4.5: Wiener-Khinchine Relation

Recall from our discussion of the Fourier transform that convolution in the time domain of the impulse response \(h(t)\) and an arbitrary system input \(u(t)\), is equivalent to multiplication in the frequency domain of the Fourier transforms. This is a property in particular of linear, time-invariant systems. Now we can make some additional strong statements in the case of random processes.

If \(u(t)\) is stationary and ergodic, and the system is LTI, then the output \(y(t)\) is also stationary and ergodic. The statistics are related using the spectrum:

\[ S_y (\omega) = |H(\omega)|^2 S_u(\omega). \]

This can be seen as a variant on the transfer function from the Fourier transform. Here, the quantity \(|H(\omega)|^2\) transforms the spectrum of the input to the spectrum of the output. It can be used to map the statistical properties of the input (such as an ocean wave field) to statistical properties of the output. In ocean engineering, this is termed the response amplitude operator, or RAO.

To prove this, we will use the convolution property of LTI systems.

\begin{align*} y(t) &= \int\limits_{-\infty}^{\infty} h(\tau) u(t - \tau) \, d\tau, \textrm{ so that} \\[4pt] R_y(t, \, \tau) &= E[y(t) y(t + \tau)], \\[4pt] &= E \left\{ \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} h(\tau_1) u(t - \tau_1) h(\tau_2) u(t + \tau - \tau_2) \, d\tau_1 d\tau_2 \right\} \\[4pt] &= \int\limits_{-\infty}^{\infty} \int \limits_{-\infty}^{\infty} d\tau_1 d\tau_2 \, h(\tau_1) h(\tau_2) E[u(t - \tau_1) u(t + \tau - \tau_2)] \\[4pt] &= \int\limits_{-\infty}^{\infty} \int \limits_{-\infty}^{\infty} d\tau_1 d\tau_2 \, h(\tau_1) h(\tau_2) R_u(\tau - \tau_2 + \tau_1) \\[4pt] & \textrm{(because the input is stationary and ergodic, } R_u \textrm{ does not depend on time)} \\[4pt] S_y(\omega) &= \int\limits_{-\infty}^{\infty} R_y(\tau) e^{-i \omega \tau} \, d\tau \\[4pt] &= \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} d\tau d\tau_1 d\tau_2 \, e^{-i \omega \tau} h(\tau_1) h(\tau_2) R_u(\tau - \tau_2 + \tau_1); \textrm{ now let } \xi = \tau - \tau_2 + \tau_1 \\[4pt] &= \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} d\xi d\tau_1 d\tau_2 \, e^{-i \omega (\xi + \tau_2 - \tau_1)} h(\tau_1) h(\tau_2) R_u (\xi) \\[4pt] &= \int\limits_{-\infty}^{\infty} d\xi \, e^{-i \omega \xi} R_u(\xi) \int\limits_{-\infty}^{\infty} e^{i \omega \tau_1} h(\tau_1) \int\limits_{-\infty}^{\infty} d\tau_2 \, e^{-i \omega \tau_2} h(\tau_2) \\[4pt] &= S_u(\omega) H^* (\omega) H(\omega). \end{align*}

Here we used the \(*\)-superscript to denote the complex conjugate, and finally we note that \(H*H=|H|^2\).

Figure \(\PageIndex{1}\): graphs of the system in terms of the system input \((S_u (\omega)\)), the system in terms of the system output \((S_y (\omega)\)), and of the transformation \(|H(\omega)|^2\) by which \(S_u(\omega)\) was multiplied to obtain \(S_y(\omega)\).