Clarify correlation.

applications/lofar2/model/pfb_os/dsp_study_erko.txt

@@ -304,21 +304,30 @@
     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):
+- 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]:
+  . 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]:
+  . 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]
@@ -327,15 +336,16 @@
                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].
+               the result.
 
-- FIR system identification from input-output measurements [JOS1 8.4.5,
-  PROAKIS 12]
+  . 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
+      y = h * x <=> H Y
 
-    xy = x cross y <==> conj(X) Y = conj(X) H Y = H |X|^2, so H = Rxy / Rxx
+      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]