A signal is split into two parts and one of them is going through a filter (say, with a transfer function H(f)) and the other part stays unchanged. Then I want to know how to calculate their cross correlation function. My guess is, given the spectral density function S(f), it will be the ordinary Wiener-Khinchine theorem with an addition of the transfer function: R=Integral{S(f)H(f)*exp(i*2*pi*f*t)df}
Agreeing with Pascal Salart, but making it simple. One signal is x(t), the other is y(t), you study the covariance matrix E(x(t+i) y(t+j)), which can be estimated by low pass filtering vectors v(t, i) = (x(t+i-N), ... x(t+i))
and w(t, j) =(y(t+j-N),... y(t+j)) with N a length of observation window,
then the scalar product estimates the mean expectation E() above.
The matrix obtained is a Gram matrix, hence it diagonalises with eigenvectors, eigenvalues obtained by Gram-Schmidt algorithm. QED...
Ok ?
Hello,
if we assume the input sequence be modeled as :
x[n] = x1[n] + x2[n]
such that "the input sequence is plit in 2 parts". Then, if the first part x1[n] is "going through a filter" with inpulse response h[n], then the output signal of the filter is given by:
y[n) = h[n ] * x1[n] (where * stands for the linear convolution operator)
The cross-correlation function between the output/input signal is :
Ryx[k] = E{ y[n] x[n-k] } (E(.) stands for the expected value operator))
= h[n] * E{ x1[n] x[n-k] }
= h[n] * E{ x1[n] (x1[n-k] + x2[n-k]) }
= h[n] * ( Rx1x1[k] + Rx1x2[k] )
Equivalently, taking the discrete Fourier transform of both sides, we get :
Syx(f) = H(f) ( Sx1x1(f) + Sx1x2(f) )
A specific case is given when x[n] is composed of two independent components x1[n] and x2[n] (with one of these two being with zero-mean), then Rx1x2[k] = 0 whatever k, and we get in this case :
Ryx[k] = h[n] * Rx1x1[k] Syx(f) = H(f) Sx1x1(f)
Sincerely,
Pascal.
Agreeing with Pascal Salart, but making it simple. One signal is x(t), the other is y(t), you study the covariance matrix E(x(t+i) y(t+j)), which can be estimated by low pass filtering vectors v(t, i) = (x(t+i-N), ... x(t+i))
and w(t, j) =(y(t+j-N),... y(t+j)) with N a length of observation window,
then the scalar product estimates the mean expectation E() above.
The matrix obtained is a Gram matrix, hence it diagonalises with eigenvectors, eigenvalues obtained by Gram-Schmidt algorithm. QED...
Ok ?
The correlation coefficient is determined by dividing the covariance by the product of the two variables' standard deviations. Standard deviation is a measure of the dispersion of data from its average.
Dear authors
Thank you for sharing this excellent question.
I agree with Prof. Di Francesco.
kind regards...
The simples way is use of Matlab: r = xcorr(x,y) . Here x is source signal, and y is filtered one.
Following the second question of Elnur Hajiyev , you are looking for the cross-correlation between y(t)=h(t)*x2(t) with x2(t) = a x1(t) with a being a constant value.
In your question, there is an error : you cannot assume that Sx1x1(f)=Sx2x2(f)
because we have :
Sx2x2(f) = TF{Rx2x2(tau}} = (a^2) TF{Rx1x1(tau} = (a^2) Sx1x1(f).
The correct reasoning is as follow :
Ryx2(tau) = E{ y(t) x2(t-tau) }
= a E{ y(t) x1(t-tau) }
= a E{ Integral{ h(u) x2(t-u) du} x1(t-tau) }
= (a^2) E{ Integral{ h(u) x1(t-u) du} x1(t-tau) }
= (a^2) Integral{ h(u) E{ x1(t-u) x1(t-tau) } du}
= (a^2) Integral{ h(u) Rx1x1(tau-u) du}
= (a^2) h(tau) * Rx1x1(tau)
Taking the Fourier transform, we get : Syx2(f) = (a^2) H(f) Sx1x1(f)
Pascal Scalart
Pascal Scalart thank you for the very clear explanation. I learned a lot from this explanation. Really appreciate it!
- Badges
- Science topic
- Similar topics
- Engineering
- Electrical Engineering
- Signal Processing