On May 17, 11:43�pm, "Michael P." <m...@home.com> wrote:> Hi Group > > I have deduced/discovered some formulas > for interpolating between FFT bins when > calculating frequency and magnitude. > > I have written a "formal" pdf document. > For posting I have made this text version. > I hope it still makes sense. > > I would appreciate you comments and thoughts > very much. Do you agree with my reasoning? > > TIA, Michael > > Here is the document: (Sorry, it's a bit long) > > --------------------- > > � � � � � � � � � � T h e � S i n c � E s t i m a t o r...> All calculations and graphs have been made using a computer program.Why??? The estimation of the sinc must be just about the easiest estimation ever, concocted - why the hect start writing theses about it??? Rune
My Sinc Estmator
Started by ●May 17, 2011
Reply by ●May 23, 20112011-05-23
Reply by ●May 23, 20112011-05-23
"Rune Allnor" <allnor@tele.ntnu.no> wrote in message news:73674982-5695-47ba-847d-d6d135b92901@c41g2000yqm.googlegroups.com...> On May 17, 11:43 pm, "Michael P." <m...@home.com> wrote: >> Hi Group >> >> I have deduced/discovered some formulas >> for interpolating between FFT bins when >> calculating frequency and magnitude. >> >> I have written a "formal" pdf document. >> For posting I have made this text version. >> I hope it still makes sense. >> >> I would appreciate you comments and thoughts >> very much. Do you agree with my reasoning? >> >> TIA, Michael >> >> Here is the document: (Sorry, it's a bit long) >> >> --------------------- >> >> T h e S i n c E s t i m a t o r > ... >> All calculations and graphs have been made using a computer program. > > Why??? > > The estimation of the sinc must be just about the easiest > estimation ever, concocted - why the hect start writing > theses about it??? > > Rune > >If it is so easy then why haven't we seen the result of it in any books or webpages. E.g. I have never seen my formula for magnitude anywhere. Michael
Reply by ●May 23, 20112011-05-23
On 5/23/2011 9:35 AM, Michael P. wrote:> If it is so easy then why haven't we seen the result of it in any books > or webpages. > E.g. I have never seen my formula for magnitude anywhere. > > MichaelI think folks are trying to be supportive and have pointed out some rather obvious things: - A formula for a single, noiseless sinusoid isn't very useful. So, why bother? - That the approach is purely analytical isn't much of a defense and suggests other purely analytical treatments as below. One might suggest that the "formula" is a restatement of a truism: This is an analytical treatment. Here, I'm going to sample in frequency without sampling in time .. just to make it easier and to make an important point: Take a continuous, infinite sinusoid. Chop it with a rectangular window of length W. Assume that the content of the rectangular window is one cycle of a periodic waveform. (This is what makes frequency "sampled" / discrete) Compute the Fourier Series. Now you have a set of Fourier Coefficients which represent the magnitude and frequency of the chopped/periodic version of the sinusoid. To me, this is obvious and is a restatement of the notion that one can find the magnitude and frequency of a sinusoid from the Fourier Coefficients - which, with judicious treatment of conventions, is the same as a Fourier Transform of a periodic waveform. Merely sampling in time and using the DFT doesn't change things that much - so the point is still made. Look familiar? I'd say that this is found in books. Fred Notes: If the Window chops the sinusoid so that there is exactly some integer number of cycles of the sinusoid in the window, then the assumed periodic waveform is the original sinusoid and the Fourier Series has but one term - all but one of the harmonics are zero-valued. This harmonic is related to the number of cycles in the Window and the "fundamental" remains at 1/W. This is the same as the trivial case where a sinusoid results in but a single sample (well, 2 samples) in a DFT. If the Window chops the sinusoid differently, then there are discontinuities at the "edges" in the periodic waveform and the sinc shape in frequency represents the additional frequencies that are created by the discontinuity. This is the same as the more interesting case you study where the underlying sinusoid is at a frequency where there are no samples - because the underlying sinusoidal frequency is not a multiple of 1/W.
