Consider a single input single output discrete-time system with x[n] as input and y[n] as output. Where the two are related as

Which one of the following statements is true about the system?
a)
It is casual and stable
b)
It is causal but not stable
c)
It is not causal but stable
d)
It is neither causal nor stable
Correct answer is option 'A'. Can you explain this answer?

Question

Consider a single input single output discrete-time system with x[n] as input and y[n] as output. Where the two are related as

Which one of the following statements is true about the system?

Correct answer is option 'A'. Can you explain this answer?

Answer

Concept:

For a system to be causal, the Present output should depend on the present or past input only.

For a system to be Stable, a Bounded input should produce a Bounded Output.

Analysis:

For 0 < n < 10,

y(n) = n|x(n)|

Let x(n) is bounded, i.e. for -∞ ≤ n ≤ ∞, x(n) ≤ M, where M is finite.

So, for -∞ ≤ n ≤ ∞, |n.x(n)| will also approach a finite value. Hence in this interval y(n) is bounded. i.e. the system is stable.

Since the output y(n) is depending on the present value of the input only, the system in this interval is also causal.

For n < 0 and n > 10:

y(n) = x(n) - x(n-1).

Let input x(n) is bounded. ∴ y(n) = x(n) - x(n-1) will also be bounded, i.e. it will go to a finite value.

Hence in this interval, the system is said to be stable.

Also, because the output y(n) depends on the present and the past input values only, the system is causal as well in this interval.

∴ After considering both the intervals we can conclude that the system is both stable and causal.