The signal y(t) = T{x(t)} = sin(2πt)^⋆x(t) + u(t - 2) is _______.

Concept

Linearity

It is the combination of additivity and homogeneity.

Another method is by observation we can say that the given signal is linear or not.

x₁(t) → y₁(t)

x2(t) → y₂(t)

x₁(t)+x₂(t) → y₁(t)+y₂(t)

α× x(t) → α × y(t)

Causal

A system is said to be causal if the present output depends on the present input and past values of the input but not on the future values.

x(t) = 0 for t < 0

Time invariant

A system is time-invariant if input and output characteristics do not change with time, time-varying nature is caused due to internal components.

if x(t) → y(t) then x(t - t₀) → y(t - t₀)

Calculation:

The given signal is multiplied with the sine function. So, the function is non-linear by observation.

Shift the signal by t₀ 

For Time invariance

y(t) = sin(2πt - t₀)*x(t - t₀) + u(t - 2 - t₀)

Replace the 't' by t - t₀ in the output signal

y(t) = sin(2π(t - t0))*x(t - t0) + u(t - 2 - t0)

Both the equations are the not same. So, the system given is Time-variant.

For Causality

y(t) = T{x(t)} = sin(2πt)⋆x(t) + u(t - 2)

y(1) = T{x(1)} = sin(2π×1)*x(1) + u(-1)

As the signal contains only present and past values of the input, hence the given signal is causal.