On Wed, 12 Jan 2011 18:07:44 -0800 (PST), Rune Allnor <allnor@tele.ntnu.no> wrote:>On Jan 12, 1:49�am, "steveu" <steveu@n_o_s_p_a_m.coppice.org> wrote: > >> That would imply that the validity of the answer from the DFT is not >> dependant on the values outside the length N, which is clearly not the >> case. > >With the caveat that it's 3AM, I've pushing 24 hrs, haven't >cleaned my glasses for days and English is not my native >language: > >Are you really suggesting that the computed result of the DFT >depends on numbers that never enter the computations? > >RuneHi Rune, You worded your question perfectly. [-Rick-]
Appendix A: Types of Fourier Transforms
Started by ●January 10, 2011
Reply by ●January 13, 20112011-01-13
Reply by ●January 13, 20112011-01-13
On Thu, 13 Jan 2011 01:02:46 -0800 (PST), Chris Bore <chris.bore@gmail.com> wrote: ...>I find this difference of viewpoint remains stark: some regard the DFT >as independent of contionuous FT theory, and some take Brigham's >viewpoint in order to relate the discrete and continuous transforms. I >think I like Brigham's approach not only because I am used to it but >also because it realtes well to Shannon's derivationof Sampling Theory >which he proves using the continuous Fourier transform (although I am >familiar with proofs of Sampling Theory that rely only on definitions >of the DFT). When I teach the DFT and FFT, or explain their >application in practical implementations, I do find colleagues who >disagree (sometimes vigorously) with the Brigham-style interpretation: >in those cases I tend to 'win' the argument because the Brigham >approach seems to appeal to those who seek an 'intuitive' >understanding of 'what really happens' so less specialised colleagues >tend to outvote the less intutive interpretation. However, for myself >I am very aware that the more 'pure' and 'mathematical' view is >equally valid, and especially that it is vital always to be very clear >about any assumptions or models that one is applying - sometimes you >have to work hard to realise what you are assuming. On the other hand >one cannot afford to debate endlessly, since no system of logic can be >consistent anyway and we often have to make stuff in a finite time. > >Chris >=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D >Chris Bore >BORES Signal Processing >www.bores.comThanks. That's pretty much what I was trying to say earlier, that understanding multiple viewpoints on this is beneficial. I heartily agree with your last sentence as well. Eric Jacobsen Minister of Algorithms Abineau Communications http://www.abineau.com
Reply by ●January 13, 20112011-01-13
On Jan 13, 4:02�am, Chris Bore <chris.b...@gmail.com> wrote:> On Jan 12, 8:57�pm, Clay <c...@claysturner.com> wrote: > > > > > On Jan 12, 2:36�pm, Fred Marshall <fmarshall_xremove_the...@xacm.org> > > wrote: > > > > On 1/10/2011 11:38 PM, Rune Allnor wrote: > > > > > And that's where the problem occurs: The condition for the > > > > FT of a function x(t) to exist (CT infinite domain) is that > > > > > integral |x(t)|^2 dt< �infinite > > > > > With x(t) = sin(t) that breaks down. Note that this has nothing > > > > to do with the sine being periodic, it has to do with it having > > > > infinite energy. (Try the same excercise with y(t) = sin(t)+sin(pi*t) > > > > to see why.) > > > > > To get out of that embarrasment, engineers (*not* mathematicians) > > > > came up with the ad hoc solution to express the periodic sine as > > > > a sequence of periods, repeated ad infinitum, and compute the > > > > Fourier series of one period. > > > > > It has no mathematical meaning, as the rather essential property > > > > of linearity of the FT breaks down (again, use the y(t) above to > > > > see why). > > > > > Again, this is totally trivial. > > > > > Rune > > > > I guess mathematicians over time have thus had a lot of fun using stuff > > > that engineers came up with.... �I fail to acknowledge a fundamental > > > difference between the two sets of folks where stuff like this is > > > concerned. �Obviously each have their areas of expertise that go beyond. > > > > Dale repeats his point about the DFT being (in my terms) an abstract > > > thingy that is simply a mapping of N points - having nothing whatsoever > > > to do with any imagined or real samples which may exist outside the > > > sample regions. �This is a surely a valid perspective of the FT as a > > > *mapping* but I've not reconciled how it fits in my own perspectives. > > > It seems that r b-j takes the opposite view which