Convolution Tutorial
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From: Fred Marshall on 2 Jan 2010 15:23 Rune Allnor wrote: > On 28 Des, 08:48, Tim Wescott <t...(a)seemywebsite.com> wrote: > >> Note here that I am drawing the same distinction between 'respond' and >> 'behave' that I do between convolution (which _is_ an arcane mathematical >> construct, but it's a damn useful one and not one I sneer at), and a nice >> representative differential equation. > > The *term* convolution, and how it is (not) taught, is the > problem, not the maths as such. > > It is a trivial exercise to derive the convolution sum formula > from only the impulse response and LTI properties of a system. > It ought to be reasonable to expect students to be able to do > this after - literally! - the first week in the first DSP class. > There is nothing more to it than splitting the input signal in > a sequence of scaled and delayed impulses, then expressing the > resulting impulse responses, and at last summing the contributions > from each impulse response to each sample at the output. > > Trivial, once you've figured out how to do it. > > But yes, I know what a hurdle it can be. I only figured out > the derivation a few years ago, some 15 years after I took my > first class on DSP. And - I hate to say it - after being 'inspired' > by our old friend Mr. Bean. > > Rune Yep. It seems that some here are a bit stuck on this. So, I borrow from Rune's phrase:" "...from only the impulse response and LTI properties of a system" I think the point has been made over and over here that a real system may not have LTI properties. An abstract system / model surely can or does. And, whether the real system does or does not is in the eye of the beholder. So, as I cautioned years ago here, we might do well to assert whether we're talking about theoretical situations (models, math, etc.)"system models" or real "systems". A lot of time is wasted debating when that distinction isn't made or isn't made clear. One might venture to say that a "system" (not meaning a system of equations) is a real thing and might be represented (very well or not so very well) by a "system model" which is an abstract thing and generally *not* [exactly] the same as the "system" of interest..... I say "generally" to mean "in the most general case". Often, the model is good enough to represent what we're interested in in the system. Maybe that's the rub, eh? Maybe some really smart comp.dsp dude can find a real system for which a perfect representation is LTI. But with how many caveats / simplifying assumptions will be necessary? I mean that there can be *no* simplifying assumptions allowed in meeting this challenge. Thus .. model / system model. It's interesting how this can be twisted in our peceptions. I once asked my boss which transistor model he used. He responded that he didn't use a model. It became clear after a short conversation that he was using a simple hfe model .. I don't remember exactly what we called it. But, he wasn't familiar with the *notion* although he could use it perfectly well! To many of us the model becomes the reality because it's the only workable understanding we have of the "thing". And then on occasion we're surprised because the simplifying assumptions we've made are limiting. In the olden days of ECAP and Sceptre the computational difficulty expanded rapidly with increased "fidelity" of transistor models - resulting in stiff systems, etc. So, there has always been strong motivation for simplifying assumptions that are acceptable. Fred
From: Chris Bore on 2 Jan 2010 16:17 On Dec 31 2009, 5:49 pm, Randy Yates <ya...(a)ieee.org> wrote: > Jerry Avins <j...(a)ieee.org> writes: > > Randy Yates wrote: > >> Jerry Avins <j...(a)ieee.org> writes: > >>> [...] > >>> Relying on a large brittle flange extending out from a stress raiser > >>> (abrupt change in section) is an engineering sin you wouldn't commit. > > >> So when you asked the question, "Why are manhole covers round?", you > >> expected the answer to be based on such knowledge of materials? > > > The expected answer is "So they don't fall through." Knowledge of > > materials is needed only to deal with nitpicking. :-) > > > This was dealt with here before. > > I don't think it has yet been dealt with properly until now. See my > recent (like, 2 minutes ago) post to Muzaffer Kal. > -- > Randy Yates % "She has an IQ of 1001, she has a jumpsuit > Digital Signal Labs % on, and she's also a telephone." > mailto://ya...(a)ieee.org %http://www.digitalsignallabs.com% 'Yours Truly, 2095', *Time*, ELO Manhole covers here (Surrey, England) are rectangular.
