"12 WPM Class A" <invalid@invalid.invalid> wrote in message news:g4iada$pm$1@news.albasani.net...> "Andor" <andor.bariska@gmail.com> wrote in message > news:fc0eb3cf-cddd-4c6b-af7c-6b146d6b220e@j22g2000hsf.googlegroups.com... >>On 3 Jul., 02:23, "maxplanck" <erik.bo...@comcast.net> wrote: >>> >>> The FT is the one DSP technique that I'm most interested in applying, so >>> I'd like to understand as much about it as may be useful. But I have >>> more >>> to learn aside from any proofs, so I'll try to learn that and see if it >>> can >>> do what I hope it can do. If anything seems too mysterious I'll come >>> back >>> to trying to understand the proof. >>> >>> Sufficient proof for my level of understanding of the FT at this point >>> would probably just be to understand why correlation works. >> What you are writing is this: >> Proposition: The Fourier Transform. >> Proof: ???. >> Since The Fourier Transform is not a proposition, but really a >> definition, it cannot be proven. > > It is clear that Max is asking for the proof that the Fourier series and > transforms do what they say on the tin, and that is to _CORRECTLY_ > represent an incoming waveform by an infinite bunch of cisoidals. > > As such, Max is a budding engineer of future repute and not just some > button-pushing technician blindly applying formulae by rote as many > of the self-styled "experts" present on this NG.I take it that you can't or won't answer the question yourself.> > How do any of the experts prove that what they claim about Fourier > is true? For myself I have only seen a derivation from the initial axiom > that repetetive waveforms can be represented by an infinite bunch > of cisoids, but I've never seen a proof of the initial axiom. I suspect > that is what Max seeks.Assuming someone presented you with a proof of that "axiom" (Jerry Avins is correct - one does not prove axioms), there would a further "axiom" - keep quibbling if it makes you happy... Or better yet, take a few math courses and see if you can figure it out.> > Keep searching, Max, but I doubt that you'll get your answer in this NG. > > My own experience of them is that they'll blindly quote formulae verbatim > at you but when you challenge them on the derivations, they'll bluster in > the first > instance and resort to rather silly and infantile outbursts inorder to try > to mask > that despite being the "experts", they haven't a clue about the > fundamentals > and thus present as uppity technicians and not professional engineers. > > > > > >
Can you recommend a good explanation of the proof of the Fourier Transform?
Started by ●July 1, 2008
Reply by ●July 3, 20082008-07-03
Reply by ●July 3, 20082008-07-03
Sorry for the confusion in reading my book! Take a look at the first two paragraphs on page 158, which discuss Figure 8-8 on page 159. This is intended to answer the question you have about why correlation is the correct algorithm to explain the DFT. (at least one way). Please let me know if you don't think this answers the question. If it isn't clear, I certainly want to rewrite the section for the next edition. Also, please feel free to e-mail me directly on questions about the book. Good luck in your DSP studies! Regards, Steve Smith http://www.dspguide.com/ch8/6.htm
Reply by ●July 4, 20082008-07-04
> gets into the proof of the FT. Fourier himself had a problem with the proof: http://www.embeddedforth.de/temp/four.pdf MfG JRD
Reply by ●July 7, 20082008-07-07
Thanks very much Steve! The section of your book that you're referring to demonstrates numerically that correlation, when implemented as it is in the DFT, produces the desired results, i.e. what we want the output of the DFT to be. This is enough proof for me to believe that it works. I'm curious about a more rigorous proof of why the operation of correlating a signal with a basis function, then multiplying by an appropriate scaling factor, produces (after conversion to polar coordinates) the magnitude & initial phase of that basis function's presence in the signal. I realize that understanding such a proof may not have much practical utility, in much the same way that an understanding of the derivation of Calculus' derivative power rule will not be useful to a scientist who is simply attempting to model physical phenomena using the existing body of math knowledge. (As long as he knows how to apply the rule, the proof of why the rule is true doesn't have any practical application for the scientist who creates models using only math techniques that are already known and developed.) However, since the FT is the heart of just about every DSP technique that I'm interested in (I'm interested in sound analysis/resynthesis), and since correlation is the heart of the FT, I would like to understand correlation at least a bit more. It's just difficult to find well written, easily understandable explanations of topics such as this, so I was hoping that someone can recommend a good one.>Sorry for the confusion in reading my book! Take a look at the firsttwo>paragraphs on page 158, which discuss Figure 8-8 on page 159. This is >intended to answer the question you have about why correlation is the >correct algorithm to explain the DFT. (at least one way). Please let me >know if you don't think this answers the question. If it isn't clear, I >certainly want to rewrite the section for the next edition. Also,please>feel free to e-mail me directly on questions about the book. Good luckin>your DSP studies! >Regards, >Steve Smith > >http://www.dspguide.com/ch8/6.htm
