diff --git a/applications/lofar2/model/pfb_os/dsp_study_erko.txt b/applications/lofar2/model/pfb_os/dsp_study_erko.txt index 200e946ae8546a00345f189aaca65d1d23703b00..16f6a0ba4eb874270a3c67f577feaecf3cd5194c 100644 --- a/applications/lofar2/model/pfb_os/dsp_study_erko.txt +++ b/applications/lofar2/model/pfb_os/dsp_study_erko.txt @@ -304,38 +304,48 @@ that the output blocks overlap, to keep the output and input sample rate the same. -- Correlation equation [JOS1 7.2.5] (for comparison with convolution): - - N-1 - xy(n) = sum conj(x(k)) y(n + k), time shift n is correlation lag - k=0 - - The cross power spectrum is [JOS1 8.4]: - - HXY(w_k) = DFT_k(xy(n)) = 1 / N conj(X(w_k)) Y(w_k) - - Correlation is a measure of similarity between two function x(k) and y(k), - for different shifts (lag) in time. Autocorrelation can show the periodicity - of a signal, because they it has similarity for some k > 0. To prove that - correlation can be expressed as convolution use a helper function - [WOLFSOUND]: - - xh[n] = sum_k x[n + k] h[k], with sum_k for k = -inf to +inf - = sum_k x[-(-n - k)] h[k] - = sum_k x1[-n - k] h[k], with x1[p] = x[-p] - = (x1[n] * h[n])[-n], because convolution equation is [LYONS Eq. 5.6, - JOS4 7.2.4]: y(n) = sum_k x(n - k) h(k) = x(n) * h(k) - = x[-n] * h[n])[-n], again with x[-p] = x1[p], so correlation can be - calculated by convolving the time flipped x and then time flip - the result. Beware correlation is not commutative, so xh[n] != - hx[n]. - -- FIR system identification from input-output measurements [JOS1 8.4.5, - PROAKIS 12] - - y = h * x <==> H Y - - xy = x cross y <==> conj(X) Y = conj(X) H Y = H |X|^2, so H = Rxy / Rxx +- Correlation + . Correlation is a measure of similarity between two function x(k) and y(k), + for different shifts (lag) in time. Autocorrelation can show the periodicity + of a signal, because then it has similarity for some k > 0. + . Difference between correlation and convolution is that convolution flips + one input, so corr(x, y) = conv(x, flip(y)). Hence if ne input is + symmetrical then correlation and convolution are the same. + * The purpose of convolution is to determine the output of a filter with + impulse response h. + * The purpose of correlation is to determine how much signal y is present + in x for different time delays (lags). + + . Correlation equation [JOS1 7.2.5]: + + N-1 + xy(n) = sum conj(x(k)) y(n + k), time shift n is correlation lag + k=0 + + . The cross power spectrum is [JOS1 8.4]: + + HXY(w_k) = DFT_k(xy(n)) = 1 / N conj(X(w_k)) Y(w_k) + + . To prove that correlation can be expressed as convolution use a helper + function [WOLFSOUND]: + + xh[n] = sum_k x[n + k] h[k], with sum_k for k = -inf to +inf + = sum_k x[-(-n - k)] h[k] + = sum_k x1[-n - k] h[k], with x1[p] = x[-p] + = (x1[n] * h[n])[-n], because convolution equation is [LYONS Eq. 5.6, + JOS4 7.2.4]: y(n) = sum_k x(n - k) h(k) = x(n) * h(k) + = x[-n] * h[n])[-n], again with x[-p] = x1[p], so correlation can be + calculated by convolving the time flipped x and then time flip + the result. + + . Convolution is commutative so x * y = y * x, but correlation is not + commutative, so xy != yx + . Use correlation for system identification from input-output measurements + [JOS1 8.4.5, PROAKIS 12] + + y = h * x <=> H Y + + xy = x cross y <=> conj(X) Y = conj(X) H Y = H |X|^2, so H = Rxy / Rxx 4) Hilbert transform (HT) and analytic signal [LYONS 9]