Question

Determine if the following system is linear and/or time-invariant y(t) = cos(3t) x(t)

Answer 1

The following system is not linear.

The following system is time-invariant

Explanation:

To determine whether a system is linear, the following condition must be satisfied:

`f(a) + f(b) = f(a+b)`

For
`y(t) = cos(3t)`, we have

`y(a) = cos(3at)`

`y(b) = cos(3bt)`

`y(a+b) = cos(3(a+b)t) = cos(3at + 3bt)`

In trigonometry, we have that:

`cos(a+b) = cos(a)cos(b) - sin(a)sin(b)`

So

`cos(3at + 3bt) = cos(3at)cos(3bt) - sin(3at)sin(3bt)`

`y(a) + y(b) = cos(3at) + cos(3bt)`

`y(a+b) = cos(3(a+b)t) = cos(3at + 3bt) = cos(3at)cos(3bt) - sin(3at)sin(3bt)`

Since
`y(a) + y(b) \\eq y(a+b)`, the system
`y(t) = cos(3t)` is not linear.

If the signal is not multiplied by time, it is time-invariant. So
`y(t) = cos(3t)`. Now, for example, if we had
`y(t) = t*cos(3t)` it would not be time invariant.