4.7. Autocorrelation and PSD

The following figure shows the autocorrelation functions and Power Spectral Densities (PSDs) for three different random noise processes in order to tie the previous chapter on Signal Processing and the current one on Time Series analysis together.

We cover here white noise and random constant, which are two limiting cases. In practice, you’ll often find a random process in between and hence close to the exponentially correlated AR(1).

In the first row, we observe the theoretical autocorrelation of white noise, which is a Dirac delta function \(\delta (\tau)\) located at the zero lag \(\tau = 0\) with amplitude equal to the noise variance \(\sigma^2\). This reflects that white noise is uncorrelated, meaning that its autocorrelation is zero for all non-zero lags \(\tau \neq 0\). In reality, when computing the autocorrelation of a realization of white noise, we typically see a non zero value of autocorrelation at lag zero, and values close to zero at other lags. The corresponding PSD of white noise is constant across all frequencies, indicating equal power distribution over the frequency spectrum.

In the second row, we observe a random constant function with value \(A\). The autocorrelation of this function is also a constant with a value of \(A^2\) at all lags. The fact that the autocorrelation is flat indicates that a constant function is perfecly correlated with itself at any time shift \(\tau\). Every sample has an identical value (only the first one is really the random one). The corresponding PSD is a Dirac delta function located at the zero frequency \(f=0\), indicating that the function is not varying in time.

In the third row, an AR(1) process is shown. We can conclude that the AR(1) is time-correlated since its autocorrelation at lags different from zero is not negligible. Additionally, we can observe that the autocorrelation decays exponentially with lag \(\tau\), reflecting that the influence of past values decreases with time (actually a continuous-time lag \(\tau\) is used here). In this case, the PSD shows that more energy is concentrated at the lower frequencies. Note that the shape of the ACF and PSD depends on the noise variance \(\sigma^2\) and the AR(1) parameter \(\phi\).

Fig. 4.27 Time series, theoretical autocorrelation, and theoretical PSD for white noise, a constant function, and AR(1) process.

Attribution

This chapter was written by Alireza Amiri-Simkooei, Christiaan Tiberius and Sandra Verhagen. Find out more here.