Phase MLE of single sinusoid

Hi,

In the MLE estimation for the phase angle of a single sinusoid
embedded in WGN, different cases arise depending on what signal
parameters you assume are known. When the frequency is assumed known,
then there are the two cases that are typically considered in the
literature for estimating the phase of the signal amplitude being
known and not being known.

I am curious how significant/practically important the case of phase
estimation when the signal amplitude is (assumed) known really is. It
appears to be of some theoretical interest (Kay in Signal Processing
devotes about a page to this case and it is handled in the literature
- see for example Hing-Cheung So, IEICE Trans Fundamentals, 2006), but
I am wondering how many practical problems it really applies to. Since
I am a mathematician and not an engineer, I am wondering if anyone has
some examples where one encounters this situation. My intuition tells
me that the case where the amplitude is assumed "known" is most
typically the case where in reality the amplitude is not known, but
the SNR is so high that the amplitude can be estimated well.

Thank you very much for any information you can provide me,

Matt Brenneman

On 27 Feb, 18:32, "junoexpress" <MTBrenne...@gmail.com> wrote:
> Hi,
>
> In the MLE estimation for the phase angle of a single sinusoid
> embedded in WGN, different cases arise depending on what signal
> parameters you assume are known. When the frequency is assumed known,
> then there are the two cases that are typically considered in the
> literature for estimating the phase of the signal amplitude being
> known and not being known.
>
> I am curious how significant/practically important the case of phase
> estimation when the signal amplitude is (assumed) known really is. It
> appears to be of some theoretical interest (Kay in Signal Processing
> devotes about a page to this case and it is handled in the literature
> - see for example Hing-Cheung So, IEICE Trans Fundamentals, 2006), but
> I am wondering how many practical problems it really applies to. Since
> I am a mathematician and not an engineer, I am wondering if anyone has
> some examples where one encounters this situation. My intuition tells
> me that the case where the amplitude is assumed "known" is most
> typically the case where in reality the amplitude is not known, but
> the SNR is so high that the amplitude can be estimated well.

I know of exatly one application where the phase of a sinusoidal
is important. Once upon a time I worked with some seismic signals
where the horizontl and vertical particle velocities were 90 degrees
out of phase, some times -90 degrees, sometimes +90 degrees.
I attempted to measure the relative phase between the signal
components when the frequency estimates were noisy and the
amplitude was not known. I assume I was some 4 to 6 months
away from cracking that one, when I ran out of $$$...

Again, that's the only application I know of where the phase
of the sinusoidals is important.

Rune

On Feb 27, 1:22 pm, "Rune Allnor" <all...@tele.ntnu.no> wrote:
> On 27 Feb, 18:32, "junoexpress" <MTBrenne...@gmail.com> wrote:
>
>
>
>
>
>
>
> > Hi,
>
> > In the MLE estimation for the phase angle of a single sinusoid
> > embedded in WGN, different cases arise depending on what signal
> > parameters you assume are known. When the frequency is assumed known,
> > then there are the two cases that are typically considered in the
> > literature for estimating the phase of the signal amplitude being
> > known and not being known.
>
> > I am curious how significant/practically important the case of phase
> > estimation when the signal amplitude is (assumed) known really is. It
> > appears to be of some theoretical interest (Kay in Signal Processing
> > devotes about a page to this case and it is handled in the literature
> > - see for example Hing-Cheung So, IEICE Trans Fundamentals, 2006), but
> > I am wondering how many practical problems it really applies to. Since
> > I am a mathematician and not an engineer, I am wondering if anyone has
> > some examples where one encounters this situation. My intuition tells
> > me that the case where the amplitude is assumed "known" is most
> > typically the case where in reality the amplitude is not known, but
> > the SNR is so high that the amplitude can be estimated well.
>
> I know of exatly one application where the phase of a sinusoidal
> is important. Once upon a time I worked with some seismic signals
> where the horizontl and vertical particle velocities were 90 degrees
> out of phase, some times -90 degrees, sometimes +90 degrees.
> I attempted to measure the relative phase between the signal
> components when the frequency estimates were noisy and the
> amplitude was not known. I assume I was some 4 to 6 months
> away from cracking that one, when I ran out of $$$...
>
> Again, that's the only application I know of where the phase
> of the sinusoidals is important.
>
> Rune

