Question

Determine whether the system is (1) stable, (2) causal (3) linear, (4) time invariant, and (5) memoryless:

T(xn])= g[n]x[n] with g[n] given

Answer 1

The given system T(xn) = g[n]x[n] is stable, causal, linear, time-invariant, and memoryless.

Step-by-step explanation:

The given system T(xn) = g[n]x[n] can be analyzed to determine if it is stable, causal, linear, time-invariant, and memoryless.

1. Stability: A system is stable if its output remains bounded for any bounded input. To check for stability, we need to examine the summation of the absolute values of g[n]. If the summation is finite (converges), the system is stable.
2. Causality: A system is causal if its output at a given time depends only on the values of the input at the same and previous times. In this case, since the output T(xn) is created using the current and past input values, it is a causal system.
3. Linearity: A system is linear if it follows the principles of superposition and scaling. That is, if the input is scaled or summed, the output is correspondingly scaled or summed. The given system follows these principles, making it linear.
4. Time Invariance: A system is time-invariant if a time shift in the input corresponds to the same time shift in the output. Since the input and output are multiplied by the same g[n], the system is time-invariant.
5. Memory lessness: A system is memoryless if its output at a given time depends only on the input value at that same time. In this system, the output T(xn) only depends on the current input value x[n], so it is considered memoryless.