# Wiener-Khintchin theorem, autocorrelation fundamentals

Started by July 29, 2005
```Hi, I am working on a summer project which requires basic knowledge of
DSP, specifically I need to compute the Autocorrelation function (ACF) and
PSD, which I have no problem computing.  I am rather posting here to get
some REASONING from the DSP and signal gurus about the basic theories of
signals to expand my rather weak knowledge on DSP.

I know that the ACF and PSD are fourier transform pair and they are
interrelated by the Wiener-Khintchin formula as:

PSD(f) = 4*Integral[cos(2*pi*f*t)*ACF(t)*dt}]  (1)

I am looking for someone to give me a reasoning as to W H Y the facts
below are TRUE. Please be patient with me, as I am having hardtime
articulating my question:

QUESTIONS:

(i) My ACF function above has a finite duration Tau (say for example
Tau= 2 picosecond) Why would if I try to PLOT my PSD, my obtained results
for the PSD for frequencies below 1/(Tau) Hz DOES NOT MAKE SENSE?  Is this
something like the Heisenberg uncertaintity principle where there is always
an uncertaintinty in the ACF/PSD since realistically we can never have an
ACF function which is infinite in duration.  For example if we had a
superficical ACF which has an infinite
duration, (1/tau =1/(infinity)=0 HZ), then we can ONLY describe
frequencies ABOVE 0 Hz, but since there is never an ACF function that is
infinite in duration, there will always be uncertainity.  Where did this
fundamental rule come from, I tried to look everywhere for a reasoning but
I could not find an answer?

(ii)The ACF I am working with has an initial downswing which is then
followed by aperiodic oscillations that attenuates at longer time (similar
to the positive half of a sinc function--i.e a decaying exponential with
some oscillatory behavior).  From what I WAS TOLD, "the psuedo-period of
the autocorrelation function will correspond to a HUMP in the PSD plot."
However, I dont quiet understand :
(a)what the "pseudo-period" of an ACF is??  The ACF that I am
working with has MORE THAN ONE  period due to its aperiodic oscillatory
behavior.  In short, is the psuedo-period the time it takes for the
initial downswing to occur or what?
(b)WHY would the frequnecy CORRESPONDING to the psuedo-period of the
ACF would yield a hump in the PSD plot?  What about the subsquent periods
of the minor oscillatory periods (side lopes)of the ACF? Would the
frequnecy correspnonding to those periods be manifested as humps in the
PSD plot? How exactly is the dominant frqunecy is captured in the PSD from
the
time sequence of  ACF?

Lebann

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```
```"lebann" <Lebann@gmail.com> wrote in message
news:YdidnV7VN8_zJ3TfRVn-pg@giganews.com...
> I know that the ACF and PSD are fourier transform pair and they are
> interrelated by the Wiener-Khintchin formula as:
>
> PSD(f) = 4*Integral[cos(2*pi*f*t)*ACF(t)*dt}]  (1)

First, note that this just says that the ACF and PSD are related by Fourier
transform. This simpler form using only cos() is valid because the ACF is
symmetric.

>  (i) My ACF function above has a finite duration Tau (say for example
> Tau= 2 picosecond) Why would if I try to PLOT my PSD, my obtained results
> for the PSD for frequencies below 1/(Tau) Hz DOES NOT MAKE SENSE?

The PSD for frequencies below 1/Tau makes fine sense.

> Is this
> something like the Heisenberg uncertaintity principle where there is
> always
> an uncertaintinty in the ACF/PSD since realistically we can never have an
> ACF function which is infinite in duration. [...] Where did this
> fundamental rule come from, I tried to look everywhere for a reasoning but
> I could not find an answer?

The rule is that a filter T seconds long cannot generally/fully between
frequencies that are less than 1/T Hz apart.  It's easy to see how this
works if someone tells you:

1) You probably already know that if a signal is bandlimited to B Hz of
(two-sided) bandwidth, then it has a certain degree of smoothness, so that
it cannot take on arbitrary independent values at times less than 1/B
seconds apart.

2) The inverse Fourier transform is the same as the Fourier transform,
except for a time reversal.

3) Time-limiting a signal therefore imposes a smoothness on its spectrum in
the same way that bandlimiting a signal imposes a smoothness in the time
domain.  A signal limited to length T can't take on arbitrary independent
values in the frequency domain at frequencies less than 1/T apart.

