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Sns mid term-test2-solution

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Sns mid term-test2-solution

  1. 1. PSDC ENGINEERING DIPLOMA ASSESSMENT Signals And Systems – Mid-Term Test 2 (B16 CE/B17 EEA&EEB) 7th January 2009, Wednesday 2:00pm – 5:00pm (3 hours)Question 1 (Signals and Systems) (a) Determine whether the following signals are periodic or aperiodic. If it is a periodic signal, determine its power. (i) x[n] = 2cos(⅓πn) + sin(½πn) (ii) x(t) = Odd{cos(2πt)} (b) If x(t) = (t2 – 1) [u(t) – u(t–4)], (i) Sketch x(t) (ii) Sketch x(t – 2) (iii) Sketch x(–3t – 2) (iv) Sketch Even{x(t – 2)} (c) Determine the following systems are casual, linear and time-invariant. (i) y[n] = 3x2[–2n] (ii) y(t) = cos (t+ ½ π) x(t)Question 2 (Convolution) (a) Evaluate the convolution sum for y[n] = x[n]∗h[n], where x[n] and h[n] are shown in Fig. Q2 (a). Fig. Q2 (a) (b) Define the convolution of two signals x(t) and h(t). Then, compute the convolution integral for y(t) = x(t)∗h(t) of the following signals: h(t) = e–3tu(t) and x(t) = e–3tu(t–1). 1
  2. 2. PSDC ENGINEERING DIPLOMA ASSESSMENTQuestion 3 (Laplace Transform) (a) Define Bilateral (two-sided) Laplace Transform. (b) Consider the signal, x(t) = e–3tu(t). Find the Laplace transform of x(t) with the associated region of convergence (ROC). (c) Consider a continuous-time LTI system described by d 2 y( t ) dy( t ) dx( t ) 2 −3 − 2 y( t ) = −3 + x( t ) dt dt dt Using the shifting property, find the unit impulse response h(t).Question 4 (z-Transform) (a) Define Bilateral (two-sided) z-Transform. (b) Consider the signal, x[k] = aku[k]. Find the z-transform of x[k] with the associated region of convergence (ROC). (c) Consider a discrete-time LTI system described by y[k] – y[k–1] – 2 y[k–2] = x[k] + 2 x[k–1] + 2 x[k–2] Using the shifting property, find the unit impulse response h[k]. "Learning without thought is useless; thought without learning is dangerous." 学而不思则罔,思而不学则殆. 2

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