Consider a signal-plus-noise process of the formz(t) = A cos

Consider a signal-plus-noise process of the form

z(t) = A cos 2π(f₀ + f_d)t + n(t)                                       (D.29)

where n(t) is given by

n(t) = n_c(t) cos 2πf₀t – n_s(t) sin 2πf₀t                         (D.30)

Assume that n(t) is an ideal band limited white-noise process with double-sided power spectral density equal to 1/2 N₀, for -1/2 B ≤ f ± f₀ ≤ 1/2 B, and zero otherwise. Write z(t) as

(a) Express n’_c(1) and n’_s (t) in terms of n_c(t) and n_s (t). Find the power spectral densities of n’_c(t) and n’_s(t), S’_nc(f) and S_n’s(f). 

(b) Find the cross-spectral density of n’_c(t) and n’_s(t), S_n’cn’s (f), and the cross-correlation function, R_n’cn’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?