The GATE Grind

GATE 2020 EE – Question 49

Signals and Systems · Linear time invariant and causal systems · 2 marks · Multiple choice

Which of the following options is true for a linear time-invariant discrete time system that obeys the difference equation:

$$y[n]-ay[n-1]=b_0x[n]-b_1x[n-1]$$

  1. $y[n]$ is unaffected by the values of $x[n-k];\ k>2$.
  2. The system is necessarily causal.
  3. The system impulse response is non-zero at infinitely many instants.
  4. When $x[n]=0,n<0$, the function $y[n];n>0$ is solely determined by the function $x[n]$.

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Show answer and explanation

Correct answer: (C) The system impulse response is non-zero at infinitely many instants.

Explanation

With feedback through $ay[n-1]$ ($a\ne0$) the system has an infinite impulse response, $h[n]=a^nb_0-b_1a^{n-1}$ for $n\ge1$, which is non-zero at infinitely many instants. The output depends on all past inputs (A is false), the equation alone does not force causality (B is false), and $y[n]$ also depends on initial conditions (D is false).