Azazello wrote: ...> A good book will certainly help but when I was starting DSP a > professor (or someone like a professor), was an absolute must. As > mentioned above, be very careful with equating different transforms to > fundamentally different sets of information. A lot of the classes > that I've taken in DSP functioned on a nearly intuitive level. > Understanding fundamentals is priority numero uno.I learned Laplace transforms originally from a skinny book in the Methuen's Monographs series, and convolution to do the homework in Misha Schwartz's IT, M, & N. (first edition), all with professorial help. Transistors were known then, but hadn't yet reached undergrad academia. Long before that, I learned about Shannon et.al. from a science article in Astounding Science Fiction (before it became Analog, I think). I actually did some successful signal processing with techniques of my own devising to solve an otherwise intractible control problem. (Plating glass beads onto metal sheets; written about a few years ago.) The point of this ramble is that I want to declare that excepting those instances, all of my "someone like a professor" mentors have been respondents here at comp.dsp, which I began to haunt a few years into retirement, and to thank them all collectively. They know who they are. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
CTFT of a discrete signal
Started by ●August 13, 2007
Reply by ●August 14, 20072007-08-14
Reply by ●August 14, 20072007-08-14
On Tue, 14 Aug 2007 21:12:48 +0000, robert bristow-johnson wrote:> On Aug 14, 4:06 pm, Jerry Avins <j...@ieee.org> wisely wrote: >> >> It's perfectly reasonable to sample a continuous /signal/ and a Dirac >> comb is a reasonable way to think or write about that. > > i want to affirm that. in fact, i do not see how Tim establishes the > rules for going between the discrete-time and continuous-time domain > without the use of the weighted dirac comb.Carefully, in a way that has limited application for analysis and that involves some hand-waving. I'm slowly working on addenda for the book, and would like to include a discussion of the dirac delta and it's uses, or a reference to a book that does use it. In fact, the relevant discussion in the book is footnoted with "One can unify the sampling and reconstruction processes with careful use of the Fourier transform. Such analysis can be a useful tool, but would require a lengthy digression from the core content of this book" So the motivated reader can study further. -- snip -- -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Reply by ●August 14, 20072007-08-14
On Aug 14, 8:37 pm, Tim Wescott <t...@seemywebsite.com> wrote:> On Tue, 14 Aug 2007 21:12:48 +0000, robert bristow-johnson wrote: > > On Aug 14, 4:06 pm, Jerry Avins <j...@ieee.org> wisely wrote: > > >> It's perfectly reasonable to sample a continuous /signal/ and a Dirac > >> comb is a reasonable way to think or write about that. > > > i want to affirm that. in fact, i do not see how Tim establishes the > > rules for going between the discrete-time and continuous-time domain > > without the use of the weighted dirac comb. > > Carefully, in a way that has limited application for analysis and that > involves some hand-waving. I'm slowly working on addenda for the book, > and would like to include a discussion of the dirac delta and it's uses, > or a reference to a book that does use it.i hope you don't dwell on that "dirac delta is a distribution and not a function" crap that gets me in fights with worthy mathematicians or with folk like Beanie. for the purpose of signal processing of not- necessarily-continuous but not-pathological signals, the Neanderthal engineering concept of the dirac impulse function is sufficient.> In fact, the relevant discussion in the book is footnoted with "One can > unify the sampling and reconstruction processes with careful use of the > Fourier transform. Such analysis can be a useful tool, but would require > a lengthy digression from the core content of this book" > > So the motivated reader can study further.okay, but i still think that it's easier to explain the connection of the continuous Fourier Transform to the DTFT by use of the dirac comb function. it's just a boneheadedly direct application of an operator that has the effect of discretizing the information in the time domain, and a reasonably simple operation in the frequency domain such that the CFT and DTFT are the same (perhaps needing a simple substitution of variable or scaling of frequency). r b-j
Reply by ●August 14, 20072007-08-14
Tim Wescott <tim@seemywebsite.com> writes:> On Tue, 14 Aug 2007 21:12:48 +0000, robert bristow-johnson wrote: > >> On Aug 14, 4:06 pm, Jerry Avins <j...@ieee.org> wisely wrote: >>> >>> It's perfectly reasonable to sample a continuous /signal/ and a Dirac >>> comb is a reasonable way to think or write about that. >> >> i want to affirm that. in fact, i do not see how Tim establishes the >> rules for going between the discrete-time and continuous-time domain >> without the use of the weighted dirac comb. > > Carefully, in a way that has limited application for analysis and that > involves some hand-waving. I'm slowly working on addenda for the book, > and would like to include a discussion of the dirac delta and it's uses, > or a reference to a book that does use it. > > In fact, the relevant discussion in the book is footnoted with "One can > unify the sampling and reconstruction processes with careful use of the > Fourier transform. Such analysis can be a useful tool, but would require > a lengthy digression from the core content of this book" > > So the motivated reader can study further.Tim, I'm not sure I respect your attempt to make this such an issue. First of all, you act like you're the only mathematician around here and the rest of us