I'm reading Steven Smith's "Scientist and Engineer's Guide to Digital Signal Processing," which is excellent imo, and I've read Lyons' "Understanding Digital Signal Processing," great as well. However, neither of these books gets into the proof of the FT. Smith goes through a brief qualitative description of correlation, and points out that the Real DFT operates on this principle. This is evident in the equation describing the Real DFT. However, I'd like to understand why correlation works for this application.. I've googled and searched this forum, and haven't found a satisfactory explanation. Can anyone recommend a good one?
Can you recommend a good explanation of the proof of the Fourier Transform?
Started by ●July 1, 2008
Reply by ●July 1, 20082008-07-01
"maxplanck" <erik.bowen@comcast.net> writes:> I'm reading Steven Smith's "Scientist and Engineer's Guide to Digital > Signal Processing," which is excellent imo, and I've read Lyons' > "Understanding Digital Signal Processing," great as well. > > However, neither of these books gets into the proof of the FT. Smith goes > through a brief qualitative description of correlation, and points out that > the Real DFT operates on this principle. This is evident in the equation > describing the Real DFT. However, I'd like to understand why correlation > works for this application.. > > I've googled and searched this forum, and haven't found a satisfactory > explanation. Can anyone recommend a good one?What do you need a proof of? Regards, Peter K. -- "And he sees the vision splendid of the sunlit plains extended And at night the wondrous glory of the everlasting stars."
Reply by ●July 1, 20082008-07-01
maxplanck wrote:> I'm reading Steven Smith's "Scientist and Engineer's Guide to Digital > Signal Processing," which is excellent imo, and I've read Lyons' > "Understanding Digital Signal Processing," great as well. > > However, neither of these books gets into the proof of the FT. Smith goes > through a brief qualitative description of correlation, and points out that > the Real DFT operates on this principle. This is evident in the equation > describing the Real DFT. However, I'd like to understand why correlation > works for this application.. > > I've googled and searched this forum, and haven't found a satisfactory > explanation. Can anyone recommend a good one?What do you want to prove? Start with proving that sinusoids form an orthogonal set. In symbols, t / (1/t) * | sin(nt)*f(t)dt as t-> infinity gives the magnitude of / -t that component of f(t) that is sin(nt), and zero for anything else. The same for cos(nt) and for other (not necessarily integer values of n. So th integral acts as a filter, extracting the frequency and phase component specified in it. Simplifications occur for periodic signals, but I think you can take it from there. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 1, 20082008-07-01
On Jul 1, 8:29�pm, p.kootsoo...@remove.ieee.org (Peter K.) wrote:> "maxplanck" <erik.bo...@comcast.net> writes: > > I'm reading Steven Smith's "Scientist and Engineer's Guide to Digital > > Signal Processing," which is excellent imo, and I've read Lyons' > > "Understanding Digital Signal Processing," great as well. > > > However, neither of these books gets into the proof of the FT. �Smith goes > > through a brief qualitative description of correlation, and points out that > > the Real DFT operates on this principle. �This is evident in the equation > > describing the Real DFT. �However, I'd like to understand why correlation > > works for this application.. > > > I've googled and searched this forum, and haven't found a satisfactory > > explanation. �Can anyone recommend a good one? > > What do you need a proof of? >there are a few different neat properties or theorems of the F.T., but i presume what Max wants is a proof of the inversion relationship +inf x(t) = integral{ X(f) exp(+j*2*pi*f*t) df} -inf given the definition of the forward F.T. +inf X(f) = integral{ x(t) exp(-j*2*pi*f*t) dt} -inf send x(t) to X(f) with the definition (nothing to prove here) and that back to x(t) with the inverse mapping, is the x(t) that comes out the same as the x(t) that went in? if no one is too anal about how one handles the order of limits, you can start with Fourier Series, let the period go to infinity, and show that this sets up the Fourier Series to look like a Riemann sum that becomes the inversion formula above. Max, would that suffice as a proof to you? or are ya gonna get all anal retentive with me if i show that to you? i'm not gonna bother otherwise. r b-j
Reply by ●July 1, 20082008-07-01
robert bristow-johnson wrote: ...> if no one is too anal about how one handles the order of limits, you > can start with Fourier Series, let the period go to infinity, and show > that this sets up the Fourier Series to look like a Riemann sum that > becomes the inversion formula above. Max, would that suffice as a > proof to you? or are ya gonna get all anal retentive with me if i show > that to you? i'm not gonna bother otherwise.You seem to get all hung up on this anal-retentive bit. Just you wait! the time is coming when penile retention will increasingly occupy your attention! :-) Jerry, who's been there. -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 1, 20082008-07-01
