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Option 2 : Non-linear, time variant, causal

CT 1: Current Affairs (Government Policies and Schemes)

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10 Mins

__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.

x_{1}(t) → y_{1}(t)

x2(t) → y_{2}(t)

x_{1}(t)+x_{2}(t) → y_{1}(t)+y_{2}(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 - t_{0}) → y(t - t_{0})

__Calculation:__

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

Shift the signal by t_{0}

**For Time invariance**

y(t) = sin(2πt - t_{0})*x(t - t_{0}) + u(t - 2 - t_{0})

Replace the 't' by t - t_{0} 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.**