Cross-correlation of two signals

21.2.3 Cross-correlation of two signals

The cross correlation of two complex vectors v=[v₁,…,v_n] and w=[w₁,…,w_m] is the complex vector z=v⋆ w (note the difference between ⋆ and the convolution operation ∗, see Section 21.2.5) of length N=n+m−1 given by

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]

and V^∗ denotes the complex conjugate of V. Cross-correlation is typically used for measuring similarity between signals.

The cross_correlation command computes the cross correlation of two vectors.

cross_correlation takes two arguments: v,w, two vectors (not necessarily the same length).
cross_correlation(v,w) returns the cross correlation v⋆ w.

Examples

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

⎡
⎣

6.0,11.0,14.0,5.0

⎤
⎦

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

⎡
⎣

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.