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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.

If X(t) and Y(t) are WSS random processes, then the maximum value of R_XY(τ) = E[X(t)Y(t+τ)] is?

Explore how to calculate the maximum value of the cross-correlation function R_XY(τ) for two wide-sense stationary (WSS) random processes. The solution involves applying the Cauchy-Schwarz inequality and understanding autocorrelation functions.

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