# Rigorous definition of the Spectral Density of a random signal?

Started by October 25, 2003
```Hi,

I know this question is not 100% relevant to this newsgroup,
as I'm talking about continuous-time signals, and not DSP...
But I couldn't find a more appropriate place to discuss this.

Anyway, there was a big discussion in one of my classes about
some characteristics of white noise, and the disagreement
seemed to be driven by a discrepancy between our concepts
of what the Power-spectral density of a random signal is.

So, I'm looking for a *rigurous* definition of what the
values of the PSD really represent.  (notice that I don't
want the PSD defined as the Fourier transform of the
auto-correlation of the signal -- what I need is a
definition of what the value of the PSD at a specific
frequency means/represents)

When I say a "rigurous definition", I mean is that the
discussion will not be solved with an informal  "it
represents the spectral contents at the given frequency",
or the typical notion that "white noise has a flat spectral
density, which means that it was equal contents at all
frequencies".

No, I'm looking for the *actual* meaning (in a rigurous
mathematical sense) of the value of the PSD at a certain
frequency.

For instance, if we are talking about the definition of
what the values of a probability density function mean,
this would be examples of what I want, and what I don't
want:

What I woud NOT want:  "The value of the PDF tells you
if it is very likely for the variable to take a value
of x, relatively to other values"

(this definition -- besides incorrect -- is purely
intuitive, and thus may lead to ambiguity and multiple
(mis)interpretations)

What I would want:  "The value of the PDF at x is the
limit as dx approaches to zero of the probability that
the variable takes values in (x, x+dx), divided by dx"
Or:  "The pdf function is such that the probability
that the variable takes values within a region R is
given by the integral of the pdf over the region R"

These are both valid examples of what I would call an
actual/rigurous definition of what the meaning of the
value of the PDF is.  (I don't know if that is *the*
right way of defining what a PDF is, but the statements
are true and unambiguous, and they describe what a value
of the PDF really tells us)

So, can someone help me with finding a rigurous definition
of what the value at a given frequency of the PSD of a
random signal means?

Thanks!

Carlos
--

```
```Hi,

I know this question is not 100% relevant to this newsgroup,
as I'm talking about continuous-time signals, and not DSP...
But I couldn't find a more appropriate place to discuss this.

Anyway, there was a big discussion in one of my classes about
some characteristics of white noise, and the disagreement
seemed to be driven by a discrepancy between our concepts
of what the Power-spectral density of a random signal is.

So, I'm looking for a *rigurous* definition of what the
values of the PSD really represent.  (notice that I don't
want the PSD defined as the Fourier transform of the
auto-correlation of the signal -- what I need is a
definition of what the value of the PSD at a specific
frequency means/represents)

When I say a "rigurous definition", I mean is that the
discussion will not be solved with an informal  "it
represents the spectral contents at the given frequency",
or the typical notion that "white noise has a flat spectral
density, which means that it was equal contents at all
frequencies".

No, I'm looking for the *actual* meaning (in a rigurous
mathematical sense) of the value of the PSD at a certain
frequency.

For instance, if we are talking about the definition of
what the values of a probability density function mean,
this would be examples of what I want, and what I don't
want:

What I woud NOT want:  "The value of the PDF tells you
if it is very likely for the variable to take a value
of x, relatively to other values"

(this definition -- besides incorrect -- is purely
intuitive, and thus may lead to ambiguity and multiple
(mis)interpretations)

What I would want:  "The value of the PDF at x is the
limit as dx approaches to zero of the probability that
the variable takes values in (x, x+dx), divided by dx"
Or:  "The pdf function is such that the probability
that the variable takes values within a region R is
given by the integral of the pdf over the region R"

These are both valid examples of what I would call an
actual/rigurous definition of what the meaning of the
value of the PDF is.  (I don't know if that is *the*
right way of defining what a PDF is, but the statements
are true and unambiguous, and they describe what a value
of the PDF really tells us)

So, can someone help me with finding a rigurous definition
of what the value at a given frequency of the PSD of a
random signal means?

Thanks!

Carlos
--

