# Proof regarding autocorrelation of a pulse in white noise

Started by June 30, 2003
```Let  x[n] = s[n] + p[n]
where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
complex signal in the for Ae^{j2pi w + phi} where w is radian
frequency and phi is a phase factor.
I want to show that if I do the autocorrelation of x[n]  (ie
x[n]x[n-1]* , where * denotes the complex conjugate) that the argument
will more closely approach the actual frequency of p[n],
so, if I did  the autocorrelation of x[n] and then took the argument
of the result, it would be with in some % of  the true frequency.
You can assume that the mean of the GWN is zero and the variance is
0.1 or whatever is convenient.

I have taken x[n]x[n-1]* and multiplied it out...and you can make a
few assumptions to limit terms, but I am not seeing a good way to show
what is happening w/ the argument.

Thanks
Craig
```
```crrea2@umkc.edu (Craig) writes:

> Let  x[n] = s[n] + p[n]
> where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
> complex signal in the for Ae^{j2pi w + phi} where w is radian
> frequency and phi is a phase factor.

I assume you mean Ae^{j2pi w n + phi} wotherwise w would not be a
frequency term.

> I want to show that if I do the autocorrelation of x[n]  (ie
> x[n]x[n-1]* , where * denotes the complex conjugate)

That's not really the autocorrelation, it's just the signal times
itself conjugate delayed by one sample.

> that the argument will more closely approach the actual frequency of
> p[n], so, if I did the autocorrelation of x[n] and then took the
> argument of the result, it would be with in some % of the true
> frequency.  You can assume that the mean of the GWN is zero and the
> variance is 0.1 or whatever is convenient.
>
>
> I have taken x[n]x[n-1]* and multiplied it out...and you can make a
> few assumptions to limit terms, but I am not seeing a good way to show
> what is happening w/ the argument.

x[n]x[n-1]* = (Ae^{j2pi w n + phi} + p[n])(Ae^-{j2pi w (n-1) + phi} + p[n-1]*)

= A^2e^{j2pi w n + phi - (j2pi w (n-1) + phi)}
+ p[n] Ae^-{j2pi w (n-1) + phi}
+ p[n-1]*Ae^{j2pi w n + phi}
+ p[n]p[n-1]*

Taking the expectation:

E{ x[n]x[n-1]* } = A^2e^{j2pi w} + 0 + 0 + 0

because E{p[k]} = 0 (p[k] is zero mean) and E{p[k]p[k-1]*} = 0 (p[k]
is white, i.e. uncorrelated).

More generally, the frequency estimator:

argument ( sum over all n ( W[n] x[n]x[n-1]* ) )

is a phase-weighted averager, of which Kay's estimator  is the most
well-known.  There is a matlab implementation of several of these:

http://www.itee.uq.edu.au/~kootsoop/wlp.m

The differences between all of them just involve how you choose the
weights W[n].  Kay  chooses the weights to be parabolic, which is the
best (minimum variance) answer if your noise power goes to zero. Lank,
Reed and Pollon  choose the weights to be uniform, which is the best

For more on this (and other) frequency estimators, have a look at .

Ciao,

Peter K.

REFERENCES
==========

  V. Clarkson, P. J.  Kootsookos and B. G.  Quinn,
``Variance Analysis of Kay's Weighted Linear Predictor
Frequency Estimator,'' IEEE Transactions on Signal
Processing, Vol. 42, pp. 2370-2379, 1994.

 S. M.  Kay, `` A Fast and Accurate Single Frequency
Estimator,'' IEEE Transactions on Acoustics, Speech
and Signal Processing, Vol. 37(12), pp. 1987-1989,
1989.

  B.C. Lovell and R.C. Williamson,
"The Statistical Performance of Some Instantaneous
Frequency Estimators," IEEE Trans. on Acoustics,
Speech and Signal Processing, Vol. 40, pp. 1708-1723,
1992.

 G. W. Lank, I. S.  Reed and G. E.  Pollon,
``A Semicoherent Detection Statistic and Doppler
Estimation Statistic,'' IEEE Transactions on
Aerospace and Electronic Systems, Vol. AES-9(2),
pp. 151-165, 1973.

 P. J. Kootsookos, ``A Review of the Frequency Estimation and
Tracking Problems,'' CRASys Technical Report, last updated
21/02/1999, URL:
http://www.itee.uq.edu.au/~kootsoop/comparison-t.pdf, last
accessed: 1/07/2003.

--
Peter J. Kootsookos

"Na, na na na na na na, na na na na"
- 'Hey Jude', Lennon/McCartney
```
```p.kootsookos@remove.ieee.org (Peter J. Kootsookos) wrote in message news:<s68isqngl9y.fsf@mango.itee.uq.edu.au>...
