (f) x[2n] + y[n — 4] 3-2-1 12 3 (a) (b) Simplify the following expressions: (a) ( sint )so P+2 jo+2)\. (b) (es = ie) (c) [e~'cos (3t — 60°)]5(t) sin [Z(r—2 @ (mE N sa» P+4 1 e) (; )oo+3) jo+2 ay (= aww oO Evaluate the following integrals: (a) ri 5(t)x(t— t) dt °° w | x(T)d(t—) dt so © | b(the! dt —0° @ | 5(2t— 3) sin xtdt (e) [ 5(t+3)e' dt —0° ioe} of (8 +4)5(1 —dadt @ | x(2 —1)5(3 —t) dt oe (h) il e) cos[¥(x—5)]8(x— 3) de —0o (a) Find the energies of the pair of signals x(t) and y(t) depicted in Figs. a andb. Sketch and find the energies of signals x(t) + y(t) and x(t) — y(t). Can you make any observation from these results? (b) Repeat part (a) for the signal pair illustrated in Fig.c. Is your observation in part (a) still valid? 9. 10. 11. Determine the power and the rms value for each of the following signals: (a) 5+ 10cos(100/ + 27/3) (b) 10cos(100t + 2 /3)+ 16 sin(150f+ 2/5) (c) (10+ 2sin3r) cos 10t (d) 10cos5rtcos 10 (e) 10sinS5tcos 10r (f) e cos wot There are many useful properties related to signal energy. Prove each of the following statements. In each case, let energy signal x (4) have energy E[x)(4)], let energy signal x2 (1) have energy E[.x2(f)], and let T be a nonzero, finite, real-valued constant. (a) (b) (c) (d) Prove E[Tx(t)] = T?E[x(1)]. That is, amplitude scaling a signal by constant T scales the signal energy by T?. Prove E[x,(t)] = Ely(t — T)]. That is, shifting a signal does not affect its energy. If (x(t) 4 0) = (a(t) = 0) and (2() F 0) => (x(t) = 0), then prove E[xy (1) + X2(f)] = Elxy(t)] + Efva()]. That is, the energy of the sum of two nonoverlapping signals is the sum of the two individual energies. Prove Ely, (7t)] = (1/|T)EL 4 ()]. That is, time-scaling a signal by T reciprocally scales the signal energy by 1/|T]. Consider the signal x(t) = 27", where u(t) is the unit step function. (a) Accurately sketch x() over (—1 <1 <1). (b) Accurately sketch y(t) = 0.5x(1 — 2f) over (-l