From: Eric Jacobsen on 2 Jan 2010 17:29 On 1/2/2010 1:23 PM, Fred Marshall wrote: > Rune Allnor wrote: >> On 28 Des, 08:48, Tim Wescott <t...(a)seemywebsite.com> wrote: >> >>> Note here that I am drawing the same distinction between 'respond' and >>> 'behave' that I do between convolution (which _is_ an arcane >>> mathematical >>> construct, but it's a damn useful one and not one I sneer at), and a >>> nice >>> representative differential equation. >> >> The *term* convolution, and how it is (not) taught, is the >> problem, not the maths as such. >> >> It is a trivial exercise to derive the convolution sum formula >> from only the impulse response and LTI properties of a system. >> It ought to be reasonable to expect students to be able to do >> this after - literally! - the first week in the first DSP class. >> There is nothing more to it than splitting the input signal in >> a sequence of scaled and delayed impulses, then expressing the >> resulting impulse responses, and at last summing the contributions >> from each impulse response to each sample at the output. >> >> Trivial, once you've figured out how to do it. >> >> But yes, I know what a hurdle it can be. I only figured out >> the derivation a few years ago, some 15 years after I took my >> first class on DSP. And - I hate to say it - after being 'inspired' >> by our old friend Mr. Bean. >> >> Rune > > Yep. > > It seems that some here are a bit stuck on this. > So, I borrow from Rune's phrase:" > "...from only the impulse response and LTI properties of a system" > > I think the point has been made over and over here that a real system > may not have LTI properties. An abstract system / model surely can or > does. And, whether the real system does or does not is in the eye of the > beholder. > > So, as I cautioned years ago here, we might do well to assert whether > we're talking about theoretical situations (models, math, etc.)"system > models" or real "systems". A lot of time is wasted debating when that > distinction isn't made or isn't made clear. > > One might venture to say that a "system" (not meaning a system of > equations) is a real thing and might be represented (very well or not so > very well) by a "system model" which is an abstract thing and generally > *not* [exactly] the same as the "system" of interest..... > I say "generally" to mean "in the most general case". > Often, the model is good enough to represent what we're interested in in > the system. Maybe that's the rub, eh? > > Maybe some really smart comp.dsp dude can find a real system for which a > perfect representation is LTI. But with how many caveats / simplifying > assumptions will be necessary? I mean that there can be *no* simplifying > assumptions allowed in meeting this challenge. > > Thus .. model / system model. > > It's interesting how this can be twisted in our peceptions. I once asked > my boss which transistor model he used. He responded that he didn't use > a model. It became clear after a short conversation that he was using a > simple hfe model .. I don't remember exactly what we called it. But, he > wasn't familiar with the *notion* although he could use it perfectly > well! To many of us the model becomes the reality because it's the only > workable understanding we have of the "thing". And then on occasion > we're surprised because the simplifying assumptions we've made are > limiting. In the olden days of ECAP and Sceptre the computational > difficulty expanded rapidly with increased "fidelity" of transistor > models - resulting in stiff systems, etc. So, there has always been > strong motivation for simplifying assumptions that are acceptable. > > Fred Some systems exist, for the most part, in a linear region of overall behavior enough to make LTI models very workable, down to pretty high orders of accuracy. In communications, for example, some consider AWGN the most aggressive channel model (I don't, but I understand the perspective), and it's been possible for well over a decade to build an entire modulator and demodulator pair, inluding analog IF/RF front ends, hook them back to back with a noise generator in between, and measure performance to within a few tenths of a dB of theoretical. This is for fairly sophisticated digital modems with a lot of processing, using typical "non-linear" elements in the analog electronics operating in their "linear" region. A few tenths of a dB is a pretty doggone small error (and a very respectable figure for "implementation loss"). Most of my career has been in comm, and I've been surprised at how much correlation one can get between simulated performance and the real world if one if just careful about what they're doing. It comes down to what you're saying, though, that one has to understand how far the "linear region" of behavior extends and how to deal with things when the system departs from that region. A lot of the world is pretty linear, enough so that for many cases the amount of error due to the difference isn't consequential. Lots of systems aren't so linear, so YMMV, etc. -- Eric Jacobsen Minister of Algorithms Abineau Communications http://www.abineau.com
From: Jerry Avins on 2 Jan 2010 21:27 Chris Bore wrote: > On Dec 31 2009, 5:49 pm, Randy Yates <ya...(a)ieee.org> wrote: >> Jerry Avins <j...(a)ieee.org> writes: >>> Randy Yates wrote: >>>> Jerry Avins <j...(a)ieee.org> writes: >>>>> [...] >>>>> Relying on a large brittle flange extending out from a stress raiser >>>>> (abrupt change in section) is an engineering sin you wouldn't commit. >>>> So when you asked the question, "Why are manhole covers round?", you >>>> expected the answer to be based on such knowledge of materials? >>> The expected answer is "So they don't fall through." Knowledge of >>> materials is needed only to deal with nitpicking. :-) >>> This was dealt with here before. >> I don't think it has yet been dealt with properly until now. See my >> recent (like, 2 minutes ago) post to Muzaffer Kal. >> -- >> Randy Yates % "She has an IQ of 1001, she has a jumpsuit >> Digital Signal Labs % on, and she's also a telephone." >> mailto://ya...(a)ieee.org %http://www.digitalsignallabs.com% 'Yours Truly, 2095', *Time*, ELO > > Manhole covers here (Surrey, England) are rectangular. How frequently do they need to be fished out of the sewer? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
From: robert bristow-johnson on 2 Jan 2010 23:21
On Jan 2, 9:27 pm, Jerry Avins <j...(a)ieee.org> wrote: > Chris Bore wrote: .... > > Manhole covers here (Surrey, England) are rectangular. > > How frequently do they need to be fished out of the sewer? maybe they just replace them (is an iron foundry nearby?). r b-j |