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From: rge11x on 15 Jan 2010 20:31
Hello

I have run into the following real life problem.

I am measuring the complex exponentials exp(i.w_k.a.t) for some fixed
and given 'a' and given set of values 'w_k' and all 't', (actually we
are measuring cos(w_k.a.t) and sin(w_k.a.t) simultaneously) in
independent noise whose standard deviation is <0.01, so it is
reasonably clean, effectively noiseless. The frequencies are not
harmonically related. What I need is the magnitude of the sum of these
phasors:

b= |exp(i.w_1.a.t) + exp(i.w_2.a.t) + exp(i.w_3.a.t) + .....|

In the actual signal generation and processing context for reasons not
important here it is difficult to generate exp(i.w.a.t), it would be
very easy to measure exp(i.w.t). There comes the obvious suggestion to
take the a-th exponent after measurement but unfortunately while 'a'
is given it is some real not necessarily integer number anywhere in
[-4, 4], thus taking the exponent brings up the problem of which
branch of the potentially infinitude of branches? If I select one
branch from say the first exp(i.w_1.t)^a, how can I add the rest of
them to the first to get 'b' ? Is there a systematic way to get the
right result? Can this be done when the measurement is noiseless, and
assume that 'a' is rational? I have at least 20, or so, frequencies
(w_k), so a direct search through all the numerator values of 'a' is
not likely to be practical if I want to restrict the phase error to be
under a few degrees.

Thanks
From: David Bernier on 15 Jan 2010 23:05
rge11x wrote:
> Hello
>
> I have run into the following real life problem.
>
> I am measuring the complex exponentials exp(i.w_k.a.t) for some fixed
> and given 'a' and given set of values 'w_k' and all 't', (actually we
> are measuring cos(w_k.a.t) and sin(w_k.a.t) simultaneously) in
> independent noise whose standard deviation is <0.01, so it is
> reasonably clean, effectively noiseless. The frequencies are not
> harmonically related. What I need is the magnitude of the sum of these
> phasors:
>
> b= |exp(i.w_1.a.t) + exp(i.w_2.a.t) + exp(i.w_3.a.t) + .....|
>
> In the actual signal generation and processing context for reasons not
> important here it is difficult to generate exp(i.w.a.t), it would be
> very easy to measure exp(i.w.t). There comes the obvious suggestion to
> take the a-th exponent after measurement but unfortunately while 'a'
> is given it is some real not necessarily integer number anywhere in
> [-4, 4], thus taking the exponent brings up the problem of which
> branch of the potentially infinitude of branches? If I select one
> branch from say the first exp(i.w_1.t)^a, how can I add the rest of
> them to the first to get 'b' ? Is there a systematic way to get the
> right result? Can this be done when the measurement is noiseless, and
> assume that 'a' is rational? I have at least 20, or so, frequencies
> (w_k), so a direct search through all the numerator values of 'a' is
> not likely to be practical if I want to restrict the phase error to be
> under a few degrees.

Computing b as a function of time t can be done with
exp(i.w.a.t) = cos(w.a.t) + i*sin(w.a.t), then adding up
the sines and cosines for the various w_k, and then taking the
modulus. Maybe this is hard to do in real-time.

What range of frequencies are there corresponding to the various
w_k ?

David

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