> "the psuedo-period of
> the autocorrelation function will correspond to a HUMP in the PSD plot."

That just means that, since the ACF and PSD are related by Fourier
transform, the PSD will show a peak for every significant sinusoidal
component in the ACF.

--
Matt

```
```lebann wrote:
> Hi, I am working on a summer project which requires basic knowledge of
> DSP, specifically I need to compute the Autocorrelation function (ACF) and
> PSD, which I have no problem computing.  I am rather posting here to get
> some REASONING from the DSP and signal gurus about the basic theories of
> signals to expand my rather weak knowledge on DSP.
>
> I know that the ACF and PSD are fourier transform pair and they are
> interrelated by the Wiener-Khintchin formula as:
>
> PSD(f) = 4*Integral[cos(2*pi*f*t)*ACF(t)*dt}]  (1)
>
> I am looking for someone to give me a reasoning as to W H Y the facts
> below are TRUE. Please be patient with me, as I am having hardtime
> articulating my question:
>
> QUESTIONS:
>
>   (i) My ACF function above has a finite duration Tau (say for example
> Tau= 2 picosecond) Why would if I try to PLOT my PSD, my obtained results
> for the PSD for frequencies below 1/(Tau) Hz DOES NOT MAKE SENSE?  Is this
> something like the Heisenberg uncertaintity principle where there is always
> an uncertaintinty in the ACF/PSD since realistically we can never have an
> ACF function which is infinite in duration.  For example if we had a
> superficical ACF which has an infinite
> duration, (1/tau =1/(infinity)=0 HZ), then we can ONLY describe
> frequencies ABOVE 0 Hz, but since there is never an ACF function that is
> infinite in duration, there will always be uncertainity.  Where did this
> fundamental rule come from, I tried to look everywhere for a reasoning but
> I could not find an answer?

It is not a theoretical rule, it's a practical rule. I did attempt
the seasonal variations of some property over a year, you need to
observe your system over a time period that is comparable to a year.
Observation over one day is not enough.

>   (ii)The ACF I am working with has an initial downswing which is then
> followed by aperiodic oscillations that attenuates at longer time (similar
> to the positive half of a sinc function--i.e a decaying exponential with
> some oscillatory behavior).  From what I WAS TOLD, "the psuedo-period of
> the autocorrelation function will correspond to a HUMP in the PSD plot."
> However, I dont quiet understand :
>       (a)what the "pseudo-period" of an ACF is??

"Pseudo period" is a basic term in DSP. A "periodic" signal is one
where the following holds,

x(t) = x(t+nT)                                               

for some T where n is all integers. Examples of periodic signals
are

x(t) = sin(2*pi*f*t)                                        [2.a]
x(t) = cos(2*pi*f*t)                                        [2.b]

The spectra of these signals will show peaks at the frequency f.

"Pseudo periodic signals" are signals that are "almost periodic",
like e.g.

x(t) = sin(2*pi*f*t)*exp(-at)                                

where 0 < a << 1. The signal in  is not periodic according to ,
but it is "close enough" for us not to make the distinction, if the
parameter 'a' is small enough. The spectrum of the signal will show
a peak at frequency f. The peak will be a bit more "fuzzy" than the
peaks of the signals [2.a] and [2.b], but it will be there.

> The ACF that I am
> working with has MORE THAN ONE  period due to its aperiodic oscillatory
> behavior.  In short, is the psuedo-period the time it takes for the
> initial downswing to occur or what?

No, see above. But you are right in that initial transients make
problems when dealing with spectra, though. But that's a completely
different story.

>       (b)WHY would the frequnecy CORRESPONDING to the psuedo-period of the
> ACF would yield a hump in the PSD plot?

See above. "Pseudo periodic" means "almost periodic", if the signal
is "close enough to periodic" you will see a peak near the dominant
frequency.

> What about the subsquent periods
> of the minor oscillatory periods (side lopes)of the ACF? Would the
> frequnecy correspnonding to those periods be manifested as humps in the
> PSD plot?

No, not necessarily. There will be some sort of effects in the
spectrum, though.

> How exactly is the dominant frqunecy is captured in the PSD from
> the
> time sequence of  ACF?

Again, when we are talking about pseudo periodic signals, it is
a basic property of the Fourier transform.

Rune

```