Question

For each of the following systems, determine whether the system is (1) stable, (2) causal, (3) linear, and (4) time invariant.

(a) Tx[n]) = (COS An)x[n]

Answer 1

The system represented by Tx[n] = (cos(An))x[n] is stable, causal, linear, and time invariant as it meets the criteria for each of these properties.

Step-by-step explanation:

System Characteristics

For a system represented by Tx[n] = (cos(An))x[n], we need to evaluate its characteristics based on the definitions provided for stability, causality, linearity, and time invariance.

1. Stable: A system is stable if, when displaced from equilibrium, it experiences a net force or torque in the direction opposite the displacement. Since the output is a bounded cosine function of the input, the system is at stable equilibrium.
2. Causal: A causal system depends only on present and past inputs. This system is causal because the output at any time n depends solely on the input at that same time n.
3. Linear: A system is linear if it satisfies both homogeneity and additivity. The cosine function is a linear operation on the input signal; therefore, the system is linear.
4. Time Invariant: A system is time invariant if a time shift in the input signal results in an equivalent time shift in the output signal. Since multiplying by a cosine of the same frequency does not change with time, the system is time invariant.