Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Mr Andrew on 11/11/2007 03:28:25
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What are they? What are they used for?...they seem to pop up a lot in physics.
This thread sprung from the discussion on light quanta at
http://www.thenakedscientists.com/forum/index.php?topic=11068.0 .
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Fourier transforms enable you to analyse non sinusoidal waveforms into their sine wave components.
This can give you an insight into how they are generated and the bandwidth required for their transmission
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Fourier analysis is a very useful and powerful tool particularly in communications but it is improtant to remeber with any form of integral transform process you may be breaking something down into component parts that can be put together to create the object but that does not mean that the parts have any relationship to the real components that made the object in the first place.
For example if I mage a loud bang by dropping a plank on a hard floor in a quiet room and record the waveform and include say a second of silence before the bang and a few seconds after the bang as the sound dies away I can break this noise down into fourier components using continuous sine waves and it will be recreated with as much accuracy as I desire but all the sine waves will continue for the entire period of the window over which I have decided to reproduce the recording it's just that the ones before the bang happens will all add up to silence! Now nature is not like that. Before I dropped the plank there was nothing, not a load of sine waves adding up to nothing! so clearly the model is accurate but it is non causal and not the way the sound was made.
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This is just another instance of Maths interfacing with the physical world. You can't conceive of ' two negative buns' but if you have five buns and then take away two, you will have three left. The (abstract) maths gives you the right answer to the problem by stating it as 5-2=3.
The Fourier transform stands on its own as a mathematical operation. Because it involves integration between plus and minus infinity, it does not correspond exactly to anything physical. BUT it gives you the right answers when you use it properly; just the same way as for all maths.
Before I dropped the plank there was nothing,
But only if you accept causality. The 'about to be dropped' ness might have always been there, along with the growing of the tree and the cutting into planks AND, more contentiously, the intention of the person to drop it.
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BEFORE IBM & UNISIS -
These were what all physics calculations were done with ----
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fhome.vicnet.net.au%2F%7Ewolff%2Fcalculators%2FBurroughs%2FBurroPortable-801-IMG_2533-5.jpg&hash=2c69b2eb587690ea178a7db3b85b7677)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fi38.photobucket.com%2Falbums%2Fe111%2Fgeezer69%2FMechanicalCalculator.jpg&hash=38ba35d955c010bf44c535381b08f0a7)
About the war, Andrew, I don't remember anything. This is a picture of me during the war.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fi38.photobucket.com%2Falbums%2Fe111%2Fgeezer69%2Fbaby.jpg&hash=a39d63fdfd4b1b4294068746eb04441b)
- but in college these were standard fare - the ones I used were a cross between the two designs above, having the handle to enter numbers but the older key board - 10 keys across. These are both 8 key. After a couple of hours you had significant arm strain if you were not a weight lifter.
Doing the math for the transforms on this thing with the aid of a slide rule (below) was more than one could endure if you were of my ilk. But there were people who made their living doing this full time.
nutters I say, nutters.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fi38.photobucket.com%2Falbums%2Fe111%2Fgeezer69%2FPostversalogsliderule.jpg&hash=a7b02447c1d45d22081a8d3458e03acb)
(Andrew - is this a good enough answer for you question in the quanta thread? - Daddy)
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Dear Daddy
Good stuff.
At School, (circa 1960) they bought two 'hand calculators'. They had 'chains' and gears inside and you keyed your numbers in by pulling down on the chains. We were able to do long division - mechanically and without knowing what was going on. Then there were little ones with a handle to crank.
At Uni, (circa 1965) they had a couple of steaming things with motors. (Mc Murdo, too) These when topokatopoka topokata as they did long division. You could get them to 'stick; in that mode but you got shouted at!
We really fancied ourselves with sliderules at School. It certainly kept you on your toes getting the decimal point in the right place. I still have the Faber 'Darmstadt' which my dad passed on to me - boxwood with ivory (type of) scales.
Then, one day, I bought a Sinclair Cambridge calculator; halleluya.
I was not alive (but conceived) during wartime. Hitler just gave up when he heard about it! What price causality?
You looked 'real cute' as they say.
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The Post Versalog II was slide rule of choice here in the States. It is picture above. Bamboo and Ivory. I still have a small, less elaborate version and a K&E aluminum one I keep in my brief case for those occasions out on a well when the generator goes kaput and the power pack in the laptop is out.
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Rather boringly moving back to the original topic, if you add enough waveforms of different phases and amplitudes together you can create any wiggle that you can concieve of. A fourier transform does the opposite of this telling you the amplitudes and phases of the sine waves you would have to use to make a certain wiggle.
