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 $$y\left[ n \right] = \left\{ {\matrix{ {n\left| {x\left[ n \right]} \right|,} & {{\rm{for}}\,0 \le n \le 10} \cr {x\left[ n \right] – x\left[ {n – 1} \right],} & {{\rm{otherwise}}} \cr } } \right.$$ Which one of the following statements is true about the system?

[amp_mcq option1=”It is causal and stable” option2=”It is causal but not stable” option3=”It is not causal but stable” option4=”It is neither causal nor stable” correct=”option1″]

The correct answer is: A. It is causal and stable.

A causal system is a system whose output depends only on the present and past inputs, not on future inputs. In this case, the output $y[n]$ is defined only for $n \ge 0$, so it is causal.

A stable system is a system whose output does not grow without bound as the input goes to infinity. In this case, the output $y[n]$ is bounded for all $n$, so it is stable.

The other options are incorrect. Option B is incorrect because the system is not stable. Option C is incorrect because the system is causal. Option D is incorrect because the system is both causal and stable.