Theo Verelst Local Diary Page 15

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Wed Dec 17 18:26:03     2003

After some efforts, I got around using a modern XP machines' parallel port to exchange data with the outside world. Sort of funny, because that machine also has an internet connection, and a webcam at times, so hardware which is connected to it and can communicate with any piece of electronics can in principle be seen, interacted with,  and activated over the internet.

I made a display (3 times 7 segment) display connected with the parallel port, and for instance made a working prototype where the number of page request of a webserver is shown on it, live, and could (maybe I'll set it up again) be seen over the internet using a webcam with streaming video. Some people watched their own page counts on the server from both ends of the earth. Fun

A light control over a relais I've tried, which also worked fine, except the circuits aren't shielded, so switching on a heavy electricity consumer in its vincinity can make it take on another state... Time for shielded cables and aluminium (foil).

Now I'm into making random numbers with Tcl/Tk and graphs of them, and I'll probably make the random generator based on real random external noise (electronically) work web-wise as a test, see what the interest is. But first some analysis of important fundamental statistics.

The printer port can also read data, and I've connected a very fast AD converter to it, which can read live data in which can be shown on a Bwise canvas, and shown in one of its simulated oscilloscope blocks. A processor prototype based on a fast Z80, and a DMA unit to read in signals at up to 50 MHz is in the making.

Also, I made the combined bwise/database/webserver work again, I probably in some time set it up for internet use with a selected set of  images, which makes live graphical editing of database backed web pages possible, which workes for a database with a few thousand images, and live searching (full search) at normal server rate (though not at hundreds of hits per second...)

A USB AD/DA converter of high quality

For some time I had some sample chips laying around from Analog Devices and Texas Instruments (known by their calculators, too) and I happened to have capies of a Elektuur article (electronics magazine) describing a project of using a recent TI codec chip for a usb (universal serial bus, versions 1 and 2, available on most newer computers) interfaced analog to digital and digital to analog convertor for 16 bits audio signals. The article used a simular chip as I had a sample from (slightly different) and the project got quite excellent overall DA specifications and quite good AD disortion figures. The DA THD (total harmonic distortion) and noise figures, as well as by and large the filtering specs match the quality of the amplifier I built, which cannot be said about cheap soundcards or AC97 codecs and their average use...

So I felt challenged to do my own connect-up-the-sample chips project to make them tick in the intended way as at least a minimal working circuit, and maybe for the first time dtive the amp I built and coupled major speakers with a seriously high quality signal.

That sounds simpler than it technically is, a usb interface isn't trivial, and neither is a high quality DA converter, let alone an even more sensitve (electronically speaking) AD converter, but the chip has got all the digital and most of the analog circuitry built in, so its a nice challenge to put all that to work.

One major prob is in that picture though: the chips are  SMD's (surface mounted devices) with 28 little pins needing to be connected neatly, which are spaced 0.5 millimeter (half a mm) apart...

That's pretty tough to solder, lets say impossible one a one by one basis using normal soldering techniques, which usually works with pins spaced 1/10 of an inch, about 2.54 mm apart, and there are no experimentors printed circuit boards around (much) which have copper connection tracks with 0.5 mm pitch on them, and if they would, it would still be hard to use them.

So I though I'd make an adapter circuit board, where I cut, with a small knife, the little copper tracks out of an all copper surface, extending them to broader tracks in a star around the chip to connect other parts. I used one of those (cheap) small grinding  machines to sharpen a small hobby knife, and started making small cuts between the chips pins while keeping it in place on the copper surface of the unetched board. After removing the chip, the cuts were deepened until they would isolate seperate tracks, and also the inside of the chips' circumference was cut to make the tracks extending from each of the pads connecting with each of the chips' pins electrically seperated.

Then I made the traks bend apart at increasing distance from the chip, in a wafer like prtern, by carefull cutting of the copper surface repeatedly to make the cuts go through the copper layer completely.After a  few tests and failures, I had a seemingly correct pattern, which would fit the chips' pins pattern without loose contact or short circuit, and had cut the broadening copper tracks to the edge of the little board, and tested with a multimeter wether they were electrically completely isiolated from eachother.

