# Can spectral density be a complex quantity?

I have a signal ($$S(t)$$) which is product of a Gaussian ($$G(t)$$) and a random phase function ($$e^{i\theta(t)}$$, here $$\theta(t)$$ is a random function), as shown below

$$S(t)=G(t).e^{i\theta(t)}$$

If I calculate the auto correlation of such a signal ($$E[S^*(t)S(t−τ)]$$) it turns out to be a complex quantity and the same goes with the spectral density (Fourier transform of the auto-correlation function). My questions are the following.

1. If the above analysis valid? as the process is not wide sense stationary.
2. If the analysis is not valid, is there some way to handle this kind of situation?
3. If the analysis is valid, What does the complex spectral density signifies?

and the same goes with the spectral density (Fourier transform of the auto-correlation function).

No, that's not the case.

Since the autocorrelation is a hermitian symmetric function for any $$S$$, its Fourier transform is always real.

If the above analysis valid? as the process is not wide sense stationary.

If the process is not WSS, then you can't just proclaim $$E[S^*(t)S(t-\tau)]$$ to be dependent on only one variable (usually, $$\tau$$), and hence, a (1D) Fourier transform doesn't make much sense.

If the analysis is not valid, is there some way to handle this kind of situation?

Depends! You might want to define/find coherency times and do Short-Time Fourier Transforms within that.

Your system, in fact, is just a phase shifted impulse response – as such, a phase-delay representation, which might be derived from a Frequency Shift-Delay plane, might be more helpful in analyzing things. You'll find such a Frequency Shift-Delay plane in what is called scatter function in wireless communications, representing the Doppler and path coefficients of a wireless channels in motion.

But in your case: Is trying to understand PSD or PSD-equivalents really useful? Don't you just want to build a parametric estimator for $$\mu_G$$, $$\sigma_G^2$$ and $$\theta$$ instead?

• Thanks Marcus for your reply.. I am specifically interested in Spectral density of such signal. Actually, I am new to the subject can you suggest some good reference for the "Frequency Shift-Delay plane" analysis. Because what i understand the "Short-Time Fourier Transforms" results depends on the size of coherency window. – Rosh Jan 3 at 17:21
• There's no PSD for non-WSS signals. I gave you "scatter function" to google for. – Marcus Müller Jan 3 at 17:21