Question

A) Determine (theoretically) if the following system is linear.

y(t) = (x(t), if x(t)>0 0, otherwise
y(t) = x(t)u(t)
x,(t) y₁(t) = x(t)u(t)
x2(t) y₂(t) = x(t)u(t)
Yₜₒₜₐₗ(t) = y,(t) + y,(t) = [x,(t) + x(t)]u(t)
y(t) = x(t)u(t) y(t) = kx(t)u(t) = ky(t) so system is linear

Answer 1

The question asks whether a system with output y(t) = x(t)u(t) is linear. The system is not linear because it fails to satisfy the additivity and homogeneity properties due to its dependency on a step function u(t).

Step-by-step explanation:

To determine if the system described is linear, we must check if it satisfies the two properties of linearity: additivity and homogeneity. For a system to be linear, the principle of superposition must apply. This means that:

1. The output for the sum of inputs should be equal to the sum of the outputs for the individual inputs (additivity).
2. The output for a scaled input should be equal to the scale factor times the output for the original input (homogeneity).

In the question, we have a system with output y(t):

• y(t) = x(t)u(t) where u(t) is a step function that is 1 for x(t) > 0 and 0 otherwise.

This system is not linear because the output depends on a step function, which does not satisfy additivity or homogeneity over the entire range of x(t). Specifically, if x1(t) > 0 and x2(t) < 0, the output will not be the sum of the individual outputs. Similarly, scaling x(t) by a negative constant can change u(t) from 1 to 0, violating homogeneity.