I put the chip in place and pressed it on the little copper places to let it make contact, and measured pin for pin wether there was contact between the pins and the pads, and especially if there was no short circuit between the neighbouring pins. After more than a little work, that proved OK, so I could start thinking about soldering the chip in place, and especially which strategy would be appropriate, because normal soldering is out of the question.

A possible SMD soldering technice, apart from special means like conducting glue ribbons and flame burner techniques, is to simply flood all the pins in a bath of soldering tin, and then remove all excessive solder by tin remover lint, or suction litze. Considering my cuts may not be a nicely shaped and of equal depth as decent printed and etched circuit board, I found that too big a challenge, so I used a more carefull variation, putting solder on each of the copper pads first using that techniqe, so that after applying the litze there would be a very thin layer of solder on each of the connection points.

Again I fitted the chip in position and measured if still all 28 pins made contact with the copper tracks and not with any neighbour, and when that chore had been done successfully, I decided to simply solder each pin in order by applying pressure on it with the soldering iron with small tip. After a few minutes, that had worked fine, and measurement confirmed the chip was properly connected...

There are machine who do all this in seconds, but considering I don't own a big lab or production plant, at least I could get my free sample to be usable!

But that is only connecting the chip in a useable way, more electronics are needed to make it tick, so I looked at the typicaly application diagram from the data sheet, and at the article, decided on a circuit, and soldered the clock circuit, the supply capacitors, the coupling capacitors, the audio plugs, the wiring and the USB connector in place in an experimental fashion, and of course was anxious to try the result on a computer.

After a double checking and measuring to see there are no short circuits or other clear errors, I connected the whole board up, to see what would happen, and it sort of failed, but not miserably. The computer said 'unrecognized Usb device' and nothing much happened. Oops.

I measured the voltage on the various pins and signals, which turned out ok, and in the end though the clock might not tick, so I measured the cristal, and when I did that (on purpose) the connected machine suddenly went 'ding dong' and reported a new usb audio device and human interface which it could recognize (as I had read) without further installations being needed.

That was not too strange, the measurement wires and device change the capacity 'seen' by the cristal circuit, so its frequency changes a very little bit, which appearently was needed to be right for the USB spec. I soldered a small correction capacitor in place, which solved the problem, and now the circuit would start up fine when connected, and appear in the equipment list of the computer just fine.

The biggest test now is of course: does the audio work?! I connected a smplifier to the outputs of the DA converter, without extra filtering, and put on a CD, and was pleased to hear music! clicking up a mixer and setting myself in the centre of the speakers I could immedeately hear what I never heard before: a very clear and controlled, seemingly uncolored sound, and a thus far unmatched stereo image, uncomparable with the cheap soundcard stuff.

Next time more about the AD converter.

New Years Day 2004

This all may seem a bit like a scientific sort of yournal, but so what, that's fine too. At least this stuff really works, which I cannot say for some science I've been in touch with in the past...

Some about the AD converter

I've also tested the analog to digital converter, to record a signal on the computer, via this Usb connected converter board, which is a lot better then all others I've tried thus far (for instance an AC'97 chip), it even sounded nice, which is rarely the case, though modern converters of course work fine. Recording a CD or for instance a synthesizer on a normal or standard AD works fine, but not for audiofile application at all. One can use a standard DA on a good HiFi system, that's worth a try, but doomed to fail, but an AD is yet another story.

This one currently isn't quiet, that is some high frequency noise (and probably some hard to notice hum) is present, showing up in a spectrum analyzer program.

Bouncing Bombs

Watching national geographic channel a bit I saw a program on british Lancaster bomber pilots in second world war planning on getting to the nazi (industry) water supply dams by making bombs which are dropped from very low altitude (less then 20 meter) in the vincinity of the dam, to bounce on the surface of the water until they would hit the dam wall, and then after hitting it they would sink, and explode at a set depth under water, with the intent to break the dam.

