"Greg Heath" <heath@alumni.brown.edu> wrote in message
news:2n9vmiFupk20U1@uni-berlin.de...

> "Greg Heath" <heath@alumni.brown.edu> wrote in message
> news:2n0hjlFr99v0U1@uni-berlin.de...
> > When I was analyzing some time-series data, I obtained the
> >autocorrelation function and corresponding estimate of the
> >1/e (=0.3679) decorrelation time by ifft-ing the power
> >spectral density. As a check, I used the MATLAB function
> >xcorr. As can be seen below and in the plot for the test
> >function z = (x-mean(x))/std(x), x = t.*exp(-t), The
> >functions and decorrelation time estimates were
> >significantly different .
> -----SNIP
> > On the other hand, if the statement, z = x;, is
> > uncommented, the spectral and temporal functions
> > and decorrelation time estimates are almost identical.
> >
> > Can anyone explain this?
>
> Hey guys, it's not fair for me to have to answer my own
> questions!
>
> The basic reason is that xcorr is obtained by zero-padding.
>
> If the temporal estimate is given by
>
> Azt = xcorr(z);, % then
> M = length(Azt) % = 2*N-1, where
> N = length(z)
>
> The corresponding spectral estimate can be obtained from
>
> Z1 = fft(z,M);
> PSD1 = abs(Z1).^2;
> Azf1 = fftshift(real(ifft(PSD1))); .
>
> Notice that fftshift ( and not ifftshift) is used to get a
> completely symmetric function about the middle index
> N.
>
> In general, these results will be different from those
> obtained from the nonzero-padded estimate
>
> Z2 = fft(z);
> PSD2 = abs(Z2).^2;
> Azf2 = fftshift(real(ifft(PSD1))); ,
>
> unless zero-padding doesn't change the shape of the
> function.
>
> For example, the plot of x and [x;zeros(M,1)] are
> sufficiently similar whereas the plots of z and
> [z;zeros(M,1)] are significantly different.
>
> I guess the lesson here is that you must be aware of
> the zero-padding whenever you use xcorr!

Actually, there is one more point to make. First note
the decorrelation time estimate in the following table
for N = 100:
length x z
-------- -------- -------
N 2.16 1.32
M 2.14 1.70
Now, notice what happens when the hanning window
is applied:
length x z
-------- -------- -------
N 2.62 1.82
M 2.62 1.83
Finally, the consistency I was looking for!
However,...
since MATLAB contains 16 different window
functions, a precise definition of decorrelation
time for a general sampled data time series may
not exist.
Hope this helps,
Greg

Reply by Greg Heath●August 3, 20042004-08-03

"Greg Heath" <heath@alumni.brown.edu> wrote in message
news:2n0hjlFr99v0U1@uni-berlin.de...

> When I was analyzing some time-series data, I obtained the
>autocorrelation function and corresponding estimate of the
>1/e (=0.3679) decorrelation time by ifft-ing the power
>spectral density. As a check, I used the MATLAB function
>xcorr. As can be seen below and in the plot for the test
>function z = (x-mean(x))/std(x), x = t.*exp(-t), The
>functions and decorrelation time estimates were
>significantly different .

-----SNIP

> On the other hand, if the statement, z = x;, is
> uncommented, the spectral and temporal functions
> and decorrelation time estimates are almost identical.
>
> Can anyone explain this?

Hey guys, it's not fair for me to have to answer my own
questions!
The basic reason is that xcorr is obtained by zero-padding.
If the temporal estimate is given by
Azt = xcorr(z);, % then
M = length(Azt) % = 2*N-1, where
N = length(z)
The corresponding spectral estimate can be obtained from
Z1 = fft(z,M);
PSD1 = abs(Z1).^2;
Azf1 = fftshift(real(ifft(PSD1))); .
Notice that fftshift ( and not ifftshift) is used to get a
completely symmetric function about the middle index
N.
In general, these results will be different from those
obtained from the nonzero-padded estimate
Z2 = fft(z);
PSD2 = abs(Z2).^2;
Azf2 = fftshift(real(ifft(PSD1))); .
unless zero-padding doesn't change the shape of the
function.
For example, the plot of x and [x;zeros(M,1)] are
sufficiently similar whereas the plots of z and
[z;zeros(M,1)] are significantly different.
I guess the lesson here is that you must be aware of
the zero-padding whenever you use xcorr!
Hope this helps.
Greg
-----SNIP

