Signal and Systems — DU B. Sc. (Hons) Electronics Previous Year Question Papers

1. (a) Input-output relation of a system is given by the following equation y = ax + b where x and y are the input and output of the system respectively, and a and b are constants. Is the system linear? (b) Consider a discrete-time LTI system with impulse response given by h[n] = alu[n] Is the system stable? (c) Show that the impulse response h[n] of a discrete-time LTI system is the first difference of its step response s[n] i.e. Find Fourier transform of x(1) = 1/(a? + 12) where a > 0. (e) Prove the frequency-shifting property of Fourier transform. (f) Show that for Laplace transform dx (t) + sX(s)-x (0) (6x3) (a) For the signal (t) shown below (6) Sketch the following signals: (b) (i) Determine the fundamental period of the signal x(t) = 2 cos(10t + 1) - sin(4t - 1) (4) (ii) Find the even and odd components of the signal x(t) = sin2t + sin2t cost + cos2t (4) (c) Sketch the following signal and determine whether the signal is power signal or energy signal or neither. (4) 3. (a) Find the Convolution of two continuous time signals (8) f(t) = el and g(t) = 312, for all t. (b) Show that convolution of an even and an odd (4) function is an odd function. (c) A system has the input-output relation given by y|n] = nx[n] Determine and explain whether the system is (i) memoryless (ii) causal (iii) linear. (6) 4. (a) Consider a system represented by the following equation y(t) = x(t)cos(ot) where x(t) and y(t) are respectively the input and output of the system. Determine whether the system is (i) stable (ii) causal(iii) linear (iv) time-invariant. (8) (b) Write a difference equation that relates the output y[n] and the input ×[n] for the discrete system shown below. (6) Unit delay (c) Consider a continuous-time LTI system whose step response is given by s(t) = e'u(t) Find (1) the impulse response (ii) the output for input xt) given by x(t) = u(t- 1) - u(t-3) (4) 5. (a) Find the complex exponential Fourier series expansion of the following periodic signal. (10) (b) Find Fourier transform of (a) Consider the signal x(t) = e-a(1+1) u(t+ 1). Find the Laplace transform of xt). (5) (b) Find inverse Laplace transform of X(8) - (83-3)/ (c) Let f(t) = teat (i) Show that f(t) satisfies df (t) = af (t) + eat dt (ii) Use part (a) to find Laplace transform of f(t). (7) 7. (a) For the continuous-time signal (10) Determine the fundamental frequency c, and the Fourier series coefficients a, such that (b) Prove the duality property of Fourier transform. (c) Find Laplace transform of x(t) = sin(t)cost).