A little marble experiment was shown to proof that when a water surface is hit by a marble at exactly the right angle, the marble, like a flintstone, bounces back on the water surface a few times before it gets in the water.

To hit the dam, major bombs were planned to first bounce up to it, and then sink because otherwise it would be very hard to get the bombs in the right position to actually blow up the dam.

Outloud: a short video clip of mine on TV

Check it out for yourself, a DivX version of my 'highway to hell' video clip with selfmade music and video is regularly being broadcast on a Amsterdam cable channel, It's not a very popular one as in ratings, I'm sure, but probably has about a million possible viewers, and at least some take it serious enough, according to some information, it is the largest cable channel in the world, whatever that means...

If you have flash installed, you can watch it here.

Demolishion times come and go, but who times them?

I pictured a demolishion project in progress of not so old houses, being broken down for reasons unknown to me.


This doubtlessly with 'modern' technology could be done a lot faster.


Nothing much in there anymore, why bother making such a mess?


Digital signal processing, revisited

I flunked my (better put, one of the) digital signal processing courses in university. As most wil know, I graduated to become an electrical engineer in 90 (or 91 I don't remember by heart), with probably one of the most wide as possible course selections, and in what in that time still was a top level section of electrical engineering, and in fact of the whole university, which als in that time still was amoung the most highly regarded in europe and the world.

I did some courses twice (and occasionally three times) before I eventually passed them, which was weard because I passed easily and with good results some quite hard and advanced ones, but the only one I did exam in three times, and failed all times (...) was called something like 'digital signal processing', and was taught (if that is the word) by the same whatever it actually was as I later had to work under as project leader, while I was responsible for picking the project, and even actually writing the first project proposal, and while I was the only serious leading and pratical force in it, and the only one who got his actual job done, and even in time.

Anyhow, there must have been some great insight, still not understood by anyone much except a small incrowd (who I later worked with myself...) that I failed to see so thoroughly that I simply wasn't permitted to pass the course.

Well, you know, it wasn't that good anyway... Really, the handwritten dictate was almost illegible, it contained mostly well known basis theory about graphs and S-like transforms, and just one or two appendices with indeed interesting  and up to standard applications, without question mainly originating from a scientist who later worked at AT&T labs who I met much later in a lecture of his, and not from the wannabee professor type.

The point should have been a good application and a good summary of the theory and well chosen examples and questions, because other courses completely covered
because I doubt the lecturor had ever written much of a program or even switched the supply of a serious digital signal processing problem or piece of electronics. Himself, I mean. Doubt it. And I'm sure because I was closest-up, first hand personal withness for years of it that projects of that person in the direction of a software project for digital design were completely and utterly unsuccesfull and an expensive failure which never served any real purpose in practice, ever. It contained essential theoretical errors, realy essential ones, considering it was supposed to be a correctnes-by-design type of system which fails provably already with a two nand-gate digital circuit, about as simple as things can get.., and its user interface would crash every ten minutes if you would be lucky, and I know, I had to use it in practice, and it was worse than draconic in its design, complete garbage, unless someone who would know what they were doing (like me...) would redo pretty much everything of it, but then right.

So can I back up that point of view, contentwise, and is that important? Well, it is the one who got me sort of fired after promising me (reasonable, historically speaking) followup project funding (being the logical person in the system to arrange that, and after I had worked with good results for years), and being reference and arranger (logically as project leader) of me going to the US to continue my work. Ended in complete treasury. I wasn't even honoured for the work I'd obviously done, and people were lined up to grab my work and results. Not that that worked, ha, I was just to damned good for that, even by far. But that's the only reason I can produce myself in public without having been betrayed to the core. And holland is not a nice place to be when being above the average is a necessary proof. Jeez, this country sucked that way already as almost a rule of thumb since I remember. 'Be normal, that's crazy enough', yeah right, unless I want it differently, I would hope, that's at least a constituational right of freedom. Or I guess when you're a foreign jesuit who can claim to drag in a lot of indirect catholic money streams, and when you have an acceptable anti-christal attitude of lying, stealing, and betraying. Interesting trading possiblities for the dutch wannabee culture of science reverents, I guess. Bit naive, Verelst? Not realy, I was that good, and I would have won easily in any reasonable country, normally. Honest. Where else in the world would an electrical engineer of my proven record not get job offers by the hand full?

