Topic: Signals and Systems
Determine the correlation coefficient between the pulses $x(t)$ and $y(t)$ shown in the figure below:
(The figure shows $x(t)$ as a rectangular pulse from $t=0$ to $t=2$ with amplitude 1, and $y(t)$ as a sine wave $\sin(2\pi t)$ from $t=0$ to $t=2$).
- A. 1
- B. -1
- C. 0
- D. $2\pi$
Answer: C. 0
Explanation: The cross-correlation at $\tau=0$ is given by $\int_{-\infty}^{\infty} x(t)y(t) dt$.
Here, $\int_{0}^{2} (1)\sin(2\pi t) dt = \left[ \frac{-\cos(2\pi t)}{2\pi} \right]_0^2 = \frac{-\cos(4\pi) + \cos(0)}{2\pi} = \frac{-1 + 1}{2\pi} = 0$. Because the integral of their product is zero, the signals are orthogonal, and the correlation coefficient is 0.