Signals correlation

The Fourier Transform can be interpreted as a cross-correlation between sine waves and the signal being analyzed. Understanding the cross-correlation concept is essential for understanding the Fourier Transform.

Cross-correlation is a mathematical operation used extensively in signal processing to measure the similarity between two signals as a function of the time lag applied to one of them.

This chapter aims to introduce the fundamental concept of cross-correlation and establish its mathematical foundation.

The topic of cross-correlation is far more extensive than what is covered in this text. However, to maintain focus, this chapter will concentrate on the cross-correlation of real periodic signals. Understanding this aspect of cross-correlation is essential for grasping the Fourier Transform, as it forms its foundational basis.

If both signals are real periodic and their fundamental periods share a finite least common period, then their cross-correlation for a discrete-time signal is given by:

\[ R_{xy} [k] = \sum_{n=k}^{N-1} x[n] \, y[n-k] \]

Or for a continuous-time signal:

\[ R_{xy} (\tau) = \frac{1}{T} \int\limits_{-T/2}^{T/2} f(t) g(t-\tau) dt \]

If these equations seem intimidating or unclear, don't worry. By the end of this chapter, you'll understand them fully. We'll go through examples together, and you'll be able to perform cross-correlation with just a pen and paper.

Let us start with a measure of similarity.