Consider a signal-plus-noise process of the form z(t) = A cos 2(f 0 + f d )t

Consider a signal-plus-noise process of the form

z(t) = A cos 2π(f₀ + f_d)t + n (t)

where ω₀ = 2πf₀, with

n (t) = n_c (t) cos ω₀t - n_s (t) sin ω₀t

an ideal band limited white-noise process with double- sided power spectral density equal to 1/2 N₀, for f₀ – B/2 ≤ |f| ≤ f₀ + B/2, and zero otherwise. Write z(t) as z(t) = A cos[2π(f₀ + f_d)t] + n′_c(t) cos[2π(f₀ + f_d)t] – n′_s(t) sin[2π(f₀ + f_d)t]

(a) Express n′_c(t) and n′_s (t) in terms of n_c(t) and n_s(t). Using the techniques developed in Section 7.5, find the power spectral densities of n′_c(t) and n′_s(t), S_n′c(f) and S_n′s(f).

(b) Find the cross-spectral density of n′_c(t) and n′_s(t), S_n′cn′s and the cross-correlation function, R_n′c′s(τ). Are n′_c(t) and n′_s(t) correlated? Are n′_c(t) and n′_s(t), sampled at the same instant independent?