Cross-correlation of two signals : cross

5.25.1 Cross-correlation of two signals : cross_correlation

cross_correlation takes two arguments, a complex vector v of length n and a complex vector w of length m . The returned value is the complex vector z=v⋆w of length N=n+m−1 which is the cross-correlation of the two input vectors, i.e. such that the following holds :

z_k=

N−1

∑

i=k

v_i−k^∗

w_i^∗, k=0,1,…,N−1,

where

v^∗=[v₀,v₁,…,v_n−1,


0,0,…,0
◥	▼	◤

m−1

] and w^∗=[


0,0,…,0
◥	▼	◤

n−1

,w₀,w₁,…,w_m−1].

Cross-correlation is typically used for measuring similarity between signals.

For example, input :

cross_correlation([1,2],[3,4,5])

Output :

[6.0,11.0,14.0,5.0]

Input :

v:=[2,1,3,2]:; w:=[1,-1,1,2,2,1,3,2,1]:; round(cross_correlation(v,w))

Output :

[2,1,0,8,9,12,15,18,13,11,5,2]

Observe that the cross-correlation of v and w is peaking at position 8 with the value 18, indicating that the two signals are best correlated when the last sample in v is aligned with the eighth sample in w. Indeed, there is an occurrence of v in w precisely at that point.