> crrea2@umkc.edu (Craig) writes:
>
> > Let  x[n] = s[n] + p[n]
> > where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
> > complex signal in the for Ae^{j2pi w + phi} where w is radian
> > frequency and phi is a phase factor.
>
> I assume you mean Ae^{j2pi w n + phi} wotherwise w would not be a
> frequency term.
>
> > I want to show that if I do the autocorrelation of x[n]  (ie
> > x[n]x[n-1]* , where * denotes the complex conjugate)
>
> That's not really the autocorrelation, it's just the signal times
> itself conjugate delayed by one sample.

This is an interesting discussion. Now, the signal model includes
the noise p[n], but it doesn't show up in Peter's estimate for the
autocorrelation, which I think it should. I can't find any flaw in
Peter's reasoning.

What did I miss?

Rune
```
```allnor@tele.ntnu.no (Rune Allnor) writes:

> p.kootsookos@remove.ieee.org (Peter J. Kootsookos) wrote
> > crrea2@umkc.edu (Craig) writes:
> >
> > > Let  x[n] = s[n] + p[n]
> > > where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
> > > complex signal in the for Ae^{j2pi w + phi} where w is radian
> > > frequency and phi is a phase factor.
> >
> > I assume you mean Ae^{j2pi w n + phi} wotherwise w would not be a
> > frequency term.
> >
> > > I want to show that if I do the autocorrelation of x[n]  (ie
> > > x[n]x[n-1]* , where * denotes the complex conjugate)
> >
> > That's not really the autocorrelation, it's just the signal times
> > itself conjugate delayed by one sample.
>
> This is an interesting discussion. Now, the signal model includes
> the noise p[n], but it doesn't show up in Peter's estimate for the
> autocorrelation, which I think it should. I can't find any flaw in
> Peter's reasoning.
>
> What did I miss?

Hi Rune,

I'm not actually finding an estimate of the autocorrelation, just the
autocorrelation at a lag of 1 sample.

To find the autocorrelation, I'd have to find:

E{x[n]x[n-m]*}

where m varies.  In my analysis, I just set m=1.

Ciao,

Peter K.

--
Peter J. Kootsookos

"Na, na na na na na na, na na na na"
- 'Hey Jude', Lennon/McCartney
```
```Hey guys, thanks a lot, that makes perfect sense, I should have seen that!
It is percisely what I was looking for.
Craig

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0306302225.10fbd0bc@posting.google.com>...
> p.kootsookos@remove.ieee.org (Peter J. Kootsookos) wrote in message news:<s68isqngl9y.fsf@mango.itee.uq.edu.au>...
> > crrea2@umkc.edu (Craig) writes:
> >
> > > Let  x[n] = s[n] + p[n]
> > > where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
> > > complex signal in the for Ae^{j2pi w + phi} where w is radian
> > > frequency and phi is a phase factor.
> >
> > I assume you mean Ae^{j2pi w n + phi} wotherwise w would not be a
> > frequency term.
> >
> > > I want to show that if I do the autocorrelation of x[n]  (ie
> > > x[n]x[n-1]* , where * denotes the complex conjugate)
> >
> > That's not really the autocorrelation, it's just the signal times
> > itself conjugate delayed by one sample.
>
> This is an interesting discussion. Now, the signal model includes
> the noise p[n], but it doesn't show up in Peter's estimate for the
> autocorrelation, which I think it should. I can't find any flaw in
> Peter's reasoning.
>
> What did I miss?
>
> Rune
```
```p.kootsookos@remove.ieee.org (Peter J. Kootsookos) wrote in message news:<s683chqbulr.fsf@mango.itee.uq.edu.au>...
> allnor@tele.ntnu.no (Rune Allnor) writes:
>
> > p.kootsookos@remove.ieee.org (Peter J. Kootsookos) wrote
> > > crrea2@umkc.edu (Craig) writes:
> > >
> > > > Let  x[n] = s[n] + p[n]
> > > > where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
> > > > complex signal in the for Ae^{j2pi w + phi} where w is radian
> > > > frequency and phi is a phase factor.
> > >
> > > I assume you mean Ae^{j2pi w n + phi} wotherwise w would not be a
> > > frequency term.
> > >
> > > > I want to show that if I do the autocorrelation of x[n]  (ie
> > > > x[n]x[n-1]* , where * denotes the complex conjugate)
> > >
> > > That's not really the autocorrelation, it's just the signal times
> > > itself conjugate delayed by one sample.
> >
> > This is an interesting discussion. Now, the signal model includes
> > the noise p[n], but it doesn't show up in Peter's estimate for the
> > autocorrelation, which I think it should. I can't find any flaw in
> > Peter's reasoning.
> >
> > What did I miss?
>
> Hi Rune,
>
> I'm not actually finding an estimate of the autocorrelation, just the
> autocorrelation at a lag of 1 sample.