```
```Carlos Moreno wrote:
> Actually, I (doubly) disagree on this point -- there is non-zero
> energy component *in a frequency band of non-zero length*.  Notice
> that the term I used, "energy density", refers to (informally
> speaking) the "amount of energy per unit of frequency  (i.e.,
> density in the above definition would be defined as the limit as
> delta-omega approaches 0 of the energy contained in the band
> (omega, omega + delta-omega) divided by delta-omega).
>
> Notice that with this definition, the energy of a particular
> frequency component is zero.

Why is that?  If I sweep with a delta function for my filter,
then delta-omega is always zero, but the energy of a particular
frequency component is not.

I think you are confusing band*width* with *height*.  The PSD
of noise will change with time, it should be noisy after all!
The actual time and frequency resolution you have in the
real world is a harder problem than just defining something
mathematically.

That could be another point of confusion - time and frequency
are both being used.  If you plot a 3D surface of amplitude
with time on one axis and frequency on another, you can convert
that to a PSD surface plot.  You can look at this integrated
along the time axis to get the average PSD of the whole history,
or you can integrate it along the frequency axis to see total
power as a function of time.  If you pick out one point of
frequency and time, it tells you something useful.

All these things come from the raw data tho.  The real world
makes getting the raw data harder, but you can still interpret
it as a mathematical object.

Patience, persistence, truth,
Dr. mike

--
Mike Rosing
www.beastrider.com                   BeastRider, LLC
SHARC debug tools

```
```Mike Rosing wrote:
> Carlos Moreno wrote:
>
>> So, can someone help me with finding a rigurous definition
>> of what the value at a given frequency of the PSD of a
>> random signal means?
>
> howdy Carlos,
>
> What's a rigorous definition of PSD to begin with?  What you
> apply the definition to should not matter.  If interpretation
> of the application seems to change with the function being
> measured, then it would seem that the original interpretation
> is wrong.  Not the definition.

I'm not claiming or suggesting that the definition is wrong.
I just need/want to know what is the definition.

I think I understand it, but since some of the fellow students
were drawing conclusions different from mine, I am sure that
either I or them have a wrong idea of what the PSD of a random
signal is.

I'm trying to find a strict definition of what the value of
the PSD at a given frequency represents.  For instance, if you
give me the PSD of a random signal, and I see that the value
at f = 60Hz is 2.5, what I want to know is:  what exactly
does that 2.5 represent?

Is it the power of the 60Hz component?  (obviously not -- if
it is a random signal, we can not get a measure of power from
the PSD).

Is it the probability that the signal has energy at that
frequency?  (obviously not -- the PSD values are not
restricted to the interval 0 to 1)

Is it the average of the total engergy of the 60Hz component
over different realizations of the random signal?

Is it the average of energy density at 60Hz over different
realizations of the random signal?

Is it the average of the power of the 60Hz component over
different realizations of the random signal?

(I'm not sure if one of the above is the correct definition;
but that's what I'm trying to know -- that's what I meant
when I said that I'm looking for a "rigurous" definition of
what the PSD values represent)

Cheers,

Carlos
--

```
```Carlos Moreno <moreno_at_mochima_dot_com@x.xxx> wrote in message news:<ZGenb.23423\$He4.751856@wagner.videotron.net>...
> Mike Rosing wrote:
> > Carlos Moreno wrote:
> >
> >> So, can someone help me with finding a rigurous definition
> >> of what the value at a given frequency of the PSD of a
> >> random signal means?
> >
> > howdy Carlos,
> >
> > What's a rigorous definition of PSD to begin with?  What you
> > apply the definition to should not matter.  If interpretation
> > of the application seems to change with the function being
> > measured, then it would seem that the original interpretation
> > is wrong.  Not the definition.
>
> I'm not claiming or suggesting that the definition is wrong.
> I just need/want to know what is the definition.
>
> I think I understand it, but since some of the fellow students
> were drawing conclusions different from mine, I am sure that
> either I or them have a wrong idea of what the PSD of a random
> signal is.

In my humble opinion, the terms "rigorous" and "statistics" are almost
contradictions in terms. While the formalities of maths also applies
when used in statistsics, I think the interpretations may not be as
strict as with other mathemathical diciplines.

Actually, from the statement of the question I think the interpretation,
not the definition, may be the problem.

> I'm trying to find a strict definition of what the value of
> the PSD at a given frequency represents.  For instance, if you
> give me the PSD of a random signal, and I see that the value
> at f = 60Hz is 2.5, what I want to know is:  what exactly
> does that 2.5 represent?
>
> Is it the power of the 60Hz component?  (obviously not -- if
> it is a random signal, we can not get a measure of power from
> the PSD).

If you mean "the power at 60 Hz in one particular realization",
I agree.

> Is it the probability that the signal has energy at that
> frequency?  (obviously not -- the PSD values are not
> restricted to the interval 0 to 1)

Agreed.

> Is it the average of the total engergy of the 60Hz component
> over different realizations of the random signal?

Nope. It's the "Power Spectral Density" we are talking about. If there
was a nonzero *energy* component in random noise, electrical energy
would be available for free: Just mount a reciever and drain energy out
of the blue.

> Is it the average of energy density at 60Hz over different
> realizations of the random signal?

Nope, due to the energy/density thing commented in the previous point.

> Is it the average of the power of the 60Hz component over
> different realizations of the random signal?

Almost: It's the average power density of all realizations.

> (I'm not sure if one of the above is the correct definition;
> but that's what I'm trying to know -- that's what I meant
> when I said that I'm looking for a "rigurous" definition of
> what the PSD values represent)

That's the kind of questions you learn the most from. Test as many
interpretations you can, discuss them with others, and make up your
own mind about what works for you.

Rune
```
```Rune Allnor wrote:

> Carlos Moreno <moreno_at_mochima_dot_com@x.xxx> wrote in message
> news:<ZGenb.23423\$He4.751856@wagner.videotron.net>...
>> ...
>> (I'm not sure if one of the above is the correct definition;
>> but that's what I'm trying to know -- that's what I meant
>> when I said that I'm looking for a "rigurous" definition of
>> what the PSD values represent)
>
> That's the kind of questions you learn the most from. Test as many
> interpretations you can, discuss them with others, and make up
>
> Rune

Hi Rune,
that's a sort of brilliant philosophical answer.
It remembers me to a piece of German literature: Lessing's "Nathan
the Wise"  or the famous words of Zarathoustra...

SCNR,
Bernhard