Why is this important? For a start it is closely related to how your ears work, they seperate out the sound into it's different pitches or waveforms, so they are used extensively in mp3 compression.
They are also very useful because a lot of systems alter how they behave depending on the frequency of the input, for example if you hit an oven shelf high frequencies will get transmitted from the shelf better than low frequencies. So it is useful to consider the sound as collection of frequencies rather than a vibration.
There are loads of other applications.
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I still have my Faber Castell "engineers guessing stick" (slide rule ) in my office but haven't used it in anger for years now but In the days before calculators and computers I was quite a dab hand. Mind you It's only good to about three figures but that's quite good enough for most practical purposes.
Integral transforms of various types including Fourier, slant and wavelet transforms are very useful for the audio and video data compression that is so commonplace nowadays. Around 15 to 20 years ago when these standards were being researched and set up I was very involved with the process.
I also invented a type of transform that I called the Cochlear transform which I believe fits the process by which the ear separates out the frequencies a lot better than Fourier transforms but unfortunately I could not find a computationally efficient version of it and although the ear modelling experts thought it was very interesting the mathematicions wanted the computational efficiency of FFTs because of the limited resources of the computers that they had to work with.
Big advantage of the cochlear transform is that as well as separating out the frequency components in a waveform it keeps all the timing components exactly in sync and is totally causal. ie no great load of signals adding up to nothing maybe with the improvements in computer power available it might be worthwhile to revisit the idea but everyone's happy with the messy transforms used in their MP3/4 processors now.
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Wow, thanks for the explanations...anyone have the mathematical formula? This sounds pretty neat!
Oh, and on the topic of slide rules, I have my uncle's old one from high school and I think that it's amazing how it works. Too bad mine is a simple one (it doesn't even have log-log scales) [:(]. My calculator ran out of batteries once during a physics test and when I whipped out my slide rule, the teacher paused the test and made me explain how it worked to the class [;D]. I don't think one has even been on our campus for thirty years or more!
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Wow, is right as there are different formulas for different purposes.
See http://en.wikipedia.org/wiki/Fourier_transform
I cannot begin to remember the simplest form so I whip out my geophysics books and pick the one that works best in the situation - or I did this before I had a geophysicist as a partner. I have forgotten most of 'HOW TO' do it. I can now pay other people to do it for me so why should I worry??
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One other neat place Fourier transforms show up is in classical optics. The way a light beam changes as it propagates can be calculated by Fourier transforms. Even more interesting is that if you have a lens, and you place a source at one focal point, the Fourier transform of the source will show up at the other focal point. This can be used to construct the Fourier transform of things without needing to use a computer.
http://en.wikipedia.org/wiki/Fourier_optics#Fourier_Transforming_Property_of_Lenses
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Before Burroughs's, Unisys, IBM we had another method of "simplifying" calculations which was by the use of Mr Napier's new fangled logarithms ( we spent much time at school on this ) this consisted on looking up the 'Log' of the relevent numbers and then division and multiplication etc could been done by the easier process of addition or subtraction and 'Anti logging'.
Slide rules are of course an analogue way of doing this.
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... breaking something down into component parts that can be put together to create the object but that does not mean that the parts have any relationship to the real components that made the object in the first place.
For example if I mage a loud bang by dropping a plank on a hard floor in a quiet room and record the waveform and include say a second of silence before the bang and a few seconds after the bang as the sound dies away I can break this noise down into fourier components using continuous sine waves and it will be recreated with as much accuracy as I desire but all the sine waves will continue for the entire period of the window over which I have decided to reproduce the recording ...
Perfectly true, of course.
However the whole concept of Fourier transforms assumes you're working with a periodic or repetitive waveform (like a continuous tone, trumpet sound, violin etc) - not a one-off crash. Furthermore the waveform should be periodic within the timeslice you analyse - there should be an exact whole-number of wavecycles in the analysis interval (otherwise you get some spurious results).
For analysing arbitrary (non-strictly periodic) signals, you use a "Short Term Fourier transform" which uses a "window function" to gracefully fade out the signal towards the ends of the sampling interval. It's a fudge, but very useful in practice.
I've got a bit of graphical and mathematical info on Fourier transforms etc on my DSP webpage at http://www.techmind.org/dsp/
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Point of order, jp -
Edited quote ----place a source at one focal point, the (spacial) Fourier transform of the[/quote]
spacial arrangement of the source will show up at the other focal plane.