Anyhow, contentwise, the course was about networks, which for every electrical engineering student are compulsary learning in a three semester course, one of the longest of all, from the same section, where I also graduated, about graphs, kirchhof equations, and then the serious stuff that follows, complex circuit matrices (MNA matrices) and their solution and application. In comparison with which the stuff in the aforementioned course is childsplay, and that course I had passed a time before  with results up to 9 (about A). So that couldn't be the problem.

Then the majority of the hard-core digital signal processing is covered by a course called something like circuit and systems, after a book by Papoulis, also part of the obbligatory curriculum which I at the time had passed, tought by the electromagnetics section. Fourier and S transform, Z transform, detailed enough proofs of the transforms and their boundary conditions, and many, many excercises of various types of applications. That should be taken as the main basis for the theory (it concerned Network Theory section at the time..) behind digital signal processing, apart from another standard Ee course for all students in as I remember the second year, information theory, which of course I also would have had to pass, and indeed did.

In information theory, signals and their processing aren't necessarily taken as causal, and various sides of the fourier transform and backtranform are discussed, and of course shannon's theorem is central, where the sampling frequency must supersede twice the maximum frequency of the sampled signal or the fourier transform isn't umambiguous, aliasing noise fill fold back in the signal spectrum and distort the result, and the backtransform wouldn't be well defined and correct anymore. And we here, like in theoretical physics, which is not necessarily taught much to every EE student, we can rewrite the signal in various sample constructed forms, for instance as a composed vector of a infinite row of impulse functions convolved with the to be sampled, continuous signal.

And than we would know that when we either have a frequency spectrum limited input signal or result of digital signal processing function, we can tranform the result back from the frequency domain to a time signal by some function and an appropriate filter or eventually, theoretically using the sync function for every sample to generate the inbetween sample interpolated values as a possibly infinite sum.

All that is important basic EE stuff, incredibly complicated in all its theory and facets, and known to everyone who passes their second year. The 'digital signal processing' course didn't relate all that much to that important fundament, and certainly didn't honour it much, or take its limitations into account. I remember sitting in office later as graduated engineer and getting a sort of tète a tète with that particular person (well, person..) where one of the great truths of the matter would seem to be unveiled a bit, where I was informed that when we relate to filters on signal sequences, the bookkeeping of the additions on a row (in the profession, I know, called the multiply/add components) is important and can be seen as a 'great way', for all really chosen individuals, to 'rrrrule the worlt', well I'm exagerating on the dramatics, but  jeez, I thought I'd see the real light soon or something, which was absolute crap, just fiddling around with the associative and distributive properties of the addition operator doesn't get my scientific imagination juices running, sorry, if that's the great signal I missed and made me not pass my exam I'd prefer to go to court for a completely malfuncting impostor as a professor...

But of course, I knew my first and second year, hell I even knew all my years of preperatory science to become an engineer, even fairly well, and I know that under the Z tranform, the comparable (though not equal!) equivalent of an analog filter (lowpass, highpass and possibly bandpass, depending on your formulation of the problem) is a polynomial, I mean, eh, did he invent something relevant here? Nope, few glances on his PhD thesis (think and pretty unreadable like most other stuff probably for similar reasons to hide real sensible content) and previous suggest the coworkers must have the inspiration, the balls, the knowledge, the scientific leadership and very much the implementation and testing skills, and that co worker could leave the rest,  to him. Lets, see, that leaves us with, eehr, well, uuhh, the civil servant attitude and ambitions (I come from civil servant family, don't start me on that, I knew that before I went to highschool), everything that sucks or distracts, and the REAL inportant things like the errors (I could by heart probably point out a few fundamental ones in key name-making essential artibles, and I mean fundamental, no typoos), the scientific messups, the taking of money for positions never correctly fulfilled, anyhow, anything that could be slightly improved by a sjageraar, except for honorable scientific positions. Maybe to become some second rate teacher, like the spouse could bring in the picture but than for jonger youth than early students, but hey, than you'd at least have to be actually right about the essential subjects, or the students or the school would sue you in the end...

