Convolution Tutorial
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From: HardySpicer on 27 Dec 2009 16:04 On Dec 28, 8:31 am, Tim Wescott <t...(a)seemywebsite.com> wrote: > On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: > > "brent" <buleg...(a)columbus.rr.com> wrote in message > > news:0fd6f825-e7ad-4642- > > a5fe-83de8ff8f...(a)x18g2000vbd.googlegroups.com... > > >>I have created a tutorial on the convolution integral. It uses an > >> interactive flash program with embedded audio files. It is located > >> here: > >>http://www.fourier-series.com/Convolution/index.html > > > You start off by saying that convolution is a mathematical operation, at > > which point I switched off. > > > Convolution is the way that real systems in the real world (such as > > pianoforte strings) > > respond to stimuli that are continuous (such as a sine wave from a > > loudspeaker in close proximity) > > Convolution is _not_ the way that real systems in the real world respond > to stimuli of any sort. Convolution is just a _mathematical operation_ > that _approximates_ what real systems do. Sometimes it even does it well. > > All real systems are nonlinear. The convolution operation is one way to > implement a linear model of a system. Thus, the convolution operation > does not model any real system with 100% accuracy. As a model, the > convolution operation is only as good as the fit between its bedrock > assumption of linearity and the system's actual conformity to linear > behavior. > > For many systems, using convolution is a horribly indirect way to > implement what should be a simple, limited-state, ordinary linear > differential equation. > > > and not just impulses (such as when hit > > with a hammer). I had difficulty with Convolution for years until it was > > explained to me in this practical way at which point it became > > meaningful > > instead of being some arcane mathematical operation which I did not > > really trust. > > > Unless you introduce the student to the practical basis of why you would > > want to undertake such a weird operation, then you might as well give > > up. > > > Mathematical analysis should come after practical experience and not > > before. > > I do agree that mathematical analysis should be kept firmly in the > context of what is real -- when I teach control systems I try to draw > examples from the real world as often as possible, and I try to keep a > clear distinction between the thing you're interested in and the > mathematical model that you've made of it. > > But then, you've already wandered away from reality if you're claiming > that real systems convolve their input signals with unfailing accuracy. > > In today's world I don't think you can ask for practical experience > before theoretical knowledge, though -- with that assumption, engineering > schools would only take technicians who had already been through an > apprenticeship, which severely cuts down on the available candidate pool. > > --www.wescottdesign.com Agreed but it's pretty dammed close. I mean how linear is an R-C network? Try adding two sine waves and passing them through an RC network. What cross-spectral terms to you get percentage wise? I would be interested to know. Hardy
From: Vladimir Vassilevsky on 27 Dec 2009 16:33 HardySpicer wrote: > I mean how linear is an R-C > network? Try adding two sine waves and passing them through an RC > network. RC can have substantial nonlinearity as any real life RC depends on voltage and temperature. Ceramic capacitors are particurlarly bad in this regard. > What cross-spectral terms to you get percentage wise? I would be > interested to know. It depends. How much of distortion do you need? Always check with the datasheets. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
From: Jerry Avins on 27 Dec 2009 16:50 HardySpicer wrote: ... > Agreed but it's pretty dammed close. I mean how linear is an R-C > network? Try adding two sine waves and passing them through an RC > network. > What cross-spectral terms to you get percentage wise? I would be > interested to know. Electrolytics are notoriously non-linear. High-value ceramic capacitors -- calcium titanate, for example -- are also non-linear. Dielectric absorption is a continuing problem with sample-and-hold capacitors. Non-linear resistors abound. Hewlwtt and Packard had the brilliance to use a tungsten resistor to stabilize the amplitude of a Wien-bridge oscillator and founded a company on the parent. (The movie Fantasia was an early user of their tunable audio oscillators.) The components we commonly use are explicitly selected to be adeqquately linear under their normal conditions of use. Most of the world doesn't work that way. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
