To solve this problem, let's analyze the given information step-by-step.

We have two wide-sense stationary (WSS) random processes, $X(t)$ and $Y(t)$. We are asked to find the maximum value of the cross-correlation function $R_{XY}(\tau) = \mathbb{E}[X(t)Y(t + \tau)]$.

Understanding Cross-Correlation

For WSS processes, the cross-correlation function between $X(t)$ and $Y(t)$ at lag $\tau$ is defined as:

$R_{XY}(\tau) = \mathbb{E}[X(t)Y(t + \tau)].$

The maximum value of the cross-correlation $R_{XY}(\tau)$ is obtained when $X(t)$ and $Y(t)$ are perfectly correlated. The Cauchy-Schwarz inequality states:

$|R_{XY}(\tau)| \leq \sqrt{R_{XX}(0) \cdot R_{YY}(0)},$

where $R_{XX}(0)$ and $R_{YY}(0)$ are the autocorrelation functions of $X(t)$ and $Y(t)$ at zero lag, respectively. The equality holds when $X(t)$ and $Y(t)$ are linearly dependent.

Conclusion

From the options provided, the maximum value of $R_{XY}(\tau)$ is:

$\sqrt{R_{XX}(0) \cdot R_{YY}(0)}.$

This corresponds to option (b).

Answer

The correct answer is (b).

Would you like more details or have any other questions? Here are 5 related questions that might interest you:

1. What is the significance of the Cauchy-Schwarz inequality in probability theory?
2. How do we compute autocorrelation functions for different types of random processes?
3. What are some applications of cross-correlation in signal processing?
4. How do WSS properties simplify the analysis of random processes?
5. What is the difference between cross-correlation and convolution in the context of signals?

Tip: When dealing with random processes, always check if they are WSS (Wide-Sense Stationary), as this can significantly simplify calculations involving autocorrelation and cross-correlation functions.