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Thread Subject:
Signal Reconstruction

Subject: Signal Reconstruction

From: kk KKsingh

Date: 4 Mar, 2010 08:43:06

Message: 1 of 12

Is there any code available which i can study ! where signal is reconstructed in frequency domain using Least sqaure...Like one of file submisson was Fast gridding method.....they calculate the frequency domain spectrum or i should say that they calculated A^(H)WY (which is dft with weights) now i want to apply least square on this ..fr reconstruction so that i can apply ifft and get back my signal..

Any help

kk

Subject: Signal Reconstruction

From: Rune Allnor

Date: 4 Mar, 2010 10:12:20

Message: 2 of 12

On 4 Mar, 09:43, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> Is there any code available which i can study ! where signal is reconstructed in frequency domain using Least sqaure...Like one of file submisson was Fast gridding method.....they calculate the frequency domain spectrum or i should say that they calculated A^(H)WY (which is dft with weights) now i want to apply least square on this ..fr reconstruction so that i can apply ifft and get back my signal..

If you want your oiginal signal back, don't do anytrhing to it
in frequency domain:

y = ifft(fft(x));
max(abs(x-y)) == eps;

if one disregards numerical round-off errors.

Do anything at all to your signal in frequency domain, and you
will not get back the original signal. You might get back *a*
time-domain signal, but it will be a different signal than
the one you started out with.

Rune

Subject: Signal Reconstruction

From: kk KKsingh

Date: 4 Mar, 2010 10:24:04

Message: 3 of 12


But the files like
http://www.mathworks.com/matlabcentral/fileexchange/25135-nufft-nfft-usfft

calculates the approximate fourier spectrum now key to come back in the time domain is to do least sqaure reconstruction of this spectrum and get the corrected spectra and finally...fft and we r back in time domain on regular grid

For example purpose i can take dft for irregular samples which will give us approximate spectrum.........the files above can just replace the dft but that is not a issue ! issue is how to get the regular grid time domain signal back from dft spectrum


kk


Rune Allnor <allnor@tele.ntnu.no> wrote in message <9bc78e6f-443c-4b3c-979e-25ae8bba5745@g11g2000yqe.googlegroups.com>...
> On 4 Mar, 09:43, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> > Is there any code available which i can study ! where signal is reconstructed in frequency domain using Least sqaure...Like one of file submisson was Fast gridding method.....they calculate the frequency domain spectrum or i should say that they calculated A^(H)WY (which is dft with weights) now i want to apply least square on this ..fr reconstruction so that i can apply ifft and get back my signal..
>
> If you want your oiginal signal back, don't do anytrhing to it
> in frequency domain:
>
> y = ifft(fft(x));
> max(abs(x-y)) == eps;
>
> if one disregards numerical round-off errors.
>
> Do anything at all to your signal in frequency domain, and you
> will not get back the original signal. You might get back *a*
> time-domain signal, but it will be a different signal than
> the one you started out with.
>
> Rune

Subject: Signal Reconstruction

From: Rune Allnor

Date: 4 Mar, 2010 10:43:34

Message: 4 of 12

On 4 Mar, 11:24, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> But the files likehttp://www.mathworks.com/matlabcentral/fileexchange/25135-nufft-nfft-...
>
> calculates the approximate fourier spectrum now key to come back in the time domain is to do least sqaure reconstruction of this spectrum and get the corrected spectra and finally...fft and we r back in time domain on regular grid
>
> For example purpose i can take dft for irregular samples which will give us approximate spectrum.........the files above can just replace the dft but that is not a issue ! issue is how to get the regular grid time domain signal back from dft spectrum

If you have questions about files you found on the file
exchange, ask the authors of those fles.

Rune

Subject: Signal Reconstruction

From: kk KKsingh

Date: 4 Mar, 2010 11:09:21

Message: 5 of 12

Sir,

My question is a general one not with the author of the file.

KK


Rune Allnor <allnor@tele.ntnu.no> wrote in message <4058b34d-2feb-46b6-b992-0ce110a65b8f@y17g2000yqd.googlegroups.com>...
> On 4 Mar, 11:24, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> > But the files likehttp://www.mathworks.com/matlabcentral/fileexchange/25135-nufft-nfft-...
> >
> > calculates the approximate fourier spectrum now key to come back in the time domain is to do least sqaure reconstruction of this spectrum and get the corrected spectra and finally...fft and we r back in time domain on regular grid
> >
> > For example purpose i can take dft for irregular samples which will give us approximate spectrum.........the files above can just replace the dft but that is not a issue ! issue is how to get the regular grid time domain signal back from dft spectrum
>
> If you have questions about files you found on the file
> exchange, ask the authors of those fles.
>
> Rune