matches better with my > > > working framework. �And, to be clear, I think that framework is less > > > abstract and related to real-world signals - which is handy at the least. > > > > Let's see then, here is a "constructive" approach to the topic: > > > (I will likely use the "engineers" convention of "believing in" Diracs > > > either explicitly or implied). > > > BE AWARE: the functions mentioned are only discrete when so defined!! > > > > If one takes an infinite continuous function and samples it then I guess > > > we should say that it should start out "bandlimited" just to be safe > > > (although I'm not sure that latter caution is necessary here - see > > > Footnote #1). > > > > Now, we compute the Fourier Transform of this function > > > f(t) > F(w) ... continuous/infinite > > > > And, we compute the Fourier Transform of the sampled version: > > > f(nT) > F'(w) with F'(w) continuous/infinite > > > And, we recognize that the Fourier Transform in this case can be > > > simplified into a discrete (infinite) Fourier Transform .. that is the > > > integral becomes a sum over the discrete samples in time. �But that's > > > only a trivial simplification so far so we still have: > > > f(nT) > F'(w) with F'(w) continuous/infinite > > > > Now I will assert that our continuous/infinite F'(w) is periodic with > > > period 1/T. [If this assertion is warranted, I could use some help with > > > that right now]. > > > > We can now compute the Inverse Fourier Transform of F'(w). �And, because > > > F'(w) is periodic, we recognize that the Fourier Transform can be > > > simplified from an infinite integral to a finite integral over one > > > period which becomes a finite (not discrete) Fourier Transform which we > > > recognize as the computation of the Fourier Series coefficients which > > > should be the same as f(nT). �But just to be careful, let's call this > > > f'(nT), OK? �[see Footnote #1] > > > > At this point we have infinite, discrete f'(nT) and continuous, periodic > > > F'(w). �And, so far, I think this is in a context that we can all > > > understand. > > > > But wait! �Having continuous F'(w) is really inconvenient isn't it? > > > And, having infinite f'(nT) is also really inconvenient isn't it? > > > What we'd really like is for f'(nT) to be finite. > > > And, what we'd really like is for F'(w) to be discrete so we can > > > represent it with numbers instead of some mathematical functional > > > expression. �Where to start? > > > > If we time-limit f'(nT) we also convolve F'(w) with a sinc. �So that > > > introduces spectral spreading or a type of aliasing. > > > > If we sample the infinite F'(w), we make f'(nT) periodic. > > > > Of course, in the end we want to do both but I wonder if folks don't > > > often think of this as being one or the other - or just don't think > > > about it at all? > > > > OK. �So let's start out by sampling F'(w). �In order to avoid aliasing, > > > we would like to pick the frequency sample interval W and in order to > > > avoid temporal aliasing or overlap at all, we need the extent of f'(nT) > > > to be less than 1/2W. > > > > So, it appears that making F'(w) discrete and making f'(nT) time-limited > > > really amount to the same thing. �We have to accept some spectral > > > spreading if indeed f'(nT) starts out being infinite and we have to > > > accept some temporal aliasing if F'(w) is going to be sampled. > > > > What has happened of course is that we all accept these potential > > > "problems". > > > - In fact, we don't encounter temporal aliasing because we *never* start > > > with an infinite f(nT) in the real world. �This means that the samples > > > f'(nT) are perfect over the interval NT and that F'(w) is a perfect > > > mapping that can be inversed (still continuous, peridic here). �We only > > > have to deal with the potential for temporal aliasing when doing > > > circular convolution. �This aspect aligns with Dale's view. > > > - And we're all used to dealing with the spectral spreading caused by > > > time limiting f'(nT) to the range of n being limited to N. > > > > So, let's NOT start out by sampling F'(w) then. > > > Let's start by time-limiting f(nT) with the range of n being limited to > > > N. �That shouldn't bother anyone too much because it's what we almost > > > always do anyway! > > > Now that f'(nT) is time-limited in the normal fashion, we can consider > > > sampling F'(w). �We just need to pick the sample interval W. > > > > Well, we have decided that f'(nT) is going to be time-limited already. > > > And, we