Spectral density of modulated noise process

Question

Suppose we have a noise process $F(t)$. Suppose $F$ is stationary and we know its autocorrelation function $$\rho_F(\tau) \equiv \langle F(0)F(\tau) \rangle$$ where the average is over realizations of the process. We then construct a new noise process $G(t) \equiv \cos(\Omega t)F(t)$. Obviously, $G$ is not stationary. The autocorrelation function of $G$ is \begin{align} \langle G(t_1) G(t_2)\rangle &= \langle F(t_1) \cos(\Omega t_1)F(t_2) \cos(\Omega t_2) \rangle \\ &= \langle F(t_1) F(t_2) \rangle \cos(\Omega t_1) \cos(\Omega t_2) \\ &= \rho_F(t_2 - t_1) \cos(\Omega t_1) \cos(\Omega t_2) \, .\\ \end{align}

The spectral density of $F$ can be usefully defined as $$S_F(\omega) = \int \frac{d\omega}{2\pi} \rho_F(\tau) e^{-i \omega \tau} \, . $$ However, since $\langle G(t_1) G(t_2) \rangle$ depends on $t_1$ and $t_2$ and not just their difference, we cannot easily define $S_G(\omega)$. This is distressing because for deterministic signals we expect modulation by a sinusoid to just shift the spectral density, so we expect there should be a simple formula for $S_G(\omega)$.

What, if any, is a meaningful way to construct the spectral density of $G$?

Answer 1

This is a standard problem in (among others) the theory of analog or digital communication systems. If $F(t)$ is real-valued, then the standard gimmick is indeed - as already pointed out in a comment by Dilip Sarwate - to add a random phase to the carrier phase. This is not only a trick but it actually reflects reality in the sense that we often do not know (and do not care to know) the exact phase of the carrier with respect to the signal $F(t)$. So what we actually want is NOT the power spectrum of $F(t)\cos(\omega_0 t)$ (which does not exist), but the power spectrum of the process $F(t)\cos(\omega_0 t+\theta)$ with some unknown phase $\theta$, the obvious model of which is a random variable with a uniform distribution in $[0,\pi)$. This random phase is of course constant over time, but its value is chosen at random reflecting our uncertainty about the origin of the time axis. With this random phase the resulting process is wide-sense stationary (WSS) (if $F(t)$ is), and the power spectrum looks exactly as one would expect.

Now comes the interesting part. If $F(t)$ is a WSS complex-valued process and if we're interested in the power spectrum of the process

$$G(t)=\text{Re}\left\{F(t)e^{j\omega_0 t}\right\}\tag{1}$$

then we could also use the random phase trick, but under some reasonable assumptions we don't even need it. The process $G(t)$ in (1) occurs e.g. in systems using quadrature modulation. It is straightforward to show that the autocorrelation of $G(t)$ is

$$\begin{align}R_G(t,\tau)&=E\left\{G(t+\tau)G^*(t)\right\}=\\&=\frac12\text{Re}\left\{R_F(\tau)e^{j\omega_0\tau}\right\}+\frac12\text{Re}\left\{R_{FF^*}(\tau)e^{j\omega_0(\tau+2t)}\right\}\tag{2}\end{align}$$

which becomes independent of $t$ if $R_{FF^*}(\tau)=0$, i.e. if the cross-correlation of $F(t)$ and $F^*(t)$, vanishes. If $F(t)=A(t)+jB(t)$, it can be shown that this is the case if

$$R_A(\tau)=R_B(\tau)\tag{3}$$

and

$$R_{AB}(\tau)=-R_{AB}(-\tau)\tag{4}$$

So if conditions (3) and (4) are satisfied, we don't even need a random carrier phase in order to define a power spectrum of $G(t)$ as defined in (1). Conditions (3) and (4) mean that the real and imaginary parts of $F(t)$ have the same autocorrelation, i.e. the same power spectrum, and that their cross-correlation is an odd function, i.e. their cross-power spectrum is purely imaginary. This last condition also implies $R_{AB}(0)=0$, i.e the real and imaginary part are uncorrelated when sampled at the same time. It turns out that for many practical problems these conditions are at least approximately satisfied.

Given that conditions (3) and (4) are satisfied, the autocorrelation function of $G(t)$ is given by

$$R_G(\tau)=\frac12\text{Re}\left\{R_F(\tau)e^{j\omega_0\tau}\right\}\tag{5}$$

and its power spectral density is given by the Fourier transform of (5):

$$S_G(\omega)=\frac14\left[S_F(\omega-\omega_0)+S^*_F(-\omega-\omega_0)\right]\tag{6}$$

where $S_F(\omega)$ is the power spectral density of $F(t)$.