Reply by ●May 23, 20112011-05-23
"Fred Marshall" <fmarshallxremove_the_x@acm.org> wrote in message news:iKwCp.274$AU4.220@en-nntp-07.dc1.easynews.com...> On 5/23/2011 9:35 AM, Michael P. wrote: > >> If it is so easy then why haven't we seen the result of it in any books >> or webpages. >> E.g. I have never seen my formula for magnitude anywhere. >> >> Michael > > I think folks are trying to be supportive and have pointed out some rather > obvious things: > > - A formula for a single, noiseless sinusoid isn't very useful. So, why > bother? > - That the approach is purely analytical isn't much of a defense and > suggests other purely analytical treatments as below. > > One might suggest that the "formula" is a restatement of a truism: > > This is an analytical treatment. Here, I'm going to sample in frequency > without sampling in time .. just to make it easier and to make an > important point: > > Take a continuous, infinite sinusoid. > Chop it with a rectangular window of length W. > Assume that the content of the rectangular window is one cycle of a > periodic waveform. (This is what makes frequency "sampled" / discrete) > Compute the Fourier Series. > Now you have a set of Fourier Coefficients which represent the magnitude > and frequency of the chopped/periodic version of the sinusoid. > > To me, this is obvious and is a restatement of the notion that one can > find the magnitude and frequency of a sinusoid from the Fourier > Coefficients - which, with judicious treatment of conventions, is the same > as a Fourier Transform of a periodic waveform. > Merely sampling in time and using the DFT doesn't change things that > much - so the point is still made. > > Look familiar? I'd say that this is found in books. > > Fred > > Notes: > If the Window chops the sinusoid so that there is exactly some integer > number of cycles of the sinusoid in the window, then the assumed periodic > waveform is the original sinusoid and the Fourier Series has but one > term - all but one of the harmonics are zero-valued. This harmonic is > related to the number of cycles in the Window and the "fundamental" > remains at 1/W. > This is the same as the trivial case where a sinusoid results in but a > single sample (well, 2 samples) in a DFT. > > If the Window chops the sinusoid differently, then there are > discontinuities at the "edges" in the periodic waveform and the sinc shape > in frequency represents the additional frequencies that are created by the > discontinuity. > This is the same as the more interesting case you study where the > underlying sinusoid is at a frequency where there are no samples - because > the underlying sinusoidal frequency is not a multiple of 1/W. >Hi Fred Thank you for being honest and telling me how it is. I was beginning to wonder why people didn't comment on the formulas themselves. I am not an expert, so I didn't know that I was stating the obvious. Maybe I can use the fact that a triangle window has sinc squared as it's impulse response in a new formula. As I say, I am not that experienced with dsp. So far I have only used the FFT to analyse music and make effects by altering the bins in the freq. domain and doing the IFFT. Best wishes, Michael
Reply by ●May 23, 20112011-05-23
Michael P. wrote:> I am not an expert, so I didn't know that I was stating the obvious. > Maybe I can use the fact that a triangle window has sinc squared > as it's impulse response in a new formula.Here is the other idea: Sinusoid: X(t) = A sin(Wt + Fi) Now, take three points (x,t) on the sinusoid and solve the system of equations to find A, W and Fi. As simple as that. No need for FFTs, windows and such. Just three points and basic math. :-) Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●May 24, 20112011-05-24
On May 23, 6:35�pm, "Michael P." <m...@home.com> wrote:> "Rune Allnor" <all...@tele.ntnu.no> wrote in message > > news:73674982-5695-47ba-847d-d6d135b92901@c41g2000yqm.googlegroups.com... > > > > > > > On May 17, 11:43 pm, "Michael P." <m...@home.com> wrote: > >> Hi Group > > >> I have deduced/discovered some formulas > >> for interpolating between FFT bins when > >> calculating frequency and magnitude. > > >> I have written a "formal" pdf document. > >> For posting I have made this text version. > >> I hope it still makes sense. > > >> I would appreciate you comments and thoughts > >> very much. Do you agree with my reasoning? > > >> TIA, Michael > > >> Here is the document: (Sorry, it's a bit long) > > >> --------------------- > > >> � � � � � � � � � � T h e � S i n c � E s t i m a t o r > > ... > >> All calculations and graphs have been made using a computer program. > > > Why??? > > > The estimation of the sinc must be just about the easiest > > estimation ever, concocted - why the hect start writing > > theses about it??? > > > Rune > > If it is so easy then why haven't we seen the result of it in any books or > webpages.Two reasons: 1) The sinc is a function that appears in one very spesific context. 2) The sinc is never actually computed.> E.g. I have never seen my formula for magnitude anywhere.Then read the books more carefully. Contemplate the material the text attempts to communictae rather than just look for formulas. Rune
