# Any statistical information in the phase of a signal?

Started by February 16, 2012
```
In basic random signal modeling, in general we toss away any information
that could be assertained from the phase of a signal. Maybe we compute the
autocorrelation and work with the PSD of a signal. The phase is forever
lost and assumed to be random... but we assume the amplitude squared (hence
the amplitude) of the F.T. on average can tell us something. (assuming
stationarity).

Ok, heres a question...

Could we say anything about the converse?  Lets say we discarded the
amplitude information of the F.T. of a signal and assumed it was random,
but we assumed we knew the phase of the F.T of the signal on average. Could
we get any statistical information out of that?
```
```On Thu, 16 Feb 2012 10:29:21 -0600, westocl wrote:

> In basic random signal modeling, in general we toss away any information
> that could be assertained from the phase of a signal. Maybe we compute
> the autocorrelation and work with the PSD of a signal. The phase is
> forever lost and assumed to be random... but we assume the amplitude
> squared (hence the amplitude) of the F.T. on average can tell us
> something. (assuming stationarity).
>
> Ok, heres a question...
>
> Could we say anything about the converse?  Lets say we discarded the
> amplitude information of the F.T. of a signal and assumed it was random,
> but we assumed we knew the phase of the F.T of the signal on average.
> Could we get any statistical information out of that?

A phase, measured precisely at an unknown time, means nothing.  Since
stationary random signals happen at unknown times, phase means nothing.

_Relative_ phase, between two signals or between one time and another in
a given signal, may mean something.

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
```
```On Feb 16, 11:29&#2013266080;am, "westocl" <cweston_@n_o_s_p_a_m.hotmail.com>
wrote:

> but we assumed we knew the phase of the F.T of the signal on average. Could
> we get any statistical information out of that?

What is the definition of the phase of the Fourier transform?
What is the definition of the average phase of the Fourier
transform?

What can you deduce from the knowledge that the
Fourier transform \$X(f)\$ of a signal \$x(t)\$  is real-valued
(which would be one possible definition of the phase
being 0 for all f; a much stronger condition than the
*average* phase being 0)?

For many signals, the phase could be interpreted
as an odd function of frequency f, and so the *average*
phase might be thought of as being 0. What signals
have the property that the *average* phase is 0?

What if the phase was a linear function of frequency?
What kind of signals have linear phase?

```
```>On Feb 16, 11:29=A0am, "westocl" <cweston_@n_o_s_p_a_m.hotmail.com>
>wrote:
>
>> but we assumed we knew the phase of the F.T of the signal on average.
Cou=
>ld
>> we get any statistical information out of that?
>
>What is the definition of the phase of the Fourier transform?
>What is the definition of the average phase of the Fourier
>transform?
>
>What can you deduce from the knowledge that the
>Fourier transform \$X(f)\$ of a signal \$x(t)\$  is real-valued
>(which would be one possible definition of the phase
>being 0 for all f; a much stronger condition than the
>*average* phase being 0)?
>
>For many signals, the phase could be interpreted
>as an odd function of frequency f, and so the *average*
>phase might be thought of as being 0. What signals
>have the property that the *average* phase is 0?
>
>What if the phase was a linear function of frequency?
>What kind of signals have linear phase?
>

Ok, sorry of the vaugness.
Let me ask a different way. Say we want to synthesize a sample domain
signal x[n] from the fourier domain X[k]. We would need the magnitude and
angle of every frequency bin (assuming conjugate symmetry) take the ifft
and we get the real signal x[n].

In random modeling, we could do somthigng like, assume the phase of X[k] to
be zero, or any random value, but assume we know the magnitude of the
bins.. take the ifft and get a signal x[n]. The only thing we really know

In my question starting from the X[k] domain... say if I made the
amplitudes of the frequency bins random, but made the phases of the
frequency bins known... would we know anything statistical about the x[n]?