Anyhow, there are others who tried to (of course?) put me down and get me out of the way bad, but I can't agree with the the idea that I must sucker my way into a course which has no good foundation and tries to teach me something essential which is either wrong, irrelevant, or subdued to other logic than normal one. (I did follow and pass, a higher year, heavy course 'mathematical logic'...) Where you go to a course to learn something corresponding to the expected level and esteem of the course, and the books and materials teach you that and challenge you to learn some more or practice with examples, and the exam tests what you have normally learned.

And not like shopping for groceries in a shabby greengrocers when the sales are gone and what's left is starting to rot, where on top of that you only get something at all if you smouze a bit with the greengrocer. Or better put: you go to your local antichrist and only stay alive when you sell your soul to the devil and his own and learn to bow for the pope better.
Instead of learning how to become a respectable and honorable engineer with corresponding knowledge and skill level and a good idea of the current state of affairs in the respective area of science and work.

Naive ? Not realy, one of the major drives in science apart from power and riches is commonly known to be honour, and as any sensible even teacher, let alone (wannabee) professor in one of the most upfront and popular areas of contemporary electrical engineering would know that. I guess, isn't it. Jeez. And every reasonably developed and only moderately morally acceptable person would at least understand honour is typically something you cannot realy steal, and have to deserve. I guess unless you are sort of a nazi, or nazi father/führer wannabee.

Anyhow, lets leave political considerations aside, for now.

And finally, HAR HAR, .... electronics.....

Well, ELECTRONICS, yes yes, all ruling (almost) , more complex than most everything except quantum physics, serious, it realy works, nobody without being serious can debug it, impossible to design for most electrical engineers even, hardcore, big business, world wide absolute top market, good old, powerfull and usefull, diversivied and specialized, digital and analog, electronics.

And I, even officially enough early on, and even long before I entered univisity happened to be an excellent electronician. In highschool earlier classes I'd make working and not so slow, cheap and well-cased analog to digital converters for a microcomputer before that was fashionable or even understood, which I heard much later on were still used by the school I made it for, while at home that was relatively basic hardware compared to what I'd make for hobby. Later on in highschool I'd for a weekend project make a TV signal generator with a game and such, hand built from bare ttl chips and electronical components, that is, seriously (if that rings no bells with you, don't bother thinking you're expert enough to even judge my capabilities). I mean I was good at advanced electronics before I ever entered university, period.

Never worked much with them for years when I left home for a student house, or much after when I worked, because I liked to do non-technical things for relaxation and lets say cultural purposes (like play in a jazz/fusion band), but I didn't need to, I after many years could easily pick up on that knowledge and field, no problem.

I guess just a lot of people didn't realy know that about me, during much of my study and work time, not realy on purpose, and, seriously, I was a electronics practicum assistant from my second year on, which was a year earlier than the official possibility, to physics students a year younger, and never had any problems, and that was known to people, so I guess I was known to know my electronics, but not many of my later acquintances ever saw me do many complicated electronics projects.

I knew what a anti-aliasing filter was before most anyone had a computer, let alone a sound card, I made all that myself, I knew what anti-aliasing noise from wrapped back spectrum parts sounded like, even, also in comparison with a amplitude modulated signal (for instance through a multiplier / ring modulator), and a frequency modulated signal. I could even long before I met the 'teacher' have told him what would be reasonable modulation indices for a frequency modulation of a signal, for instance to keep the bessel function ruled spectrum within niquist rate bounds, before introducing distortion in  the back tranformation part of for instance a pitch track loop. Physicists know bessel transforms, too, as do FM transmitter electronicists with theoretical background. As soon as a sample rate is modulated (at least sinusodially), the resulting spectrum is theoretically unbounded to begin with, which is important and fundamental to know before doing a back transformation, for instance, because that cause aliasing noise, at least. Never heard of in any course.