Reply by ●May 24, 20112011-05-24
On May 23, 2:40�pm, "Michael P." <m...@home.com> wrote:> "Fred Marshall" <fmarshallxremove_th...@acm.org> wrote in message > > news:iKwCp.274$AU4.220@en-nntp-07.dc1.easynews.com... > > > > > > > On 5/23/2011 9:35 AM, Michael P. wrote: > > >> If it is so easy then why haven't we seen the result of it in any books > >> or webpages. > >> E.g. I have never seen my formula for magnitude anywhere. > > >> Michael > > > I think folks are trying to be supportive and have pointed out some rather > > obvious things: > > > - A formula for a single, noiseless sinusoid isn't very useful. �So, why > > bother? > > - That the approach is purely analytical isn't much of a defense and > > suggests other purely analytical treatments as below. > > > One might suggest that the "formula" is a restatement of a truism: > > > This is an analytical treatment. �Here, I'm going to sample in frequency > > without sampling in time .. just to make it easier and to make an > > important point: > > > Take a continuous, infinite sinusoid. > > Chop it with a rectangular window of length W. > > Assume that the content of the rectangular window is one cycle of a > > periodic waveform. (This is what makes frequency "sampled" / discrete) > > Compute the Fourier Series. > > Now you have a set of Fourier Coefficients which represent the magnitude > > and frequency of the chopped/periodic version of the sinusoid. > > > To me, this is obvious and is a restatement of the notion that one can > > find the magnitude and frequency of a sinusoid from the Fourier > > Coefficients - which, with judicious treatment of conventions, is the same > > as a Fourier Transform of a periodic waveform. > > Merely sampling in time and using the DFT doesn't change things that > > much - so the point is still made. > > > Look familiar? �I'd say that this is found in books. > > > Fred > > > Notes: > > If the Window chops the sinusoid so that there is exactly some integer > > number of cycles of the sinusoid in the window, then the assumed periodic > > waveform is the original sinusoid and the Fourier Series has but one > > term - all but one of the harmonics are zero-valued. �This harmonic is > > related to the number of cycles in the Window and the "fundamental" > > remains at 1/W. > > This is the same as the trivial case where a sinusoid results in but a > > single sample (well, 2 samples) in a DFT. > > > If the Window chops the sinusoid differently, then there are > > discontinuities at the "edges" in the periodic waveform and the sinc shape > > in frequency represents the additional frequencies that are created by the > > discontinuity. > > This is the same as the more interesting case you study where the > > underlying sinusoid is at a frequency where there are no samples - because > > the underlying sinusoidal frequency is not a multiple of 1/W. > > Hi Fred > > Thank you for being honest and telling me how it is. > I was beginning to wonder why people didn't comment on the > formulas themselves. > I am not an expert, so I didn't know that I was stating the obvious. > Maybe I can use the fact that a triangle window has sinc squared > as it's impulse response in a new formula. > As I say, I am not that experienced with dsp. So far I have only > used the FFT to analyse music and make effects by altering > the bins in the freq. domain and doing the IFFT. > > Best wishes, Michael- Hide quoted text - > > - Show quoted text -yikes tough crowd mike, most of the "new schemes" presented in DSP articles in IEEE are rehash of old stuff so don't sweat it, (except in those articles you have to penetrate the obscure language before you realize, oh, that is all is saying)
Reply by ●May 24, 20112011-05-24
On May 24, 7:19�am, steve <bungalow_st...@yahoo.com> wrote:> On May 23, 2:40�pm, "Michael P." <m...@home.com> wrote: > > > > > "Fred Marshall" <fmarshallxremove_th...