Reply by ●July 7, 20082008-07-07
On Jul 7, 8:14 pm, "maxplanck" <erik.bo...@comcast.net> wrote:> Thanks very much Steve! > > The section of your book that you're referring to demonstrates numerically > that correlation, when implemented as it is in the DFT, produces the > desired results, i.e. what we want the output of the DFT to be. This is > enough proof for me to believe that it works. > > I'm curious about a more rigorous proof of why the operation of > correlating a signal with a basis function, then multiplying by an > appropriate scaling factor, produces (after conversion to polar > coordinates) the magnitude & initial phase of that basis function's > presence in the signal.Because the basis functions are orthogonal. With any orthogonal transform, correlating against the basis functions and then multiplying the coefficients with the conjugate basis functions, you're always going to get back to where you started (save for edge cases, discontinuities and other mathematical niggles). -- Oli
Reply by ●July 7, 20082008-07-07
On 7 Jul, 20:14, "maxplanck" <erik.bo...@comcast.net> wrote:> Thanks very much Steve! � > > The section of your book that you're referring to demonstrates numerically > that correlation, when implemented as it is in the DFT, produces the > desired results, i.e. what we want the output of the DFT to be. �This is > enough proof for me to believe that it works.Wrong. That's smoke and mirrors that might look reasonable up front, but only obfuscates in the long run. Both the the DFT/IDFT pair and correlation is based on the inner product, but that's where the similarity ends. Claiming that the study of either the DFT/IDFT pair or correlation provides insight about the other, makes as much sense as to claim that there is a deeper relation between a wood rowboat and a wood house. While both wood houses and wood boats are built from planks (and thus are based on the same fundamentals), actually building the two (not to mention the properties of the end products) are so different that one is better off forgetting the fact that both share wood planks as their fundamentals, and rather view each separately on their own terms.> I'm curious about a more rigorous proof of why the operation of > correlating a signal with a basis function, then multiplying by an > appropriate scaling factor, produces (after conversion to polar > coordinates) the magnitude & initial phase of that basis function's > presence in the signal. �Then look up 'inner product' and 'coordinate tranform' or 'basis transform' in a text on linear algebra. The DFT is nothing more than a coordinate transform which is implemented as a set of inner products between the signal and a set of basis functions, which can be expressed as a matrix-vector product. Similarly, the IDFT is the inverse transform, which also can be expressed as a matrix-vector product. Get a good book on DSP and look for the matrix W in the chapter that discusses the DFT. Rune
Reply by ●July 7, 20082008-07-07
On Jul 7, 2:14�pm, "maxplanck" <erik.bo...@comcast.net> wrote:> Thanks very much Steve! � > I'm curious about a more rigorous proof of why the operation of > correlating a signal with a basis function, then multiplying by an > appropriate scaling factor, produces (after conversion to polar > coordinates) the magnitude & initial phase of that basis function's > presence in the signal. �I'm not going to pretend I can answer your question that will satisfy you, but, in my experience when I have trouble like this it is because the operation I am trying to understand was simply to far removed from the fundamental concept it was implementing that it became too difficult to comprehend alone by itself (like FFT and DFT. ) Something that was initially understandable may become incomprehensible when transformed, even when examined at a "atomic level" (i.e., looking at the FFT equations). To your situation, I don't think there will be any explanation that will satisfy you why correlating a basis function with scaling gets you what you get. The only way to get out of this mess is to take a step back, I think Steve's DFT by simultaneous equations may help you. And then a �dry� mathematical proof why one is equivalent to the other will be the bridge you need to satisfy you.
