<p>Hey, if you have the time and energy, could you explain the math behind fourier transforms or point me to a good website that does? I understand the concept of breaking down a wave into component trigonometric functions, but I can't wrap my head around the math. I just don't understand how to calculate it by hand or calculator. (As reference, I know math up to differential equations and physics up to calculus based physics.) Thanks :)</p>
<p>You can take about any "Advanced Calculus for Scientists & Engineers" book out of the library and it should be in there.</p>
<p>I think monydad is refering to Greenberg. </p>
<p>Equally good in some cases would be an EE intro signals and systems book (i.e. Oppenheim). </p>
<p><a href="http://www.mathworld.com%5B/url%5D">www.mathworld.com</a> is generally a pretty good math reference website</p>
<p>Bump. I'm currently taking Calc BC but have seen references to Fourier transforms every once in a while (not in class, pretty much just surfing the web). In plain English, what are Fourier Transforms useful for? I checked out the mathworld link and didn't really get much out of it.</p>
<p>If you have a signal that seems random, but in fact is the sum of a bunch of sin and cos functions, the Fourier transform is the way to deconstruct the signal into it's constituent functions. To do this, you first:
- Discretize the signal: convert the continuous function into a series of bins.
- Treat the new data as a vector, and apply the transform (which has the form of a matrix) on that vector.</p>
<p>A classical example has been done to sunspot numbers. That is, the number of sunspots in a given year. (Already discrete). You would use an FT to determine the two primary sin functions (amplitude and period) that 'explain' the sunspot numbers.</p>
<p>yeah, as it has been said, any signal can be broken down into sine wave components. </p>
<p>for instance, a square wave follows a bunch of sine waves compiled in the form of a degenerative e^-x function for a harmonic decay series (harmonics 1,3,5,...etc). if you use harmonics in a different fashion (1, 2, 3) you can create saw waves. something like 1, 5, 9, creates triangle..oh i forget.</p>
<p>it will be useful if you ever take systems and signals engineering (a required course for every person who attends harvey mudd).</p>
<p>Not only can it be used in signals, but as a structural gal, I use Fourier transforms to simplify earthquake records.</p>
<p>My boyfriend, a composer/conductor who also does research in electronic music, uses Fourier transforms for signal processing. Reverse Fourier transforms can be used to take audio signals and break them down into the original notes... Lots of applications for this... identifying overtones, "dictating" music with your voice, piano tuning software... Cool stuff.</p>
<p>Essentially, any squiggly line can be broken down into sine wave components, and you can put them back together again. Lots of applications. Very very useful. If you've done anything like Taylor series expansions, it's kinda-sorta reminiscent of that.</p>
<p>Frequency space (Fourier transforms and Laplace transforms, for example) has some really nice properties for some really complicated things in the time domain. Convolution (<em>the</em> most important thing to hit linear system theory since, well, scalar multiplication, I suppose) turns into plain-old everyday multiplication. Linear, ordinary differential equations become mere algebra. Both are based on the idea that exp(st) is your friend when it comes to integration, taking derivatives, and the like. If you can figure out what happens to a generic exp(st), then build a signal as a linear combination of constants and exp(st) for different values of s (purely imaginary for Fourier, complex as you'd like for Laplace), the you can figure out what happens to your signal.</p>
<p>fourier transform can convert any signal into a sum of sinusoidal signals</p>
<p>Be careful - not any signal :) Can't do sin(1/t), for example.</p>
<p>Fourier Transforms or Fourier Series, they're two different (though related) things.</p>
<p>I've totally never seen an earthquake that exhibits a frequency response spectrum of sin(1/t). ;)</p>
<p>aibarr - that's the nice thing about measurable, versus hypothetical, signals. You can <em>always</em> do a fourier transform post-hoc on a measured discrete signal - even if the measurements are saturated or flat out wrong :)</p>
<p>I wrote a very nice post relating Fourier Transforms to NMR in detail with links and apparently it didn't transfer. That sucks. It was basically explaining proton flip with an electromagnetic force. The large pusle at the beginning excites all and the resonance frequencies (small time, more varied the frequencies, a drum beat vs a held note) are found by breaking the wave down into components and those resonance frequencies are exhibited as spikes in the new version of the graph (transforms the time graph into a frequency graph). The frequencies vary by how many other hydrogen (for H1NMR) are near, other chemicals. For instance a hydrogen near a halide (like fluorine, chlorine) will have a higher chemical shift due to deprotection of the hydrogen (the magnetic force will hit it harder) because the halides are so electronegative they pull the electrons closer and deshield the proton on the hydrogen. Great stuff.</p>