Hi Rune,

Thanks for the reply. In this case, finding a problem involving phase
estimation of a sinusoid is not that difficult for me. For example,
the angle of arrival calculations one does with signals obtained from
an antenna array is one such example that readily comes to mind, and
in the lit on this problem "phase angle recovery" is often cited.

But, my problem is a bit more than that: it is estimating the phase of
a sinusoid where both the frequency and amplitude are assumed to be
known. (Actually, the problem can be extended to multiple signals if
they are at different freqs, I believe).

As your answer suggests though, perhaps such a real-life problem is
not easy to come by, and this problem is of mostly theoretical
interest. I posted to this group in fact, for exactly this reason: I
figured that I could perhaps get a consensus on whether or not this
problem is of any real practical significance.

Matt

Not sure why you might possibly need a phase of a single sinusoid ?...

But some people might need to estimate a phase difference between two
sinusoids of the same frequency, say, between an original sine wave
and a delayed and amplitude modulated version of it (plus some noise,
of course).
Sounds like a very useful practical task to me.
Or, a more general formulation: how to estimate a phase difference
between two periodic signals (not necessarily sine waves) with same
(unknown) period and same (but unknown) harmonic structure, but
different (unknown) amplitudes plus some noise added to each signal ?

I see nothing impossible here: a robust and very accurate solution can
be achieved without breaking the bank :-)
Still, it's gonna cost you some $$$$...


On Feb 27, 12:32 pm, "junoexpress" <MTBrenne...@gmail.com> wrote:
> Hi,
>
> In the MLE estimation for the phase angle of a single sinusoid
> embedded in WGN, different cases arise depending on what signal
> parameters you assume are known. When the frequency is assumed known,
> then there are the two cases that are typically considered in the
> literature for estimating the phase of the signal amplitude being
> known and not being known.
>
> I am curious how significant/practically important the case of phase
> estimation when the signal amplitude is (assumed) known really is. It
> appears to be of some theoretical interest (Kay in Signal Processing
> devotes about a page to this case and it is handled in the literature
> - see for example Hing-Cheung So, IEICE Trans Fundamentals, 2006), but
> I am wondering how many practical problems it really applies to. Since
> I am a mathematician and not an engineer, I am wondering if anyone has
> some examples where one encounters this situation. My intuition tells
> me that the case where the amplitude is assumed "known" is most
> typically the case where in reality the amplitude is not known, but
> the SNR is so high that the amplitude can be estimated well.
>
> Thank you very much for any information you can provide me,
>
> Matt Brenneman

On Feb 27, 9:32 am, "junoexpress" <MTBrenne...@gmail.com> wrote:
> Hi,
>
> In the MLE estimation for the phase angle of a single sinusoid
> embedded in WGN, different cases arise depending on what signal
> parameters you assume are known. When the frequency is assumed known,
> then there are the two cases that are typically considered in the
> literature for estimating the phase of the signal amplitude being
> known and not being known.
>
> I am curious how significant/practically important the case of phase
> estimation when the signal amplitude is (assumed) known really is. It
> appears to be of some theoretical interest (Kay in Signal Processing
> devotes about a page to this case and it is handled in the literature
> - see for example Hing-Cheung So, IEICE Trans Fundamentals, 2006), but
> I am wondering how many practical problems it really applies to. Since
> I am a mathematician and not an engineer, I am wondering if anyone has
> some examples where one encounters this situation. My intuition tells
> me that the case where the amplitude is assumed "known" is most
> typically the case where in reality the amplitude is not known, but
> the SNR is so high that the amplitude can be estimated well.