Reply by Greg Heath●July 31, 20042004-07-31

When I was analyzing some time-series data, I obtained the autocorrelation
function and corresponding estimate of the 1/e (=0.3679) decorrelation time
by ifft-ing the power spectral density. As a check, I used the MATLAB
function xcorr. As can be seen below and in the plot for the test function
z = (x-mean(x))/std(x), x = t.*exp(-t), The functions and decorrelation
time estimates were significantly different .
Autocorrelation Value
Lag(sec) Temporal Spectral Difference
---------------------------------------------------------
1.2 0.5694 0.4383
*1.3 0.5273 *0.3790 0.1483
1.4 0.4862 0.3212
1.5 0.4461 0.2650
1.6 0.4072 0.2106
*1.7 *0.3695 0.1584 0.2111
1.8 0.3331 0.1082
On the other hand, if the statement, z = x;, is uncommented, the spectral
and temporal functions and decorrelation time estimates are almost
identical.
Can anyone explain this?
TIA,
Greg
************************************************************
clear, close all, clc
display('***** Standardized Time Function **********')
dt = 0.1, N = 100, T = N*dt, t = dt*(0:N-1)';
x = t.*exp(-t); meanx = mean(x), stdx = std(x)
z = (x-meanx)/stdx;
%z = x;
display('* Autocorrelation Function (Temporal Estimate) *')
Az = xcorr(z); M = length(Az), em1 = exp(-1)
tAz = [-t(end:-1:2); t];
[Azmax iAzmax] = max(Az), Az = Az/Azmax;
Azcheck = [M - (2*N-1);Azmax - sumsqr(z);iAzmax - N]
display('* 1/e Decorrelation Time (Temporal Estimate) *')
i = 1:M-1; i1 = find( Az(i) >= em1 & Az(i+1) < em1)
t1 = tAz(i1), t2 = t1+dt, A1 = Az(i1), A2=Az(i1+1)
tauAz = t1 + [(A1-em1)/(A1-A2)]*dt
I = (i1-5:i1+2)';
tauAzcheck = [ I tAz(I) Az(I) ]
display('********** Power Spectral Density **********')
Z = fft(z); PSDz = abs(Z/N).^2; N2 = N^2
PSDzcheck = [ PSDz(1) - 0 ; sum(PSDz) - (N-1)/N ]
display('* Autocorrrelation Function (Spectral Estimate) *')
Azf = ifftshift(real(ifft(PSDz))); Q = fix((N+2)/2)
tAzf = [-t(Q:-1:2); t(1:Q-1)];
[Azfmax iAzfmax] = max(Azf), Azf = Azf/Azfmax;
Azfcheck = [length(Azf) - N; Azfmax - sumsqr(z)/N2;...
iAzfmax - Q]
display('* 1/e Decorrelation Time (Spectral Estimate) *')
j = 1:N-1;j1 = find( Azf(j) >= em1 & Azf(j+1) < em1)
t1 = tAzf(j1),t2 = t1 + dt,A1 = Azf(j1),A2 = Azf(j1+1)
tauAzf = t1 + [(A1-em1)/(A1-A2)]*dt
J = (j1-1:j1+5)';
tauAzfcheck = [ J tAzf(J) Azf(J) ]
display('******* Autocorrelation Comparison Plots ********')
figure, plot(tAz, Az), hold on, plot(tAzf, Azf,'r')
hold on, plot(tauAz, em1,'b*'), hold on
plot(tauAzf, em1,'r*'), hold on
plot([-T T],[em1 em1],'r--'), hold on
plot([-T T],[0 0],'k')
legend('Temporal Estimate','Spectral Estimate')
xlabel('Time (sec)')
title('Comparison of Autocorrelation Estimates')