don't want to cause temporal aliasing or overlap by sampling > > > F'(w) too sparsely. �What is the sample interval that will *just* avoid > > > such overlap? > > > If the length of f'(nT) is NT then the temporal period introduced by > > > sampling F'(w) must be >= NT. �[And, as we discussed recently, this > > > means that the period is of duration => NT and is, if you will, > > > "spanned" by at least N+1 samples where the end samples are equal). > > > This means that W=>1/NT and we normally choose W=NT. > > > > Thus sampling, we have: F'(kW) = F'(k/NT) where F'(w) was already > > > periodic over 1/T so we have a range of k limited to N. > > > Having sampled F'(w), we now have a periodic version of the assumed > > > time-limited f'(nT) which we'll call f''(nT). > > > > Thus, we have taken an acceptable time-limited sample in time and > > > converted it to a sampled and periodic "time function" in order to also > > > be able to have a sampled "frequency function" which happens to also be > > > periodic. > > > > As above, one could decide using Dale's framework that there is a finite > > > sequence in time ... which is something that we're all very used to > > > anyway ... and it has a corresponding finite sequence in frequency. �I > > > see no big problem with that but it's not the way I like to *think* > > > about it. �And, I've been prone to saying that something *is* periodic > > > when perhaps I should say that *I prefer to think of it* as periodic. > > > There's certainly a connection to the literature, physical systems, etc. > > > > I see very little difference looking at a finite sequence on a line and > > > mapping that sequence over the finite length into a circle. �One can > > > choose to traverse a function plotted on the circle just once (which is > > > equivalent to it being finite on an infinite line) or consider it to be > > > representative of a periodic function and traverse the circle > > > continuously. �There is precedent for this: > > > In antenna and array design we can plot the beam pattern as a periodic > > > function of the look angle in a polar plot OR we can plot the beam > > > pattern as a function of an infinite-ranged look angle. �The historic > > > van der Maas function for antenna patterns was done on the latter and > > > has infinite extent. �In this case the antenna is continuous of finite > > > length and the beam pattern is continuous and infinite. �Then we get > > > into terminology like "the visible region" etc. etc. > > > > Fred > > > > Footnote #1: > > > If f(t) is *not* strictly bandlimited to B < 1/2T: > > > then the computation of continuous/infinite F'(w) will involve some > > > overlap / "aliasing". > > > Thus, the Inverse Fourier Transform of F'(w) will not match the original > > > f(nT) so we call it f'(nT). > > > > I think this must be what Rune was referring to re: linearity..... > > > > Anyway, given f'(nT) we can compute its Fourier Transform to get F'(w). > > > So, now we have a consistent transform pair. > > > f'(nT) and F'(w) > > > but f'(nT) here is no longer necessarily a perfect replica of anything > > > that may have existed at the "beginning". > > > > Footnote #2: > > > When we're dealing with real-world signals there's no such thing as > > > strictly bandlimited nor infinite extent including infinite periodic. > > > But there is such thing as strictly time-limited which analytically > > > means infinite bandwidth. �We have to live with the discrepancy here and > > > do so by accepting effective time spans and effective bandwidths.- Hide quoted text - > > > > - Show quoted text - > > > Fred, > > > Thanks for your summing up the two different philosophies here. > > > I think one issue is viewing the DFT as some sort of limiting form of > > a Fourier Series which in turn can be derived from Fourier Transforms. > > A program to calculate a DFT does not perform > > ... > > read more �- Hide quoted text - > > - Show quoted text -Chris, I agree that you can derive fancy ways of arriving at the DFT from FTs. But since that is not the only way to get there, then what we don't know is if the properties associated with the FT get carried forward to the DFT. Sure in some cases they do to some extent. I have and like O. Brigham's book. But if one looks at a Linear Algebra approach, you will see that DFTs have complete sets of basis vectors so nothing is left out of their representations and you can DFT random or other non bandlimited data. And Linear Algebra doesn't require sampling or bandlimiting to achieve its results. So bandlimiting is not required for DFTs. And sampling only comes in if you are sample something. Some data by its nature is discrete to begin with and we can DFT that data. I think what needs to be taught are the assumptions and most importantly when they do and don't apply. Clay
