complex exponential embedded in white noise

Hi all,
       I have a doubt regarding complex exponential in white noise. Almost
all of the statistical DSP books state the following.
       The signal is of the form 
        x(n) = |A|exp(j(nw + phi)) + w(n)  and we have to estimate 'w' &
'A'
 
             Here 'A'(amplitude) is not a random variable nor is
'w'(freq)
        phi is mostly taken to be a random variable uniformly distributed
between -pi to +pi while w(n) is white gaussian noise with some variance.

               My doubt is :
1)why phi is always taken as a Uniform RV between limits -pi to +pi? Why
phi cannot be a gaussian RV like w(n). Further if phi is a URV between -pi
to +pi then the height of pdf is 1/2*pi ie all the phase values between -pi
to +pi  are equiprobable with probability 1/2*pi.Is this the case with real
world signals also or is it only a theoretical convenience? How can a phase
error  of -pi+/-delta (or +pi+/-delta) be having same probablility as
0+/-delta in a real world signal?

2)Is there any method to check the underlying pdf ( or at least estimate
it ) of 'phi' if i have some sample realisations of x(n) with me?

Regards,
Yogesh P.G.     
 

On Aug 20, 9:29 am, "ypg" <yogesh_ghar...@yahoo.com> wrote:
> Hi all,
>        I have a doubt regarding complex exponential in white noise. Almost
> all of the statistical DSP books state the following.
>        The signal is of the form
>         x(n) = |A|exp(j(nw + phi)) + w(n)  and we have to estimate 'w' &
> 'A'
>
>              Here 'A'(amplitude) is not a random variable nor is
> 'w'(freq)
>         phi is mostly taken to be a random variable uniformly distributed
> between -pi to +pi while w(n) is white gaussian noise with some variance.
>
>                My doubt is :
> 1)why phi is always taken as a Uniform RV between limits -pi to +pi? Why
> phi cannot be a gaussian RV like w(n). Further if phi is a URV between -pi
> to +pi then the height of pdf is 1/2*pi ie all the phase values between -pi
> to +pi  are equiprobable with probability 1/2*pi.Is this the case with real
> world signals also or is it only a theoretical convenience? How can a phase
> error  of -pi+/-delta (or +pi+/-delta) be having same probablility as
> 0+/-delta in a real world signal?

Typically that model is used for communication applications where
the transmitter and the receiver have different clocks.  So the phase
difference is pretty much uniformly distributed, because it's
dictated by the relative clocks and the propagation delay.

Your model seems to be more for a measurement or acquisition
system where there's a lot more synchronicity.

>
> 2)Is there any method to check the underlying pdf ( or at least estimate
> it ) of 'phi' if i have some sample realisations of x(n) with me?

Possible, assuming that w[n] is sufficiently small such that the
effect of noise on your estimate of \phi is small enough.

This is a common problem in modeling, which is that your estimation
system has to be sufficiently accurate that it does not dilute the
model.

On Wed, 20 Aug 2008 08:17:49 -0700 (PDT), julius <juliusk@gmail.com>
wrote:

>On Aug 20, 9:29 am, "ypg" <yogesh_ghar...@yahoo.com> wrote:
>> Hi all,
>>        I have a doubt regarding complex exponential in white noise. Almost
>> all of the statistical DSP books state the following.
>>        The signal is of the form
>>         x(n) = |A|exp(j(nw + phi)) + w(n)  and we have to estimate 'w' &
>> 'A'
>>
>>              Here 'A'(amplitude) is not a random variable nor is
>> 'w'(freq)
>>         phi is mostly taken to be a random variable uniformly distributed
>> between -pi to +pi while w(n) is white gaussian noise with some variance.
>>
>>                My doubt is :
>> 1)why phi is always taken as a Uniform RV between limits -pi to +pi? Why
>> phi cannot be a gaussian RV like w(n). Further if phi is a URV between -pi
>> to +pi then the height of pdf is 1/2*pi ie all the phase values between -pi
>> to +pi  are equiprobable with probability 1/2*pi.Is this the case with real
>> world signals also or is it only a theoretical convenience? How can a phase
>> error  of -pi+/-delta (or +pi+/-delta) be having same probablility as
>> 0+/-delta in a real world signal?
>
>Typically that model is used for communication applications where
>the transmitter and the receiver have different clocks.  So the phase
>difference is pretty much uniformly distributed, because it's
>dictated by the relative clocks and the propagation delay.
>
>Your model seems to be more for a measurement or acquisition
>system where there's a lot more synchronicity.

