On Jan 13, 8:31�pm, Fred Marshall <fmarshall_xremove_the...@xacm.org> wrote:> 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.Maybe human thought. What maths is concerned, the mapping idea is the sole POV.> >> 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?We might. But I don' see it as relevant to discussing the mapping properties of the DFT. If we were struggling to understand how well the DFT represents the FT of a CT signal, then yes, sampling is a key issue. But as far as I am aware, we were discussing properties of the DFT.> >> OK. �So let's start out by sampling F'(w)....> For example, it seems like you have tried to dismiss sampling a > continuous function in frequency. �I welcome that. �But where's the > rationale?The rationale is that 'sampling' is a physical activity usually performed by a tangible device like the ADC. All ADCs I am aware of work in time domain. If you want spectra instead of TD you need to convert TD data to frequency domain somehow. For discrete samples one can use the DFT, in which case the FD coefficients are *computed*, not sampled. In CT one can use somes sort of filter. In which case the FD data are *processed*, not sampled. Rune
Appendix A: Types of Fourier Transforms
Started by ●January 10, 2011
Reply by ●January 14, 20112011-01-14
Reply by ●January 14, 20112011-01-14
On Jan 13, 4:27�pm, Clay <c...@claysturner.com> wrote:> 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. > > ClayThat is exactly right and is I thought what I said. :-)
Reply by ●January 14, 20112011-01-14
On Jan 13, 7:31�pm, Fred Marshall <fmarshall_xremove_the...@xacm.org> wrote:> 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>> 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.Whilst I totally agree with this, I do think one can bear in mind that 'signal' processing in many cases does intend to process a 'signal' - which I interpret to be a real-w-rold thing. And such a signal does bring with it assumptions, as we have all agreed. Those assumptions, and the eventual relating of the results of computation back to some real-world thing, cannot be avoided. They are either explicit (great) or implicit in the context. So while I agree that the DFT may be presented as an abstract mapping that has no 'assumptions' outside its own scope, I also suggest that when discussing signal processing as opposed to abstract math, those assumptions must be included. Some choose to do so by suggesting that the DFT itself 'makes' such assumptions, while others prefer to apply the assumptions external to the DFT. Both, I think, are valid and lead to similar conclusions. However, what I do disagree with is the idea that the 'abstract' approach is somehow devoid of context. First, we can show that no logical system can be proven without introducing external assumptions. Second, we can argue that no symbolic framework (including math) can be presented as being independent of an external viewpoint that frames it and provides its context. I think the extremes of the arguments presented here could be categorized (accurately and not insultingly) as 'naive idealist' and 'naive realist'. By which I mean the formal philosophical and logical positions of those names. The problem of trying to argue that one or the other position is true, is that we can show that truth cannot be proven and so we land in an endless debate. I suggest that signal processing is more of a pragmatic than pure abstract field, and so I incline often towards the pragmatic realist approach that what works is useful and that I agree with everybody but I do what I find easiest. Chris --- Chris Bore BORES Signal Processing www.bores.com
Reply by ●January 14, 20112011-01-14
On Jan 13, 7:31�pm, Fred Marshall <fmarshall_xremove_the...@xacm.org> wrote:> 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>> 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.Whilst I totally agree with this, I do think one can bear in mind that 'signal' processing in many cases does intend to process a 'signal' - which I interpret to be a real-w-rold thing. And such a signal does bring with it assumptions, as we have all agreed. Those assumptions, and the eventual relating of the results of computation back to some real-world thing, cannot be avoided. They are either explicit (great) or implicit in the context. So while I agree that the DFT may be presented as an abstract mapping that has no 'assumptions' outside its own scope, I also suggest that when discussing signal processing as opposed to abstract math, those assumptions must be included. Some choose to do so by suggesting that the DFT itself 'makes' such assumptions, while others prefer to apply the assumptions external to the DFT. Both, I think, are valid and lead to similar conclusions. However, what I do disagree with is the idea that the 'abstract' approach is somehow devoid of context. First, we can show that no logical system can be proven without introducing external assumptions. Second, we can argue that no symbolic framework (including math) can be presented as being independent of an external viewpoint that frames it and provides its context. I think the extremes of the arguments presented here could be categorized (accurately and not insultingly) as 'naive idealist' and 'naive realist'. By which I mean the formal philosophical and logical positions of those names. The problem of trying to argue that one or the other position is true, is that we can show that truth cannot be proven and so we land in an endless debate. I suggest that signal processing is more of a pragmatic than pure abstract field, and so I incline often towards the pragmatic realist approach that what works is useful and that I agree with everybody but I do what I find easiest. Chris --- Chris Bore BORES Signal Processing www.bores.com
Reply by ●January 14, 20112011-01-14