Reply by ●January 13, 20112011-01-13
> > And Clay, in your earlier post, which one of us were you saying is the > cow? :) >I wasn't saying anyone was bovine. But we certainly don't want all to be ovine, and that is helped along by some who are caprine ;-) Clay
Reply by ●January 13, 20112011-01-13
On Thu, 13 Jan 2011 08:27:46 -0800 (PST), Clay <clay@claysturner.com> wrote:>On Jan 13, 4:02=A0am, Chris Bore <chris.b...@gmail.com> wrote:,,,>> >> > Thanks for your summing up the two different philosophies here. >> >> > I think one issue is viewing the DFT as some sort of limiting form of >> > a Fourier Series which in turn can be derived from Fourier Transforms. >> > A program to calculate a DFT does not perform >> >> ... >> >> read more =BB- Hide quoted text - >> >> - Show quoted text - > >Chris, I agree that you can derive fancy ways of arriving at the DFT >from FTs. But since that is not the only way to get there, then what >we don't know is if the properties associated with the FT get carried >forward to the DFT. Sure in some cases they do to some extent. I have >and like O. Brigham's book. But if one looks at a Linear Algebra >approach, you will see that DFTs have complete sets of basis vectors >so nothing is left out of their representations and you can DFT random >or other non bandlimited data. And Linear Algebra doesn't require >sampling or bandlimiting to achieve its results. So bandlimiting is >not required for DFTs. And sampling only comes in if you are sample >something. Some data by its nature is discrete to begin with and we >can DFT that data.One can break it down further to the idea that each bin is computed as the dot product of the input and the basis function for that bin. That's essentially a correlation operation, and the linear algebra matrix multiplication just implements a correlator bank. That fact and the fact that the basis functions are orthogonal helps one to construct a foundation for interpreting the output. So there are lots of ways to look at this, and for the most part they don't contradict each other. I think that's a beautiful and interesting thing about it, that it can be viewed many different ways and each can potentially reveal its own insights. Since many people have different ways of thinking about things, and FTs and DFTs and all that stuff are not necessarily easy to comprehend for a student, the idea that there are multiple ways to gain a reasonable understanding of the fundamentals is, IMHO, a good thing.>I think what needs to be taught are the assumptions and most >importantly when they do and don't apply. > >ClayThat's a key point, as it is with many things. Eric Jacobsen Minister of Algorithms Abineau Communications http://www.abineau.com
Reply by ●January 13, 20112011-01-13
On 1/12/2011 6:13 PM, Rune Allnor wrote: ....snip.....>> Dale repeats his point about the DFT being (in my terms) an abstract >> thingy that is simply a mapping of N points - having nothing whatsoever >> to do with any imagined or real samples which may exist outside the >> sample regions. > > I garee with Dale on this.***And so did I Rune. I just gave Dale credit for illuminating that viewpoint.> >> This is a surely a valid perspective of the FT as a >> *mapping* but I've not reconciled how it fits in my own perspectives. > > It's the *only* perspective of the FT, in any of its > shades, shapes or forms. >***I'm disappointed to hear that there is only a single viewpoint or framework possible. That seems too restrictive in human thought.> >> If one takes an infinite continuous function and samples it then... > > Why do you bring sampling into this? We were discussing > the FT up till this point, not sampling.***If I had not intended to discuss sampling then I guess I wouldn't have brought it up. But I did. It seems obvious to me. Surely if we can discuss abstract mathematical relationships then we can discuss the conversion from a particular continuous function to a discrete sequence, no?