Of course. One of these days I think I'll have to start actually reading
the posts...

Rune

> To find the autocorrelation, I'd have to find:
>
> E{x[n]x[n-m]*}
>
> where m varies.  In my analysis, I just set m=1.
>
> Ciao,
>
> Peter K.
```
```Craig,

Ae^{j2pi w n + phi} should be
Ae^{j(2pi w n + phi)}

throughout this discussion.

Dirk

Dirk A. Bell
DSP Consultant

crrea2@umkc.edu (Craig) wrote in message news:<82396605.0307010443.961f275@posting.google.com>...
> Hey guys, thanks a lot, that makes perfect sense, I should have seen that!
> It is percisely what I was looking for.
> Craig
>
>
> allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0306302225.10fbd0bc@posting.google.com>...
> > p.kootsookos@remove.ieee.org (Peter J. Kootsookos) wrote in message news:<s68isqngl9y.fsf@mango.itee.uq.edu.au>...
> > > crrea2@umkc.edu (Craig) writes:
> > >
> > > > Let  x[n] = s[n] + p[n]
> > > > where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
> > > > complex signal in the for Ae^{j2pi w + phi} where w is radian
> > > > frequency and phi is a phase factor.
> > >
> > > I assume you mean Ae^{j2pi w n + phi} wotherwise w would not be a
> > > frequency term.
> > >
> > > > I want to show that if I do the autocorrelation of x[n]  (ie
> > > > x[n]x[n-1]* , where * denotes the complex conjugate)
> > >
> > > That's not really the autocorrelation, it's just the signal times
> > > itself conjugate delayed by one sample.
> >
> > This is an interesting discussion. Now, the signal model includes
> > the noise p[n], but it doesn't show up in Peter's estimate for the
> > autocorrelation, which I think it should. I can't find any flaw in
> > Peter's reasoning.
> >
> > What did I miss?
> >
> > Rune
```
```"Dirk Bell" <dirkman@erols.com> wrote

> Craig,
>
> Ae^{j2pi w n + phi} should be
> Ae^{j(2pi w n + phi)}
>
> throughout this discussion.
>
> Dirk
>
> Dirk A. Bell
> DSP Consultant
>

D'oh!  Thanks, Dirk.

Ciao,

Peter K.

--
Peter J. Kootsookos

"Na, na na na na na na, na na na na"
- 'Hey Jude', Lennon/McCartney

```
```dirkman@erols.com (Dirk Bell) wrote in message news:<6721a858.0307011007.31849acd@posting.google.com>...
Think Dirk,
I noticed that as well, no biggie, once I saw how he began, I was able
to put it together readly.  I just couldn't find a good starting
point.  Maybe too much coffee!

> Craig,
>
> Ae^{j2pi w n + phi} should be
> Ae^{j(2pi w n + phi)}
>
> throughout this discussion.
>
> Dirk
>
> Dirk A. Bell
> DSP Consultant
>
>
> crrea2@umkc.edu (Craig) wrote in message news:<82396605.0307010443.961f275@posting.google.com>...
> > Hey guys, thanks a lot, that makes perfect sense, I should have seen that!
> > It is percisely what I was looking for.
> > Craig
> >
> >
> > allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0306302225.10fbd0bc@posting.google.com>...
> > > p.kootsookos@remove.ieee.org (Peter J. Kootsookos) wrote in message news:<s68isqngl9y.fsf@mango.itee.uq.edu.au>...
> > > > crrea2@umkc.edu (Craig) writes:
> > > >
> > > > > Let  x[n] = s[n] + p[n]
> > > > > where s[n] is complex Gaussian white noise (GWN)  and p[n] is a
> > > > > complex signal in the for Ae^{j2pi w + phi} where w is radian
> > > > > frequency and phi is a phase factor.
> > > >
> > > > I assume you mean Ae^{j2pi w n + phi} wotherwise w would not be a
> > > > frequency term.
> > > >
> > > > > I want to show that if I do the autocorrelation of x[n]  (ie
> > > > > x[n]x[n-1]* , where * denotes the complex conjugate)
> > > >
> > > > That's not really the autocorrelation, it's just the signal times
> > > > itself conjugate delayed by one sample.
> > >
> > > This is an interesting discussion. Now, the signal model includes
> > > the noise p[n], but it doesn't show up in Peter's estimate for the
> > > autocorrelation, which I think it should. I can't find any flaw in
> > > Peter's reasoning.
> > >
> > > What did I miss?
> > >
> > > Rune
```
```"Craig" <crrea2@umkc.edu> wrote

> Maybe too much coffee!

There is no such thing as too much coffee.

Ciao,

Peter K.

--
Peter J. Kootsookos

"Na, na na na na na na, na na na na"
- 'Hey Jude', Lennon/McCartney

```