Does cross correlation indeed give correct signal leg value?

I want to find the position of signal [3,4] in signal [1,2,3,4,5,6] using cross correlation and thus xcorr function

x = [1,2,3,4,5,6];

y = [3,4];

[a,b] = xcorr(x,y);

a = [0,0,0,0,4,11,18,25,32,39,18];

for leg values b = [-5 -4 -3 -2 -1 0 1 2 3 4 5];

thus max value is 39, i.e. at right shift of 4 but signal there is [5,6] not [3,4] i.e. at the right shift of 2

This is a very basic operation, I am not getting why it's not giving correct results, could someonne please tell what I am missing here?

Answer 1

[Chunru]

Correlation will give the correct signal delay estimate when signal to noise ratio is sufficiently high.

Consider a signal x is a delayed version of the template y:

%x = [1,2,3,4,5,6];
x = [0,0,3,4,0,0];
y = [3,4];
[a,b] = xcorr(x, y)

a = 1×11

0 0 0 0.0000 0 0 12.0000 25.0000 12.0000 -0.0000 0

b = 1×11

-5 -4 -3 -2 -1 0 1 2 3 4 5

It can be observed that the peak occus at the time lag of 2, indicating the signal template y is delayed by 2 in signal x.

When small amount of noise added, the estimate of the peak is still correct:

x = [0,0,3,4,0,0] + randn(1,6)*0.2

x = 1×6

0.2324 -0.0266 2.8088 4.2078 -0.2091 0.0202

y = [3,4];
[a,b] = xcorr(x, y)

a = 1×11

0.0000 0 0 0.0000 0.9297 0.5908 11.1552 25.2576 11.7872 -0.5464 0.0606

b = 1×11

-5 -4 -3 -2 -1 0 1 2 3 4 5

However, when you keep increase the noise level, the estimate may be wrong.

In your code:

%x = [1,2,3,4,5,6];
x = [0,0,3,4,0,0] + [1 2 0 0 5 6];   % second term can be considered as noise
y = [3,4];
[a,b] = xcorr(x, y)