# construction of a signal given its autocorrelation sequence

Started by February 25, 2008
```Hello! My question relates to DSP theory: suppose that i have an
autocorrelation sequence in a closed form formula. This autocorrelation
sequence may correspond to many different signals. My question is how can
i construct a signal whose autocorrelation is the given one?
Anyone expert?

```
```On Feb 25, 10:38&#2013266080;am, "manolis" <el01...@mail.ntua.gr> wrote:
> Hello! My question relates to DSP theory: suppose that i have an
> autocorrelation sequence in a closed form formula. This autocorrelation
> sequence may correspond to many different signals. My question is how can
> i construct a signal whose autocorrelation is the given one?
>

as you said, that autocorrelation corresponds to many different
signals.  you *don't* know which one it was.

it just like knowing the frequency response of a signal, magnitude
only. (the Fourier Transform of the autocorrelation is the power
spectrum which has no phase). throwing away the phase throws away
necessary information needed to inverse F.T. back to the signal.

so, in general, you are asking for something you cannot have because
you lack sufficient information.

r b-j
```
```On Mon, 25 Feb 2008 08:22:41 -0800, robert bristow-johnson wrote:

> On Feb 25, 10:38&nbsp;am, "manolis" <el01...@mail.ntua.gr> wrote:
>> Hello! My question relates to DSP theory: suppose that i have an
>> autocorrelation sequence in a closed form formula. This autocorrelation
>> sequence may correspond to many different signals. My question is how
>> can i construct a signal whose autocorrelation is the given one?
>>
>>
> as you said, that autocorrelation corresponds to many different signals.
>  you *don't* know which one it was.
>
> it just like knowing the frequency response of a signal, magnitude only.
> (the Fourier Transform of the autocorrelation is the power spectrum
> which has no phase). throwing away the phase throws away necessary
> information needed to inverse F.T. back to the signal.
>
> so, in general, you are asking for something you cannot have because you
> lack sufficient information.
>
> r b-j

I dunno.  I agree that given an autocorrelation signal you can't
reconstruct _the_ signal, but I think the OP is just asking if there's a
way to construct _a_ signal.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
```
```On Mon, 25 Feb 2008 09:38:09 -0600, manolis wrote:

> Hello! My question relates to DSP theory: suppose that i have an
> autocorrelation sequence in a closed form formula. This autocorrelation
> sequence may correspond to many different signals. My question is how
> can i construct a signal whose autocorrelation is the given one? Anyone
> expert?

The Fourier transform of a signal's autocorrelation function is the
square of the magnitude of the Fourier transform of the signal.

So you should be able to take the Fourier transform of the
autocorrelation function, take it's square root, apply any old phase
shift you want to the result, and take the inverse Fourier transform of
that.  Then you will have a signal that will generate the desired
autocorrelation function, if not the original signal itself.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
```
```On 25 Feb, 16:38, "manolis" <el01...@mail.ntua.gr> wrote:
> Hello! My question relates to DSP theory: suppose that i have an
> autocorrelation sequence in a closed form formula. This autocorrelation
> sequence may correspond to many different signals. My question is how can
> i construct a signal whose autocorrelation is the given one?

You need to make a *choise* about what phase spectrum to use.
The usual way is to try and find a signal with the given
autocorrelation whch is also minimum phase. That's a key
ingrediense in the analysis of the standard AR models.

Rune
```