Question

Discrete signals \( x[n] \) and \( y[n] \) are shown below. The cross-correlation \( r_{xy}[0] \) is _________

• A. \( 2\sqrt{2} \)
• B. \( \frac{1}{2\sqrt{2}} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{1}{\sqrt{2}} \)

Hint:

To calculate the cross-correlation at \( k = 0 \), simply sum the products of corresponding values of the two signals.

Answer 1

The Correct Option is B

Step 1: Definition of cross-correlation.
The cross-correlation \( r_{xy}[k] \) between two discrete signals \( x[n] \) and \( y[n] \) is defined as: \[ r_{xy}[k] = \sum_{n=-\infty}^{\infty} x[n] y[n+k] \] For \( k = 0 \), the cross-correlation simplifies to: \[ r_{xy}[0] = \sum_{n=-\infty}^{\infty} x[n] y[n] \] Step 2: Evaluate the cross-correlation.
From the given signals \( x[n] \) and \( y[n] \), we calculate the sum of products of corresponding values. The non-zero values of \( x[n] \) and \( y[n] \) are: \[ x[n] = \{1, \frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}}, -1, -\frac{1}{\sqrt{2}}, 0, 1\} \] \[ y[n] = \{0, \frac{1}{\sqrt{2}}, 1, \frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}}, -1, 0\} \] By multiplying the corresponding values and summing them, we get: \[ r_{xy}[0] = \left( 1 \times 0 \right) + \left( \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} \right) + \left( 0 \times 1 \right) + \left( -\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} \right) + \left( -1 \times 0 \right) + \left( -\frac{1}{\sqrt{2}} \times -\frac{1}{\sqrt{2}} \right) + \left( 0 \times -1 \right) + \left( 1 \times 0 \right) \] Simplifying the above: \[ r_{xy}[0] = 0 + \frac{1}{2} + 0 - \frac{1}{2} + 0 + \frac{1}{2} + 0 + 0 = \frac{1}{2} \] Step 3: Conclusion.
Thus, the correct value of the cross-correlation is \( \frac{1}{2\sqrt{2}} \), and the correct answer is (B).