> >> OK. So let's start out by sampling F'(w). > > Keeping in tune with your 'practical' approach: How do > you 'sample' F(w)? What kinds of ADCs work in frequency > domain? >***Perhaps there is a flaw in the fabric.... Well, let's see: Dale mentions that he worked for companies who made spectrum analyzers. Many of those devices generate the spectrum as a continuouss function. And, I'll bet, that some of them either started with sampled data to generate that continuous function. And, I will assert, that if I want to notionally sample that continuous function then I may do it in my head at least. The *domain* really doesn't matter. Let's not take the notion of being practical all the way to "it must be demonstrated to exist physically in order to discuss it". ***What should one conclude from this comment? One can compute the FT of a continuous function/signal or of a discrete sequence. The result is a continous function. Are we to conclude that it is somehow unreasonable or improper to imagine sampling this continuous function? Whey is that any more unreasonable than imagining sampling a time function?> All you have achieved is to swap a quagmire for quick sand.***I'll leave this to others to comment on. ***What I attempted to do was: 1) create a framework for *discussion* along the lines of something that makes a lot of sense to me. The intent was to unify related thoughts and to point out possible differences on the way. 2) seek constructive comments regarding what I may have left out, jumped over, stated really incorrectly, etc. I *tried* to say when things were fuzzy in my mind and did ask for help. It appears that others have agreed by citing references of similar discussions. So far, I've not seen comments about how to improve the discussion I presented. It seems to me that this would be useful. It's presented in a step-by-step fashion so that any step might be dismissed (with rationale I'd hope) or improved or embellished on. For example, it seems like you have tried to dismiss sampling a continuous function in frequency. I welcome that. But where's the rationale? Fred
Reply by ●January 13, 20112011-01-13
On 1/12/2011 6:13 PM, Rune Allnor wrote: ....snip.....>> Dale repeats his point about the DFT being (in my terms) an abstract >> thingy that is simply a mapping of N points - having nothing whatsoever >> to do with any imagined or real samples which may exist outside the >> sample regions. > > I garee with Dale on this.***And so did I Rune. I just gave Dale credit for illuminating that viewpoint.> >> This is a surely a valid perspective of the FT as a >> *mapping* but I've not reconciled how it fits in my own perspectives. > > It's the *only* perspective of the FT, in any of its > shades, shapes or forms. >***I'm disappointed to hear that there is only a single viewpoint or framework possible. That seems too restrictive in human thought.> >> If one takes an infinite continuous function and samples it then... > > Why do you bring sampling into this? We were discussing > the FT up till this point, not sampling.***If I had not intended to discuss sampling then I guess I wouldn't have brought it up. But I did. It seems obvious to me. Surely if we can discuss abstract mathematical relationships then we can discuss the conversion from a particular continuous function to a discrete sequence, no?> >> OK. So let's start out by sampling F'(w). > > Keeping in tune with your 'practical' approach: How do > you 'sample' F(w)? What kinds of ADCs work in frequency > domain? >***Perhaps there is a flaw in the fabric.... Well, let's see: Dale mentions that he worked for companies who made spectrum analyzers. Many of those devices generate the spectrum as a continuouss function. And, I'll bet, that some of them either started with sampled data to generate that continuous function. And, I will assert, that if I want to notionally sample that continuous function then I may do it in my head at least. The *domain* really doesn't matter. Let's not take the notion of being practical all the way to "it must be demonstrated to exist physically in order to discuss it". ***What should one conclude from this comment? One can compute the FT of a continuous function/signal or of a discrete sequence. The result is a continous function. Are we to conclude that it is somehow unreasonable or improper to imagine sampling this continuous function? Whey is that any more unreasonable than imagining sampling a time function?> All you have achieved is to swap a quagmire for quick sand.***I'll leave this to others to comment on. ***What I attempted to do was: 1) create a framework for *discussion* along the lines of something that makes a lot of sense to me. The intent was to unify related thoughts and to point out possible differences on the way. 2) seek constructive comments regarding what I may have left out, jumped over, stated really incorrectly, etc. I *tried* to say when things were fuzzy in my mind and did ask for help. It appears that others have agreed by citing references of similar discussions. So far, I've not seen comments about how to improve the discussion I presented. It seems to me that this would be useful. It's presented in a step-by-step fashion so that any step might be dismissed (with rationale I'd hope) or improved or embellished on. For example, it seems like you have tried to dismiss sampling a continuous function in frequency. I welcome that. But where's the rationale? Fred