That probably would solve some 'problems' in previous dsp projects, where it is never taken into account how signals are actually behaved in terms of sampling, resampling, anti aliasing and backtransformation. Interesting things can be said about such subjects, possibly essential to break some peoples carreers, I figure when I imagine and go overa bit what some people's ambitions, thesises and plans are.

In electronics, time varying filters, un-equdistant or multiple-rate sampling, N-th order filters with resonances, sample and hold circuits, integrating circuits, analog to digital converters and v.v., and a lot of derived subjcts including oversampling were hot topics, also for me, long before I entered university, and I would have expected from a digital signal processing course to at least give the appropriate theoretical considerations to go with such considerations, but the writer didn't even leave the suggestion for them or the hope he'd much appreciate the need and relevancies of such subject. But, authority is authority, and the grade giver is per definition higher than the student, isn't it.

I did pitch tracking and real time digital microcomputer based waveform analysis and generation on my hobby table before I entered university or the person finished its thesis or did all to much in speech coding. I knew my stuff, I could teach him, apart from all that wonderfull and powerfull theory which I would have liked to learn also about more advanced subjects in the area, in a more advanced course, but nada, I'm sure there wasn't any theory worth noting, not even the regular ones much. That's bad, but its worse when I'm ruled out of a relevant course by a acedemically uncapable person who decides on God knows what logic, maybe myb grandmother is too jewish, that I just cannot pass such test.

But as I said, electronically speaking without ever having shown it much to the circuits involved, and otherwise too, I'm good, and not affraid to push for an academic comparison and especially proof game where I'm sure a lot of people wanting to claim certain theoretical grounds can be disproven to be fundamentally right, for real, contentwise. I do know quite some of that stuff, aslo provably. I could function quite capably at lets say PhD level theoretical physics level with my knowledge and skills, I see none of them do that, and apart from my officially proven EE skills, in all the necessary areas, not just some vague mathematically tinted ambitious digital signal processing 'ideas', that area, and my quite relevant electronics knowledge ARE fundamental to both the theory, practice, history and scientific history of the area. Even fundamental.

Guys praying daily to the god of hilbert spaces probably don't like very involved fock space computations for their power. And people who probably know nothing at all about century old electronic butterworth filters cannot be expected to redo all of the electronic filter theory successfully just like that. And finally non-physicists who want to talk about the state of non-linear or time varying a dynamical system on the basis of limited theory should be carefull to singlehandedly go against a vast amount of work in such areas in physics for almost a century, and in electronics likewise, and claim significant authority without being knowledgeable about these sciences, they might look a touch retarded, theoretially. And mathematical theory in all relevant progressive areas of science of the last century is probably mostly exclusively led on by very practical considerations from these areas of science, and not concocked up by some self-ascribed heavyweight and self indulgent mathematicians who would not even plead guilty to machiavellianism, though are clearly a lot worse guilty as reactionary, papal, hierarchically organized secretive, super-bourgeous and most other things I guess as certain type of sleezeballs from a long ago officially defeated class of self ascribed illuminated more-equal-than-all-others-per-definition or inherited alledged arian supremacy pig kind.

Well, I got blond hear and sort of blue yes and am of more than good enough built, I can pass that last test. And I can try to be just as pig-hearted and smart as three centuries of still frankenstein hopes inspired satanists and general power greeds. I guess. Looking at the field and trying my contentwise chances, I'd say I got a good shot at theoretically defeating their games, as long as nobody in the environment seems to bother or dare much. Well, I was a top EE student. I had passed amoung the top 10 percent in the shortest possible time my first year, without a familymember in the faculty or area, or some special information points, I just worked hard and learned my stuff. I made the second year in the shortest possible time also, being part of even smaller percentage, and then I worked on university by the side, and was quite active in more than that, for instance in synthesizer programming and playing, electronics projects, and making and researching for software, not part of university curriculum, and in fact also passing an extensive bible course. I mean, trying to write me of as second rate in terms of capabiblities and also in the relevant areas of expertise won't work, even though some later walks of life of mine might seem to have invited that. But then I was making profits and learning from my own company, learning and using professional unix, and becoming top funk/blues/and modern and classical jazz pianist/synthesist, which is outside of reach of most everybody I had to deal with in the section I'm talking about, and in most similar areas. And maybe not the fastest student anymore, but one with one of the most diversivied and relevant and heavyweight courses set around. And one who didn't stay behind his desk but liked also to travel, fix cars and stuff like that, I mean the 'honour' of a civil servant ambition only goes so far.

