MATLAB Central Newsreader - autocorrelation of sine function

autocorrelation of sine function

Nor Faizah — Thu, 13 Mar 2008 12:56:02 -0400

Hello,

I'm trying to validate the xcorr function in MATLAB with a
simple signal of a sine function. As from the theoretical
result, the autocorrelation of Asin(wt) will give A^2/2cos
(w*tau).
However, when I computed these command,

%%%%%%%% Input parameters %%%%%%%
SR=1000
T =1/SR
L=1000
TT=(-L:L-1)*T
fs=100
As=1
%%%%%%%% Creating Signal %%%%%%%
signal= As*sin(2*pi*fs*TT)
signal= signal(:);
%%%%%%%%%%%%%%%%% Plotting the created signal
figure(1); plot(TT,signal)
xlabel('Time (s)')
ylabel('Y(t)')
%%%%% Calculating the auto-correlation of the signal %%%%%
[c,lags]=xcorr(signal,'coeff');
figure(2); plot(lags,c);
xlabel('\tau (s)')
ylabel('normalised correlation, R')

The autocorrelation plot (figure(2)) results in a cosine
function multiple with some exponential function (which
tends to zero).It suppose to result in a continuous
periodic function of cosine. So, i'm not sure what is
MATLAB actually doing when determining/plotting an
autocorrelation. Could anyone help me with this?

Many thanks.

Re: autocorrelation of sine function

Malcolm Lidierth — Thu, 13 Mar 2008 13:54:02 -0400

try [c,lags]=xcorr(signal,'unbiased');

Coeff estimates are biased - for lags greater than zero the
number of valid data point pairs falls off to 1 at abs(lag)
==data length.

Re: autocorrelation of sine function

Nor Faizah — Fri, 14 Mar 2008 08:32:01 -0400

"Malcolm Lidierth" <ku.ca.lck@htreidil.mloclam> wrote in
message <frbblq$hff$1@fred.mathworks.com>...
> try [c,lags]=xcorr(signal,'unbiased');
>
> Coeff estimates are biased - for lags greater than zero
the
> number of valid data point pairs falls off to 1 at abs
(lag)
> ==data length.
>

Thanks a lot. It gives the right plot. I don't understand
what do you mean by data point pairs falls off to 1 at abs
(lag) ==data length.

Re: autocorrelation of sine function

Pekka — Fri, 14 Mar 2008 10:20:18 -0400

"Nor Faizah " <a7khawarizmi@yahoo.com> wrote in message
<frdd61$grl$1@fred.mathworks.com>...
> "Malcolm Lidierth" <ku.ca.lck@htreidil.mloclam> wrote in
> message <frbblq$hff$1@fred.mathworks.com>...
> > try [c,lags]=xcorr(signal,'unbiased');
> >
> > Coeff estimates are biased - for lags greater than zero
> the
> > number of valid data point pairs falls off to 1 at abs
> (lag)
> > ==data length.
> >
>
> Thanks a lot. It gives the right plot. I don't understand
> what do you mean by data point pairs falls off to 1 at abs
> (lag) ==data length.

Autocorrelation function r(lag) = E[x(n)*x(n+lag)]
If your data vector length is N, for lag==, you have the
full N samples to estimate that expected value.
For lag=1, (you "shift" one sample) you have only N-1
samples left for estimation.
And actually when you reach lag=N-1, you will have only one
sample left for the estimate.
The larger the abs(lag), the smaller the number of samples
in the estimate (mean of products) and therefore the
estimate will have higher variance. Therefore the biased
estimate is usually preferred, it attenuates the values for
large lags, where there is higher variance.
doc xcorr shows you the formulas plus link to one book for
further reading
Didn't plan to write this long, sorry...