@acm.org> wrote in message > > >news:iKwCp.274$AU4.220@en-nntp-07.dc1.easynews.com... > > > > On 5/23/2011 9:35 AM, Michael P. wrote: > > > >> If it is so easy then why haven't we seen the result of it in any books > > >> or webpages. > > >> E.g. I have never seen my formula for magnitude anywhere. > > > >> Michael > > > > I think folks are trying to be supportive and have pointed out some rather > > > obvious things: > > > > - A formula for a single, noiseless sinusoid isn't very useful. �So, why > > > bother? > > > - That the approach is purely analytical isn't much of a defense and > > > suggests other purely analytical treatments as below. > > > > One might suggest that the "formula" is a restatement of a truism: > > > > This is an analytical treatment. �Here, I'm going to sample in frequency > > > without sampling in time .. just to make it easier and to make an > > > important point: > > > > Take a continuous, infinite sinusoid. > > > Chop it with a rectangular window of length W. > > > Assume that the content of the rectangular window is one cycle of a > > > periodic waveform. (This is what makes frequency "sampled" / discrete) > > > Compute the Fourier Series. > > > Now you have a set of Fourier Coefficients which represent the magnitude > > > and frequency of the chopped/periodic version of the sinusoid. > > > > To me, this is obvious and is a restatement of the notion that one can > > > find the magnitude and frequency of a sinusoid from the Fourier > > > Coefficients - which, with judicious treatment of conventions, is the same > > > as a Fourier Transform of a periodic waveform. > > > Merely sampling in time and using the DFT doesn't change things that > > > much - so the point is still made. > > > > Look familiar? �I'd say that this is found in books. > > > > Fred > > > > Notes: > > > If the Window chops the sinusoid so that there is exactly some integer > > > number of cycles of the sinusoid in the window, then the assumed periodic > > > waveform is the original sinusoid and the Fourier Series has but one > > > term - all but one of the harmonics are zero-valued. �This harmonic is > > > related to the number of cycles in the Window and the "fundamental" > > > remains at 1/W. > > > This is the same as the trivial case where a sinusoid results in but a > > > single sample (well, 2 samples) in a DFT. > > > > If the Window chops the sinusoid differently, then there are > > > discontinuities at the "edges" in the periodic waveform and the sinc shape > > > in frequency represents the additional frequencies that are created by the > > > discontinuity. > > > This is the same as the more interesting case you study where the > > > underlying sinusoid is at a frequency where there are no samples - because > > > the underlying sinusoidal frequency is not a multiple of 1/W. > > > Hi Fred > > > Thank you for being honest and telling me how it is. > > I was beginning to wonder why people didn't comment on the > > formulas themselves. > > I am not an expert, so I didn't know that I was stating the obvious. > > Maybe I can use the fact that a triangle window has sinc squared > > as it's impulse response in a new formula. > > As I say, I am not that experienced with dsp. So far I have only > > used the FFT to analyse music and make effects by altering > > the bins in the freq. domain and doing the IFFT. > > > Best wishes, Michael- Hide quoted text - > > > - Show quoted text - > > yikes tough crowd mike, most of the "new schemes" presented in DSP > articles in IEEE are rehash of old stuff so don't sweat it, (except in > those articles you have to penetrate the obscure language before you > realize, oh, that is all is saying)Even in the old stuff you had to penetrate the obscure language. Consider: Use of the Discrete Fourier Transform in the Measurement of Frequencies and Levels of Tones PDF Rife, D.C.; Vincent, G.A. Bell System Technical Journal Volume 49, Issue 2, February 1970 | 11095.4K http://www.alcatel-lucent.com/bstj/vol49-1970/articles/bstj49-2-197.pdf This is a classic (but far from the first) interpolation paper. Three types of windows are discussed: class I: maximum sidelobe rolloff windows class II: (two parameter) Taylor windows class III: an ad hoc set of windows The maximum sidelobe rolloff windows are a frequent "rediscovery" in the IEEE literature. The richest collection of fft interpolation windows in the IEEE literature seems to be in the Transactions on Instrumentation and Measurement For example: http://wwwir.vub.ac.be/elec/Papers%20on%20web/Papers/JohanSchoukens/IM92Schoukens-The%20Interpolated.pdf For comparison to other methods: http://home.mit.bme.hu/~sarhegyi/pubs/msthesis.pdf The optical guys have a long history of fft interpolation, too: http://www.eng.tau.ac.il/~yaro/RecentPublications/ps&pdf/EfficientSincInterpolation_ApllOpt.pdf Dale B. Dalrymple
Reply by ●May 24, 20112011-05-24