From: brent on 27 Dec 2009 19:20 On Dec 27, 2:42 pm, Rune Allnor <all...(a)tele.ntnu.no> wrote: > On 27 Des, 11:01, "invalid" <inva...(a)invalid.invalid> wrote: > > > "brent" <buleg...(a)columbus.rr.com> wrote in message > > >news:0fd6f825-e7ad-4642-a5fe-83de8ff8f7f6(a)x18g2000vbd.googlegroups.com.... > > > >I have created a tutorial on the convolution integral. It uses an > > > interactive flash program with embedded audio files. > > > It is located here: > > >http://www.fourier-series.com/Convolution/index.html > > > You start off by saying that convolution is a mathematical operation, > > at which point I switched off. > > Then you have a problem. Like it or not, DSP is applied maths. > > > Convolution is the way that real systems in the real world (such as > > pianoforte strings) > > respond to stimuli that are continuous (such as a sine wave from a > > loudspeaker in close proximity) and not just impulses (such as when > > hit with a hammer). > > It is an *idealized* *representation* of what happens. > > > I had difficulty with Convolution for years until it > > was explained to me in this practical way at which point it became > > meaningful > > Did you pay tuition fees to anyone for teaching you DSP > before that? If so, you might have a law case for them not > delivering what you paid them for. > > > instead of being some arcane mathematical operation which I did not > > really trust. > > Do you trust that 2+2 = 4? Or that you go bankrupt if you > spend more $$$ than your income can sustain? If so you will > have to trust convolution. > > > Unless you introduce the student to the practical basis of why you would > > want to undertake such a weird operation, then you might as well give up. > > Nope. Get new students. > > Abstractions and engineering are hard intellectual work. Throughout > history only a slect small percentage of the population have turned > out to be able to handle such concepts. If people who can not cope > with these kinds of things try to learn DSP, it is *their* problem; > not the subject's. > Abstractions are hard work. For me, I have spent much of my life trying to visualize the math. Once I can visualize it, then I understand it. It is the long process of constructing the model in my head that is hard. My tutorial website is intended to help people visualize the math and concepts of various topics. In building these pages I have been able to also solidify the model and visualization in my head to a much fuller degree. To some degree, I agree with "invalid" about trying to make convolution meaningful in a real world way. In the tutorial I actually stated that convolution needs to be looked at from two distinct points of views, one point of view is the mathematical operation and the other point of view is the practical application. I agree with invalid because as an engineer I ultimately only care about math to the extent that I can get something done with the math, so it is more than a math operation and one needs to know how convolution helps solve real world problems. > > Mathematical analysis should come after practical experience and not > > before. > > No. > I agree with both of you here. Speaking for myself, I want to know how things work and how to design things. The math then becomes a required chore to get to my end goal. The nice thing about hard math is that it makes the barrier to entry more difficult, which then makes me feel all the more special about myself (ok - kidding - not completely kidding but mostly). > > IMHO. > > You are plain wrong. > > Rune
From: brent on 27 Dec 2009 19:33
On Dec 27, 2:31 pm, Tim Wescott <t...(a)seemywebsite.com> wrote: > Convolution is _not_ the way that real systems in the real world respond > to stimuli of any sort. Convolution is just a _mathematical operation_ > that _approximates_ what real systems do. Sometimes it even does it well. > I have read your thoughts on non-linear systems a couple of times. I see what you are saying, but a good understanding of the ideal linear system is first required (not an insignificant hurdle) > All real systems are nonlinear. The convolution operation is one way to > implement a linear model of a system. Thus, the convolution operation > does not model any real system with 100% accuracy. As a model, the > convolution operation is only as good as the fit between its bedrock > assumption of linearity and the system's actual conformity to linear > behavior. In the end, nothing can be modeled with absolute certainty. In the end, statistical processes are going to need to be applied. But that does not take away from the absolute need for good modeling. Part of the engineers job is to determine which model to use. > > For many systems, using convolution is a horribly indirect way to > implement what should be a simple, limited-state, ordinary linear > differential equation. > > > I do agree that mathematical analysis should be kept firmly in the > context of what is real -- when I teach control systems I try to draw > examples from the real world as often as possible, and I try to keep a > clear distinction between the thing you're interested in and the > mathematical model that you've made of it. > > But then, you've already wandered away from reality if you're claiming > that real systems convolve their input signals with unfailing accuracy. > But nobody is claiming unfailing accuracy. > In today's world I don't think you can ask for practical experience > before theoretical knowledge, though -- with that assumption, engineering > schools would only take technicians who had already been through an > apprenticeship, which severely cuts down on the available candidate pool. > For me it has been a steady growth, simultaneously of practical knowledge and theoretical knowledge. They feed off of each other. I agree with what you say, but because you seem to support global warming legislation, I just had to jump in and find something to disagree with you on. brent |