ESD, PSD and autocorrelation of white noise

Energy and power spectral density

Energy spectral density (ESD): for an energy signal $g(t)\leftrightarrow G(f)$, $\Psi_g(f)=\lvert G(f)\rvert^2$ (J/Hz). It shows how the energy is spread over frequency:

$$E=\int_{-\infty}^{\infty}\lvert g(t)\rvert^2dt=\int_{-\infty}^{\infty}\Psi_g(f)\,df$$

and $\Psi_g(f)$ is the Fourier transform of the energy autocorrelation $R_g(\tau)$.

Power spectral density (PSD): for a power signal, take a truncated version $g_T(t)$ (length $T$) with transform $G_T(f)$:

$$S_g(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}\ \ \text{(W/Hz)},\qquad P=\int_{-\infty}^{\infty}S_g(f)\,df$$

For a WSS random process, $S_X(f)=\lim_{T\to\infty}E[\lvert X_T(f)\rvert^2]/T$. Both ESD and PSD are real, even and non-negative.

Wiener–Khinchin relation (PSD $\leftrightarrow$ AC): for a power signal, $R_g(\tau)=\lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}g_T(t)g_T(t+\tau)dt$. The time-domain integral is $g_T(\tau)*g_T(-\tau)$ scaled by $1/T$, and $g_T(-t)\leftrightarrow G_T^*(f)$, so

$$\begin{aligned} \mathcal{F}\{R_g(\tau)\} &= \lim_{T\to\infty}\frac{1}{T}G_T(f)G_T^*(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}=S_g(f) \end{aligned}$$

Hence

$$S_g(f)=\int_{-\infty}^{\infty}R_g(\tau)e^{-j2\pi f\tau}d\tau,\qquad R_g(\tau)=\int_{-\infty}^{\infty}S_g(f)e^{j2\pi f\tau}df$$

and at $\tau=0$: $R_g(0)=\int S_g(f)df=P$.

Autocorrelation of white noise from its PSD

White noise is an idealised random noise whose power spectral density is flat (constant) over all frequencies, like white light containing all colours. Thermal and shot noise are nearly white up to very high frequencies ($\sim 10^{12}$ Hz), so the model is accurate for any practical receiver bandwidth. It is usually also assumed Gaussian, zero-mean (AWGN).

PSD (two-sided):

$$S_W(f)=\frac{N_0}{2}\quad\text{for all } f$$

where $N_0=kT_e$ (W/Hz), $k$ = Boltzmann constant and $T_e$ = equivalent noise temperature.

Autocorrelation (inverse Fourier transform of PSD, using $\mathcal{F}^{-1}\{1\}=\delta(\tau)$):

$$R_W(\tau)=\int_{-\infty}^{\infty}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\,\delta(\tau)$$

   S_W(f)                      R_W(tau)
     |                             ^  (N0/2) delta
 N0/2+-----------------            |
     |                             |
 ----+-----------------> f   ------+------> tau
     0                             0

Meaning: $R_W(\tau)=0$ for every $\tau\ne0$, so any two samples of white noise, however close, are uncorrelated (and independent if Gaussian). The total power $R_W(0)=\int S_W(f)df$ is infinite, which is why white noise is only a model; after any filter with finite bandwidth the power becomes finite.

Band-limited white noise (more practical): if $S(f)=\frac{N_0}{2}$ for $\lvert f\rvert\le B$ and zero elsewhere,

$$R(\tau)=\int_{-B}^{B}\frac{N_0}{2}e^{j2\pi f\tau}df=N_0B\,\frac{\sin2\pi B\tau}{2\pi B\tau}$$

Its power is finite, $R(0)=N_0B$, and samples spaced $1/2B$ apart are uncorrelated.