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IEC222 - Signals and Systems
Problem Sheet - V
IEC222 - Signals and Systems Problem Sheet - V Use the Fourier transform analysis equation to calculate the Fourier transforms of: (a) e2"Yulr— 1) (by eI Sketch and label the magnitude of each Fourier transform. Use the Fouricr transform analysis equation (4.9) to calculate the Fourier transforms ot: {a) 6(¢+1)+d6(r-1) (db) £{u(-2 — f+ u(t — 2)} Sketch and label! the magnitude of each Fourier transform. Determine the Fourier transform of each of the following periodic signals: (a) sinQ2at+ 4) (b) 1+ cos(6ar + J) Use the Fourier transform synthesis equation to determine the inverse Fourier transforms of: (a) X\(jw) = 27 6(w) + w 8(@ — 47) + 7 Sw + 477) 2 0O=w=2 (b) X.(jw) = 4-2, -2sa@<0 0, |w|>2 Given that x(t) has the Fourier transform X(jw), express the Fourier transforms of the signals listed below in terms of X(jw). You may find useful the Fourier transform properties. (a) x\(f) = x1 -) + a(-1-24) (b) x2(t) = x@Gr— 6) (c) x(t) = 4 x(t -1) (a) Determine the Fourier transform of the following signal: . 2 x(t) = (St) at (b) Use Parseval’s relation and the result of the previous part to determine the nu- merical value of +00 . 4 A= | ? (=) dt = at 10. 11. Given the relationships y(t) = x(t) * A(t) and gt) = x(3t) * ho), and given that x(¢) has Fourier transform X(jw) and h(t) has Fourier transform HA (ja), use Fourier transform properties to show that g(t) has the form 8(t) = Ay(Bi). Determine the values of A and B. Consider the Fourier transform pair ett, _2 1l+@? (a) Use the appropriate Fourier transform properties to find the Fourier transform of te7"!. (b) Use the result from part (a), along with the duality property, to determine the Fourier transform of _ at aa Determine whether each of the following statements is true or false. Justify your answers. (a) An odd and imaginary signal always has an odd and imaginary Fourier trans- form. (b) The convolution of an odd Fourier transform with an even Fourier transform is always odd. Find the impulse response of a system with the frequency response _ (sin*(3@)) cos @ H(jo) “5 Consider a causal LTI system with frequency response H(je) = ——~. Ve) jo+3 For a particular input x(¢) this system is observed to produce the output yt) = e ult) — e “ult. Determine x(f). 14. Let X(jw) denote the Fourier transform of the signal x(t) depicted in Figure . (a) Find £X( ja). (b) Find X(j0). (c) Find {%, X(jw) de. (d) Evaluate [%, X(jo)*22¢e)# de. (e) Evaluate {”, |X(jw)[? dw. (f) Sketch the inverse Fourier transform of Re{X(jo)}. Note. You should perform all these calculations without explicitly evaluating X(jw). x} —r 0 1 2 3 t 15. (a) Compute the convolution of each of the following pairs of signals x(t) and h(t) by calculating X(jw) and H(jw), using the convolution property, and inverse transforming. (i) x(t) = te~**u(t), h(t) = e~**u(t) Gi) x(t) = re~% u(t), A(t) = te“ u(t) Git) x(t) = eT u(t), ht) = e'u(—D (b) Suppose that x(7) = e~"-? u(¢— 2) and A(2) is as depicted in Figure . Verify the convolution property for this pair of signals by showing that the Fourier transform of v(t) = x(t) * AC) equals H(jw)X(jw). nit) 1 -41 3 t 16. The input and the output of a stable and causal LTI system are related by the dif- ferential equation Py) | ody) dt? dt + 8y(t) = 2x(0) (a) Find the impulse response of this system. (b) What is the response of this system if x(t) = te u(t)? (c} Repeat part (a) for the stable and causal LTI system described by the equation aed Plt) JO + y(t) = 2 2? — an “dP 17. A causal and stable LTI system S has the frequency response . jot+4 HU) ~ 5 Oy Sa (a) Determine a differential equation relating the input x(t) and output y(r) of S. (b) Determine the impulse response h(r) of S. (c) What is the output of S when the input is x(t) = e Put) — te * u(t)? 18. Consider an LTI system whose response to the input x(t) = [e' + e* u(t) is y(t) = [2e7! — 2e* Jul). (a) Find the frequency response of this system. (b) Determine the system’s impulse response. (c) Find the differential equation relating the input and the output of this system. 19. Use partial fraction expansions to determine the inverse FT for the following signals: fa) X(je) = Torta {b) X(jo) = — ae +3 le) Xt) = Pa id) Xie) = 7 $e ry (fe) X(ja) = Hier ies if) Xtjo) = 224 Ot (jw + 2) 
