>On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: > >> "brent" <bulegoge@columbus.rr.com> wrote in message >> news:0fd6f825-e7ad-4642- >a5fe-83de8ff8f7f6@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 itwell. Convolution *is* the way many real systems behave. Its not some arcane mathematical trick. Its the direct mathematical representation of the underlying physical process. How well it fits reality is generally a matter of how much the system is affected by second order effects. This is pretty much like any other area of science and engineering.>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.You must absolutely loath the entire scientific education system. Almost everything is taught as if it obeys relatively simple relationships, and that's pretty much always a first order approximation. Often the higher order elements are so small you can largely ignore them. If you want accuracy, you'd better scrap Newton's laws of motion. If you really want to complain about people being taught about stuff like its an real accurate model, look at the real villans, like how capacitors are taught. The number of engineers who treat them like they are linear devices is truly sad. They demand that the latest silicon can do A/D conversion at high speed with >16 bits precision, and then surround them with tiny surface mount capacitors who's characteristics are bizarrely funky. Steve
Convolution Tutorial
Started by ●December 26, 2009
Reply by ●December 28, 20092009-12-28
Reply by ●December 28, 20092009-12-28
On Sun, 27 Dec 2009 13:04:19 -0800, HardySpicer wrote:> On Dec 28, 8:31 am, Tim Wescott <t...@seemywebsite.com> wrote: >> On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: >> > "brent" <buleg...@columbus.rr.com> wrote in message >> > news:0fd6f825-e7ad-4642- >> >> a5fe-83de8ff8f...@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.If arguments about capacitor nonlinearities are too subtle, try doing this with a 1000 ohm resistor, a 1 microfarad, 50V cap, then plug the assembly into a 120V, 60Hz wall socket. As a thought experiment, of course. -- www.wescottdesign.com
Reply by ●December 28, 20092009-12-28
On Sun, 27 Dec 2009 22:46:35 -0600, steveu wrote:>>On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: >> >>> "brent" <bulegoge@columbus.rr.com> wrote in message >>> news:0fd6f825-e7ad-4642- >>a5fe-83de8ff8f7f6@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. > > Convolution *is* the way many real systems behave. Its not some arcane > mathematical trick.I agree.> Its the direct mathematical representation of the > underlying physical process.I disagree. The _direct_ mathematical representation of a mass being acted on by a force in one dimension in a Newtonian frame is a differential equation, with the order depending on whether you're interested in the mass's velocity or position. 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 differential equation models the reality. The convolution integral solves the differential equation. That's two different things. If you don't believe me, I'll post a convolution integral, and you can reply with the differential equation -- including variable names -- that it was generated from.> How well it fits reality is generally a > matter of how much the system is affected by second order effects. This > is pretty much like any other area of science and engineering. > >>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. > > You must absolutely loath the entire scientific education system.Actually not. My issue here is narrow, and it is that the notion that one is working with ideal models, rather than real systems, is not expressed, much less stressed. Few engineering students grasp that the model is not the reality, and many stumble as a consequence of this lack.> Almost > everything is taught as if it obeys relatively simple relationships, and > that's pretty much always a first order approximation. Often the higher > order elements are so small you can largely ignore them. If you want > accuracy, you'd better scrap Newton's laws of motion.If I want accuracy I'll use Newton's laws of motion, but if I'm going over 1% of the speed of light and I want 14-bit accuracy I'll damn well brush up on my relativity.> If you really want to complain about people being taught about stuff > like its an real accurate model, look at the real villans, like how > capacitors are taught. The number of engineers who treat them like they > are linear devices is truly sad. They demand that the latest silicon can > do A/D conversion at high speed with >16 bits precision, and then > surround them with tiny surface mount capacitors who's characteristics > are bizarrely funky.You merely find a specific example of my overall complaint to harp upon, where I harp upon the underlying problem. -- www.wescottdesign.com
