A query on Power spectral density (PSD)

Question

Say a narrow band signal $n(t)$ has Power Spectral Density (PSD) $S(f)$.

If the signal $n(t)$ got multiplied by $\cos(2\pi Ft)$, then what will the PSD of resulting signal in terms of $S(f)$ be? Will this be $\frac{S(f+F) + S(f-F)}{2}$ ?

Answer 1

Almost, but let's start at the beginning. If the random process $N(t)$ has a power spectrum then it is at least wide-sense stationary (WSS), i.e., its mean and its autocorrelation function do not depend on time. However, the process

$$Z(t)=N(t)\cos(2\pi f_ct)\tag{1}$$

is not stationary, and, consequently, it has no power spectrum (the mean of the random process $Z(t)$ is given by $E[N(t)\cos(2\pi f_ct)] = E[N(t)]\cos(2\pi f_c t)$ which clearly depends on time even if $E[N(t)]$ does not, a similar reasoning can be made for the autocorrelation function).

Luckily, we're usually not interested in the power spectrum of the process $(1)$, but rather in the power spectrum of the process

$$Y(t)=N(t)\cos(2\pi f_ct+\theta)\tag{2}$$

where $\theta$ is a random phase which is independent of $N(t)$ and which is uniformly distributed on $[0,2\pi)$. This random phase reflects the uncertainty of the carrier phase with respect to the process $N(t)$. So adding a random phase is not only a "trick" that works well, but it actually reflects the nature of a modulated random process much better than the process $(1)$, which assumes a known relationship between the carrier phase and the signal.

The process given by $(2)$ is WSS, and its autocorrelation function can be computed as follows:

$$\begin{align}R_{YY}(\tau)&=E[Y(t)Y(t+\tau)]\\&=E[N(t)\cos(2\pi f_ct+\theta)N(t+\tau)\cos(2\pi f_c(t+\tau)+\theta)]\\&=\frac12 E[N(t)N(t+\tau)(\cos(2\pi f_c\tau)+\cos(2\pi f_c(2t+\tau)+2\theta))]\\&=\frac12R_{NN}(\tau)\cos(2\pi f_c\tau)+\frac12 R_{NN}(\tau)\underbrace{E[\cos(2\pi f_c(2t+\tau)+2\theta)]}_{=0}\\&=\frac12R_{NN}(\tau)\cos(2\pi f_c\tau)\tag{3}\end{align}$$

where I've used the independence of $N(t)$ and $\theta$, and where $R_{NN}(\tau)=E[N(t)N(t+\tau)]$.

Finally, the power spectrum of $Y(t)$ is given by the Fourier transform of $(3)$:

$$S_{YY}(f)=\frac14\left[S_{NN}(f-f_c)+S_{NN}(f+f_c)\right]\tag{4}$$

where $S_{NN}(f)$ is the power spectrum of $N(t)$.

In sum, if we add a random phase to the carrier then the modulated process is also WSS (if the baseband process is WSS), and its power spectrum is given by $(4)$.