Parseval's theorem; PSD–autocorrelation relation for white noise

Parseval's theorem

Parseval's theorem states that the power (or energy) of a signal is the same whether computed in the time domain or in the frequency domain.

• Periodic power signal with Fourier coefficients $c_n$: $P=\frac{1}{T_0}\int_{T_0}\lvert g(t)\rvert^2dt=\sum_{n=-\infty}^{\infty}\lvert c_n\rvert^2$
• Energy signal (Rayleigh form): $E=\int\lvert g(t)\rvert^2dt=\int\lvert G(f)\rvert^2df$

Relation between PSD and autocorrelation

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

Example: white noise. The PSD is flat, $S_W(f)=N_0/2$. Then by the relation above:

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

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

A flat (infinitely wide) spectrum corresponds to an impulse autocorrelation: white noise is uncorrelated with itself at any non-zero shift. Conversely, if white noise is band-limited to $B$, $R(\tau)=N_0B\,\text{sinc}(2B\tau)$ spreads out (with $\text{sinc}(x)=\sin\pi x/\pi x$); the narrower the spectrum, the wider the autocorrelation.

Link with Parseval: putting $\tau=0$ in $R(\tau)=\int S(f)e^{j2\pi f\tau}df$ gives $R(0)=\int S(f)df=P$, which is exactly Parseval's theorem for power: the time-average power equals the area under the PSD.