People may be confused, if they are thinking about frequencies and time functions of light waves etc. The time / frequency thing has a parallel in geometry but all the sums are the same. Spacial frequency refers to ' lines per inch' or, more accurately, 'sinewave stripes per inch'.
Put a point at one focus and you get a 'zone plate' image at the other - the well known sin x over x pattern.
The Fourier transform, in fact, should be quoted in its complex form, to be accurate - this means that, rather than just building up an arbitrary waveform with sinewaves, you also need a corresponding set of cosine waves, too, or phase information about your sinewaves. If you don't have the cosines, too or the phases , you cannot get the shape right. It doesn't matter much for audio waveforms, which get so dispersed that the ear more or less ignores phase structure information in order to make sense of what it hears but edges and lines in pictures get distorted, in particular.
Have just read your post s-s.
NO - Fourier Analysis is really what you are describing. The Fourier Transform, in general, is defined f or ANY F(t). When the waveform repeats, the transform gives you discrete harmonics and also allows you to do DST and FFT. But it is a sub-set of the Fourier thing.
Wiki is full of it but it is a bit over the top for non-maths people. But the whole thing is a bit of a mouthful and relies on integral calculus.
To relate to the 'real world' , you have to be careful with FTs and resort to 'windowing' and other 'fudges'. but this is just like a lot of the Maths which is used in Science.
You even have to be careful when solving a quadratic equation, after all. . . .
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I am very familiar with windowing and short term Fourier transforms and agree that in the limit you can do it, but you need to start with your smallest period in the nested transforms extremely short. I was looking for a process that matches the way the ear functions, which is not like a Fourier transform, the best way of describing how the ear works is as an impulse driven active filterbank with peak extraction. I have heard amazing reconstructions of real sounds and music done using just up to six simultaneous sinewave generators. the ear just does not detect the artifacts!
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Perfectly true, of course.
However the whole concept of Fourier transforms assumes you're working with a periodic or repetitive waveform (like a continuous tone, trumpet sound, violin etc) - not a one-off crash.
You are talking about Fourier series, not Fourier transform. When the function is periodic, the Fourier transform semplify in a series instead of an integral; for a generic function you have to compute the integral.
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What are they? What are they used for?...they seem to pop up a lot in physics.
ww.thenakedscientists.com/forum/index.php?topic=11068.0 .
They map you from a "time domain" (oscilloscope trace) representation of something to a "frequency-domain" (spectrum analyser) representation eg. for audio or radiofrequency signals. In diffraction-optics (light and microwaves for example) you can use Fourier transforms to map positions/directions to spatial frequencies (sorry that's probably not very helpful - someone else give me a better definition here!).
Their technological uses are extremely important. Audio (eg MP3) and pictures (JPEG/MPEG) use Fourier transforms or (very similar) Discrete Cosine transforms to analyse and simplify details for data-compression applications. Many many "digital" radio/communication systems (DAB radio, DVB-T terrestrial televison, WiFi, ADSL modems, etc) now use Coded Orthogonal Frequency Division Multiplexing (COFDM) modulation to transmit the data, which is fundamentally dependent on Fourier transforms.
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"if you add enough waveforms of different phases and amplitudes together you can create any wiggle that you can concieve of."
Not quite.
It needs to be monatonic. You couldn't model the sort of wave that surfers seem to like where there's a pile of water with a gap under it, then the sea under that.
Like this rather pretty one here.
http://www.trueartworks.com/poster.php/0000-9521
Can I just throw the word "apodization" in here? it's from the Greek words meaning without feet.
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Agreed BC and that's a serious wave :-) but in a conventional time sequence waves that shape just don't make sense
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Actually techmind, your explanation made perfect sense. Your are in effect taking a function of time and making it a function of frequency, correct? So, over a given time interval (the integration, I'm assuming, has bounds), there are a number of waves with each frequency (the number represented by the new function of frequency). Ah, I see where this fits in with black-body radiation now! "I can see clearly now...."
Wait, so, to find a function of frequency for the light emitted from a black body at a constant temperature, could you measure the EM waves across a point in space and take E(t) and apply a Fourier Transform to it? If you do this at varying temperatures you could empirically derive Planck's Law! Oh Man! Is there a function relating light intensity to the amplitude of electromagnetic waves? Planck's Law is Intensity or Energy Density, right?
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... breaking something down into component parts that can be put together to create the object but that does not mean that the parts have any relationship to the real components that made the object in the first place.
