# Autocorrelation

Started by March 12, 2007
```Hello,

do anyone know why a cardinal sinus apear in the temporal domain when
we apply auto-correlation on a finite digital signal?

```
```Hi,

You probably have  a sinusoidal (or narrowband stochactic process) in
your originial signal. Note that the autocorrelation function is *not*
in the temporal domain (but a lag domain).

The fact that the auto-correlation magnitude decreases as you move
away from the zero lag component has to do with the method that you
use to compute the autocorrealtion and the finite size of your data.
Also, if you autocorrelation function looks very much like  a
triangle, you probably have a dc component in your signal.

```
```On 12 mar, 18:18, "Ikaro" <ikarosi...@hotmail.com> wrote:
> The fact that the auto-correlation magnitude decreases as you move
> away from the zero lag component has to do with the method that you
> use to compute the autocorrealtion and the finite size of your data.
> Also, if you autocorrelation function looks very much like  a
> triangle, you probably have a dc component in your signal.

I use an unbiased correlation :
I multiply the autocorrelation coefficient by 1/(N-lag) (with N the
maximum lag... I make an appodisation of the original data to "cut"
the high amplitudes near |lag|=N)
this is to avoid the decreases due to the number of samples stacked...
(its the idea but I use the correlation theorem... ie the product of
the fourier transform of the original signal).

My question was for a more general case... A lot of people says that
the autocorrelation is dominated by a cardinal sinus (ie something
like sin(A)/A), May be that it's related to the general expression of
the Autocorrelation in the lags domain for finite signals.

```
```>My question was for a more general case... A lot of people says that
> the autocorrelation is dominated by a cardinal sinus (ie something
> like sin(A)/A),

Which people say that and under what context??

Once again, the autocoreleation description that you mentioned above
is what I would expect for a sinusoidal (or narrowband signal) with a
biased estimator.

If you are talking about the fourier domain, that this is related to
windowing. If you don't window your signal than that is equivalent to
windowing youir signal with a rectangular function. In which case, in
the frequency domain you will be convolving your power spectral
density function with the cardinal sinus function (the fourier
transform of the rectangular window).

```
```>My question was for a more general case... A lot of people says that
> the autocorrelation is dominated by a cardinal sinus (ie something
> like sin(A)/A),

Which people say that and under what context??

Once again, the description of the autocorrelation that you mentioned
above
is what I would expect for a sinusoidal (or narrowband signal) with a
biased estimator.

If you are talking about the fourier trasform of the autocorrelation
function, than this is related to
windowing techniques. If you don't window your signal, this is
equivalent to
windowing using a rectangular window. Which in
the frequency domain means you will be convolving the signal  power
spectral
density function with the cardinal sinus function (the fourier
transform of the rectangular window).

```
```>My question was for a more general case... A lot of people says that
> the autocorrelation is dominated by a cardinal sinus (ie something
> like sin(A)/A),

Which people say that and under what context??

Once again, the description of the autocorrelation that you mentioned
above
is what I would expect for a sinusoidal (or narrowband signal) with a
biased estimator.

If you are talking about the fourier trasform of the autocorrelation
function, than this is related to
windowing techniques. If you don't window your signal, this is
equivalent to
windowing using a rectangular window. Which in
the frequency domain means you will be convolving the signal  power
spectral
density function with the cardinal sinus function (the fourier
transform of the rectangular window).

```
```thanks for your reply ...

> If you are talking about the fourier transform of the autocorrelation
> function, than this is related to
> windowing techniques. If you don't window your signal, this is
> equivalent to
> windowing using a rectangular window. Which in
> the frequency domain means you will be convolving the signal  power
> spectral
> density function with the cardinal sinus function (the fourier
> transform of the rectangular window).

ok, I agree with you ... but I speak about the autocorrelation in the
lag domain...

```
```> Which people say that and under what context??
In seismology.
I want to see the the mean autocorrelation function evolution for
signals recorded by seismometers...  sample freq=100Hz / bandwidth :
0.2-1 Hz.

```
```On Mar 13, 1:21 am, "zac" <zacharie.dupu...@gmail.com> wrote:
>
> > If you are talking about the fourier transform of the autocorrelation
> > function, than this is related to
> > windowing techniques. If you don't window your signal, this is
> > equivalent to
> > windowing using a rectangular window. Which in
> > the frequency domain means you will be convolving the signal  power
> > spectral
> > density function with the cardinal sinus function (the fourier
> > transform of the rectangular window).
>
> ok, I agree with you ... but I speak about the autocorrelation in the
> lag domain...

it might be that if the power spectrum is strictly bandlimited, then
the auto-correlation is convolved with a sinc fuction (that's what we
here in comp.dsp-land call the "cardinal sinus function").   but if
it's multiplied by the sinc() function in the "lag" domain, that means
the power spectrum is convolved with a rect() function and i have no
idea why that would happen naturally.

r b-j

```