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autocorrelation

Started by John McDermick 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