Subject: Signal Reconstruction

From: kk KKsingh

Date: 4 Mar, 2010 21:13:25

Message: 6 of 12

Even i can get the Fourier domain spectrum by Taylor approximation.....after that i need to work on it...........question is how

"kk KKsingh" <akikumar1983@gmail.com> wrote in message <hmo4d1$7pi$1@fred.mathworks.com>...
> Sir,
>
> My question is a general one not with the author of the file.
>
> KK
>
>
> Rune Allnor <allnor@tele.ntnu.no> wrote in message <4058b34d-2feb-46b6-b992-0ce110a65b8f@y17g2000yqd.googlegroups.com>...
> > On 4 Mar, 11:24, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> > > But the files likehttp://www.mathworks.com/matlabcentral/fileexchange/25135-nufft-nfft-...
> > >
> > > calculates the approximate fourier spectrum now key to come back in the time domain is to do least sqaure reconstruction of this spectrum and get the corrected spectra and finally...fft and we r back in time domain on regular grid
> > >
> > > For example purpose i can take dft for irregular samples which will give us approximate spectrum.........the files above can just replace the dft but that is not a issue ! issue is how to get the regular grid time domain signal back from dft spectrum
> >
> > If you have questions about files you found on the file
> > exchange, ask the authors of those fles.
> >
> > Rune

Subject: Signal Reconstruction

From: Greg Heath

Date: 5 Mar, 2010 10:09:33

Message: 7 of 12

On Mar 4, 3:43 am, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> Is there any code available which i can study ! where signal is reconstructed in frequency domain using Least sqaure...Like one of file submisson was Fast gridding method.....they calculate the frequency domain spectrum or i should say that they calculated A^(H)WY (which is dft with weights) now i want to apply least square on this ..fr reconstruction so that i can apply ifft and get back my signal..

If ( x , t and f are column vectors)

(1) W = exp( -2*pi*j * f * t' )

(2) X = W * ( x .* dts )

where dts is a weighted modification of diff(t),
then x should be reconstructed using

(3) x = ( W \ X ) ./ dts

instead of the more familiar

(4) x = W' * ( X .* dfs )

On the other hand, if you assume
that (4) is valid, the spectrum should
be obtained from

(5) X = ( W \ x ) ./ dfs

instead of (2).

To minimize confusion, define XFT and
XLS using (2) and (5), respectively.

Hope this helps.

Greg

Subject: Signal Reconstruction

From: Greg Heath

Date: 5 Mar, 2010 10:41:57

Message: 8 of 12

On Mar 5, 5:09 am, Greg Heath <he...@alumni.brown.edu> wrote:
> On Mar 4, 3:43 am, "kk KKsingh" <akikumar1...@gmail.com> wrote:
>
> > Is there any code available which i can study ! where signal is reconstructed in frequency domain using Least sqaure...Like one of file submisson was Fast gridding method.....they calculate the frequency domain spectrum or i should say that they calculated A^(H)WY (which is dft with weights) now i want to apply least square on this ..fr reconstruction so that i can apply ifft and get back my signal..
>
> If ( x , t and f are column vectors)
>
> (1)   W = exp( -2*pi*j * f * t' )
>
> (2)   X = W * ( x .* dts )
>
> where dts is a weighted modification of diff(t),
> then x should be reconstructed using
>
> (3)   x = ( W  \ X ) ./ dts
>
> instead of the more familiar
>
> (4)   x = W' * ( X .* dfs )
>
> On the other hand, if you assume
> that (4) is valid, the spectrum should
> be obtained from
>
> (5)  X = ( W \ x ) ./ dfs
>
> instead of (2).
>
> To minimize confusion, define XFT and
> XLS using (2) and (5), respectively.

Now if X is defined on a uniform grid using
(5), you should be able to interpolate x on a
uniform grid, t2, using

W2 = exp( -2*pi*j * f * t2' )

x2 = W2' * ( XLS .* dfs )

When

length(t2) = N = length(f)

x2 = C(Fs,N) * ifft( XLS )

where C is a constant that depends on Fs and N

(I'm too sleepy to figure it out now ... Good Night!)

Hope this helps.