Reply by ●July 7, 20082008-07-07
maxplanck wrote:> Thanks very much Steve! > > The section of your book that you're referring to demonstrates numerically > that correlation, when implemented as it is in the DFT, produces the > desired results, i.e. what we want the output of the DFT to be. This is > enough proof for me to believe that it works. > > I'm curious about a more rigorous proof of why the operation of > correlating a signal with a basis function, then multiplying by an > appropriate scaling factor, produces (after conversion to polar > coordinates) the magnitude & initial phase of that basis function's > presence in the signal. > > I realize that understanding such a proof may not have much practical > utility, in much the same way that an understanding of the derivation of > Calculus' derivative power rule will not be useful to a scientist who is > simply attempting to model physical phenomena using the existing body of > math knowledge. (As long as he knows how to apply the rule, the proof of > why the rule is true doesn't have any practical application for the > scientist who creates models using only math techniques that are already > known and developed.) > > However, since the FT is the heart of just about every DSP technique that > I'm interested in (I'm interested in sound analysis/resynthesis), and since > correlation is the heart of the FT, I would like to understand correlation > at least a bit more. It's just difficult to find well written, easily > understandable explanations of topics such as this, so I was hoping that > someone can recommend a good one.To understand the FFT, you need to understand what a basis function is. The powers of x form a basis set. You can approximate any function with a power series, just as you can with sines and cosines. Just as a function separates uniquely as a power series, so is there a unique trigonometric series that can describe it. Trig functions are actually easier to separate out, but the point is that of a separation procedure gives a result, it is the correct and only one. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 7, 20082008-07-07
Rune Allnor wrote:> On 7 Jul, 20:14, "maxplanck" <erik.bo...@comcast.net> wrote: >> Thanks very much Steve! >> >> The section of your book that you're referring to demonstrates numerically >> that correlation, when implemented as it is in the DFT, produces the >> desired results, i.e. what we want the output of the DFT to be. This is >> enough proof for me to believe that it works. > > Wrong. That's smoke and mirrors that might look reasonable > up front, but only obfuscates in the long run. Both the > the DFT/IDFT pair and correlation is based on the inner > product, but that's where the similarity ends. > > Claiming that the study of either the DFT/IDFT pair or > correlation provides insight about the other, makes as > much sense as to claim that there is a deeper relation > between a wood rowboat and a wood house. While both > wood houses and wood boats are built from planks (and > thus are based on the same fundamentals), actually > building the two (not to mention the properties of the > end products) are so different that one is better off > forgetting the fact that both share wood planks as > their fundamentals, and rather view each separately > on their own terms. > >> I'm curious about a more rigorous proof of why the operation of >> correlating a signal with a basis function, then multiplying by an >> appropriate scaling factor, produces (after conversion to polar >> coordinates) the magnitude & initial phase of that basis function's >> presence in the signal. > > Then look up 'inner product' and 'coordinate tranform' or > 'basis transform' in a text on linear algebra. > > The DFT is nothing more than a coordinate transform which > is implemented as a set of inner products between the signal > and a set of basis functions, which can be expressed as a > matrix-vector product. Similarly, the IDFT is the inverse > transform, which also can be expressed as a matrix-vector > product. > > Get a good book on DSP and look for the matrix W in the > chapter that discusses the DFT.Rune, What is the correlation between a sinusoid and an arbitrary function? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 8, 20082008-07-08
On 7 Jul, 22:41, Jerry Avins <j...@ieee.org> wrote:> Rune Allnor wrote: > > On 7 Jul, 20:14, "maxplanck" <erik.bo...@comcast.net> wrote: > >> Thanks very much Steve! � > > >> The section of your book that you're referring to demonstrates numerically > >> that correlation, when implemented as it is in the DFT, produces the > >> desired results, i.e. what we want the output of the DFT to be. �This is > >> enough proof for me to believe that it works. > > > Wrong. That's smoke and mirrors that might look reasonable > > up front, but only obfuscates in the long run. Both the > > the DFT/IDFT pair and correlation is based on the inner > > product, but that's where the similarity ends....> > Get a good book on DSP and look for the matrix W in the > > chapter that discusses the DFT. > > Rune, > > What is the correlation between a sinusoid and an arbitrary function?It's the inner product between the sine and the function. Introducing the term 'correlation' is a red herring if the main objective is to study the DFT. Rune
