# Relation between probability of a signal and its magnitude spectrum

I am a newbie. In these days I am searching if there are reference about properties that can relate the probability of the signal and the magnitude spectrum. I know what are both. But I was curious wheter there are connection between them. Can you help me

Given a signal y(t), the magnitude spectrum is $$Y(f)=\left\lvert\int y(t)e^{-2\pi ft}\,\mathrm dt\right\rvert,$$ in other words the absolute value of the FT of the signal.

• Hi! What do you refer to wehen you say "Probability of a signal"? – Marcus Müller May 8 at 12:39
• And to make it easier to explain exactly such that it fits your wishes: When you say "I know what the magnitude spectrum is", could you write that down (and add that to your question by editing it) as formula? (if you can't write that as a formula, no problem, but then we really need to understand how you define it then) – Marcus Müller May 8 at 12:48
• Yes, sorry. I was meaning the probability density function – Sonia Bellicchi May 8 at 13:07
• great! But then, the definition of magnitude spectra directly involves the probability density function of the signal, so to help you we'd really need to know how your formula for the magnitude spectrum looks like! – Marcus Müller May 8 at 13:12
• @SoniaBellicchi that's what I asked you to add to the question by editing it. Done this for you! – Marcus Müller May 8 at 15:49

Given a signal y(t), the magnitude spectrum is $$Y(f)=\left\lvert\int y(t)e^{-2\pi ft}\,\mathrm dt\right\rvert,$$ in other words the absolute value of the FT of the signal.

But that's only applicable for deterministic signals¹. If your signal itself is random, then you can't know $$y(t)$$ – you only know some stochastic properties of it. There's no $$y(t)$$ to transform if you don't actually know $$y(t)$$. For random signals, $$y(t)$$ can't be given, otherwise the signal wouldn't be random. You might observe a realization, but that realization doesn't describe the full stochastic process!

The definition of the power spectral density for random signals like your $$y$$ is the Fourier transform of the autocorrelation function. For that to be sensibly defined, you need your autocorrelation function to depend only on one variable, the time shift, which means your random signal needs to be wide-sense stationary. But, if that's the case, then the autocorrelation function (ACF) is defined as

$$\phi_{yy}(\tau) := \mathbb E \left\{ y(t) y^*(t+\tau) \right\},$$

and then the power spectral density, which gives your magnitude spectrum is its Fourier transform, $$\Phi_{yy}(f) := \int_{-\infty}^{\infty} \phi_{yy}(\tau) e^{-i2\pi f\tau}\,\mathrm d\tau$$.

Now, in the expectation $$\mathbb E \left\{ y(t) y^*(t+\tau) \right\}$$, there's your random signals probability density function:

\begin{align} \phi_{yy}(\tau) &:= \mathbb E \left\{ y(t) y^*(t+\tau) \right\} \\ &= \int_{-\infty}^\infty \int_{-\infty}^\infty y'_1\,y'_2 \,\cdot\, f_{ (y({t_1}),y({t_1+\tau})) } (y'_1,y'_2) \, \mathrm d y'_2 \; \mathrm d y'_1 \end{align}

Notice that here, the density is a function of two values, and parameterizes over the time shift.

¹ you're also missing imaginary unit in the exponent and the integration bounds, but that's not the problem here. But, really, make sure to be able to write down the Fourier transform correctly.

• I think the key point being missed in all the above is that the density function determines the autocorrelation function which in turn determines the power spectral density but the reverse connection breaks down. The reverse connection allows us to get the autocorrelation function from the power spectral density but one cannot get from the autocorrelation function to the density, and I think that this last step is what the OP was asking about. – Dilip Sarwate May 8 at 20:06
• @DilipSarwate hm. Sonia ask how to relate pdf and magnitude spectrum. "Relate" imho doesn't imply "direction from spectrum to pdf". – Marcus Müller May 8 at 20:08
• Just a little request of advice. I followed a short course of signal processing, but I always have more complex question compared to the topic that I have studied in the cours. Which can be a good book, that in your experience is complete? – Sonia Bellicchi May 8 at 20:26
• @MarcusMüller is the probability density function $f_{y(t_{1}),y(t_{1}+\tau))}?$ what do the variables $y'_{1}$ and $y'_{2}$ represent ? – Mardy Owens May 8 at 20:47
• @SoniaBellicchi excellent question! Signal processing often starts with the assumption that the signal is deterministic, right. Things like stochastic processes (which your random signal is) were introduced to me in my probability theory course at university (I studied EE) – but I recon that might be not that common for engineering students. I've recommended Steven M. Kay's book before, but it might be overkill. – Marcus Müller May 9 at 11:46