Lets look at the interesting domain of digital signal processing, and some of the theoretical prerequisites of a looooot of work in that area.

You see, when you are doing signal processing in the digital domain, most peope will at some point or another assume a time discrete signal, and often forgotten, also a quantized signal, meaning AD converted with non-infinite accuracy.

Also, when doing so, the assumtion is that the signal in question is frequency content limited, either by itself, or by applying  pre-sampling anti-aliasing filtering. Those assumtions can be worked with, but it is rare that they in practice are met theoretically, at all, completely. Error analysis which is needed as counter measure is usually completely absent.

Finally, the fourier transformation is an integral from t=-oo to +oo, which needs to be honoured for theoretic 'proofs' which need and use it as  prerequisite or fundamental theorem. That excludes a lot of practical signals from the picture, unless completely proper measures are taken, which is usually not the case. Also, the transformation from the fourier domain back to the time domain is a infinite integral, where depending on the conditioning of the transform the result can easily be a signal which is non-zero from time minus through plus infinity!

Any theoretical proof which doesn't take this into account, errs in the boundary conditions, and is to begin with fundamentally wrong. And useless as the basis for further mathematical proofs.

The well known and widely used fourier transform is a very limited and not theoretically strong example. Any proof or analysis of a system on the basis of it is bound to be theoretically unsound to begin with, unless one knows and defines exactly what one is doing, and understand the difference between a fft related frequency component, and a proof-usable fourier tranform frequency component. Of course I'm not saying the fft is essentially 'wrong' or undesirable or misunderstood, it's a practically quite usefull transform, but has weighing integrals and theoretical properties unlike the continuous or correctly sampled fourier integral with infinite integration interval. And the results can therefor no be taken as useable in the same way as full fourier transforms, and in practical sense be quite different, and suggestive of things many people I'm sure are not very aware of.

To start a theoretical point about this, lets take the common case of wanting to make a fourier transform about a limited piece of a signal, for instance like a window in a fft, but trying to take a complete (not fast) fourier transform. When the signal is non-zero for only within a certain interval, this is easier, but otherwise we may want to cut a piece out of the signal by multiplying it with a time apperture 'block' function of convenient size, 1 within the chosen interval and 0 outside.

This is mathematically perfectly sound, but it has an immedeate and devastating effect for the fourier transform and for electronicists: the result has in almost all cases immedeately an infinite spectrum, and a pretty nasty one, even. So that when we require the signal to be niquist frequency bounded before sampling it and subsequently (computer) transforming it, that condition to begin with is never met anymore, and can therfore not ever be used in any subsequent proof which requires it, including the essential sampling theorem.Oops.

The practical or electronical point of this is that to fourier transform correctly, or sample with sufficient resolution, the signal cannot just be sampled for a short period, which removes long term information from it, which in the normal analysis covers its progression. Apart from the obvious difficulty of detecting frequencies with longer equivalent wavelength than the window we have chosen.

Another 'full blown' solution to this whole problem is to take a transformation basis of spatially confined functions, gaussians, combined with a frequency dependent basis, like in quantum physics, where the whole computation for each regarded measurement sample includes the measurement uncertainly and the combination and the effect of combined spatial and frequency properties. Evidently this is an excellent orthogonizable basis, which in limited form is is similar to the reknown wavelet basis. The advantage of the quantum theory approach is that the whole computation is on the basis of the spatial and frequency properties, right until the answer, so that at the point of the outcome, conclusions about that combination can be drawn.