Question

Determine whether the LTI systems with given impulse response are causal, memory-less, and/or stable. Justify your answers.

(a) h[n] = 8[2n]

Answer 1

The LTI system with impulse response h[n] = δ[2n] is causal because it does not depend on future input, memoryless because the output depends only on the input at the same time, and stable because it produces bounded output for any bounded inputs.

Step-by-step explanation:

The subject at hand involves analyzing an LTI (Linear Time-Invariant) system's impulse response to determine if the system is causal, memoryless, or stable. Given an impulse response h[n] = δ[2n], we can make the following observations:

• A system is causal if the output at any time depends only on values of the input at the present time and in the past, not on future inputs. As δ[2n] is not defined for negative values of n, this system is causal since it outputs zero for any time index n before the impulse occurs.
• A system is memoryless if the output at any time depends only on the input at that same time. Since δ[2n] is only non-zero when n = 0, and it does not depend on any other values of n, this system can be considered memoryless.
• A system is stable if every bounded input to the system produces a bounded output. As the impulse response is a simple delta function scaled in time, which is itself bounded, the system is stable.

Therefore, the given LTI system with impulse response h[n] = δ[2n] is causal, memoryless, and stable.