One hypothetical example is where an AGC prescales your
signal amplitude to a known value, and with equal or higher
accuracy than your estimation circuit is capable. (e.g. pre-
calibrated high quality low noise analog AGC followed by
a relatively low bit-resolution ADC before extracting phase).


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M

Matt wrote:

> I am curious how significant/practically important the case of phase
> estimation when the signal amplitude is (assumed) known really is.

PSK decoding [1] seems to be about phase estimation. I'm no modulation
or sinusoidal estimation expert, so I don't know what algorithms are
actually used.

Regards,
Andor

[1] http://en.wikipedia.org/wiki/Phase-shift_keying

hi all,
      whatever Matt Brenneman (juno express) has asked..let me put
this way..

in upcoming Wimax base stations, calibration plays a biggest
role..when you are going for advanced options like MIMO or AAS. in
that case..using OFDM (a plurality of sinusoidal signals) we need to
estimate the phase infliced by each antenna at EACH FREQUENCY along
the COMPLETE OFDM signal (WHICH CORRESONDS TO ESTIMATION OF PHASE FOR
INDIVIDUAL SINUSOIDAL SIGNAL) both in uplink and downlink case..

As you told you are a mathematician..As an R&D engineer..i could think
in this way of estimation..of phase for sinusoidal signalss

but more relevant to your question are cases of GSM and GPRS
systems..where single carrier communication based BASE STATION
calibration are carried out ..which indeed have knowledge of both
amplitude and  frequency but..we need to estimate phase..inspite of
nonlinearities added by RF Components

ParticleFilter Reddy

On Feb 28, 12:26 pm, "Andor" <andor.bari...@gmail.com> wrote:
> Matt wrote:
> > I am curious how significant/practically important the case of phase
> > estimation when the signal amplitude is (assumed) known really is.
>
> PSK decoding [1] seems to be about phase estimation. I'm no modulation
> or sinusoidal estimation expert, so I don't know what algorithms are
> actually used.
>
> Regards,
> Andor
>
> [1]http://en.wikipedia.org/wiki/Phase-shift_keying

On 27 Feb, 20:01, "junoexpress" <MTBrenne...@gmail.com> wrote:
> On Feb 27, 1:22 pm, "Rune Allnor" <all...@tele.ntnu.no> wrote:
>
>
>
>
>
> > On 27 Feb, 18:32, "junoexpress" <MTBrenne...@gmail.com> wrote:
>
> > > Hi,
>
> > > In the MLE estimation for the phase angle of a single sinusoid
> > > embedded in WGN, different cases arise depending on what signal
> > > parameters you assume are known. When the frequency is assumed known,
> > > then there are the two cases that are typically considered in the
> > > literature for estimating the phase of the signal amplitude being
> > > known and not being known.
>
> > > I am curious how significant/practically important the case of phase
> > > estimation when the signal amplitude is (assumed) known really is. It
> > > appears to be of some theoretical interest (Kay in Signal Processing
> > > devotes about a page to this case and it is handled in the literature
> > > - see for example Hing-Cheung So, IEICE Trans Fundamentals, 2006), but
> > > I am wondering how many practical problems it really applies to. Since
> > > I am a mathematician and not an engineer, I am wondering if anyone has
> > > some examples where one encounters this situation. My intuition tells
> > > me that the case where the amplitude is assumed "known" is most
> > > typically the case where in reality the amplitude is not known, but
> > > the SNR is so high that the amplitude can be estimated well.
>
> > I know of exatly one application where the phase of a sinusoidal
> > is important. Once upon a time I worked with some seismic signals
> > where the horizontl and vertical particle velocities were 90 degrees
> > out of phase, some times -90 degrees, sometimes +90 degrees.
> > I attempted to measure the relative phase between the signal
> > components when the frequency estimates were noisy and the
> > amplitude was not known. I assume I was some 4 to 6 months
> > away from cracking that one, when I ran out of $$$...
>
> > Again, that's the only application I know of where the phase
> > of the sinusoidals is important.
>
> > Rune
>
> Hi Rune,
>
> Thanks for the reply. In this case, finding a problem involving phase
> estimation of a sinusoid is not that difficult for me. For example,
> the angle of arrival calculations one does with signals obtained from
> an antenna array is one such example that readily comes to mind, and
> in the lit on this problem "phase angle recovery" is often cited.