need you to enlighten us. Some of us have had some analysis as well. I think we all must admit that there is always more to be learned. Secondly, let me present a gedanken to try to convince you that avoiding expressing a Dirac delta function "out in the open" is not as clearcut as you might think. Let's say that we use the old model of sampling as "modulation" (really just multiplying) of the input signal by the infinite impulse train, but we only do so within the Fourier transform, so that the Dirac delta functions are within an integral. There, now you and the rest of the mathematicians should be happy, right? OK, but guess what - we do get an actual Fourier transform out of this. So this Fourier transform represents some signal. Just becuase the signal is *represented* in the frequency domain doesn't mean that it isn't a signal. It's a "thing" - an entity - and it exists apart from any representation of it. So my question is, what is this signal when represented in the time domain? I'll bet it's a bunch of scaled Diracs "out in the open" ... -- % Randy Yates % "I met someone who looks alot like you, %% Fuquay-Varina, NC % she does the things you do, %%% 919-577-9882 % but she is an IBM." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://home.earthlink.net/~yatescr
Reply by ●August 14, 20072007-08-14
robert bristow-johnson wrote:> On Aug 14, 8:37 pm, Tim Wescott <t...@seemywebsite.com> wrote: >> On Tue, 14 Aug 2007 21:12:48 +0000, robert bristow-johnson wrote: >>> On Aug 14, 4:06 pm, Jerry Avins <j...@ieee.org> wisely wrote: >>>> It's perfectly reasonable to sample a continuous /signal/ and a Dirac >>>> comb is a reasonable way to think or write about that. >>> i want to affirm that. in fact, i do not see how Tim establishes the >>> rules for going between the discrete-time and continuous-time domain >>> without the use of the weighted dirac comb. >> Carefully, in a way that has limited application for analysis and that >> involves some hand-waving. I'm slowly working on addenda for the book, >> and would like to include a discussion of the dirac delta and it's uses, >> or a reference to a book that does use it. > > i hope you don't dwell on that "dirac delta is a distribution and not > a function" crap that gets me in fights with worthy mathematicians or > with folk like Beanie. for the purpose of signal processing of not- > necessarily-continuous but not-pathological signals, the Neanderthal > engineering concept of the dirac impulse function is sufficient. >I won't. I promise. I may put it into a footnote to keep my readers from being mobbed, but that's about it.>> In fact, the relevant discussion in the book is footnoted with "One can >> unify the sampling and reconstruction processes with careful use of the >> Fourier transform. Such analysis can be a useful tool, but would require >> a lengthy digression from the core content of this book" >> >> So the motivated reader can study further. > > okay, but i still think that it's easier to explain the connection of > the continuous Fourier Transform to the DTFT by use of the dirac comb > function. it's just a boneheadedly direct application of an operator > that has the effect of discretizing the information in the time > domain, and a reasonably simple operation in the frequency domain such > that the CFT and DTFT are the same (perhaps needing a simple > substitution of variable or scaling of frequency). >If I wanted to explain that connection you can count on me invoking the Dirac delta function(al). _All_ I wanted to do in the book is explain aliasing, and even there it was mostly in the context of "don't worry about aliasing in a control system, it's different from other DSP apps". -- 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
Reply by ●August 15, 20072007-08-15
Randy Yates wrote:> Tim Wescott <tim@seemywebsite.com> writes: > >> On Tue, 14 Aug 2007 21:12:48 +0000, robert bristow-johnson wrote: >> >>> On Aug 14, 4:06 pm, Jerry Avins <j...@ieee.org> wisely wrote: >>>> It's perfectly reasonable to sample a continuous /signal/ and a Dirac >>>> comb is a reasonable way to think or write about that. >>> i want to affirm that. in fact, i do not see how Tim establishes the >>> rules for going between the discrete-time and continuous-time domain >>> without the use of the weighted dirac comb. >> Carefully, in a way that has limited application for analysis and that >> involves some hand-waving. I'm slowly working on addenda for the book, >> and would like to include a discussion of the dirac delta and it's uses, >> or a reference to a book that does use it. >> >> In fact, the relevant discussion in the book is footnoted with "One can >> unify the sampling and reconstruction processes with careful use of the >> Fourier transform. Such analysis can be a useful tool, but would require >> a lengthy digression from the core content of this book" >> >> So the motivated reader can study further. > > Tim, > > I'm not sure I respect your attempt to make this such an issue. First > of all, you act like you're the only mathematician around here and the > rest of us need you to enlighten us. Some of us have had some analysis > as well. I think we all must admit that there is always more to be > learned. >I'm sorry if it reads that way, that's not how it was intended. Mostly I want the beginners to understand that sampling by the Dirac delta function is the _model_; the _reality_ is numbers in a register. As a model, the Delta function is a great thing. If I do make an issue of anything it is all the places where I was taught the model as reality, then tripped up once I got into the real world. Anyone who's experienced already understands this stuff; I trust you to run anything you read on USENET through your own BS filter. (stuff I more or less agree with snipped) -- 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