<p>so what is the benefit of being able to break down a "squiggly line" signal into a bunch of sin and cos waves at different frequencies? Lets take earthquake records, for example. There must be all sorts of crazy physical phenomena going on to create weird looking vibration patterns or signals. So why does being able to break the signal down into its components have any value? I've read all about Fourier Transforms capabilities to simplify signals, but I don't get why that is so important. Also don't see the relationship between the math and the physics...That's what I'm failing to understand, I guess. Meh, maybe it'll become clear when I actually take the classes.</p>
<p>eng_dude - here's my shot at it:
If you have a linear, time-invariant system, then there are some rules that apply (assume x and y are functions of, oh, time):
(1) zero input yields zero output
(2) if x->y, a<em>x->a</em>y for some scalar a
(3) if x1->y1 and x2->y2, x1+x2->y1+y2
(4) if x->y at some time, x->y for all times</p>
<p>Given that, for those kinds of systems, it is often "easy" to figure out what happens to an exponential (either exp(jwt) or exp(st) where s is made up of a real part and an imaginary part). If you can do that easily, and you can build up an input as a linear combination of constants and exp(jkw_0 t) for Fourier Series, exp(jwt) for Fourier Transforms, or exp(st) for Laplace Transforms, then you can predict an output using the rules above. Furthermore, you can categorize the system by what it does to particular frequencies (or complex frequencies).</p>
<p>((now I see where LaTeX would be such a nice part of this))</p>
<p>Lemme see if I can find a good handout and post a link :)</p>
<p>As an aside: I still think exp(j pi) + 1 = 0 is the most amazing equation ever.</p>
<p>As another aside: yeah, I am part Electrical Engineering, part Mechanical Engineer, so sqrt(-1) is j...</p>
<p>this is the most pointless topic ever</p>
<p>Eng_dude, I know fourier transforms are critical to hearing implants because it breaks down complex sound waves.</p>
<p>Quote: so what is the benefit of being able to break down a "squiggly line" signal into a bunch of sin and cos waves at different frequencies? Lets take earthquake records, for example. There must be all sorts of crazy physical phenomena going on to create weird looking vibration patterns or signals. So why does being able to break the signal down into its components have any value?</p>
<hr>
<p>Seeing as how I've spent two years doing seismic engineering, I can field this one....
Each component of that Fourier transform affects a structure in a certain way. When you get to dynamics, you'll learn a reasonably simple relationship for motion arising from harmonic excitation, which is just an engineering-y way of saying that the force applied to the structure (in the case of structural engineering) varies in a sine wave over time. So you take all those sine waves from the Fourier transform, collect all the results from those simple relationships for each of those the sine waves, aggregate them all into a single result, and from that very complex engineering data, you can get a very complex structural motion without a whole lot of trouble.</p>
<p>Basically, you take something complex, feed it into a computer, the computer will then break it down into a ton of very simple but very numerous chunks, do all those simple calculations, and spit out a very complex result.</p>
<p>So, that's how I use Fourier transforms... I'm a fan, and I don't think it's pointless at all:
<a href="http://web.ics.purdue.edu/%7Ebraile/edumod/eqphotos/eqphotos1_files/image018.jpg%5B/url%5D">http://web.ics.purdue.edu/~braile/edumod/eqphotos/eqphotos1_files/image018.jpg</a></p>
<p>(Excessive displacement and insufficient flexibility in conjunction with soft story effects yields suckage. The main problem was probably insufficient tie-in between columns and beams or floor membranes. If a seismic analysis like the one I just talked about had been run based upon what's considered a "large" earthquake for the area, engineers would have known that those joints weren't up to snuff. Fourier transforms might have saved this building!)</p>
<p>
[quote]
I wrote a very nice post relating Fourier Transforms to NMR in detail with links and apparently it didn't transfer. That sucks. It was basically explaining proton flip with an electromagnetic force. The large pusle at the beginning excites all and the resonance frequencies (small time, more varied the frequencies, a drum beat vs a held note) are found by breaking the wave down into components and those resonance frequencies are exhibited as spikes in the new version of the graph (transforms the time graph into a frequency graph). The frequencies vary by how many other hydrogen (for H1NMR) are near, other chemicals. For instance a hydrogen near a halide (like fluorine, chlorine) will have a higher chemical shift due to deprotection of the hydrogen (the magnetic force will hit it harder) because the halides are so electronegative they pull the electrons closer and deshield the proton on the hydrogen. Great stuff.
[/quote]
yeah NMR is great, basically revolutionized organic chem</p>