Question

Discrete-Time Multiple Choice:

Show your work for each question as much as you can

(a) Which of the following systems is linear?) (B) y[n] 2 cos(x (n) (D) y[n] x(n) (b) Which of the following systems is time-invariant?() (C) yfn] n[n] (D) yIn] - x-2n] (B) y[n] - cos(nx(n)) (c) Which of the following systems is causal? () (A) yln]-2x[-n] (d) Which of the following systems is NOT stable? () (C) yln] vx(n) (B) yinl 50 sin(rIn)+ 10 cos(2x())D)h(n) (0.2)"u(r) discrete-time LTI system with the impulse response h(n). If (e) Consider a relaxed then the output is theinput signal is δ(n-5) + 26(n + 2),

Answer 1

[A] implies Y[n] = Y[n - 1] - 2 X[n] Reason implies By rearranging above equation, we get implies Y[n] - Y[n - 1] = -2 X[n] (1) For discrite time Linear system T[a x_1(n) + b x_2 (n)] = a T [x(n)] + bT [x_2 (n)] Hence for input x_1 (n) o/p will be y_1 (n) and for x_2 (n) y_2 (n) then above will become = Y_1 (n) - Y_1 (n - 1) = -2 x_1 (n) for x_1 (n) i/p implies Y_2 (n) - y_2 (n - 1) = -2 x_2 (n) for x_2 (n) i/p Adding eq (i) and (ii) we get implies y_1 (n) - y_1 (n - 1) + y_2 (n) - y_2 (n - 1) = -2 x_1(n) - 2 x_2 (n) Now i/p x(n) replaced by [x_1 (n) + x_2 (n)] in eq (i) Then it become y_2 (n) - y_2 (n - 1) = -2 [x_1 (n) + x_2 (n)] = -2 x_1 (n) 2 x_2 (n)
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