# How does spectral density of a power signal change when it is multiplied by $\cos(2\pi f_ct)$?

Suppose we have a power signal $$n(t)$$ whose power- auto-correlation function is: $$R_{n}(\tau)$$.

Now we multiply the signal by a cosine as $$\cos(2\pi f_ct) \cdot n(t)$$ whose auto-correlation is $$R_{n_i}(\tau)$$.

How is $$R_{n}(\tau)$$ mathematically related to $$R_{n_i}(\tau)$$ ?

$$R_{n}(\tau) = E(n(t)n(t-\tau))$$ $$R_{n_i}(\tau) = \cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))E(n(t)n(t-\tau))=\cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))R_{n}(\tau)$$ Using $$\cos(a-b) = \cos a \cos b + \sin a \sin b$$, we get $$R_{n_i}(\tau) = \cos(2\pi f_ct)[ \cos(2\pi f_ct ) \cos(2\pi f_c\tau)+ \sin(2\pi f_ct) \sin(2\pi f_c\tau)]R_{n}(\tau)$$ or $$R_n(\tau) = \frac{R_{n_i}(\tau)}{\cos(2\pi f_ct)[ \cos(2\pi f_ct ) \cos(2\pi f_c\tau)+ \sin(2\pi f_ct) \sin(2\pi f_c\tau)]}$$
• but $\cos(f(t))$ is not a random process. Hence seen as constant with respect to $E(.)$ – Ahmad Bazzi Jul 8 at 10:25