For example if I mage a loud bang by dropping a plank on a hard floor in a quiet room and record the waveform and include say a second of silence before the bang and a few seconds after the bang as the sound dies away I can break this noise down into fourier components using continuous sine waves and it will be recreated with as much accuracy as I desire but all the sine waves will continue for the entire period of the window over which I have decided to reproduce the recording ...
Perfectly true, of course.
However the whole concept of Fourier transforms assumes you're working with a periodic or repetitive waveform (like a continuous tone, trumpet sound, violin etc) - not a one-off crash. Furthermore the waveform should be periodic within the timeslice you analyse - there should be an exact whole-number of wavecycles in the analysis interval (otherwise you get some spurious results).
For analysing arbitrary (non-strictly periodic) signals, you use a "Short Term Fourier transform" which uses a "window function" to gracefully fade out the signal towards the ends of the sampling interval. It's a fudge, but very useful in practice.
I've got a bit of graphical and mathematical info on Fourier transforms etc on my DSP webpage at http://www.techmind.org/dsp/
Seismic processing is NOT of a uniform source of data signal. It is the addition of many seismic reflection signals from different surface locations, the averaging of these and then takes the idea of deconvolution then uses Fourier transform to "clean" the data received for this very non-coherent source.
Stolen from Wiki - http://en.wikipedia.org/wiki/Deconvolution
The concept of deconvolution had an early application in reflection seismology. In 1950, Enders Robinson was a graduate student at MIT. He worked with others at MIT, such as Wiener, Norman Levinson, and economist Paul Samuelson, to develop the "convolutional model" of a reflection seismogram. This model assumes that the recorded seismogram s(t) is the convolution of an Earth-reflectivity function e(t) and a seismic wavelet w(t) from a point source, where t represents recording time. Thus, our convolution equation is
s(t) = e(t) * w(t).
The seismologist is interested in e, which contains information about the Earth's structure. By the convolution theorem, this equation may be Fourier transformed to
S(ω) = E(ω)W(ω)
in the frequency domain.
- end of Wiki quote.
I may be wrong but isn't this the use of Fourier transforms to analyze non-coherent data?
(please note - I pulled out my first, undergraduate seismic reflection geophysics book and it didn't even mention deconvolution. My graduate book did. God, I'm a fossil!)
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Some of you are confusing Fourier transforms, which do not strictly require periodicity, with Fourier series, which does. For example, one can calculate the FT of the unit step function, u(t), which is: U(ω)= πδ(ω) + 1/(iω). I'm currently reading an excellent book on the subject: Dr. Euler's Fabulous Formula, by Paul J. Nahin, ISBN 0-691-11822-1. It stars out discussing Euler's formula for exp(ix)=cos(x)+i*sin(x) and then gradually moves on to Fourier's formulas.
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Can I just throw the word "apodization" in here?
Name - dropper!
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Actually techmind, your explanation made perfect sense...
...Ah, I see where this fits in with black-body radiation now! "I can see clearly now...."
Wait, so, to find a function of frequency for the light emitted from a black body at a constant temperature, could you measure the EM waves across a point in space and take E(t) and apply a Fourier Transform to it? If you do this at varying temperatures you could empirically derive Planck's Law! Oh Man! Is there a function relating light intensity to the amplitude of electromagnetic waves? Planck's Law is Intensity or Energy Density, right?
Errr.... [???]
Maybe - but for measuring blackbody radiation you just use a diffraction-grating (or prism) and measure the energy spectrum directly. We don't have the technology to measure a light-waveform and computationally crank the Fourier transform! You could argue that the diffraction-grating does the Fourier transform for you?
If you were measuring the Cosmic Microwave Background Radiation (CMBR) -at a blackbody temperature of just 3Kelvin- then a computed Fourier transform might be the way to go, I suppose.
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Yeah, I never said that was the best way to do it. Although, maybe we do have the technology...all you would need is an antenna and something to graph current through the wire as a function of time. Isn't that how a radio works, EM waves make the electrons in an antenna oscillate? You'd just need an antenna with very low resistance (superconductor, maybe not so extreme) so that high frequency oscillations can be picked up.
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I's not the resistance of the antenna that's the problem; its having fast enough electronics to measure the signal. We could do it for radiowaves up to microwaves and just about into the mm wave regions but most of the IR, light, UV Xray and gamma ray spectra would elude us (not a problem for the cosmic microwave background obviously).
It's easier to do it differently and use a Fourier transform spectrometer- that will get us up to the UV.
http://en.wikipedia.org/wiki/Fourier_transform_spectroscopy