On Jul 3, 8:23�am, "maxplanck" <erik.bo...@comcast.net> wrote:> Sufficient proof for my level of understanding of the FT at this point > would probably just be to understand why correlation works. �I can see that > it works by running some signals through an FT then an inverse FT, but > understanding at a conceptual level why it works is what I would be > interested in.If you are only interested in correlation, why are you asking about Fourier transforms instead of correlation integrals. I'm confused. :-\ The correlation of two signals is nothing more than the inner product of the two, suitably scaled. If you reverse the direction of one of the signals, find the inner product, and suitably scale, you have the convolution of the two signals. These are probably the two most fundamental operations in signal processing. More basic and widespread than even a Fourier transform. Any good maths book will show you how those processes are derived, and what their outputs mean. Regards, Steve
Can you recommend a good explanation of the proof of the Fourier Transform?
Started by ●July 1, 2008
Reply by ●July 2, 20082008-07-02
Reply by ●July 3, 20082008-07-03
On 3 Jul., 02:23, "maxplanck" <erik.bo...@comcast.net> wrote:> Sure any math must first take some axioms for granted in order to be > derived, but it can still be useful to understand why certain things are > so, to a certain degree. � > > Knowing about the necessity of axioms makes it easier to take all kinds of > things for granted, but taking too many things for granted can limit a > person's understanding and ability to apply and/or modify techniques. > > 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.You are probably not writing what mean. A "proof" is something that follows a "proposition". For example: Proposition: sqrt(2) is irrational. Proof: By contradiction: let p/q = sqrt(2), where p and q are natural numnbers, and q > 1. Then .... 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.>�I can see that it works ....What do you mean by "it works" ? It works in doing what exactly?> ... by running some signals through an FT then an inverse FT, ...You won't see much by doing that. Putting x through a FT and then through an inverse FT just gives you x again.> ... but > understanding at a conceptual level why it works is what I would be > interested in.What part of the Fourier Transform works in doing what? Regards, Andor
Reply by ●July 3, 20082008-07-03
"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. 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. 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.
Reply by ●July 3, 20082008-07-03
"12 WPM Class A" <invalid@invalid.invalid> wrote in message news:g4iada$pm$1@news.albasani.net...> > 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.PS. Part of their bluster is to demand that you go back and study your text books in greater detail but I have no doubt that you will have appeared here having already done so and not found your answer.
Reply by ●July 3, 20082008-07-03
Reply by ●July 3, 20082008-07-03
"Andor" <andor.bariska@gmail.com> wrote in message news:a9767a55-eaaa-43c1-b386-bac6ba087e42@l64g2000hse.googlegroups.com...>> ... silly and infantile ... > uh oh: these words, where have I seen them before?Perhaps when you yourself resorted to childish remarks when you were mathematically challenged?
Reply by ●July 3, 20082008-07-03
12 WPM Class A wrote:> ... 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. ...Axioms don't get proven. Come back when you know a little more. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 3, 20082008-07-03
On Jul 2, 7:23 pm, "maxplanck" <erik.bo...@comcast.net> wrote: [snip]> > Sufficient proof for my level of understanding of the FT at this point > would probably just be to understand why correlation works. I can see that > it works by running some signals through an FT then an inverse FT, but > understanding at a conceptual level why it works is what I would be > interested in.I'm also not sure what you are looking for (and you probably aren't sure, either). But consider for example the Discrete Fourier Transform of a *periodic* discrete-time signal x[n]: X[k] = \sum_n x[n] exp(-j 2\pi/N k n). You can consider this to be a set of projections onto a basis given by \exp(-j 2 \pi/N k n) So you can write the transform as a matrix multiplication of the vector x with the so-called DFT matrix given by the basis. I don't know what you mean when you say that you understand what correlation is, but hopefully this brings things a bit closer. Now you can consider two things: 1. You can prove all sorts of properties about this matrix (it's full- rank, it can be normalized, it's unitary, etc etc etc). 2. You can try to use the same insight on the continuous-time version, by replacing the vector projection by integration with a function. Then you may realize that this is much harder to do very formally. Does that help? Julius
Reply by ●July 3, 20082008-07-03
"Jerry Avins" <jya@ieee.org> wrote in message news:ptqdnTHfA7g6TvHVnZ2dnUVZ_hqdnZ2d@rcn.net...> 12 WPM Class A wrote: >> ... 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. ... > Axioms don't get proven. Come back when you know a little more.Max, As I said, they resort to infantile remarks.
Reply by ●July 3, 20082008-07-03
On Jul 3, 7:21�am, Andor <andor.bari...@gmail.com> wrote:> > ... silly and infantile ... > > uh oh: these words, where have I seen them before? > > :-)sounds like Beanie. which is fine. welcome Gareth. about the topic at hand, i've been waiting for a little more from Max, because i don't wanna bother proving something that wasn't asked for. i think that to "prove the Fourier Transform", there are two *big* steps (assuming you're already past calculus). first is to "prove the Fourier Series". that is to begin with a periodic function, x(t+T) = x(t) for all t and show that this series +inf x(t) = SUM{ c_k exp(j*2*pi*k/T*t) } k=-inf actually converges to the periodic x(t) when the coefficients are calculated as t0+T c_k = (1/T) integral{ x(u) exp(-j*2*pi*k/T*u) du} t0 for any t0. clearly the summation above is periodic with the same period T. knowing Euler's formula exp(j*v) = cos(v) + j*sin(v) and substituting the c_k integral into the c_k of the summation is a expresses a proposition of equality that needs to be proven which, if you switch the order of summation and integration (put the integral onto the outside), is not too hard to hammer out. or you can substitute the summation in for x(u) in the integral and see if you get c_k coming out. that's what the textbooks do. the second big step is to use the above Fourier Series facts (if you already trust them) and the Riemann integral to come up with the non- periodic, non-discrete counterpart that we usually call the Fourier Integral or the Fourier Transform. we let t0=-T/2 and then let T go to infinity. the 1/T becomes a "dt" and the c_k becomes X(f). if Max wants me to do either big steps, as long as he doesn't harass me about how i throw around expressions to set up the Riemann integral, i'll do it. but i'm not going to bother unless i understand that this is what he wants "proven". i can't tell yet. r b-j