Reply by ●December 28, 20092009-12-28
On 28 Des, 08:48, Tim Wescott <t...@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
Reply by ●December 28, 20092009-12-28
>On Sun, 27 Dec 2009 13:04:19 -0800, HardySpicer wrote: > >> On Dec 28, 8:31 am, Tim Wescott <t...@seemywebsite.com> wrote: >>> On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: >>> > "brent" <buleg...@columbus.rr.com> wrote in message >>> > news:0fd6f825-e7ad-4642- >>> >>> a5fe-83de8ff8f...@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 mathematicaloperation,>>> > 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 iteven>>> does it well. >>> >>> All real systems are nonlinear. The convolution operation is oneway>>> 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 betweenits>>> 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 mein>>> > 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 andnot>>> > 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 keepa>>> 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'reclaiming>>> that real systems convolve their input signals with unfailingaccuracy.>>> >>> 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. > >If arguments about capacitor nonlinearities are too subtle, try doing >this with a 1000 ohm resistor, a 1 microfarad, 50V cap, then plug the >assembly into a 120V, 60Hz wall socket. > >As a thought experiment, of course.That thought experiment misses what is interesting about capacitor non-linearity. Its mostly a breakdown issue. Try looking at the data sheet for some physically small SMD capacitors, and you'll notice some interesting graphs. Things like capacitance vs applied DC voltage. You'll find the capacitance of some devices varies greatly with the applied DC, while that DC is well within the device's ratings. This is definitely not the capacitor you learn about in high school physics. Steve
Reply by ●December 28, 20092009-12-28
On Dec 27, 12:42�pm, Rune Allnor <all...@tele.ntnu.no> wrote:> > 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. >( Insert EE instead of DSP above) I have been thinking about your comment here. I disagree. As I think about it, the people who actually taught me EE were not my professors but the people that wrote the books (and on the job mentors). I would say I pretty much self taught myself everything through the books or learned on the job, even though I have a degree. Now as you got me thinking about it, what EE school did was to let me know what I did not know. I remember Donald Rumsfeld said that the greates dangers are not knowing what you do not know. I certainly became aware of many things that I did not know through EE school. I pretty much left EE school still not knowing much of anything, but at least now I knew what I did not know and have been able to spend the last 25 years with good books, on the job mentors, and more time to learn the things I did not know. So In hindsight, I wonder if the value of the EE degree is that it gives you the credentials that you might have enough brains to actually learn something later and might actually be creative as you tool at your job.
Reply by ●December 28, 20092009-12-28
steveu wrote:>> On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: >> >>> "brent" <bulegoge@columbus.rr.com> wrote in message >>> news:0fd6f825-e7ad-4642- >> a5fe-83de8ff8f7f6@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. > > Convolution *is* the way many real systems behave. Its not some arcane > mathematical trick. Its the direct mathematical representation of the > underlying physical process. How well it fits reality is generally a matter > of how much the system is affected by second order effects. This is pretty > much like any other area of science and engineering. > >> 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. > > You must absolutely loath the entire scientific education system. Almost > everything is taught as if it obeys relatively simple relationships, and > that's pretty much always a first order approximation. Often the higher > order elements are so small you can largely ignore them. If you want > accuracy, you'd better scrap Newton's laws of motion. > > If you really want to complain about people being taught about stuff like > its an real accurate model, look at the real villans, like how capacitors > are taught. The number of engineers who treat them like they are linear > devices is truly sad. They demand that the latest silicon can do A/D > conversion at high speed with >16 bits precision, and then surround them > with tiny surface mount capacitors who's characteristics are bizarrely > funky.I had one guy with a Ph.D. in some electrical branch of physics tell me that the curved line on the schematic representation of a 'lytic was a "mere visual embellishment". To prove that a polar capacitor was a contradiction in terms, he wrote out the defining equation. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●December 28, 20092009-12-28
