As per time displacement theorem in Laplace transformation, displacement in the time domain by T becomes

A linear system is one in which the relationship between input and output is linear.

All linear system possess the property of superposition i.e., if input u₁ gives output y₁ and input u₂ gives output y₂ then input u₁ + u₂ will give output y₁ + y₂.

Superposition implies homogencity i.e., if input u gives output y, then input ku gives output ky where k is a rational number.

In general a system is linear if a transformation T satisfies the relation

T(au₁ + ?u₂) = aT(u₁) + ?T(u₂)

where a and ? are constant.

Because no.of zero is more than Pole, therefore when you find transfer function.

Numerator power will higher than denominator, and such transfer functions is not realizable.

I_C = V(j?C).

By applying time shifting and scaling property.

It is a pulse lasting for t = a.

Energy density spectrum = |G(f)|² = |10|² = 100.

It is a modulation process. The resultant has (f + f_c) and (f - f_c) terms.

m (t) = 4 cos 100 pt + 8 sin 200 pt + cos 300 pt

Nyquist sampling freq f_s ? 2f_m where f_m is highest frequency component in given signal and highest f_m in 3^rd part

2pf_mt = 300 pt

f_m = 150 Hz

f_s = 2 x 150 p 300 Hz

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For linearity

y₁[n] = x₁[n] + x₂[n - 10] ...(1)

y₂ = x₂[n] + x₂[n - 10] ...(2)

y₁[n] = x₁[n] + x₂[n] + x₂[n - 10] + x₂[n - 10] ...(3)

Now find y₁[n] + y₂[n]

Corresponding to x₁[n] + x₂[n]

It is same as equation (3) hence linear.

But in part (c) y[n] = x²[n]

? y₁[n] = x²₁[n], y₂[n] = x²₂[n]? y₁[n] + y₂[n] = x²₂[n]...3

But y₁[n] + y₂[n]

Corresponing x₁[n] + x₂[n] is y₁[n] + y₂[n] = {x₁[n] + x₂[n]}²

= x₁²[n] + x₂²[n] + 2x₁[n] x₂[n]....4

Equations (3) and (4) are not same hence not linear.

In transfer function, no of zeros = no. of poles, hence it is all pass fliter.