# autocorrelation

Started by January 3, 2011
```Is there a signal x for which the autocorrelation sequence r would be
a 'flat' line ?

For example r = [0.5 0.5 0.5 0.5 0.5 0.5]

My guess is no. Can somebody confirm?

```
```On 01/03/2011 03:40 PM, John McDermick wrote:
> Is there a signal x for which the autocorrelation sequence r would be
> a 'flat' line ?
>
> For example r = [0.5 0.5 0.5 0.5 0.5 0.5]
>
> My guess is no. Can somebody confirm?

I think that you could prove that for a signal of finite energy there'd
be a strict maximum at t = 0.  But I'm too lazy to do it right now...

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
```
John McDermick wrote:

> Is there a signal x for which the autocorrelation sequence r would be
> a 'flat' line ?

X = 1.

> For example r = [0.5 0.5 0.5 0.5 0.5 0.5]
>
> My guess is no. Can somebody confirm?

Idiot.
```
```On Jan 3, 7:08=A0pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> John McDermick wrote:
> > Is there a signal x for which the autocorrelation sequence r would be
> > a 'flat' line ?
>
> X =3D 1.
>
> > For example r =3D [0.5 0.5 0.5 0.5 0.5 0.5]
>
> > My guess is no. Can somebody confirm?
>
> Idiot.

Thank you, but the autocorrelation of 1 is a point not a line.
```
```On Jan 3, 6:40=A0pm, John McDermick <johnthedsp...@gmail.com> wrote:
> Is there a signal x for which the autocorrelation sequence r would be
> a 'flat' line ?
>
> For example r =3D [0.5 0.5 0.5 0.5 0.5 0.5]
>
> My guess is no. Can somebody confirm?

Try a Kronecker Delta function.
```
```On Jan 4, 8:26=A0am, John McDermick <johnthedsp...@gmail.com> wrote:
> On Jan 3, 7:08=A0pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
>
> > John McDermick wrote:
> > > Is there a signal x for which the autocorrelation sequence r would be
> > > a 'flat' line ?
>
> > X =3D 1.
>
> > > For example r =3D [0.5 0.5 0.5 0.5 0.5 0.5]
>
> > > My guess is no. Can somebody confirm?
>
> > Idiot.
>
> Thank you, but the autocorrelation of 1 is a point not a line.

John,

If one point doesn't satisfy, try two:

a=3D(sqrt(3)-j)/2 and b=3D(sqrt(3)+j)/2

and the auto correlation of {a,b} is {1,1,1}

Does that help?

Clay

```
```On Jan 4, 12:09=A0pm, Clay <c...@claysturner.com> wrote:
> On Jan 4, 8:26=A0am, John McDermick <johnthedsp...@gmail.com> wrote:
>
> > On Jan 3, 7:08=A0pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
>
> > > John McDermick wrote:
> > > > Is there a signal x for which the autocorrelation sequence r would =
be
> > > > a 'flat' line ?
>
> > > X =3D 1.
>
> > > > For example r =3D [0.5 0.5 0.5 0.5 0.5 0.5]
>
> > > > My guess is no. Can somebody confirm?
>
> > > Idiot.
>
> > Thank you, but the autocorrelation of 1 is a point not a line.
>
> John,
>
> If one point doesn't satisfy, try two:
>
> a=3D(sqrt(3)-j)/2 and b=3D(sqrt(3)+j)/2
>
> and the auto correlation of {a,b} is {1,1,1}
>
> Does that help?
>
> Clay

Nice :o)

I didn't consider complex signals. In this particular scenario the
signals are not complex, but it's nice to know ...so thank you
again...

```
```On Jan 4, 11:35=A0am, Clay <c...@claysturner.com> wrote:
> On Jan 3, 6:40=A0pm, John McDermick <johnthedsp...@gmail.com> wrote:
>
> > Is there a signal x for which the autocorrelation sequence r would be
> > a 'flat' line ?
>
> > For example r =3D [0.5 0.5 0.5 0.5 0.5 0.5]
>
> > My guess is no. Can somebody confirm?
>
> Try a Kronecker Delta function.

A Kronecker Delta function is a function of two variables?? But I
guess you mean an impulse??

The autocorrelation of an impulse is not a sequence of identical
values as far as I can tell (?)
```
```
Clay wrote:
> On Jan 3, 6:40 pm, John McDermick <johnthedsp...@gmail.com> wrote:
>
>>Is there a signal x for which the autocorrelation sequence r would be
>>a 'flat' line ?
>>
>>For example r = [0.5 0.5 0.5 0.5 0.5 0.5]
>>
>>My guess is no. Can somebody confirm?
>
>
> Try a Kronecker Delta function.

There could be complex sequences like:

1 +/- j sqrt(3)
[ 1,  ----------------  ]
2

And so on.

VLV
```
```On Jan 4, 11:09=A0am, Clay <c...@claysturner.com> wrote:

>
> a=3D(sqrt(3)-j)/2 and b=3D(sqrt(3)+j)/2
>
> and the auto correlation of {a,b} is {1,1,1}
>

This must be using a different definition
of autocorrelation than the usual one for
complex-valued sequences, viz.,

R_{x}(k) =3D sum x_i . (x_{i+k})*

where * means complex conjugation and
i+k is taken modulo the length of the sequence
(for periodic autocorrelation).

With x =3D (a, b) where a and b are as specified
by Clay, we have that |a| =3D |b| =3D 1, and so
R_{x}(0) =3D |a|^2 + |b|^2 =3D 2, not 1.  The periodic
autocorrelation *does* have value 1 for offset 1
(that is, R_{x}(1) =3D 1) as stated by Clay (and
verified by the OP?) but the in-phase value is 2.

Tim Wescott's assertion that the autocorrelation
function has a strict maximum at t =3D 0 is correct
(except for the trivial case which Vladimir
Vassilevsky pointed out and the OP rejected).
It *is* possible for a signal to have *all* the
nonzero-offset (i.e., out-of-phase) periodic
autocorrelation values to be the same (though
different from the zero-offset (or in-phase) value).
The well-known PN sequences with their
"thumb-tack" autocorrelation functions are the
classic example of this kind of signal.  Young
folks who have seen only push-pins used on
bulletin boards may have trouble with this last
statement....

Hope this helps

Dilip Sarwate

```