4.6: Spectrum Interpretation

Let us consider now the stationary and ergodic random process with description:

\[ y(t) = \sum_{n=1}^N a_n \cos (\omega_n t + \psi_n), \]

where \(\psi_n\) is a random variable with uniform distribution in the range \([0, \, 2 \pi]\). As mentioned previously, this process has autocorrelation

\[ R(\tau) = \dfrac{1}{2} \sum_{n=1}^N a_n^2 \cos \omega_n \tau; \]

and then

\[ S(\omega) = \dfrac{1}{2} \sum_{n=1}^N a_n^2 \pi [\delta(\omega - \omega_n) + \delta(\omega + \omega_n)]. \]

As with the Fourier transform, each harmonic in the time domain maps to a pair of delta functions in the frequency domain. However, unlike the Fourier transform, there is no phase angle associated with the spectrum - the two delta functions are both positive and both real.

Furthermore, a real process has infinitely many frequency components, so that the spectrum really become a continuous curve. For example, the Bretschneider wave spectrum in ocean engineering is given by

\[ S^+ (\omega) = \dfrac{5}{16} \dfrac{\omega_m^4}{\omega^5} H_{1/3}^2 e^{-5 \omega_m^4 / 4 \omega^4} \]

where \(\omega\) is frequency in radians per second, \(\omega_m\) is the modal (most likely) frequency of any given wave, \(H_{1/3}\) is the significant wave height. The \(+\) superscript on \(S(\omega)\) indicates a ”one-sided spectrum,” wherein all the energy at positive and negative frequencies has been collected into the positive frequencies. We also take into account a factor of \(1/2 \pi\) (for reasons given below), to make the formal definition

\begin{align} S^+ (\omega) = \dfrac{1}{\pi} &S(\omega), && \text{for \(\omega \geq 0\), and} \\[4pt] &0, && \text{for \(\omega < 0\).} \end{align}

What is the justification for the factor of \(1/2 \pi\)? Consider that

\begin{align} R(\tau) &= \dfrac{1}{2 \pi} \int\limits_{-\infty}^{\infty} S(\omega) e^{i \omega \tau} \, d\omega \longrightarrow \\[4pt] R(0) &= \dfrac{1}{2 \pi} \int\limits_{-\infty}^{\infty} S(\omega) \, d\omega \\[4pt] &= \dfrac{2}{2 \pi} \int\limits_{0}^{\infty} S(\omega) \, d\omega, \end{align}

and therefore that

\[ \omega^2 = R(0) = \int\limits_{0}^{\infty} S^+ (\omega) \, d\omega.\]

In words, the area under the one-sided spectrum is exactly equal to the variance, or the square of the standard deviation of the process.