Sample Power Spectral Density

The Fourier transform of the sample autocorrelation function $\hat{r}_{v,N}$ (see (6.6)) is defined as the sample power spectral density (PSD):

$${\hat S}_{v,N}(\omega) \isdef \hbox{\sc DTFT}_\omega\{\hat{r}_{v,N}\} \isdef \sum_{n=-\infty}^\infty \hat{r}_{v,N}(n)e^{-j\omega n}$$ (7.11)

This definition coincides with the classical periodogram when $\hat{r}_{v,N}$ is weighted differently (by a Bartlett window).

Similarly, the true power spectral density of a stationary stochastic processes $v(n)$ is given by the Fourier transform of the true autocorrelation function $r_v(l)$ , i.e.,

$$S_{v}(\omega) = \hbox{\sc DTFT}_\omega\{r_v\}.$$ (7.12)

For real signals, the autocorrelation function is always real and even, and therefore the power spectral density is real and even for all real signals.

An area under the PSD expressed as a function of frequency $f$ in Hz, $S_x(2\pi f)\cdot\Delta f$ , comprises the contribution to the variance of $x(n)$ from the frequency interval $[f,f+\Delta f]$ . The total integral of the PSD then gives the total variance:

$$\int_{-\pi}^\pi S_v(\omega) \frac{d\omega}{2\pi} = \int_{-0.5}^{0.5} S_v(2\pi f) df = r_v(0) = \sigma_v^2,$$ (7.13)

again assuming $v(n)$ is zero mean.^7.5

Since the sample autocorrelation of white noise approaches an impulse, its sample PSD approaches a constant, as can be seen in Fig.6.1. Its true PSD is precisely constant (variance divided by $2\pi$ ). This means that white noise contains all frequencies in equal amounts. Since white light is defined as light of all colors in equal amounts, the term ``white noise'' is seen to be analogous.