```
```Bernhard Holzmayer <holzmayer.bernhard@deadspam.com> wrote in message news:<3305646.PMFrtvB4JK@holzmayer.ifr.rt>...
> Rune Allnor wrote:
>
> > Carlos Moreno <moreno_at_mochima_dot_com@x.xxx> wrote in message
> > news:<ZGenb.23423\$He4.751856@wagner.videotron.net>...
> >> ...
> >> (I'm not sure if one of the above is the correct definition;
> >> but that's what I'm trying to know -- that's what I meant
> >> when I said that I'm looking for a "rigurous" definition of
> >> what the PSD values represent)
> >
> > That's the kind of questions you learn the most from. Test as many
> > interpretations you can, discuss them with others, and make up
> >
> > Rune
>
> Hi Rune,
> that's a sort of brilliant philosophical answer.
> It remembers me to a piece of German literature: Lessing's "Nathan
> the Wise"  or the famous words of Zarathoustra...

I don't know any of those (except for that intro music by one of the
Strauss'es in "2001 - a Space Oddyssey")... but thanks for the warning.
We don't want any "Oracle of Delphi"-type answers, do we... ;)

> SCNR,

???

Rune
```
```Rune Allnor wrote:

> Bernhard Holzmayer <holzmayer.bernhard@deadspam.com> wrote in
> message news:<3305646.PMFrtvB4JK@holzmayer.ifr.rt>...
>> Rune Allnor wrote:
>>
>> > Carlos Moreno <moreno_at_mochima_dot_com@x.xxx> wrote in
>> > message news:<ZGenb.23423\$He4.751856@wagner.videotron.net>...
>> >> ...
>> >> (I'm not sure if one of the above is the correct definition;
>> >> but that's what I'm trying to know -- that's what I meant
>> >> when I said that I'm looking for a "rigurous" definition of
>> >> what the PSD values represent)
>> >
>> > That's the kind of questions you learn the most from. Test as
>> > many interpretations you can, discuss them with others, and
>> > make up your own mind about what works for you.
>> >
>> > Rune
>>
>> Hi Rune,
>> that's a sort of brilliant philosophical answer.
>> It remembers me to a piece of German literature: Lessing's
>> "Nathan
>> the Wise"  or the famous words of Zarathoustra...
>
> I don't know any of those (except for that intro music by one of
> the Strauss'es in "2001 - a Space Oddyssey")... but thanks for the
> warning. We don't want any "Oracle of Delphi"-type answers, do
> we... ;)

You're right. However, sometimes, it's close together...
>
>> SCNR,
>
> ???
= _S_orry, _C_ould _N_ot _R_esist
>
> Rune

Bernhard

```
```Carlos Moreno wrote:

> Is it the power of the 60Hz component?  (obviously not -- if
> it is a random signal, we can not get a measure of power from
> the PSD).

Why not?

I mean, that's _statistical_ power, not power as
in physics... Sometimes they're not the same...

bye,

--

piergiorgio

```
```How about going to the library and looking at any one of many good books
on the statistics of time series?  Vol. 1 of Priestly is a good one for
this question.

-Tom

--

To respond by email, replace "somewhere" with "astro" in the