# Autocorrelation and Power Spectral Density (Discrete)

The Autocorrelation, $\phi_{aa}[\kappa]$, of a discrete time random process, $a[k]$, is defined as:

$$\phi_{aa}[\kappa] = \mathrm{E}\left\{ a[k+\kappa]a^*[k] \right\}$$

Taking its fourier transform would give the Power Spectral Density, which is given in my book as:

$$\mathscr{F}\big\{ \phi_{aa}[\kappa] \big\} = \Phi_{aa}\left(e^{j2\pi fT} \right) = \sum\limits_{\kappa = -\infty}^{\infty} \phi_{aa}[\kappa] e^{-j2\pi\kappa fT}$$

Now, why don't they denote simply as: $\Phi_{aa}[\kappa]$?

Is the form above with the exponential signify something related to sampling interval? I'm quite confused with this form and any explanation to better understand this form would be helpful.

• let $\omega \triangleq 2 \pi f T$ and you'll have this expression in terms of normalized angular frequency, and it will also be clear that the variant of Fourier Transform implied is the DTFT. – robert bristow-johnson Jul 13 '17 at 0:34
• Are you aware that the DTFT of a discrete process is actually continuous? – Envidia Jul 13 '17 at 0:36
• Well, for a start, the summation is done over $\kappa$ so it will not appear in the result. The only variable that remains after summation is $f$ (or $2\pi f T$). -1 for a complete misunderstanding of how summations work. – Peter K. Jul 13 '17 at 12:53

First of all lets state more correctly that a discrete-time 1D auto-correlation sequence (ACS), $\phi_{XX}[\kappa]$, of a single parameter $\kappa$, of a discrete-time random process $X[n,s]$ will exist as long as the process is stationary or at least WSS (wide sense stationary).

And therefore the power spectral density (PSD), $\Phi_{XX}(e^{j2\pi fT})$, of the process will also exist iff the random process is stationary (or WSS at least) and its ACS is stable, in the sense that its Fourier transform will exist. (In case ACS is not stable, generalized definition of the Fourier transform is used to represent nonconvergent integrals with impulses or with discontinuties at those particular frequencies...)

Note that the notation changes slightly and I prefer to use $\phi_{XX}[m]$ to denote the ACS of the WSS random process $X[n,s]$ and use $\Phi_{XX}(e^{j\omega})$ to indicate its PSD, which is indeed given by the discrete-time Fourier transform (DTFT) of its ACS : $$\Phi_{XX} (e^{j\omega}) = \mathcal{F} \{ \phi_{XX}[m] \} = \sum\limits_{m=-\infty}^{\infty} \phi_{XX}[m] e^{-j\omega m}$$

Since the PSD of a WSS random process $X[n,s]$ is a function returned by the DTFT of a sequence, it must be a periodic and continuous function of frequency $\omega$ , as all other Fourier transforms. We are only interested in the first period from $\omega=-\pi$ to $\omega=\pi$ (or from $\omega=0$ to $\omega=2\pi$)

Coming to your confusion, in most practical applications, a discrete-time random process $X[n,s]$ is represented by a discrete-time random sequence $x[n]$ which is created by the process of sampling a particular realization, $x(t)$, of a continuous-time random process $X(t,s)$ with a sampling period $T_s$. In such a case $\omega = 2\pi f T_s$ could be used as indication of the sampling relationship between the continuous-time and the discrete-time random processes involved. Also in communications terminology, frequency in Hertz $f$ is preferred over frequency in radians per second $\Omega$.

Note also that I've deliberately used a capital letter $X$ , instead of a lowercase $x$, to denote the ownership of those ACS and PSD functions; They belong to the random process $X[n,s]$ and not to a single particular realisation sequence denoted by $x[n]$. The sequence $x[n]$ belongs to the family of all functions that constitude the ensamble of the random process $X[n,s]$ ($n$ denotes the (time) index and $s$ denotes the outcome of the experiement for those indexed random variables $X_n(s)$ associated with the process $X$)

The random process $X[n,s]$ exists as a theoretical entity, an abstract being with its own characterisation that generates all those possible realizations $x[n]$ being observed. In practice we can only observe one or more $x[n]$ and therefore we have to estimate the true (but unobservable) quantities of ACS and PSD that belong to random process $X[n,s]$ from those realisations by means of statistical estimation methods. This process is not exact, hence in practice we can only compute an approximation to those true theoretical entities of ACS and PSD from particular obsrevations of instances $x[n]$ of the random process $X[n,s]$.