Question

An important concept in many communications applications is the correlation between two signals. In the problems at the end of Chapter 2, we will have more to say about this topic and will provide some indication of how it is used in practice.

For now, we content ourselves with a brief introduction to correlation functions and some of their properties. Let x(t) and y(t) be two signals; then the correlation function is defined as Rxy(t) = ∫x(1 + τ)y(τ)dτ. The function Rxx(t) is usually referred to as the autocorrelation function of the signal x(t), while Rxy(t) is often called a cross-correlation function.

(a) What is the relationship between Ryy(t) and Rxx(t)?

Answer 1

The relationship between Ryy(t) and Rxx(t) depends on the symmetry properties of the original signals x(t) and y(t). The time shift introduces a negative sign due to the anti-symmetry.

If both x(t) and y(t) are symmetric with respect to the origin (meaning their values are equal for t and -t), then: Ryy(t) = Rxx(t). This is because the time shift (τ) in the integral definition of Rxy(t) doesn't affect the result for symmetric signals.

If both x(t) and y(t) are anti-symmetric with respect to the origin (meaning their values are negated when switching between t and -t), then Ryy(t) = -Rxx(t). In this case, the time shift introduces a negative sign due to the anti-symmetry.