can anybody explain a fourier transform to me in REALLY simple terms? it’s been three years since my last calculus class, and i am very rusty. i have tried looking this up, but all encyclopedias etc. make me feel real dumb what with their big words and esoteric jargon.
a little help folks?
Well, It’s been 28 years since my last calculus class, but as I recall, a simple way of expressing what a Fourier transform is would be to think of it as spreading a function out in the frequency domain. For example, a Fourier transform of a function that represents a squarewave would be an infinite series of odd harmonics with decreasing amplitude co-efficients.
A continuous function can be represented as a sum of sinusoids. The Fourier transform is the amplitude of all the sinusoids you’d need to add together to make that function.
Your ears perform a Fourier transform. If a sound wave is vibrating very fast, you hear a high pitch. If it is vibrating more slowly, you hear a lower pitch.
It’s been over a Month since My last two courses which involved heavy Calculus ended, but I’ve been brushing up since then, and I’d like to give it a shot. I’m not going to tell you everything about a Fourier transform, but I hope this lets you understand the basic concept.
Do you know what a Taylor Series is? With a Taylor Series, you can write a function which is not usually defined in terms of polynomials, in terms of an infinite sum of polynomials. The simplest example:
e[sup]x[/sup] = 1 + x + x[sup]2[/sup]/2! + x[sup]3[/sup]/3! + x[sup]4[/sup]/4! ···
A Fourier Series is a similar idea, except it only works on periodic functions. Do you know what a periodic function is? One that repeats itself, like Sine or Cosine. The theory behind Fourier transforms is, that a periodic function which is not usually defined in terms of sines and cosines, can be written in terms of an infinited sum of sines and cosines. The example Diver was talking about is a good one, but I’ll have to do it from memory, so I’m bound to get it wrong. (Someone please correct me on this.) The squarewave function is a periodic function that alternates between having a value of +1 and having a value of -1. It’s like a squared-off sine wave, looking something like this:
___ ___ ___
___ ___ ___
It can, however, be represented like this:
Sin(x) + Sin(3x)/3! + Sin(5x)/5! + Sin(7x)/7! ···
There are some functions, that when transformed into a Fourier Series, become more workable. I hope that helps some.
Once again, I’m apologizing for my hasty formula-spewing. That should read:
Sin(x) + Sin(3x)/3 + Sin(5x)/5 + Sin(7x)/7 ···
No factorials involved. And, to explain it a little better, I should point out (if you don’t remember this from your Math courses) a couple of terms. These Sin() functions are called harmonics because they all take the sine of values (x, 3x, 5x, 7x) which are multiples of the same thing, in this case x. They’re called odd harmonics, of course, because they’re only the odd multiples. The amplitudes of these functions are, in this case, 1, 1/3, 1/5, 1/7, and so on. For sine and cosine functions, the coefficient is the amplitude. So, this series can be expressed, as Diver put it, as “an infinite series of odd harmonics with decreasing amplitude co-efficients”. Simple, no?
Two other interesting tidbits to throw into the mix here:
First off, if you apply a Fourier transform to a Gaussian distribution (the “Bell Curve”), you get another Gaussian, but of different width-- Wide distributions turn into narrow ones, and vice versa. If I recall correctly, the product of the two characteristic widths is 2.
Secondly, in quantum mechanics, the momentum of a particle can be expressed as being proportional to the Fourier transform of the position (remember, nothing’s definite in QM, it’s all distributions). This means that if a particle’s position distribution is very narrow (i.e., the position is well-known), the momentum distribution will be very wide, i.e., the momentum is not well-known. Voila! Heisenberg’s Uncertainty Principle.
Let’s suppose you created a sort of electric organ in which the sound was produced by a bunch of different tuning forks. Each tuning fork creates a pure tone with a different frequency (you know: “oooooooooooo”.) Each tuning fork also has a dial which controls how loud it sounds.
Now, if you listen to the sound of someone playing a continuous, drawn-out note on a violin, you’ll notice that it has a distinctive resonance and timer which makes it sound very different from the sound of a note from a trumpet or a piano. It also sounds very different from the “oooooo” of a single tuning fork.
Fourier discovered that it’s possible to combine the sounds of some of the tuning forks to make it sound exactly like a violin or any other instrument. A Fourier transform is a mathematical analysis of the sound of an instrument which tells you what combination of tuning forks you would need, and how loud they would have to be.
As has already been mentioned, your ear can tell the difference between a violin and a trumpet because it performs a Fourier transform, breaking up the sound into its individual, pure components.
Bonus information: Fourier transforms are very important in quantum mechanics, because of the whole particle-wave duality thing. This is why you can’t know the position and momentum of a particle at the same time.
-Ben
Superb expositions by all! (And, Ben - are you a teacher? If not, you should be.)
Achernar said “A Fourier Series is a similar idea, except it only works on periodic functions.” If so, and I’m not disputing, then how do you “you apply a Fourier transform to a Gaussian distribution (the “Bell Curve”)” as Chronos discussed?
I don’t know what they are, but the Seti@home software sure computes a ton of Fouriers.