Greg

Subject: Signal Reconstruction

From: kk KKsingh

Date: 13 Mar, 2010 06:29:07

Message: 9 of 12

Hey greg! Its not a signal reconstruction, its the same grid on which we are getting a signal


Greg Heath <heath@alumni.brown.edu> wrote in message <d257682b-ea25-4528-a907-71f9672ce4c3@z35g2000yqd.googlegroups.com>...
> On Mar 5, 5:09 am, Greg Heath <he...@alumni.brown.edu> wrote:
> > On Mar 4, 3:43 am, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> >
> > > Is there any code available which i can study ! where signal is reconstructed in frequency domain using Least sqaure...Like one of file submisson was Fast gridding method.....they calculate the frequency domain spectrum or i should say that they calculated A^(H)WY (which is dft with weights) now i want to apply least square on this ..fr reconstruction so that i can apply ifft and get back my signal..
> >
> > If ( x , t and f are column vectors)
> >
> > (1)   W = exp( -2*pi*j * f * t' )
> >
> > (2)   X = W * ( x .* dts )
> >
> > where dts is a weighted modification of diff(t),
> > then x should be reconstructed using
> >
> > (3)   x = ( W  \ X ) ./ dts
> >
> > instead of the more familiar
> >
> > (4)   x = W' * ( X .* dfs )
> >
> > On the other hand, if you assume
> > that (4) is valid, the spectrum should
> > be obtained from
> >
> > (5)  X = ( W \ x ) ./ dfs
> >
> > instead of (2).
> >
> > To minimize confusion, define XFT and
> > XLS using (2) and (5), respectively.
>
> Now if X is defined on a uniform grid using
> (5), you should be able to interpolate x on a
> uniform grid, t2, using
>
> W2 = exp( -2*pi*j * f * t2' )
>
> x2 = W2' * ( XLS .* dfs )
>
> When
>
> length(t2) = N = length(f)
>
> x2 = C(Fs,N) * ifft( XLS )
>
> where C is a constant that depends on Fs and N
>
> (I'm too sleepy to figure it out now ... Good Night!)
>
> Hope this helps.
>
> Greg

Subject: Signal Reconstruction

From: Greg Heath

Date: 13 Mar, 2010 06:44:03

Message: 10 of 12

On Mar 13, 1:29 am, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> Heygreg! Its not a signal reconstruction, its the same grid on which we are getting a signal
]
 Reconstruction means to construct again.
Therefore, I have no idea what you mean.
Please explain.

and

PLEASE DO NOT TOP.POST.

It is very annoying to have to scroll down to make
sense of what is said above.

Greg

Subject: Signal Reconstruction

From: kk KKsingh

Date: 13 Mar, 2010 07:06:04

Message: 11 of 12

Greg Heath <heath@alumni.brown.edu> wrote in message <e6377112-14e1-4364-85f0-5bb43c2da63c@q21g2000yqm.googlegroups.com>...
> On Mar 13, 1:29 am, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> > Heygreg! Its not a signal reconstruction, its the same grid on which we are getting a signal
> ]
> Reconstruction means to construct again.
> Therefore, I have no idea what you mean.
> Please explain.
>
> and
>
> PLEASE DO NOT TOP.POST.
>
> It is very annoying to have to scroll down to make
> sense of what is said above.
>
> Greg

Sorry for top posting ! Let suppose i have a decimated image now I will reconstruct my image (original image) same applies for signal if i have some signal which has missing samples then i would prefer to get its approximate spectrum by taylor method or by yours dftgh and then try to construct the correct spectrum by weighted least squares and finally apply backward fourier kernal or ifft to get signal back in short I am regularizing the signal

Subject: Signal Reconstruction

From: Greg Heath

Date: 13 Mar, 2010 09:35:26

Message: 12 of 12

On Mar 13, 2:06 am, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> Greg Heath <he...@alumni.brown.edu> wrote in message <e6377112-14e1-4364-85f0-5bb43c2da...@q21g2000yqm.googlegroups.com>...
> > On Mar 13, 1:29 am, "kk KKsingh" <akikumar1...@gmail.com> wrote:
> > > Heygreg! Its not a signal reconstruction, its the same grid on which we are getting a signal
> > ]
> >  Reconstruction means to construct again.
> > Therefore, I have no idea what you mean.
> > Please explain.
>
> > and
>
> > PLEASE DO NOT TOP.POST.
>
> > It is very annoying to have to scroll down to make
> > sense of what is said above.
>
> > Greg
>
> Sorry for top posting ! Let suppose i have a decimated image now I will reconstruct my image (original image) same applies for signal if i have some signal which has missing samples then i would prefer to get its approximate spectrum by taylor method or by yours dftgh and then try to construct the correct spectrum by weighted least squares and finally apply backward fourier kernal or ifft to get signal back in short I am regularizing the signal

You cannot use the ifft unless both both t and f are
uniformly spaced with length N and df*dt = 1/N.

Obtain the spectrum using LS. Try zeropadding
and using the IDFT to interpolate missing
original data.

Hope this helps.

Greg

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