"steve" <bungalow_steve@yahoo.com> wrote in message news:806aba6c-3965-4613-b359-a706b32d7e29@e17g2000prj.googlegroups.com...> On May 18, 2:02 pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote: >> Michael P. <m...@home.com> wrote: >> >> (snip) >> >> > Well put Dave. >> > All bin magnitudes are points on a scaled Sinc curve. >> > As I say in the document now window must be used. >> > That is because only a rectangular window (= no window) is tranformed >> > to a Sinc curve. >> >> Yes, but why the sinc in the first place? >> >> The Sinc comes out as the Fourier transform of Rect, the result >> of reconstructing an infinitely long periodic signal. >> >> If one does an FFT on a finite duration signal, then adds the >> constraint that all the points not sampled (to plus and minus infinity) >> are zero, then one finds Sinc as the inverse transform. > > The sinc response is purely a filter characteristic of a process of > truncating a signal. It has nothing to do with an FFT or assumptions > of what the signal is doing before or after the truncated record. >This is one very important piece of information that I have learned from this discussion. Thank you Steve. I thought the sinc response came from the FFT. But now that I understand it, it's obvious. Truncating a sine and considering the result as one cycle of a periodic signal will not result in a periodic pure sine signal. Unless you are lucky. Michael
Reply by ●May 24, 20112011-05-24
"steve" <bungalow_steve@yahoo.com> wrote in message news:d1e050d4-02ce-4c54-9527-e4a7194289c6@y31g2000vbp.googlegroups.com...> On May 23, 2:40 pm, "Michael P." <m...@home.com> wrote: >> "Fred Marshall" <fmarshallxremove_th...@acm.org> wrote in message >> >> news:iKwCp.274$AU4.220@en-nntp-07.dc1.easynews.com... >> >> >> >> >> >> > On 5/23/2011 9:35 AM, Michael P. wrote: >> >> >> If it is so easy then why haven't we seen the result of it in any >> >> books >> >> or webpages. >> >> E.g. I have never seen my formula for magnitude anywhere. >> >> >> Michael >> >> > I think folks are trying to be supportive and have pointed out some >> > rather >> > obvious things: >> >> > - A formula for a single, noiseless sinusoid isn't very useful. So, >> > why >> > bother? >> > - That the approach is purely analytical isn't much of a defense and >> > suggests other purely analytical treatments as below. >> >> > One might suggest that the "formula" is a restatement of a truism: >> >> > This is an analytical treatment. Here, I'm going to sample in >> > frequency >> > without sampling in time .. just to make it easier and to make an >> > important point: >> >> > Take a continuous, infinite sinusoid. >> > Chop it with a rectangular window of length W. >> > Assume that the content of the rectangular window is one cycle of a >> > periodic waveform. (This is what makes frequency "sampled" / discrete) >> > Compute the Fourier Series. >> > Now you have a set of Fourier Coefficients which represent the >> > magnitude >> > and frequency of the chopped/periodic version of the sinusoid. >> >> > To me, this is obvious and is a restatement of the notion that one can >> > find the magnitude and frequency of a sinusoid from the Fourier >> > Coefficients - which, with judicious treatment of conventions, is the >> > same >> > as a Fourier Transform of a periodic waveform. >> > Merely sampling in time and using the DFT doesn't change things that >> > much - so the point is still made. >> >> > Look familiar? I'd say that this is found in books. >> >> > Fred >> >> > Notes: >> > If the Window chops the sinusoid so that there is exactly some integer >> > number of cycles of the sinusoid in the window, then the assumed >> > periodic >> > waveform is the original sinusoid and the Fourier Series has but one >> > term - all but one of the harmonics are zero-valued. This harmonic is >> > related to the number of cycles in the Window and the "fundamental" >> > remains at 1/W. >> > This is the same as the trivial case where a sinusoid results in but a >> > single sample (well, 2 samples) in a DFT. >> >> > If the Window chops the sinusoid differently, then there are >> > discontinuities at the "edges" in the periodic waveform and the sinc >> > shape >> > in frequency represents the additional frequencies that are created by >> > the >> > discontinuity. >> > This is the same as the more interesting case you study where the >> > underlying sinusoid is at a frequency where there are no samples - >> > because >> > the underlying sinusoidal frequency is not a multiple of 1/W. >> >> Hi Fred >> >> Thank you for being honest and telling me how it is. >> I was beginning to wonder why people didn't comment on the >> formulas themselves. >> I am not an expert, so I didn't know that I was stating the obvious. >> Maybe I can use the fact that a triangle window has sinc squared >> as it's impulse response in a new formula. >> As I say, I am not that experienced with dsp. So far I have only >> used the FFT to analyse music and make effects by altering >> the bins in the freq. domain and doing the IFFT. >> >> Best wishes, Michael- Hide quoted text - >> >> - Show quoted text - > > yikes tough crowd mike, most of the "new schemes" presented in DSP > articles in IEEE are rehash of old stuff so don't sweat it, (except in > those articles you have to penetrate the obscure language before you > realize, oh, that is all is saying)Yes, it's a tough crowd :-) But I understand why and I can take it ... I hope. Michael