Reply by ●January 13, 20112011-01-13
On 1/12/2011 6:13 PM, Rune Allnor wrote: ....snip.....>> Dale repeats his point about the DFT being (in my terms) an abstract >> thingy that is simply a mapping of N points - having nothing whatsoever >> to do with any imagined or real samples which may exist outside the >> sample regions. > > I garee with Dale on this.***And so did I Rune. I just gave Dale credit for illuminating that viewpoint.> >> This is a surely a valid perspective of the FT as a >> *mapping* but I've not reconciled how it fits in my own perspectives. > > It's the *only* perspective of the FT, in any of its > shades, shapes or forms. >***I'm disappointed to hear that there is only a single viewpoint or framework possible. That seems too restrictive in human thought.> >> If one takes an infinite continuous function and samples it then... > > Why do you bring sampling into this? We were discussing > the FT up till this point, not sampling.***If I had not intended to discuss sampling then I guess I wouldn't have brought it up. But I did. It seems obvious to me. Surely if we can discuss abstract mathematical relationships then we can discuss the conversion from a particular continuous function to a discrete sequence, no?> >> OK. So let's start out by sampling F'(w). > > Keeping in tune with your 'practical' approach: How do > you 'sample' F(w)? What kinds of ADCs work in frequency > domain? >***Perhaps there is a flaw in the fabric.... Well, let's see: Dale mentions that he worked for companies who made spectrum analyzers. Many of those devices generate the spectrum as a continuouss function. And, I'll bet, that some of them either started with sampled data to generate that continuous function. And, I will assert, that if I want to notionally sample that continuous function then I may do it in my head at least. The *domain* really doesn't matter. Let's not take the notion of being practical all the way to "it must be demonstrated to exist physically in order to discuss it". ***What should one conclude from this comment? One can compute the FT of a continuous function/signal or of a discrete sequence. The result is a continous function. Are we to conclude that it is somehow unreasonable or improper to imagine sampling this continuous function? Whey is that any more unreasonable than imagining sampling a time function?> All you have achieved is to swap a quagmire for quick sand.***I'll leave this to others to comment on. ***What I attempted to do was: 1) create a framework for *discussion* along the lines of something that makes a lot of sense to me. The intent was to unify related thoughts and to point out possible differences on the way. 2) seek constructive comments regarding what I may have left out, jumped over, stated really incorrectly, etc. I *tried* to say when things were fuzzy in my mind and did ask for help. It appears that others have agreed by citing references of similar discussions. So far, I've not seen comments about how to improve the discussion I presented. It seems to me that this would be useful. It's presented in a step-by-step fashion so that any step might be dismissed (with rationale I'd hope) or improved or embellished on. For example, it seems like you have tried to dismiss sampling a continuous function in frequency. I welcome that. But where's the rationale? Fred
Reply by ●January 13, 20112011-01-13
