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From: Rune Allnor on 23 Feb 2010 12:43 On 23 Feb, 17:59, "kk KKsingh" <akikumar1...(a)gmail.com> wrote: > Can you give me your comments on the below :) OK, you are close enough; I won't mess around anymore. The key is the term Wnk. In the regularly sampled case, the implicit assumption is that both n and k are integers. For the irregular case, relax that assumption such that n and/or k might be any real number. Next, express the irregular Fourier transform in terms of a matrix W, where W(i,j) = Wnk = exp(sqrt(-1)*2*pi*f(i)*k(j)) or something like that. So the crux is to come up with the W matrix that expresses the transform, which might be irregular in both time and frequency. The key is Manolakis & Proakis (4th ed., 2007) equations 7.1.21/7.1.22 where n and k are *not* integers. In that case you will be able to express the irregularly sampled FT as a matrix-vector product similar to equation 7.1.26, where you need to modify the matrix W as given for the regular case in eq 7.1.25. Rune
From: kk KKsingh on 4 Mar 2010 05:19 Thank you sir Rune Allnor <allnor(a)tele.ntnu.no> wrote in message <2818cbed-68d0-40a3-971d-75b23d2e483e(a)y26g2000vbb.googlegroups.com>... > On 23 Feb, 17:59, "kk KKsingh" <akikumar1...(a)gmail.com> wrote: > > Can you give me your comments on the below :) > > OK, you are close enough; I won't mess around anymore. > The key is the term Wnk. In the regularly sampled case, > the implicit assumption is that both n and k are integers. > For the irregular case, relax that assumption such that > n and/or k might be any real number. > > Next, express the irregular Fourier transform in terms of > a matrix W, where W(i,j) = Wnk = exp(sqrt(-1)*2*pi*f(i)*k(j)) > or something like that. > > So the crux is to come up with the W matrix that expresses > the transform, which might be irregular in both time and > frequency. The key is Manolakis & Proakis (4th ed., 2007) > equations 7.1.21/7.1.22 where n and k are *not* integers. > In that case you will be able to express the irregularly > sampled FT as a matrix-vector product similar to equation > 7.1.26, where you need to modify the matrix W as given > for the regular case in eq 7.1.25. > > Rune
From: kk KKsingh on 25 Mar 2010 00:01
Rune Allnor <allnor(a)tele.ntnu.no> wrote in message <2818cbed-68d0-40a3-971d-75b23d2e483e(a)y26g2000vbb.googlegroups.com>... > On 23 Feb, 17:59, "kk KKsingh" <akikumar1...(a)gmail.com> wrote: > > Can you give me your comments on the below :) > > OK, you are close enough; I won't mess around anymore. > The key is the term Wnk. In the regularly sampled case, > the implicit assumption is that both n and k are integers. > For the irregular case, relax that assumption such that > n and/or k might be any real number. > > Next, express the irregular Fourier transform in terms of > a matrix W, where W(i,j) = Wnk = exp(sqrt(-1)*2*pi*f(i)*k(j)) > or something like that. > > So the crux is to come up with the W matrix that expresses > the transform, which might be irregular in both time and > frequency. The key is Manolakis & Proakis (4th ed., 2007) > equations 7.1.21/7.1.22 where n and k are *not* integers. > In that case you will be able to express the irregularly > sampled FT as a matrix-vector product similar to equation > 7.1.26, where you need to modify the matrix W as given > for the regular case in eq 7.1.25. > > Rune you mean some thing like this if N=500 M=1001 (frequency domain points) f=(-M/2):(M/2-1); freq axis for i=1:length(N); for j=1:length(M); Wnk = exp(sqrt(-1)*2*pi*t2(i)*f(j)) end end so Fourier matrix will be irregular matrix , am i right sir |