Question

Given a continuous-time system with the input-output relation:


`y(t) = T\left\{x(t) \right\} = x^2(t)`

Determine whether this system is:

- Linear
- Time-invariant

Answer 1

1. The system is nonlinear.
2. The system is time-invariant.

Step-by-step explanation:

The given input-output relation is a nonlinear function of the input signal x(t), since it involves squaring the signal at every point in time. A linear system must satisfy the property of superposition, which states that the output to a linear combination of inputs must be the same as the linear combination of the outputs to each individual input.

To determine whether the system is time-invariant, we need to check if a time shift in the input signal produces the same time shift in the output signal. That is, if:
`y(t) = T{x(t)}` for all values of t. Then we want to check
`y(t - τ) = T{x(t - τ)}` for all values of
`τ` and
`t`.

We shall the consider the case where
`τ > 0`, then we have:

`y(t - τ) = (x(t - τ))^2`

`T{x(t - τ)} = (x(t - τ))^2`

Since
`y(t - τ) = T{x(t - τ)}`, we can conclude that the given system is time-invariant.

Therefore, the given system is nonlinear but time-invariant.