On Jan 13, 7:31=A0pm, Fred Marshall <fmarshall_xremove_the...@xacm.org> wrote:> 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 whatsoeve=r> >> 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. =A0I just gave Dale credit for illuminating that > viewpoint. > > > > >> =A0 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. =A0That 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. =A0But I did. =A0It seems obvious to me. =A0Surely if=we> can discuss abstract mathematical relationships then we can discuss the > conversion from a particular continuous function to a discrete sequence, =no?> > > > >> OK. =A0So 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.... =A0Well, let's see: =A0Dale > mentions that he worked for companies who made spectrum analyzers. =A0Man=y> of those devices generate the spectrum as a continuouss function. =A0And, > I'll bet, that some of them either started with sampled data to generate > that continuous function. =A0And, I will assert, that if I want to > notionally sample that continuous function then I may do it in my head > at least. =A0The *domain* really doesn't matter. =A0Let'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? =A0One can compute the FT > of a continuous function/signal or of a discrete sequence. =A0The result > is a continous function. =A0Are 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. =A0The 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. =A0So far, I've not seen comments about how to improve the > discussion I presented. =A0It seems to me that this would be useful. =A0I=t'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. =A0I welcome that. =A0But where's the > rationale? > > Fred>> 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.Whilst I totally agree with this, I do think one can bear in mind that 'signal' processing in many cases does intend to process a 'signal' - which I interpret to be a real-w-rold thing. And such a signal does bring with it assumptions, as we have all agreed. Those assumptions, and the eventual relating of the results of computation back to some real-world thing, cannot be avoided. They are either explicit (great) or implicit in the context. So while I agree that the DFT may be presented as an abstract mapping that has no 'assumptions' outside its own scope, I also suggest that when discussing signal processing as opposed to abstract math, those assumptions must be included. Some choose to do so by suggesting that the DFT itself 'makes' such assumptions, while others prefer to apply the assumptions external to the DFT. Both, I think, are valid and lead to similar conclusions. However, what I do disagree with is the idea that the 'abstract' approach is somehow devoid of context. First, we can show that no logical system can be proven without introducing external assumptions. Second, we can argue that no symbolic framework (including math) can be presented as being independent of an external viewpoint that frames it and provides its context. I think the extremes of the arguments presented here could be categorized (accurately and not insultingly) as 'naive idealist' and 'naive realist'. By which I mean the formal philosophical and logical positions of those names. The problem of trying to argue that one or the other position is true, is that we can show that truth cannot be proven and so we land in an endless debate. I suggest that signal processing is more of a pragmatic than pure abstract field, and so I incline often towards the pragmatic realist approach that what works is useful and that I agree with everybody but I do what I find easiest. Chris --- Chris Bore BORES Signal Processing www.bores.com
Reply by ●January 14, 20112011-01-14
On Jan 13, 2:13�am, Rune Allnor <all...@tele.ntnu.no> wrote:> On Jan 12, 8:36�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. > > I garee with Dale on this. > > >�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. > > > 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. > > > 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?Filterbanks? Mechanical reed frequency spectrum analyzers? Tuning forks? Ultrasound beam formers where the probe elements effectively sample the 'angular spectrum (ie frequnecy..) of plane waves' directly? Chris --- Chris Bore BORES Signal Processing> > All you have achieved is to swap a quagmire for quick sand. > > Rune
Reply by ●January 14, 20112011-01-14
On 1/12/2011 11:36 AM, Fred Marshall wrote: In Footnote #1 I said:> 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).That wasn't correct. I should have said: "If one takes an infinite continuous function and samples it, and if f(t) is *not* strictly bandlimited to B < 1/2T then the sampled version f(nT) will generally introduce some frequency overlap / "aliasing"." ....the Inverse Fourier Transform of F'(w) *will* match the original f(nT).
Reply by ●January 14, 20112011-01-14
On 1/13/2011 8:27 AM, Clay wrote:> ....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, Very good point. I find the comment "doesn't require bandlimiting" interesting and (if one is in the context of continuous signals for starters) might add: "because it's already too late for that" or "because the sequence is what it is". It follows from "doesn't require sampling". If one *does* start with a continuous function and samples it, one that isn't bandlimited, then one gets a sequence which has within it some inherent aliasing relative to the original (except for some nice special cases, depending on how one looks at it). Transforms work fine forward and inverse on this sequence. ... which is your point I believe ... and applies to DTFTs as well as DFTs. Fred
Reply by ●January 15, 20112011-01-15
On Jan 14, 10:17�am, Chris Bore <chris.b...@gmail.com> wrote:> On Jan 13, 2:13�am, Rune Allnor <all...@tele.ntnu.no> wrote:> > Keeping in tune with your 'practical' approach: How do > > you 'sample' F(w)? What kinds of ADCs work in frequency > > domain? > > Filterbanks?Nope. Filters work on temporal samples.> Mechanical reed frequency spectrum analyzers?Those use a mechanical filter prior to the ADC that records the temporal signal.> Tuning > forks?Temporal signal.> Ultrasound beam formers where the probe elements effectively sample > the 'angular spectrum (ie frequnecy..) of plane waves' directly?Those are a collection of senors that each record a temporal signal. Rune
Reply by ●January 15, 20112011-01-15
On Jan 14, 10:11�am, Chris Bore <chris.b...@gmail.com> wrote:> On Jan 13, 7:31�pm, Fred Marshall <fmarshall_xremove_the...@xacm.org> > wrote: > > > > > > > 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 > >> 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. > > Whilst I totally agree with this, I do think one can bear in mind that > 'signal' processing in many cases does intend to process a 'signal' - > which I interpret to be a real-w-rold thing.You might interpret that as you which, but once you do, you effectively limit yourself from exploiting efficient methods to extract information from data. Even data that originate in the Real World. I have more than once told the story of how I developed a passive sonar detector that worked orders of magnitude better than the usual stuff - my initial simulations indicated 10-12 dB better detection indexes - before doing any of the fancy or elaborate stuff. Of course, I had used *maths*, not *intuition* or *analogies* to arrive at my algorithms, which meant that none of those who ought to have had a huge interes in my results were anywhere near capable of understanding what I had done. But of ocurse, if you are content (proud, even...?) with passing on your own mediocricy, then there is little I or anyone else can do about it. Rune