```
```stretching the definition of "random" a bit:
My random signal happens to be randomly frequency/phase modulated (for
example FM radio, GMSK modulation as in GSM or the like)
My "phase" is the absolute phase of a complex-valued baseband equivalent
front-end when tuned to the right station.
From the phase, I can demodulate the signal, reconstruct the original
signal and of course then do just about any statistical analysis I like.

So this is example where the answer is clearly "yes".
```
```westocl <cweston_@n_o_s_p_a_m.hotmail.com> wrote:
>>On Feb 16, 11:29=A0am, "westocl" <cweston_@n_o_s_p_a_m.hotmail.com>

(snip)
> Ok, sorry of the vaugness.
> Let me ask a different way. Say we want to synthesize a sample domain
> signal x[n] from the fourier domain X[k]. We would need the
> magnitude and angle of every frequency bin (assuming conjugate
> symmetry) take the ifft and we get the real signal x[n].

> In random modeling, we could do somthigng like, assume the
> phase of X[k] to be zero, or any random value, but assume we
> know the magnitude of the bins.. take the ifft and get
> is it's statistical properties.

If you don't have any phase information, and need to reconstruct
the signal anyway, it is probably best to supply random phase.

There was a discussion some time ago, about the fact that
both a delta impulse and white noise have the same frequency
spectrum. The difference is in phase.

> In my question starting from the X[k] domain... say if I made the
> amplitudes of the frequency bins random, but made the phases of the
> frequency bins known... would we know anything statistical about
> the x[n]?

As I understand it, the ear is relatively insensitive to the
phase of signals. That is, two sounds with the same frequency
spectrum but different phase will sound, to us, fairly similar.

On the other hand, if you look on an oscilloscope they might
look completely different.

Phase is used in localizing sound sources, through a separate
neural pathway. I was told by someone who studies aural
signals that for low frequencies the nerve impulse comes out
at the peak (I believe positive pressure, but I might have
forgotten) of the sine, and for higher frequencies at the
top of every Nth (N increases with frequency) sine.

-- glen
```
```On Feb 16, 8:29&#2013266080;am, "westocl" <cweston_@n_o_s_p_a_m.hotmail.com>
wrote:
> ... In basic random signal modeling, in general we toss away any information
> that could be assertained from the phase of a signal. Maybe we compute the
> autocorrelation and work with the PSD of a signal. The phase is forever
> lost and assumed to be random...

Why make such a baseless assumption?

>
> Ok, heres a question...
>
> Could we say anything about the converse? &#2013266080;Lets say we discarded the
> amplitude information of the F.T. of a signal and assumed it was random,
> but we assumed we knew the phase of the F.T of the signal on average. Could
> we get any statistical information out of that?

Reconstruction from phase is an old question. How much you can do
depends on how much information you have and what your goals are. For
an example considering phase reconstruction see:
Title: 	Signal reconstruction from phase or magnitude
Author: 	Hayes, Monson H
Department: 	Massachusetts Institute of Technology. Dept. of
Electrical Engineering and Computer Science.
Publisher: 	Massachusetts Institute of Technology
Issue Date: 	1981
Description: 	Thesis (Sc.D.)--Massachusetts Institute of Technology,
Dept. of Electrical Engineering and Computer Science, 1981.

http://dspace.mit.edu/handle/1721.1/44246

Dale B. Dalrymple
```
```On Feb 17, 7:29&#2013266080;am, "mnentwig"
<markus.nentwig@n_o_s_p_a_m.renesasmobile.com> wrote:
> stretching the definition of "random" a bit:
> My random signal happens to be randomly frequency/phase modulated (for
> example FM radio, GMSK modulation as in GSM or the like)
> My "phase" is the absolute phase of a complex-valued baseband equivalent
> front-end when tuned to the right station.
> From the phase, I can demodulate the signal, reconstruct the original
> signal and of course then do just about any statistical analysis I like.
>
> So this is example where the answer is clearly "yes".

In FM the amplitude information is considered constant in the carrier
and the information is held in the phase with respect to
a centre carrier frequency. So if you can measure phase you can work
out instantaneous frequency by differentiating and this gives you your
amplitude
back.

Hardy
```