# autocorrelation

Started by January 3, 2011
```On Jan 4, 1:21=A0pm, dvsarwate <dvsarw...@yahoo.com> wrote:
> 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. =A0The 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. =A0Young
> folks who have seen only push-pins used on
> bulletin boards may have trouble with this last
> statement....
>
> Hope this helps
>
> Dilip Sarwate

Thanks Dilip,

For pointing out my error. My brain is still not in gear after a flu
ridden vacation. But I'm on the mend.

Clay

```