DoA estimation is similar to frequency estimation. The wavenumber
plays the same part in those equations as frequency does in
time-domain problems.

> But, my problem is a bit more than that: it is estimating the phase of
> a sinusoid where both the frequency and amplitude are assumed to be
> known. (Actually, the problem can be extended to multiple signals if
> they are at different freqs, I believe).

I have an instictive dislike for problem formulations that contain
the phrase similar to "[some parameters] can be assumed to
be known."

> As your answer suggests though, perhaps such a real-life problem is
> not easy to come by, and this problem is of mostly theoretical
> interest. I posted to this group in fact, for exactly this reason: I
> figured that I could perhaps get a consensus on whether or not this
> problem is of any real practical significance.

It depends on he exact formulation of the problem. There is such a
thing as a Phase Modulation scheme in communucations, where
phase recovery is  a significant point. This is a completely
different
animal than phase estimation of sinusoidals, though.

Rune

On 27 Feb, 21:19, "fizteh89" <d...@soundmathtech.com> wrote:
> Not sure why you might possibly need a phase of a single sinusoid ?...
>
> But some people might need to estimate a phase difference between two
> sinusoids of the same frequency, say, between an original sine wave
> and a delayed and amplitude modulated version of it (plus some noise,
> of course).
> Sounds like a very useful practical task to me.
> Or, a more general formulation: how to estimate a phase difference
> between two periodic signals (not necessarily sine waves) with same
> (unknown) period and same (but unknown) harmonic structure, but
> different (unknown) amplitudes plus some noise added to each signal ?
>
> I see nothing impossible here: a robust and very accurate solution can
> be achieved without breaking the bank :-)

Sounds like cross correlation with phase unwrapping, to me...?

> Still, it's gonna cost you some $$$$...

Bendat & Piersol's "Random Data" sells for $110 on www.amazon.com.

Rune

>Hi,
>
>In the MLE estimation for the phase angle of a single sinusoid
>embedded in WGN, different cases arise depending on what signal
>parameters you assume are known. When the frequency is assumed known,
>then there are the two cases that are typically considered in the
>literature for estimating the phase of the signal amplitude being
>known and not being known.
>
>I am curious how significant/practically important the case of phase
>estimation when the signal amplitude is (assumed) known really is. It
>appears to be of some theoretical interest (Kay in Signal Processing
>devotes about a page to this case and it is handled in the literature
>- see for example Hing-Cheung So, IEICE Trans Fundamentals, 2006), but
>I am wondering how many practical problems it really applies to. Since
>I am a mathematician and not an engineer, I am wondering if anyone has
>some examples where one encounters this situation. My intuition tells
>me that the case where the amplitude is assumed "known" is most
>typically the case where in reality the amplitude is not known, but
>the SNR is so high that the amplitude can be estimated well.
>
>Thank you very much for any information you can provide me,
>
>Matt Brenneman
>
>


I worked on a noise cancellation study about 2 years ago.  I had baseband
PCM communications signals which had a significant amount of sinusoidal
noise; in the worse case, enough to reduce the SNR about 14 dB.  I
re-modulated the hard symbols at the output of a channel equalizer
(convolving these symbols with a model of the channel to get an estimate
of my equalizer inputs) and then subtracted a model of this periodic noise
from the input of the equalizer.  This required me to track the phase,
frequency and amplitude of the N larges tones after each block was
equalized.  I went with this method because it did not invlove any
convolution with the equalizer input (such as a COMB filter would
require), which would lengthen my channel considerably.