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

User Helina
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Answer:

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.

User Krishan
by
8.2k points
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