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

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  1. Linear, time variant, non-causal
  2. Non-linear, time variant, causal
  3. Linear, time invariant, causal
  4. Non-linear, time invariant, non-causal

Answer (Detailed Solution Below)

Option 2 : Non-linear, time variant, causal
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Detailed Solution

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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.

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