# Power spectrum and autocorrelation form a fourier transform pair

Started by December 11, 2007
```Im an undergraduate in DSP having now completed the taught modules of the
course covering the usual introduction level signal processing theory such
as sampling theorm, fourier analysis, convolution, correlation. I'm now
undertaking my thesis, the detection of modern comms signals, and have
been reading up on the associated wealth of theory and papers, in
particular statistical signal processing. I've found numerous references
to the fourier transform pair;

power spectral density <=> autocorrelation function

I understand that the PSD is the spectrum of a signal, and the ACF gives a
measure of the similarity between a signal and itself at increasing lag,
but what I dont understand is how these two are related by way of the
fourier transform??

Can anybody help me to understand this relationship?

Many thanks,

```
```On 11 Des, 15:04, "amlangford" <amlangf...@qinetiq.com> wrote:
> Im an undergraduate in DSP having now completed the taught modules of the
> course covering the usual introduction level signal processing theory such
> as sampling theorm, fourier analysis, convolution, correlation. I'm now
> undertaking my thesis, the detection of modern comms signals, and have
> been reading up on the associated wealth of theory and papers, in
> particular statistical signal processing. I've found numerous references
> to the fourier transform pair;
>
> power spectral density <=> autocorrelation function
>
> I understand that the PSD is the spectrum of a signal, and the ACF gives a
> measure of the similarity between a signal and itself at increasing lag,
> but what I dont understand is how these two are related by way of the
> fourier transform??
>
> Can anybody help me to understand this relationship?

It pops straight out of the maths. The crude version
(ignoring scaling coefficients) goes something like

rxx[k] = sum_n x[n]x[n+k]

FT{rxx[k]} = sum_k sum_n x[n]x[n+k] exp(jwk)

p = n+k, k = p-n

= sum_p sum_n x[n]x[p]exp(jw(p-n))

= sum_p x[p] exp(-jwp) sum_n x[n] exp(jwn)

= |FT{x[n]}|^2

Did that help?

Rune
```
```On Dec 11, 9:53 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 11 Des, 15:04, "amlangford" <amlangf...@qinetiq.com> wrote:
>
>
>
> > Im an undergraduate in DSP having now completed the taught modules of the
> > course covering the usual introduction level signal processing theory such
> > as sampling theorm, fourier analysis, convolution, correlation. I'm now
> > undertaking my thesis, the detection of modern comms signals, and have
> > been reading up on the associated wealth of theory and papers, in
> > particular statistical signal processing. I've found numerous references
> > to the fourier transform pair;
>
> > power spectral density <=> autocorrelation function
>
> > I understand that the PSD is the spectrum of a signal, and the ACF gives a
> > measure of the similarity between a signal and itself at increasing lag,
> > but what I dont understand is how these two are related by way of the
> > fourier transform??
>
> > Can anybody help me to understand this relationship?
>
> It pops straight out of the maths. The crude version
> (ignoring scaling coefficients) goes something like
>
> rxx[k] = sum_n x[n]x[n+k]
>
> FT{rxx[k]} = sum_k sum_n x[n]x[n+k] exp(jwk)
>
>               p = n+k, k = p-n
>
>            = sum_p sum_n x[n]x[p]exp(jw(p-n))
>
>            = sum_p x[p] exp(-jwp) sum_n x[n] exp(jwn)
>
>            = |FT{x[n]}|^2
>
> Did that help?
>
> Rune

Just thought I'd add that Rune's solution is for deterministic
signals. If x(t) is a (wide-sense stationary) random process, then the
PSD is defined to be the Fourier transform of its autocorrelation
function, but the derivation is different (and much more complicated,
if you ask me). The theorem has a name:

http://en.wikipedia.org/wiki/Wiener%E2%80%93Khinchin_theorem

If you ask me, the PSD <-> correlation function relationship is just
one of those things that is best accepted and forgotten about; I don't
think delving into the details gives you much insight.

