Computing Convolution and Cross-Correlation Function Between Two Sequences

In signal processing, computing convolution and cross-correlation functions between two sequences, or calculating autocorrelation functions for single sequences represents fundamental and frequent requirements. Convolution is a crucial signal operation that combines two signals into a new composite signal, often implemented using algorithms like the overlap-add method or direct convolution using nested loops. Cross-correlation functions measure similarity between two signals and can be computed efficiently using FFT-based methods to reduce computational complexity from O(n²) to O(n log n). Autocorrelation functions analyze information repetition within signals and are particularly useful for periodicity detection, typically implemented through methods like biased or unbiased estimators. These core computational operations form the foundation of signal processing, enabling deeper understanding and more effective manipulation of signal data through functions such as MATLAB's conv(), xcorr(), and xcorr(x,'unbiased') for respective implementations.