Even if the clocks are perfect and perfectly synchronized, the phase
will changed at the receiver as the distance between the transmitter
and receiver changes (in perfect LOS conditions).  Increasing the
range by a wavelength will traverse 2pi of carrier phase spatially.

Since range, in terms of wavelength, tends to be uniformly random, phi
is random with uniform distribution.

Then there's uncertainty in clocks, LO frequencies, multipath, cable
lengths, etc., etc., and it's just more clear that phase is completely
random with uniform distribution.

Another way to look at it is that if phi weren't uniformly random it'd
mean that some phase was more likely than the rest.   What phase would
it be and why is that one more likely?

Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.ericjacobsen.org

Blog: http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php

On Aug 20, 12:29 pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote:
> On Wed, 20 Aug 2008 08:17:49 -0700 (PDT), julius <juli...@gmail.com>
> wrote:
>
>
>
> >On Aug 20, 9:29 am, "ypg" <yogesh_ghar...@yahoo.com> wrote:
> >> Hi all,
> >>        I have a doubt regarding complex exponential in white noise. Almost
> >> all of the statistical DSP books state the following.
> >>        The signal is of the form
> >>         x(n) = |A|exp(j(nw + phi)) + w(n)  and we have to estimate 'w' &
> >> 'A'
>
> >>              Here 'A'(amplitude) is not a random variable nor is
> >> 'w'(freq)
> >>         phi is mostly taken to be a random variable uniformly distributed
> >> between -pi to +pi while w(n) is white gaussian noise with some variance.
>
> >>                My doubt is :
> >> 1)why phi is always taken as a Uniform RV between limits -pi to +pi? Why
> >> phi cannot be a gaussian RV like w(n). Further if phi is a URV between -pi
> >> to +pi then the height of pdf is 1/2*pi ie all the phase values between -pi
> >> to +pi  are equiprobable with probability 1/2*pi.Is this the case with real
> >> world signals also or is it only a theoretical convenience? How can a phase
> >> error  of -pi+/-delta (or +pi+/-delta) be having same probablility as
> >> 0+/-delta in a real world signal?
>
> >Typically that model is used for communication applications where
> >the transmitter and the receiver have different clocks.  So the phase
> >difference is pretty much uniformly distributed, because it's
> >dictated by the relative clocks and the propagation delay.
>
> >Your model seems to be more for a measurement or acquisition
> >system where there's a lot more synchronicity.
>
> Even if the clocks are perfect and perfectly synchronized, the phase
> will changed at the receiver as the distance between the transmitter
> and receiver changes (in perfect LOS conditions).  Increasing the
> range by a wavelength will traverse 2pi of carrier phase spatially.
>
> Since range, in terms of wavelength, tends to be uniformly random, phi
> is random with uniform distribution.
>
> Then there's uncertainty in clocks, LO frequencies, multipath, cable
> lengths, etc., etc., and it's just more clear that phase is completely
> random with uniform distribution.
>
> Another way to look at it is that if phi weren't uniformly random it'd
> mean that some phase was more likely than the rest.   What phase would
> it be and why is that one more likely?
>
> Eric Jacobsen
> Minister of Algorithms
> Abineau Communicationshttp://www.ericjacobsen.org
>
> Blog:http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php

That's right.

Another way to argue is that even if the phenomena causing the
phase mismatch were not uniformly distributed, the variance of this
phenomena is more than likely sufficient such that the remainder
after taking modulo 2\pi yields a uniform distribution.

Julius