# How to find spectral density of a signal whose correlation depends on time?

Suppose we have two uncorrelated, real white noise $$\xi_1(t)$$ and $$\xi_2(t)$$, with zero mean, and equal variance.

Now, I create the signal $$\xi(t) = a\xi_1(t)\cos(\omega_0 t) + b\xi_2(t) \sin(\omega_0 t)$$. ($$a,b$$ are real)

I want to find the spectral density of $$\xi$$. The correlation $$<\xi(t) \xi(t+\tau)>$$ depends on both $$t$$ and $$\tau$$. Now, to calculate spectral density, should I calculate $$\int_{-\infty}^{\infty} e^{i\omega\tau} <\xi(0) \xi(\tau)> d\tau$$?

Or should it be $$\int_{-\infty}^{\infty} e^{i\omega\tau} (\text{average over all t of} <\xi(t) \xi(t+\tau)>) d\tau$$ ?

I have calculated the correlation $$<\xi(t) \xi(t+\tau)>$$

$$=<[a\xi_1(t)\cos(\omega_0 t) + b\xi_2(t)\sin(\omega_0 t)]\times [a\xi_1(t+\tau)\cos(\omega_0 (t+\tau)) + b\xi_2(t+\tau))\sin(\omega_0 (t+\tau)]>$$

$$= a^2 \cos(\omega_0 t)\cos(\omega_0 (t+\tau))<\xi_1(t) \xi_1(t+\tau)>$$

$$+ b^2 \sin(\omega_0 t)\sin(\omega_0 (t+\tau))<\xi_2(t) \xi_2(t+\tau)>$$

$$+ ab \cos(\omega_0 t)\sin(\omega_0 (t+\tau))<\xi_1(t) \xi_2(t+\tau)>$$

$$+ ab \sin(\omega_0 t)\cos(\omega_0 (t+\tau))<\xi_2(t) \xi_1(t+\tau)>$$

Since $$\xi_1(t),\xi_2(t)$$ are uncorrelated noise, this is equal to $$= a^2 \cos(\omega_0 t)\cos(\omega_0 (t+\tau)) \cdot c\delta(\tau) + b^2 \sin(\omega_0 t)\sin(\omega_0 (t+\tau)) \cdot c\delta(\tau)$$

(where $$\delta(\tau)$$ is the Dirac delta function, $$c$$ is a real number that depends on the amplitude of the noises)

• "Gaussian white noise ξ1(t) and ξ2(t), with zero mean, and equal variance" -- note that white Gaussian noise has no variance (though some would say that the variance is infinite).
– MBaz
May 23, 2019 at 14:03

Your process is not stationary. As you already correctly noted, your autocorrelation function depends on $$t$$ and $$\tau$$. Let me call it $$\varphi(\tau,t)$$.

There are multiple ways of dealing with such cases. One is to simply consider Fourier transforms with respect to each of the time variables, treating them independently: The transform with respect to $$\tau$$ gives you frequency (say, $$f$$), where as the transform with respect to $$t$$ gives you a rate of change as in how fast do your statistics change, the latter often being referred to as Doppler frequency (say $$\alpha$$).

Now you can define four functions:

• Time-varying ACF $$\varphi(\tau,t)$$
• Time-varying Power spectral density $$\Phi(f,t)$$
• Delay/Doppler cross spectral density $$\varphi(\tau,\alpha)$$: often also called scattering function
• Frequency/Doppler power spectrum $$\Phi(f,\alpha)$$

These are also called the second set of Bello functions, the concrete naming of each of them varies widely across sources.

Another way of attacking the problem is to go to the Wigner-Ville distribution and its variants, have a look here.