power spectrum density is always positive?

Hi,
I am working on hw, and to find out the result I need to assume that
the power spectrum density of a RP X(t) is always positive. Is it
always true?

Thanks

VijaKhara said the following on 26/09/2006 20:32:
> Hi,
> I am working on hw, and to find out the result I need to assume that
> the power spectrum density of a RP X(t) is always positive. Is it
> always true?

Yes.  Although I guess it's possible that rounding errors in 
calculations could lead to unintended negative values (in which case 
it's probably best to set -ve values to zero).


-- 
Oli

"VijaKhara" <VijaKhara@gmail.com> writes:

> Hi,
> I am working on hw, and to find out the result I need to assume that
> the power spectrum density of a RP X(t) is always positive. Is it
> always true?

Hi Vija,

Yes. I've tried to find a good proof but the only one I've
come across (from both [schwartz] and [hayes]) is not
really satisfying to me.

Their proof stems from the fact that the PSD is equivalent
to the expected value of the magnitude squared of the DFT of
a section of the signal in the limit as that section's extent
approaches infinity. Since the magnitude squared is always
positive, so is the PSD.

--Randy


@book{schwartz,
  title = "Signal Processing: Discrete Spectral Analysis, Detection, and Estimation",
  author = "{Mischa~Schwartz and Leonard~Shaw}",
  publisher = "McGraw-Hill",
  year = "1975"}
@BOOK{hayes,
  title = "{Statistical Digital Signal Processing and Modeling}",
  author = "{Monson~H.~Hayes}",
  publisher = "Wiley",
  year = "1996"}

-- 
%  Randy Yates                  % "Midnight, on the water... 
%% Fuquay-Varina, NC            %  I saw...  the ocean's daughter." 
%%% 919-577-9882                % 'Can't Get It Out Of My Head' 
%%%% <yates@ieee.org>           % *El Dorado*, Electric Light Orchestra
http://home.earthlink.net/~yatescr

"VijaKhara" <VijaKhara@gmail.com> wrote in message
news:1159299165.516373.60620@m73g2000cwd.googlegroups.com...
> Hi,
> I am working on hw, and to find out the result I need to assume that
> the power spectrum density of a RP X(t) is always positive. Is it
> always true?
>
> Thanks
>
If you square something the result is always positive.

M.



-- 
Posted via a free Usenet account from http://www.teranews.com

Randy Yates wrote:
> "VijaKhara" <VijaKhara@gmail.com> writes:
> 
>> Hi,
>> I am working on hw, and to find out the result I need to assume that
>> the power spectrum density of a RP X(t) is always positive. Is it
>> always true?
> 
> Hi Vija,
> 
> Yes. I've tried to find a good proof but the only one I've
> come across (from both [schwartz] and [hayes]) is not
> really satisfying to me.
> 
> Their proof stems from the fact that the PSD is equivalent
> to the expected value of the magnitude squared of the DFT of
> a section of the signal in the limit as that section's extent
> approaches infinity. Since the magnitude squared is always
> positive, so is the PSD.
> 
> --Randy

In what way is "its the sum of some squares, so it can't go negative" 
unsatisfying? Are you concerned that is doesn't cover the possibility of 
the things you are squaring being complex?

It seems to me another way of looking at this is heating and cooling. If 
the PSD could be negative, wouldn't that mean the signal could cool 
things down if it were appropriately filtered? While not exactly a 
proof, the notion that could happen seems absurd.

Steve

On Wed, 27 Sep 2006 11:45:13 +0800, Steve Underwood <steveu@dis.org>
wrote:

>Randy Yates wrote:
>> "VijaKhara" <VijaKhara@gmail.com> writes:
>> 
>>> Hi,
>>> I am working on hw, and to find out the result I need to assume that
>>> the power spectrum density of a RP X(t) is always positive. Is it
>>> always true?
>> 
>> Hi Vija,
>> 
>> Yes. I've tried to find a good proof but the only one I've
>> come across (from both [schwartz] and [hayes]) is not
>> really satisfying to me.
>> 
>> Their proof stems from the fact that the PSD is equivalent
>> to the expected value of the magnitude squared of the DFT of
>> a section of the signal in the limit as that section's extent
>> approaches infinity. Since the magnitude squared is always
>> positive, so is the PSD.
>> 
>> --Randy
>
>In what way is "its the sum of some squares, so it can't go negative" 
>unsatisfying? Are you concerned that is doesn't cover the possibility of 
>the things you are squaring being complex?

One also needs to remember that when things are complex the
calculation involves a transpose so the result is sum of two squares.

Hey theoreticians and hot air enthusiasts -

    Duh.  Key word is POWER.  Can power be negative?

Tsk.





"VijaKhara" <VijaKhara@gmail.com> wrote in message
news:1159299165.516373.60620@m73g2000cwd.googlegroups.com...
> Hi,
> I am working on hw, and to find out the result I need to assume that
> the power spectrum density of a RP X(t) is always positive. Is it
> always true?
>
> Thanks
>

VijaKhara skrev:
> Hi,
> I am working on hw, and to find out the result I need to assume that
> the power spectrum density of a RP X(t) is always positive. Is it
> always true?

The PSD is (formally) 'non-negative', which is slightly different from
'positive'. The definition of the PSD is achieved through Parseval's
identity to be

P = sum |X(k)|^2

where |X(k)|^2 is the k'th coefficient of the PSD. Be aware that
you have to use the unitary DFT or FT to get this to work,
i.e. all the scaling coefficients need to be in all the correct places.

If you are working on a theoretical justification, don't bring in the
autocorrelation of x(t) since it has nothing to do with the
*definition*
of the PSD, it only describes and *estimator* for the PSD.

As others already have commented, numerical errors might
cause a vanishing coeffcient to pop out on the negative side.

Rune

Major Misunderstanding said the following on 27/09/2006 04:20:
> "VijaKhara" <VijaKhara@gmail.com> wrote in message
> news:1159299165.516373.60620@m73g2000cwd.googlegroups.com...
>> Hi,
>> I am working on hw, and to find out the result I need to assume that
>> the power spectrum density of a RP X(t) is always positive. Is it
>> always true?
>>
>> Thanks
>>
> If you square something the result is always positive.
> 

Except if it's complex.  But that's OK, because PSD is really defined as 
magnitude squared, which is absolutely definitely always positive.

-- 
Oli

Bill wrote:

> Hey theoreticians and hot air enthusiasts -

>     Duh.  Key word is POWER.  Can power be negative?

In the sense of the question, maybe not.

Consider, though, instantaneous power (I times V) into a
reactive load.  Power can be a vector, and sometimes goes
in the opposite direction.

On the other hand, probability can never be negative.

-- glen