Rune Allnor wrote:> On 28 Des, 08:48, Tim Wescott <t...@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.I had a difficult time when introduced to convolution through the convolution integral, and that as a sidebar if it's use (in M.Schwartz). I had to get my head around the dummy variable ans all that. After struggling for a while, I worked out the discrete form we all know and love and it came clear. Much later, I taught it to a high-school kid in less than half an hour as the simplest way I knew to answer an esoteric question he had. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●December 28, 20092009-12-28
On Mon, 28 Dec 2009 04:02:44 -0600, steveu wrote:>>On Sun, 27 Dec 2009 13:04:19 -0800, HardySpicer wrote: >> >>> On Dec 28, 8:31 am, Tim Wescott <t...@seemywebsite.com> wrote: >>>> On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: >>>> > "brent" <buleg...@columbus.rr.com> wrote in message >>>> > news:0fd6f825-e7ad-4642- >>>> >>>> a5fe-83de8ff8f...@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. >> >>If arguments about capacitor nonlinearities are too subtle, try doing >>this with a 1000 ohm resistor, a 1 microfarad, 50V cap, then plug the >>assembly into a 120V, 60Hz wall socket. >> >>As a thought experiment, of course. > > That thought experiment misses what is interesting about capacitor > non-linearity. Its mostly a breakdown issue. > > Try looking at the data sheet for some physically small SMD capacitors, > and you'll notice some interesting graphs. Things like capacitance vs > applied DC voltage. You'll find the capacitance of some devices varies > greatly with the applied DC, while that DC is well within the device's > ratings. This is definitely not the capacitor you learn about in high > school physics.True, and I'm not going to argue because that's not my original point. I'm merely trying to make my original point with a sledgehammer, to avoid argument. _All_ real systems are nonlinear. You can't look at a physical system and say "here's a universally accurate linear model of this system. You don't have to look at a physical system and ask "_is_ this system nonlinear". All you need to do, and what you should do every time you set out to model, is to look at a physical system and ask "_how_ and _when_ am I going to care about this system's nonlinearities?". Someday I want to try building a VCO with one of them fancy new caps. Should be interesting. -- www.wescottdesign.com
Reply by ●December 28, 20092009-12-28
On Mon, 28 Dec 2009 11:59:52 -0500, Jerry Avins wrote:> steveu wrote: >>> On Sun, 27 Dec 2009 10:01:07 +0000, invalid wrote: >>> >>>> "brent" <bulegoge@columbus.rr.com> wrote in message >>>> news:0fd6f825-e7ad-4642- >>> a5fe-83de8ff8f7f6@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. >> >> Convolution *is* the way many real systems behave. Its not some arcane >> mathematical trick. Its the direct mathematical representation of the >> underlying physical process. How well it fits reality is generally a >> matter of how much the system is affected by second order effects. This >> is pretty much like any other area of science and engineering. >> >>> 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. >> >> You must absolutely loath the entire scientific education system. >> Almost everything is taught as if it obeys relatively simple >> relationships, and that's pretty much always a first order >> approximation. Often the higher order elements are so small you can >> largely ignore them. If you want accuracy, you'd better scrap Newton's >> laws of motion. >> >> If you really want to complain about people being taught about stuff >> like its an real accurate model, look at the real villans, like how >> capacitors are taught. The number of engineers who treat them like they >> are linear devices is truly sad. They demand that the latest silicon >> can do A/D conversion at high speed with >16 bits precision, and then >> surround them with tiny surface mount capacitors who's characteristics >> are bizarrely funky. > > I had one guy with a Ph.D. in some electrical branch of physics tell me > that the curved line on the schematic representation of a 'lytic was a > "mere visual embellishment". To prove that a polar capacitor was a > contradiction in terms, he wrote out the defining equation.You mean the defining equation of a cap? Leaving out all the nasty non- ideal features that real people have to deal with every day? An electrolytic capacitor is just a really poor diode with deceptive packaging. Nothing is more useful than a practical guy with a PhD, but having a PhD sure doesn't guarantee that practical bent. -- www.wescottdesign.com