There is a tiny difference between Fourier Series and Fourier Transform. Fourier Series is exactly as Achernar described - you add discrete sine wave components, i.e. start with the sin(x) component, the next component is sin(2x), etc. (or is it 3x?) Fourier Transform is continuous; you need to add everything fron infinitely low frequency (Sin (0.00001 x) and smaller), continuously up to infinitely high frequency components. With a Fourier Transform, you can transform any function - it doesn’t have to be periodic. As long as the function doesn’t diverge at infinity, you can take a Fourier Transform. If it does diverge at infinity (say y=e[sup]x[/sup]) you use something similar, the Laplace Transform, which is altogether much less intuitive but can be used for the same kind of things.
As Cynical said, Seti@home is a good example of why you want to use Fourier transforms. The data you get from a radio telescope is E(t), i.e. how the electrical field varies as time gos on. It’s a little like looking at a buoy or float on the sea and recording the vertical position every second. Now, this motion may seem random, but there may be strong periodic components hidden in it. So you apply Fourier transform, and now you have a plot of signal strength vs. frequency. If you see a sharp spike, it means that a very sharp and narrow sine wave is hidden in the signal. In the ocean, this may be the resonant frequency of the bay, or maybe the vibration from a passing ship’s engine. In a radio signal, it might indicate an artificial radio signal, because natural radio emissions don’t usually produce sharp spikes. (There are exceptions, but they think they can tell the difference.)
The Fourier transform you use here is yet another variant, a Discrete Fourier Transform. This is like a regular Fourier Transform, but you apply it to a limited number of data points. Since the information contained in this data is finite, you only need (you can only get out of it) a discrete number of frequency vs. power data points. No need for an infinte series. The common algorithm used for this is called Fast Fourier Transform, or FFT.
I think there is some confusion between a Fourier series and a Fourier transform. It’s been a while, so I may be way off, but this is the best I can remember:
A Fourier series approximates a function with a series of Sin functions. What you get when you are done would graph to nearly the same thing as the original function.
A Fourier transform is just outside the range of my memory banks, QM and all that groovy math in college being 7 years behind me now. IIRC, it is an operation you perform on a function to give you a different function which represents the coefficients and frequencies of the sin functions in the Fourier series for the function. So, in the case of the square wave it would look like a connect-the-dots version of the right half of a gaussian curve. (?)
As to how you can transform a non-periodic function, I’m pretty much WAGing at this point that it involves complex/imaginary numbers so you can invoke that exponential decay stuff. Or something.
I hope I haven’t embarrassed myself too badly here - geez, I need to go back to school.
Would it make sense to view Fourier transforms as a way of providing a sort of infinite dimensional basis for any continuous function?
The way Fourier transforms work with non-periodic functions is a bit complicated to do, but I can try to explain if I remember it correctly :
Basically, your typical aperiodic function can only go on so long (non-typical ones are those that diverge at infinity as mentioned by scr4.) So you pretend that it’s periodic with a really long period, if you need to. And if no goes out there checking to see if the function’s really repeating, you’ll be fine. It’s a long way to go out there and check, anyways, so I wouldn’t worry. Additionally, you can make things easier by messing with the repeating part of it, since you made that part up anyway. If you make the function even, you get only a cosine series, and if you make the function odd, you get only a sine series. (An even function is a mirror image across the y-axis (f(-x) = f(x)); an odd function is a mirror image that’s also been flipped (f(-x) = -f(x)).
Hopefully that’s not too confusing. Anyways, it’s like cheating, but it works, which is what most math is anyway. The other funky functions that don’t work are those that aren’t piecewise continous (too many discontinuities spoil the series). Even those that come in pieces have to be analysed separately, often more trouble than it’s worth.
And Ben, did Fourier actually mess around with tuning forks? AFAIK, he released this stuff in work on Heat transfer or something like that, not the audio spectrum.
panama jack
This message will repeat in 5000 years.
Sure. It’s just a Laplace transform, except with imaginary numbers. Isn’t that simple? 
Okay, I’ll be serious from now on.
As others have said, the Fourier transform and Fourier series are similar, but not quite the same. A Fourier series is calculated in terms of the period (how long it takes to repeat). If you leave the period as a variable, and then take the limit as it goes to infinity, you’ll get the Fourier transform. So basically, you’re pretending that after an infinite amount of time, the function goes back to the beginning, then does the same thing after another infinite period of time. This is just a metaphor for what you’re calculating, so don’t worry if it makes no sense to you.
Okay, that was rather long. Here’s a shorter explantion:
Bascically, what’s going on is that you’re considering the orignal function to be the result of some combination of waves. The transform represents the strength of each frequency.
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My understanding is that the big debate of the day was how to create mathematical models for vibrating strings. They knew that a vibrating string was a sine wave, but the problem was to model the string starting from the time you pull it with your finger and through all the wiggling it does before it settles down into a real sine wave. Fourier’s brilliant insight was that all their math regarding sine waves applied to all the funny shapes that the string could get bent into, so long as you used enough sine waves.
If I might expand on the Quantum mechanical aspects of all this, de Broglie decided that if light could be a particle in addition to being a wave, then electrons and other particles of matter could be waves in addition to being particles. Schrodinger realized that this meant that you could treat matter with the same math you use in treating guitar strings, and modern QM was born.
-Ben