Question

A continuous-time linear system S with input x(t) and output y(t) yields the following input-output pairs:

x(t) = eʲ²ᵗ →ˢ y(t) = eʲ³ᵗ
x(t) = e⁻ʲ²ᵗ →ˢ y(t) = e⁻ʲ³ᵗ.
if x₁(t) = cos(2t), determine the corresponding output y₁(t) for system S.

Answer 1

To find the output y₁(t) for a cosine input to a linear system, express the cosine as a sum of exponential functions, apply the system's response to each, and convert back. For the given system with the input x₁(t) = cos(2t), the output is y₁(t) = cos(3t).

Step-by-step explanation:

To determine the output y₁(t) for the system S when the input is x₁(t) = cos(2t), we can use the principle of superposition for linear systems and the given input-output pairs.

First, we express the cosine function as a linear combination of exponential functions using Euler's formula:

x₁(t) = cos(2t) = ½(eʸ²ᵗ + e⁻ʸ²ᵗ)

Using the given input-output pairs:

• eʸ²ᵗ →ˢ eʸ³ᵗ
• e⁻ʸ²ᵗ →ˢ e⁻ʸ³ᵗ

We can find the system's response to x₁(t):

y₁(t) = ½(eʸ³ᵗ + e⁻ʸ³ᵗ)

Finally, we can convert the output back to a cosine function using Euler's formula:

y₁(t) = cos(3t)

This result is consistent with the linearity and time invariance of the system S.