Reply by ●August 15, 20072007-08-15
Tim Wescott <tim@seemywebsite.com> writes:> Randy Yates wrote: >> Tim Wescott <tim@seemywebsite.com> writes: >> >>> On Tue, 14 Aug 2007 21:12:48 +0000, robert bristow-johnson wrote: >>> >>>> On Aug 14, 4:06 pm, Jerry Avins <j...@ieee.org> wisely wrote: >>>>> It's perfectly reasonable to sample a continuous /signal/ and a Dirac >>>>> comb is a reasonable way to think or write about that. >>>> i want to affirm that. in fact, i do not see how Tim establishes the >>>> rules for going between the discrete-time and continuous-time domain >>>> without the use of the weighted dirac comb. >>> Carefully, in a way that has limited application for analysis and that >>> involves some hand-waving. I'm slowly working on addenda for the book, >>> and would like to include a discussion of the dirac delta and it's uses, >>> or a reference to a book that does use it. >>> >>> In fact, the relevant discussion in the book is footnoted with "One can >>> unify the sampling and reconstruction processes with careful use of the >>> Fourier transform. Such analysis can be a useful tool, but would require >>> a lengthy digression from the core content of this book" >>> >>> So the motivated reader can study further. >> Tim, >> I'm not sure I respect your attempt to make this such an issue. First >> of all, you act like you're the only mathematician around here and the >> rest of us need you to enlighten us. Some of us have had some analysis >> as well. I think we all must admit that there is always more to be >> learned. >> > I'm sorry if it reads that way, that's not how it was intended. > Mostly I want the beginners to understand that sampling by the Dirac > delta function is the _model_; the _reality_ is numbers in a register.Sorry, I misread you badly then, Tim.> As a model, the Delta function is a great thing. If I do make an > issue of anything it is all the places where I was taught the model as > reality, then tripped up once I got into the real world.Well, yeah. Do you think anyone out of college really expects to see a Dirac generator sitting on the bench? :) -- % Randy Yates % "My Shangri-la has gone away, fading like %% Fuquay-Varina, NC % the Beatles on 'Hey Jude'" %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Shangri-La', *A New World Record*, ELO http://home.earthlink.net/~yatescr
Reply by ●August 15, 20072007-08-15
Randy Yates wrote:> Tim Wescott <tim@seemywebsite.com> writes:...>> As a model, the Delta function is a great thing. If I do make an >> issue of anything it is all the places where I was taught the model as >> reality, then tripped up once I got into the real world. > > Well, yeah. Do you think anyone out of college really expects to see a > Dirac generator sitting on the bench? :)If I am to judge by many of the questions we get here, plenty of people taking advanced degrees think that. Many have never seen a workbench (or a soldering iron) so they have no idea what might be on it. jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●August 15, 20072007-08-15
On Aug 14, 3:12 pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:> NewLine wrote: > > What I would like to do (as a first step in some reasoning) is to take the > > (continuous time) Fourier transform (CTFT) of a discrete 'signal/function'. > > I think (I am trying to proof that to myself) that this should be equal to > > the DTFT of that signal, but I am not sure if that is 100% correct. > > When working in the discrete domain and using the DTFT my signal is just a > > bunch of numbers I think. (or not?) > > However it 'feels' to me that to be able to take the CTFT of that discrete > > signal I should represent the numbers by scaled diracs. > > Among the transform pairs there is > > discrete <--> periodic > > Applying that twice, you get > > discrete periodic <--> discrete periodic > > which is the DTFT.Glen, i think that's the DFT (discrete periodic <--> discrete periodic). the DTFT is aperiodic, discrete time <--> periodic, continuous frequency no? r b-j
Reply by ●August 15, 20072007-08-15
On Aug 15, 12:21 am, Jerry Avins <j...@ieee.org> wrote:> Randy Yates wrote: > > Tim Wescott <t...@seemywebsite.com> writes: > > ... > > >> As a model, the Delta function is a great thing. If I do make an > >> issue of anything it is all the places where I was taught the model as > >> reality, then tripped up once I got into the real world. > > > Well, yeah. Do you think anyone out of college really expects to see a > > Dirac generator sitting on the bench? :) > > If I am to judge by many of the questions we get here, plenty of people > taking advanced degrees think that. Many have never seen a workbench (or > a soldering iron) so they have no idea what might be on it.i can be the first to criticize some academics in EE (i remember a prof, trying to make a simple electro-mechanical servo mechanism, using a pot instead of a shaft encoder for the position indicator, so he could just subtract that voltage from the set point), but i don't think that even academics think a box called a "signal generator" will create infinite voltages. i *do* remember, in school, turning the pulse width to a minimum and using the little spikes coming out to let me visualize the impulse response of some little circuit. but i knew that they were crude approximations to the dirac impulses and that they had their impulse strength measured in volt-sec. i was much more adept with a soldering iron when i was a jr. high school kid and building my Heathkit HW-100. except for recently treating a bad solder joint in my car stereo (that kept dropping out the left speaker, that was a bitch to find), i 've hardly touched a soldering iron since i used to "fatten my mac" (guess which year that was?). so i'm about as bad as the academics. r b-j
