7-66 Form the product of two statistically independent...

7-66 Form the product of two statistically independent jointly wide-sense stittionary random processes $X(r)$ and $Y(r)$ as $$ W(t)=X(t) Y(t) $$ Find general expressions for the following correlation functions and power spectrums in terms of those of $X(t)$ and $Y(t):(t) R_W w(t, t+\tau)$ and $s_{\mathbb{W} W}(t)$, (b) $R_{x w}(t, t+\tau)$ and $\delta_{x w}(c)$, and $(c) R_{w x}(t, t+\tau)$ and $\delta_{w x}(c)$. $(d)$ If $$ R_{x x}(\mathrm{t})=\left(W_1 / \pi\right) \operatorname{Sa}\left(W_1 \tau\right) $$ and $$ R_n(t)=\left(1 V_2 / \pi\right) \operatorname{Sa}\left(W V_2 t\right) $$ with constants $W_2>W_1$, find explicis functions for $R_{w w}(t, t+t)$ and $s_{w w}(w)$.

7-66 Form the product of two statistically independent jointly wide-sense stittionary random processes $X(r)$ and $Y(r)$ as
$$
W(t)=X(t) Y(t)
$$

Find general expressions for the following correlation functions and power spectrums in terms of those of $X(t)$ and $Y(t):(t) R_W w(t, t+\tau)$ and $s_{\mathbb{W} W}(t)$, (b) $R_{x w}(t, t+\tau)$ and $\delta_{x w}(c)$, and $(c) R_{w x}(t, t+\tau)$ and $\delta_{w x}(c)$. $(d)$ If
$$
R_{x x}(\mathrm{t})=\left(W_1 / \pi\right) \operatorname{Sa}\left(W_1 \tau\right)
$$
and
$$
R_n(t)=\left(1 V_2 / \pi\right) \operatorname{Sa}\left(W V_2 t\right)
$$
with constants $W_2>W_1$, find explicis functions for $R_{w w}(t, t+t)$ and $s_{w w}(w)$.

Key Concepts

Cross-Correlation Functions

Cross-correlation functions extend the concept of autocorrelation to pairs of different processes and reveal the degree to which one process resembles a time-shifted version of another. In systems where a process is formed as the product of two processes, cross-correlation functions between the product process and the original processes provide insights into their mutual statistical dependence and are key in understanding the overall behavior of the system.

Product of Random Processes

When two random processes are multiplied together, the resulting process has statistical properties that can often be expressed in terms of the original processes’ characteristics. The analysis typically involves calculating the autocorrelation and cross-correlation functions of the product process by leveraging the independence and stationarity properties, as well as known transforms such as the Fourier transform.

Statistical Independence

Statistical independence between processes indicates that the joint probability of any two events is the product of their individual probabilities. This property simplifies the analysis of combined systems because expectations involving products of independent random processes can be factorized into the products of expectations, making it easier to derive quantities such as correlation functions and power spectrums.

Autocorrelation Function

The autocorrelation function measures the statistical similarity between values of a process at different times, providing insight into the temporal dependencies inherent in the process. For WSS processes, this function depends solely on the lag, which greatly simplifies the calculation and interpretation, particularly when determining the output statistics of systems involving such processes.

Wide-Sense Stationarity

Wide-sense stationarity (WSS) is a property of random processes where the mean is constant over time and the autocorrelation function depends only on the time difference rather than on the specific time instants. This simplifies analysis considerably because it reduces the time dependencies in statistical measures and allows the use of Fourier transform techniques to relate time and frequency domain representations.

Power Spectral Density

The power spectral density (PSD) of a random process is the Fourier transform of its autocorrelation function. It describes how the power of a signal is distributed over frequency. In the context of WSS processes, the PSD provides an alternative perspective to the time domain autocorrelation and is essential for analyzing systems in the frequency domain.

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