Question

What is the energy sequence of the signal ax(n)+by(n-l) in terms of cross correlation and auto correlation sequences?

(a) a^2rxx(0)+b^2ryy(0)+2abrxy(0)

(b) a^2rxx(0)+b^2ryy(0)-2abrxy(0)

(c) a^2rxx(0)+b^2ryy(0)+2abrxy(1)

(d) a^2rxx(0)+b^2ryy(0)-2abrxy(1)

This question was posed to me during an interview.

The origin of the question is Correlation of Discrete Time Signals topic in section Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

Right answer is (c) a^2rxx(0)+b^2ryy(0)+2abrxy(1)

For explanation I would say: The energy signal of the signal ax(n)+by(n-l) is

\(\sum_{n=-\infty}^{\infty}[ax(n)+by(n-l)]^2\)

= \(a^2 \sum_{n=-\infty}^{\infty}x^2(n)+b^2 \sum_{n=-\infty}^{\infty}y^2(n-l)+2ab \sum_{n=-\infty}^{\infty}x(n)y(n-l)\)

= a^2rxx(0)+b^2ryy(0)+2abrxy(l)