matplotlib.pyplot as plt import numpy as np

Define the parameters

T = 1 # Assume T = 1 for simplicity a = 1 # Assume amplitude a = 1

Define the cross-correlation function analytically

tau = np.linspace(-2T, 2T, 1000) R_xy = np.piecewise(tau, [(-2T <= tau) & (tau < -T), (-T <= tau) & (tau < -T/2), (-T/2 <= tau) & (tau < T/2), (T/2 <= tau) & (tau < T), (T <= tau) & (tau <= 2T)], [0, lambda t: a2 * (t + T), lambda t: a2 * T, lambda t: a**2 * (T - t), 0])

Plot the cross-correlation function

plt.figure(figsize=(8, 6)) plt.plot(tau, R_xy, label=r"$\tilde{R}{xy}(\tau)$") plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--') plt.title("Cross-Correlation Function $~{R}{xy}(\tau)$") plt.xlabel(r"$\tau$") plt.ylabel(r"$~{R}_{xy}(\tau)$") plt.grid() plt.legend() plt.show()The cross-correlation function $\tilde{R}_{xy}(\tau)$ has been sketched above based on the signals $x(t)$ and $y(t)$. The graph shows the variation of the cross-correlation function with respect to the delay $\tau$. The function follows the shape of the overlapping regions of $x(t)$ and $y(t)$ as $\tau$ shifts.

Let me know if you would like to discuss the solution in more detail or have additional questions.

Here are five related questions to expand your understanding:

1. How is the cross-correlation function mathematically defined and what does it signify?
2. Why is the result symmetric for these specific signals $x(t)$ and $y(t)$?
3. What happens to the cross-correlation function if one signal is scaled by a constant factor?
4. How does the duration $T$ of the signals affect the width of the cross-correlation result?
5. How is the cross-correlation function different from convolution?

Tip: The cross-correlation function is a powerful tool to determine the similarity between two signals as a function of time lag; it's widely used in signal processing and communication systems.