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Consider the discrete-time systems T1 and T2 defined as follows: {T1 x}[n] = x[0] + x[1] +... + x[n] {T2 x}[n] = x[0] + 1/2 x[1] +... + 1/n x[n] Which one of the following statements is true?

1. T1 and T2 are BIBO stable.
2. T1 and T2 are not BIBO stable.
3. T1 is BIBO stable but T2 is not BIBO stable.
4. T1 is not BIBO stable but T2 is BIBO stable.

Correct answer: T1 and T2 are not BIBO stable.

Solution

Both T1 and T2 fail to be BIBO stable because their output can become unbounded for bounded input signals. T1 accumulates all previous inputs, leading to an ever-increasing output, while T2's weighted sum diverges as n increases, indicating instability.

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