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Re: maximum entropy method for deconvolution
*To*: mathgroup at smc.vnet.net
*Subject*: [mg75469] Re: maximum entropy method for deconvolution
*From*: dantimatter <dantimatter at gmail.com>
*Date*: Wed, 2 May 2007 03:49:06 -0400 (EDT)
*References*: <200704281000.GAA09123@smc.vnet.net><f146p6$mcq$1@smc.vnet.net>
Thanks to all for the help, and a special thanks to Guillermo Sanchez
for the Modeling and Simulation notebook. I'm finding that my answer
is still 'unpleasant'. Could it be that the convolution of a function
with a step-function is something that simply cannot be deconvolved?
That perhaps there's information lost in the convolution and it could
never be recovered?
Dan
On Apr 30, 2:44 am, "turnback" <turnb... at bluebottle.com> wrote:
> Hi,
>
> Your problem is a deconvolution problem. It is difficult in general. But
> since your function p is simply a rectangle function, things may not be that
> bad.
>
> Let me define t0:=33.6, g(t):=G, f(t):=F
>
> Then according to the definition of convolution, we can get:
>
> g(t)=Integrate[f(s), {s, t-t0, t}]
>
> and for any small dt, we can get:
>
> ==> g(t+dt)=Integrate[f(s), {s, t+dt-t0, t+dt}]
>
> ==> g(t+dt)-g(t)
> =Integrate[f(s), {s, t+dt-t0, t+dt}]-Integrate[f(s), {s, t-t0, t}]
> =Integrate[f(s), {s, t, t+dt}]-Integrate[f(s), {s, t-t0, t-t0+dt}]
> =Integrate[f(s)-f(s-t0), {s, t, t+dt}]
>
> define l(t):=f(t)-f(t-t0), then
>
> g(t+dt)-g(t)=Integrate[l(s), {s,t,t+dt}]
> ==> Limit[(g(t+dt)-g(t))/dt,dt->0] = Limit[(Integrate[l(s),
> {s,t,t+dt}])/dt,dt->0]
>
> Suppose g(t) is differentiable, then
>
> ==> g'(t)=l(t)=f(t)-f(t-t0)
>
> then performing Fourier transform:
>
> ==> I*w*G(w)=(1-Exp[-I*w*t0])*F(w)
>
> ==> F(w)=G(w)*I*w/(1-Exp[-I*w*t0])
>
> With inverse transform, you get f(t).
>
> By the way, According to my knowledge, p's Fourier transform is a Sinc
> function. When you claim function p has a lot of zeros in the frequency
> domain, you may implement the transform by using Discrete Fourier Transform
> and you pick up a specific sampling rate. If real time performance is not
> your concern, try to increase your sampling rate a lot. You should get
> correct result with your original approach.
>
> After all, I just realise that instead of writing
> p=UnitStep[t]*UnitStep[33.6-t], you can write it as:
> p[t]=UnitStep[t]-UnitStep[t-33.6]. You may find much easier proof.
>
> hui.
>
>
>
> ----- Original Message -----
> From: "dantimatter" <dantimat... at gmail.com>
> To: <mathgr... at smc.vnet.net>
> Sent: Saturday, April 28, 2007 6:00 AM
> Subject: maximum entropy method for deconvolution
>
> > hello all,
>
> > first off, many thanks to 'Roman' et al for all the previous help with
> > my inversion problem.
>
> > i have a convolution function G which is the convolution of F and p (G
> > = F**p). i know G and i know p, and i'd like to extract F. i can do
> > this by taking the Fourier transform of G, dividing by the Fourier
> > transform of p, and inverting the result to get F. the problem is
> > that p is a step function (p = UnitStep[t]*UnitStep[33.6-t]) which has
> > a lot of zeros in frequency space, and thus it is difficult to get at
> > F via inversion. Mathematica is happily doing the inversion but the
> > results are very noticeably wrong.
>
> > i understand from my conversations with some of you and much time
> > spent in the library that this is in general a difficult problem, but
> > there are some methods that are known to make this type of problem
> > tractable, such as the maximum entropy method (MEM) for inversion. is
> > anyone aware of an implementation of a MEM algorithm in Mathematica?
> > i have read Numerical Recipes a couple of times and i am unable to get
> > my head around the relevant chapter. If there isn't a Mathematica
> > implementation, perhaps someone could offer some advice on where else
> > to look? if it exists, a "for dummies" type book with step-by-step
> > instructions would be the best resource for me...
>
> > cheers,
> > dan
>
> ----------------------------------------------------------------------
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