The GATE Grind

GATE 2024 EE – Question 38

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

Consider the discrete-time systems $T_1$ and $T_2$ defined as follows:

$$\{T_1x\}[n]=x[0]+x[1]+\cdots+x[n]$$

$$\{T_2x\}[n]=x[0]+\frac12x[1]+\cdots+\frac1{2^n}x[n]$$

Which one of the following statements is true?

  1. $T_1$ and $T_2$ are BIBO stable.
  2. $T_1$ and $T_2$ are not BIBO stable.
  3. $T_1$ is BIBO stable but $T_2$ is not BIBO stable.
  4. $T_1$ is not BIBO stable but $T_2$ is BIBO stable.

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Correct answer: (D) $T_1$ is not BIBO stable but $T_2$ is BIBO stable.

Explanation

$T_1$ is an accumulator: for the bounded input $x[n]=1$ the output is $n+1$, which is unbounded, so it is not BIBO stable. For $T_2$, $|\{T_2x\}[n]|\le\max|x|\sum_{k=0}^n2^{-k}<2\max|x|$, so it is BIBO stable.