In a video i saw that while calculating ESD or PSD of a signal time auto correlation function was used when it can be also done by the square of FT of the signal.Why we followed that approach even when the later process is more straight-forward and easy.

How does it helps us in later stage?

  • $\begingroup$ "we" don't always prefer that. Unless you can restrict your "always" (for example by stating your source – something that's always good to do!), I'd say the assumption on which this question is based is false. Not quite sure what to answer there. Especially since, if you think about it, the autocorrelation of a stochastic signal and the square of its Fourier transform are inherently linked by the Wiener-Kinchin theorem (if applicable to the signal, needs WSS). In all honesty, I think you're misunderstanding what was said (it was probably a hint towards Wiener-Kinchin), so: $\endgroup$ – Marcus Müller Aug 23 '18 at 19:14
  • $\begingroup$ It's unclear (to me) what you're asking. Please add a source and context. $\endgroup$ – Marcus Müller Aug 23 '18 at 19:14
  • $\begingroup$ (this is by the way the same thing as with your previous question: you are assuming something "always" happens in a specific way, and ask about that, whereas it really doesn't always happen in that way) $\endgroup$ – Marcus Müller Aug 23 '18 at 19:15
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    $\begingroup$ For starters, if you square a signal, you will alias frequency content higher than fs/4. $\endgroup$ – Ben Aug 23 '18 at 19:16
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    $\begingroup$ @Ben I think OP's algorithm squares the Fourier Transform (and really, this sounds like he watched a video that introduced the Wiener-Kinchin theorem). $\endgroup$ – Marcus Müller Aug 23 '18 at 19:18

Let's distinguish theory from practice.

In the theory of discrete-time random signals, the spectral content of a given random process $X[n,s]$ (abbreviated as $x[n]$ for simplicity) is often desired, just as it's desired for deterministic signals (such as that of a single pulse of duration T). This spectral content is termed as the power spectral denisty (PSD) as explained later.

When the frequency of a signal is mentioned at once, the immediate consequent is the Fourier transform of that signal. However when the signal is random, there is a problem; random processes do not have Fourier transforms. Why? Because they have infinite energy. Yes, a random signal will have infinite energy as the sum $$ \sum_{n=-\infty}^{\infty} |x[n]|^2 $$

diverges; (as $\lim_{n \to \infty} |x[n]|^2 \neq 0$, which is sufficient to prove that the infinite series cannot converge.) . Then, the DT Fourier transform of $x[n]$ does not exist: $$X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} $$ does not converge for any $\omega$.

Then what can be done? The accepted approach is to use another signal; the auto-correlation sequence (ACS) $r_{xx}[m]$ of the random process which is defined as;

$$ r_{xx}[m] = \mathcal{E}\{x[n]x[n+m]^{*} \} ~~~,~~~\text{ for } -\infty < m < \infty$$

The most important property of this ACS (from DTFT point of view) is that it has finite energy (when being aperiodic) and infinte energy only if it is periodic. Both of these cases are sufficient to compute a valid Fourier transform ot it which is denoted as the power spectral density (PSD) of the random process $x[n]$: $$ S_{xx}(\omega) = \sum_{m=-\infty}^{\infty} r_{xx}[m] e^{-j \omega m} $$

The indicator power stems from the fact that in essence it's trying to compute a Fourier transform of a multiplication of two signals; $x[n]$ with itself $x[n+m]$ at a lag $m$. Hence it's about the square of the signal, the instantaneous power.

All these are theory. Because to compute the PSD, you need to know the ACS. But the ACS is related to the PDF (probability density/distribution function) of the random process $x[n]$. And the PDF in practice is not known. In theory (as a part of a homework assignment for example) you can propose a PDF for a given process, and then proceed analytically to compute its ACS and PSD. Yet in practice the signals that you are observing are truly random hence the analytical formulation of their PDF is not exactly known.

So in practice you must find a convenient means of computing the PSD of a partially observed random process (such as a speech given my Martin Luther King). All these approaches are stemming from what is called as Estimation Theory which delas with finding convenient methods of computing certain parameters related with random variables and processes.

One of the methods to estimate the PSD of a random process relies on estimating its ACS from the observed data and then taking the Fourier transform of it to reach an estimate of the true PSD. (The true power spectral density relies on the true ACS which is not available but estimated from data).

The other approach, and the more commonly used one, directly estimates the PSD (which is itself a random quantity) from the observed data. The most basic of such direct PSD estimation is known as the periodogram and obtained by computing the Fourier transform of the signal itself;i.e.,

$$I(\omega) = K \cdot |X(\omega)|^2 $$ where $X(\omega)$ is the DTFT of the finite length and finite energy observed random signal $x[n]$, $K$ is a proper scaling factor and $I(\omega)$ is the periodogram estimate of the PSD.

And this is shows a practical means of computing (actually estimating) the PSD associated with a random process $X[n,s]$ over one or more of its observations $x[n]$.


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