Let {X(t),-∞<t<∞} be weakly stationary with covariance function R(s)= Cov(X(t), X(t+s)) and let

step 1 Power spectral density is the Fourier transform of the auto -correlation. S -s -s -s -s of omega

Let {X(t),-∞<t<∞} be weakly stationary with covariance...

Let $\{X(t),-\infty<t<\infty\}$ be weakly stationary with covariance function $R(s)=$ $\operatorname{Cov}(X(t), X(t+s))$ and let $\widetilde{R}(w)$ denote the power spectral density of the process. (i) Show that $\widetilde{R}(w)=\widetilde{R}(-w)$. It can be shown that $$ R(s)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \widetilde{R}(w) e^{i w s} d w $$ (ii) Use the preceding to show that $$ \int_{-\infty}^{\infty} \widetilde{R}(w) d w=2 \pi E\left[X^{2}(t)\right] $$

Let $\{X(t),-\infty<t<\infty\}$ be weakly stationary with covariance function $R(s)=$ $\operatorname{Cov}(X(t), X(t+s))$ and let $\widetilde{R}(w)$ denote the power spectral density of the process.
(i) Show that $\widetilde{R}(w)=\widetilde{R}(-w)$. It can be shown that
$$
R(s)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \widetilde{R}(w) e^{i w s} d w
$$
(ii) Use the preceding to show that
$$
\int_{-\infty}^{\infty} \widetilde{R}(w) d w=2 \pi E\left[X^{2}(t)\right]
$$

Key Concepts

Weak Stationarity

A weakly stationary process has constant mean and an autocovariance function that depends only on the lag between time points, not on the absolute time. This property ensures that the second order statistical properties remain invariant over time, making the process amenable to spectral analysis and simplifying its characterization.

Autocovariance Function

The autocovariance function measures the covariance between values of a stochastic process at two different times and depends solely on the time lag for a weakly stationary process. It encapsulates the time dependence structure of the process and is crucial for connecting time-domain and frequency-domain representations.

Power Spectral Density

The power spectral density (PSD) quantifies how the variance (power) of a process is distributed across different frequency components. It is defined as the Fourier transform of the autocovariance function for a stationary process, providing insights into the process’s periodicity and frequency content.

Fourier Transform

The Fourier transform converts a time-domain function, such as the autocovariance function, into a frequency-domain representation. Its properties, including symmetry and inversion, are fundamental in linking the covariance function with the power spectral density of a stationary process.

Wiener-Khinchin Theorem

The Wiener-Khinchin theorem establishes that the power spectral density of a stationary process is the Fourier transform of its autocovariance function. This result provides a bridge between time and frequency domain analyses and underpins many methods in signal processing and time series analysis.

Parseval's Theorem

Parseval's theorem asserts that the total energy or variance of a signal in the time domain is equal to the total energy in the frequency domain. In the context of stationary processes, it implies that integrating the power spectral density over all frequencies yields a quantity proportional to the process’s variance, linking the overall power in the frequency domain back to the time-domain moment.

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