On Jul 1, 10:09�pm, Jerry Avins <j...@ieee.org> wrote:> robert bristow-johnson wrote: > > � �... > > > if no one is too anal about how one handles the order of limits, you > > can start with Fourier Series, let the period go to infinity, and show > > that this sets up the Fourier Series to look like a Riemann sum that > > becomes the inversion formula above. �Max, would that suffice as a > > proof to you? or are ya gonna get all anal retentive with me if i show > > that to you? �i'm not gonna bother otherwise. > > You seem to get all hung up on this anal-retentive bit.well, i've been bitten (more like snapped at) by math folk that don't like to see a dirac impulse outside an integral. we've been told that SUM{ delta(t-n) } = SUM{ exp(j*2*pi*k*t) } n k is not even wrong, but an expression that make no sense. you remember the slugfests we've had here a decade ago, no? i've had it also at the talk page of the Nyquist/Shannon Sampling theorem Wikipedia article. i'm reasonably anal about math, but not that anal. if the true, honest to God, infinitesimal width Dirac delta function cannot be described as the spike of unit area that we engineers do, then, for my money, define the above delta (as a function of time) to have one Planck Time in width, and then i'll say it's close enough. fuck it.> Just you wait! > the time is coming when penile retention will increasingly occupy your > attention! :-) > > Jerry, who's been there.of course, Jerry, despite the fact that i haven't run mile straight through for over a decade (in H.S. track, i did it in 4:41), despite the daily 20 mg lovastatin and niacin megadose (for pretty much the rest of my observable life), despite the sorta chronic back shit (i haven't donned a girdle or truss yet), i cannot out-codger you. you are my parents generation and, to all of our benefit and joy, you still be with us on this planet (something that is, only relatively recently, not the case regarding my parents). but i went to a club in Cambridge the other day and, at age 52, was carded. the female door "host" was a little embarrassed to see 1956 on my drivers license. r b-j (who thinks about his own mortality a lot more of late.)
Reply by ●July 2, 20082008-07-02
robert bristow-johnson wrote: ...> but i went to a club in Cambridge the other day and, at age 52, was > carded. the female door "host" was a little embarrassed to see 1956 > on my drivers license.Well, you know that the hippie look is timeless! :-) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 2, 20082008-07-02
On Jul 1, 11:21�pm, Jerry Avins <j...@ieee.org> wrote:> robert bristow-johnson wrote: > > � �... > > > but i went to a club in Cambridge the other day and, at age 52, was > > carded. �the female door "host" was a little embarrassed to see 1956 > > on my drivers license. > > Well, you know that the hippie look is timeless! :-)well, in Burlington VT there are a couple of college-aged kids with hippie hair (my daughter's guitar teacher is one), but most folks i see that have the hippie look have a shade of hair that is somewhere between yours and mine. i wouldn't expect them to get carded either. r b-j
Reply by ●July 2, 20082008-07-02
On 2 Jul, 00:41, "maxplanck" <erik.bo...@comcast.net> wrote:> I'm reading Steven Smith's "Scientist and Engineer's Guide to Digital > Signal Processing," which is excellent imo, and I've read Lyons' > "Understanding Digital Signal Processing," great as well. > > However, neither of these books gets into the proof of the FT. �Smith goes > through a brief qualitative description of correlation, and points out that > the Real DFT operates on this principle. �This is evident in the equation > describing the Real DFT. �However, I'd like to understand why correlation > works for this application.. > > I've googled and searched this forum, and haven't found a satisfactory > explanation. �Can anyone recommend a good one?I don't understand exactly what you want. The only proofs that are 'easy' with respect to FTs are those which deal with the DFT/IDFT pair, since they can be expressed in terms of matrix algebra. Once one starts messing with infinitely long sequences and coninuous spectra, one really needs tools from rather advanced maths. Unfortunately, most of the people who discuss those sorts of things (me included!) don't master those tools. Hence lots of more or less 'intuitive' explanations of things that might not be as intuitive after all. Not to mention the frequent sematics wars. If you want a basic understanding quicly, stick with the DFT/IDFT pair. If you want the thorough stuff, take a couple of maths classes on Real Analysis. Rune
Reply by ●July 2, 20082008-07-02
maxplanck wrote:> I'm reading Steven Smith's "Scientist and Engineer's Guide to Digital > Signal Processing," which is excellent imo, and I've read Lyons' > "Understanding Digital Signal Processing," great as well. > > However, neither of these books gets into the proof of the FT. Smith goes > through a brief qualitative description of correlation, and points out that > the Real DFT operates on this principle. This is evident in the equation > describing the Real DFT. However, I'd like to understand why correlation > works for this application.. > > I've googled and searched this forum, and haven't found a satisfactory > explanation. Can anyone recommend a good one?The classic for this is "Signals and Systems" by Oppenheim &c (there are various editions, with various other authors). I didn't fully understand the Fourier transform until I'd been smacked upside the head with it several times, which didn't happen until I was 4 years into my EE program. Even then I've found never ending opportunities to deepen my appreciation for it (Van Trees' "Detection & Estimation" gives good insight into the superset of which the Fourier transform is but a member, but you pretty much have to be taking a class that includes the book and a highly motivated, excellent teacher if you're going to get anything out of it -- Van Trees isn't the most accessible author). So don't expect full understanding to come from just one book, or for it to be easy. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
