A signal m(t) is multiplied by a sinusoidal waveform of frequency f_c such that
v(t)=m(t) cos 2pf_ct
If Fourier transform of m(t) is M(f), Fourier transform of v(t) will be

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.

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

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

.

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.