# inverse of autocorrelation

Started by April 13, 2006
Greetings,

Is it possible to recover a complex and periodic signal from its
autocorellation function (of course the result will be a time shifted
version, but it's OK)? If so, what's the inverse transformation.

Thank you.


"Cagdas Ozgenc" <cagdas.ozgenc@gmail.com> writes:

> Greetings,
>
> Is it possible to recover a complex and periodic signal from its
> autocorellation function (of course the result will be a time shifted
> version, but it's OK)? If so, what's the inverse transformation.

In general, no. The autocorrelation function, or equivalent, the
power spectrum, lacks any phase information about the original
signal, thus it cannot be reconstructed unless you assume some
relationship between the magnitude and phase (e.g., constant phase,
minimum phase, etc.).
--
%  Randy Yates                  % "Rollin' and riding and slippin' and
%% Fuquay-Varina, NC            %  sliding, it's magic."
%%% 919-577-9882                %
%%%% <yates@ieee.org>           % 'Living' Thing', *A New World Record*, ELO

Randy Yates wrote:
> "Cagdas Ozgenc" <cagdas.ozgenc@gmail.com> writes:
>
> > Greetings,
> >
> > Is it possible to recover a complex and periodic signal from its
> > autocorellation function (of course the result will be a time shifted
> > version, but it's OK)? If so, what's the inverse transformation.
>
> In general, no. The autocorrelation function, or equivalent, the
> power spectrum, lacks any phase information about the original
> signal, thus it cannot be reconstructed unless you assume some
> relationship between the magnitude and phase (e.g., constant phase,
> minimum phase, etc.).

understand. When the original signal is complex, correlation of the
signal with its own shifted versions retains imaginary components on
shifts except 0 shift.

What am I missing here?

Thank you.


Cagdas Ozgenc wrote:
> Randy Yates wrote:
> > "Cagdas Ozgenc" <cagdas.ozgenc@gmail.com> writes:
> >
> > > Greetings,
> > >
> > > Is it possible to recover a complex and periodic signal from its
> > > autocorellation function (of course the result will be a time shifted
> > > version, but it's OK)? If so, what's the inverse transformation.
> >
> > In general, no. The autocorrelation function, or equivalent, the
> > power spectrum, lacks any phase information about the original
> > signal, thus it cannot be reconstructed unless you assume some
> > relationship between the magnitude and phase (e.g., constant phase,
> > minimum phase, etc.).
>
> understand. When the original signal is complex, correlation of the
> signal with its own shifted versions retains imaginary components on
> shifts except 0 shift.
>
> What am I missing here?

I'm not sure, but you can confirm it's true by considering that the
Fourier transform of the autocorrelation function is the PSD (Power
Spectral Density).  PSD = |X(f)|^2, i.e. all phase information is
destroyed.

--
Oli


On 2006-04-13 10:20:22 -0300, "Cagdas Ozgenc" <cagdas.ozgenc@gmail.com> said:

> Greetings,
>
> Is it possible to recover a complex and periodic signal from its
> autocorellation function (of course the result will be a time shifted
> version, but it's OK)? If so, what's the inverse transformation.
>
> Thank you.

one dimension you get many answers. View things in terms of polynomials
in complex variables and your problem is easier to see. The autocovariance
will correspond to a polynomial with many conjugate pairs of roots. The
many answers are the result of choosing on root from each pair. This
usually goes by the name of the "zero flipping problem". So if you
knew which root to pick in each pair you would be fine, but knowing
requires more outside information than you have given.

In higher dimensions things are easier. Blame the fundemental theorem
of algebra for the problem in one dimension as it does not apply in
higher dimensions. You still have pairs of factors, just many fewer
pairs.


"Cagdas Ozgenc" <cagdas.ozgenc@gmail.com> writes:

> Randy Yates wrote:
>> "Cagdas Ozgenc" <cagdas.ozgenc@gmail.com> writes:
>>
>> > Greetings,
>> >
>> > Is it possible to recover a complex and periodic signal from its
>> > autocorellation function (of course the result will be a time shifted
>> > version, but it's OK)? If so, what's the inverse transformation.
>>
>> In general, no. The autocorrelation function, or equivalent, the
>> power spectrum, lacks any phase information about the original
>> signal, thus it cannot be reconstructed unless you assume some
>> relationship between the magnitude and phase (e.g., constant phase,
>> minimum phase, etc.).
>
> understand. When the original signal is complex, correlation of the
> signal with its own shifted versions retains imaginary components on
> shifts except 0 shift.

Yes, but \phi_{zz}(\tau) = \phi_{zz}^*(-\tau), i.e., the
autocorrelation of a complex signal (or, indeed, any signal) is
Hermitian symmetric, which means that the power spectrum is real
whether the original signal is complex or real, thus it lacks phase.
--
%  Randy Yates                  % "Midnight, on the water...
%% Fuquay-Varina, NC            %  I saw...  the ocean's daughter."
%%% 919-577-9882                % 'Can't Get It Out Of My Head'
%%%% <yates@ieee.org>           % *El Dorado*, Electric Light Orchestra

> Ask google about "spectral factorization". The problem is that in
> one dimension you get many answers. View things in terms of polynomials
> in complex variables and your problem is easier to see. The autocovariance
> will correspond to a polynomial with many conjugate pairs of roots. The
> many answers are the result of choosing on root from each pair. This
> usually goes by the name of the "zero flipping problem". So if you
> knew which root to pick in each pair you would be fine, but knowing
> requires more outside information than you have given.
>
> In higher dimensions things are easier. Blame the fundemental theorem
> of algebra for the problem in one dimension as it does not apply in
> higher dimensions. You still have pairs of factors, just many fewer
> pairs.

I got it. I experienced a similar problem before where you can easily
compute powers of complex numbers but cannot recover them by taking
roots since there are many roots to a complex number.

I read about another many to one transformation but this time you can
recover the signal.

It is called Knox-Thompson transform. However I didn't understand what
they mean by deltaOmega parameter in this transformation.

Do you know anything about it?

Thank you