Jason
```
```On Dec 12, 3:04 am, "amlangford" <amlangf...@qinetiq.com> wrote:
> Im an undergraduate in DSP having now completed the taught modules of the
> course covering the usual introduction level signal processing theory such
> as sampling theorm, fourier analysis, convolution, correlation. I'm now
> undertaking my thesis, the detection of modern comms signals, and have
> been reading up on the associated wealth of theory and papers, in
> particular statistical signal processing. I've found numerous references
> to the fourier transform pair;
>
> power spectral density <=> autocorrelation function
>
> I understand that the PSD is the spectrum of a signal, and the ACF gives a
> measure of the similarity between a signal and itself at increasing lag,
> but what I dont understand is how these two are related by way of the
> fourier transform??
>
> Can anybody help me to understand this relationship?
>
> Many thanks,
>

The Weiner Kitchen Theorem - look it up.

Hardy
```
```On 12 Dez., 06:51, HardySpicer <gyansor...@gmail.com> wrote:
> On Dec 12, 3:04 am, "amlangford" <amlangf...@qinetiq.com> wrote:
>
>
>
>
>
> > Im an undergraduate in DSP having now completed the taught modules of the
> > course covering the usual introduction level signal processing theory such
> > as sampling theorm, fourier analysis, convolution, correlation. I'm now
> > undertaking my thesis, the detection of modern comms signals, and have
> > been reading up on the associated wealth of theory and papers, in
> > particular statistical signal processing. I've found numerous references
> > to the fourier transform pair;
>
> > power spectral density <=> autocorrelation function
>
> > I understand that the PSD is the spectrum of a signal, and the ACF gives a
> > measure of the similarity between a signal and itself at increasing lag,
> > but what I dont understand is how these two are related by way of the
> > fourier transform??
>
> > Can anybody help me to understand this relationship?
>
> > Many thanks,
>
>
> The Weiner Kitchen Theorem - look it up.

Maybe it should just be called the "Hot Dog Theorem", this has less
potential for grammatic failure...

:-)
```
```Andor wrote:
> On 12 Dez., 06:51, HardySpicer <gyansor...@gmail.com> wrote:
>> On Dec 12, 3:04 am, "amlangford" <amlangf...@qinetiq.com> wrote:
>>
>>
>>
>>
>>
>>> Im an undergraduate in DSP having now completed the taught modules of the
>>> course covering the usual introduction level signal processing theory such
>>> as sampling theorm, fourier analysis, convolution, correlation. I'm now
>>> undertaking my thesis, the detection of modern comms signals, and have
>>> been reading up on the associated wealth of theory and papers, in
>>> particular statistical signal processing. I've found numerous references
>>> to the fourier transform pair;
>>> power spectral density <=> autocorrelation function
>>> I understand that the PSD is the spectrum of a signal, and the ACF gives a
>>> measure of the similarity between a signal and itself at increasing lag,
>>> but what I dont understand is how these two are related by way of the
>>> fourier transform??
>>> Can anybody help me to understand this relationship?
>>> Many thanks,
>> The Weiner Kitchen Theorem - look it up.
>
> Maybe it should just be called the "Hot Dog Theorem", this has less
> potential for grammatic failure...

Not the Hot-dog Cookery theorem?

Jerry
--
Engineering is the art of making what you want from things you can get.
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```
```On 12 Dez., 19:25, Jerry Avins <j...@ieee.org> wrote:
> Andor wrote:
> > On 12 Dez., 06:51, HardySpicer <gyansor...@gmail.com> wrote:
> >> On Dec 12, 3:04 am, "amlangford" <amlangf...@qinetiq.com> wrote:
>
> >>> Im an undergraduate in DSP having now completed the taught modules of the
> >>> course covering the usual introduction level signal processing theory such
> >>> as sampling theorm, fourier analysis, convolution, correlation. I'm now
> >>> undertaking my thesis, the detection of modern comms signals, and have
> >>> been reading up on the associated wealth of theory and papers, in
> >>> particular statistical signal processing. I've found numerous references
> >>> to the fourier transform pair;
> >>> power spectral density <=> autocorrelation function
> >>> I understand that the PSD is the spectrum of a signal, and the ACF gives a
> >>> measure of the similarity between a signal and itself at increasing lag,
> >>> but what I dont understand is how these two are related by way of the
> >>> fourier transform??
> >>> Can anybody help me to understand this relationship?
> >>> Many thanks,