On 1/12/2011 12:57 PM, Clay wrote: -> > Fred, > > Thanks for your summing up the two different philosophies here. > > I think one issue is viewing the DFT as some sort of limiting form of > a Fourier Series which in turn can be derived from Fourier Transforms. > A program to calculate a DFT does not perform some other operation > first like finding the Fourier Series or Fourier Transform and then > apply some sort of conversion (simplification or approximation) to > arrive at the Discrete Fourier Transform. It simply performs a finite > number of vector dot products with finite length vectors to find the > DFT directly from the finite length of data. So while one may view a > DFT as a limiting form of another transformation, it is not the other > transformation, therefore not all properties associated with the other > transforms are carried forward. Operationally the DFT performs a > mapping. That is how I look at it. Now as to why one wants the DFT of > a set of data maybe justified by these transform relationships. But > one should be careful as to how they differ. > > Maybe this is my mathematician background coming through, but oh well. > > It is sort of like when one says two triangles are the same. I say no > they are not the same since they are two triangles. We say they are > congruent - everything we can measure about them is the same, but > they themselves are not the same. Maybe it is a form of pedantic hair > splitting. > > ClayClay, thanks. You say:>> I think one issue is viewing the DFT as some sort of limiting form of >> a Fourier Series which in turn can be derived from Fourier Transforms. >> A program to calculate a DFT does not perform some other operation >> first like finding the Fourier Series or Fourier Transform and then >> apply some sort of conversion (simplification or approximation) to >> arrive at the Discrete Fourier Transform.It wasn't my intent to suggest exactly that - although your comment does give me pause for thought. Let's see if I can express it reasonably: 1) First, perhaps I/we should be careful in such a discussion to be clear whether a transformed discrete sequence is finite or notionally infinite. Common terminology is that DFT implies "finite" and DTFT implies "infinite". So I'll be more careful to make the distinction when necessary. 2) I found a rather interesting comment which may or may not apply directly here: "If the expression that defines the DFT is evaluate for all integers k instead of just for k=0,....,N-1, then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N." So, maybe that's all there is to that one point, that some of us prefer to think in terms of that extension....... it seems so trivial to me that I just jump over it. 3) I don't think that a program to calculate a DFT performs some other operation first..... So, that's not an issue. 4) But, what I do believe is this: One can perform a regular infinite Fourier transform on a finite, discrete sequence for what that's worth. The result is an infinite, continuous function which is also periodic. I mention this only as a point of reference. Then it can be simplified from the integral transform into a finite sum. The result is the same. (I told everyone I was going to allow Diracs without further discussion). And, the finite sum is the same as one gets by constructing a continuous, infinite, periodic function with the sequence being the coefficients of a Fourier Series - well at least within a scale factor. Per Stanford's CCRMA: "We now show that the DFT of a sampled signal x(n) (of length N), is proportional to the Fourier series coefficients of the continuous periodic signal obtained by repeating and interpolating . More precisely, the DFT of the N samples comprising one period equals N times the Fourier series coefficients." But I *really* don't want to argue the inner math stuff as long as the perspectives can be supported thereby. So, in this discussion I'm trying to find where there's something wrong with the constructive description I gave. I'm willing to defend my arguments, hope to not be defensive in doing so and hope to learn something too. Fred
Reply by ●January 14, 20112011-01-14
On Jan 13, 5:04�pm, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:> On Wed, 12 Jan 2011 18:07:44 -0800 (PST), Rune Allnor > > <all...@tele.ntnu.no> wrote: > >On Jan 12, 1:49�am, "steveu" <steveu@n_o_s_p_a_m.coppice.org> wrote: > > >> That would imply that the validity of the answer from the DFT is not > >> dependant on the values outside the length N, which is clearly not the > >> case. > > >With the caveat that it's 3AM, I've pushing 24 hrs, haven't > >cleaned my glasses for days and English is not my native > >language: > > >Are you really suggesting that the computed result of the DFT > >depends on numbers that never enter the computations? > > >Rune > > Hi Rune, > � �You worded your question perfectly.Thanks. I took those precations because I am worried i might have misunderstood Steve's statement. It's not the easiest of phrasings to untangle. Rune
