what does fast fourier transform do with a raw signal- simple pls?

Answer 1

I think a really simple way to think about this is by thinking of piano keys. Each key corresponds to a specific note (or frequency). When you touch that key you get (mostly, but we're simplifying) on pure frequency. This is what we could call the "frequency domain", imagine a big spike on a number line right where you hit your key.

But what that note is really doing is causing a string to vibrate up and down really fast. If you hit the middle "A" note that would be 440 vibrations a second, it would look like a wave on a graph of displacement versus time. We can call this the "time domain."

So basically, (and very basically), a FFT picks out individual frequencies in some signal and shows you how much of your total signal (in the time domain) is in a particular place in your frequency domain. If you did a FFT on a plot of the brightness of the sky over many weeks, you'd expect to see something of a peak around the frequency corresponding to 24 hours.

Answer 2

OK suppose you have a signal y=f(t) which is 'well-behaved' ie continuous and differentiable within the limits of interest. It can be shown that this signal is the infinite sum of harmonic functions of decreasing amplitude and increasing frequency. The fourier transform maps the signal onto inverse time or frequency space. So for a simple sine wave you would get a single vertical line of at a given value of frequency. If higher harmonics were present like a flute, piano or human voice sound, then you would get a set of lines with the maximum at the fundamental frequency and the rest as even or odd or both multiples of this frequency but with diminishing intensity.
If the function is non-integrable, then you have to sample it preferably over 128, 256 etc intervals and use a numerical analysis algorithm to evaluate the F(t). This makes the technique esp applicable to computer calculation and the various time-efficient algorithms (programs) which run in at least poynomially bounded time are called discrete fast fourier transforms (or DFFT's)

Answer 3

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Tough to answer simply, as there are a lot of versions of the FFT, but I'll give it a try: FFT is a higher speed way of calculating a Fourier Transform. The idea of the Fourier Transform is to take a wave, such as a sound, where you have heights (volumes) at different times, giving a graph of a complicated wave, and what you want instead is to know what *Frequencies* are in the sound--what musical notes make up the sound, so to speak. The FFT takes a list of volume levels and gives a list of frequencies instead. I hope that helps; not sure how basic or detailed you wanted. Good luck.

Answer 4

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what does fast fourier transform do with a raw signal- simple pls?

Answer 5

A fourier transform can convert a raw signal into a series of sine and cosine terms; this follows from Fourier's remarkable theorm. The terms can be analyzed for characteristics, such as a dominant term or some harmonic term.

Answer 6

The Fourier transform tells you what frequency components go into making a signal. the Fast Fourier Transform is a way of calculating this quickly.

Answer 7

To understand a Fast Fourier Transform (commonly called FFT) you need to understand the core mathematics behind the Fourier Transform.

The Fourier Transform has a central underlying idea behind it:

Any function can be written as a combination of sine and cosine waves of different frequencies.

This is kind of weird at first glance. If you went to the chalkboard and drew a straight line using a yardstick, there's no way you'd guess that you could re-create the line using sine and cosine waves.

After all, a straight line is straight, and sines and cosines are wavy.

But you can, you just have to be clever about how you combine the sines and cosines.

The trick is to realize that as you add sines and cosines, you can *change* how much each sine wave and each cosine wave affects the *overall sum total* of the waves by using *multiplicative factors* to control the amplitude of each wave you add.

Those multiplication factors are called the Fourier Coefficients.

If you add two functions, you simply have to go point by point along the graph, summing the height of one function with the height of another function to get the resulting function which is their sum.

When you do this with several wavy functions, like sines and cosines, you *can* get a net result that is a fairly straight line.

If your straight line started down low below the x-axis at the start of the interval you're interested in, and increased to be above the x-axis at the end of the interval you're interested in, then the multiplicative coefficient for the sine wave that starts at the beginning of the interval, crosses at the midpoint, and ends at the end of the interval, would probably have a negative coefficient. This would cause that sine wave to drop down, then come up through the midpoint of the interval, and then climb back up above the x-axis, roughly "following" the straight line, while still remaining a sine wave.

Other waves that couldn't do this would have *smaller magnitude* multiplicative coefficients, so they would not throw off the approximation of the straight line.

To get a perfect approximation of the straight line using sines and cosines, you'd need an *infinite* number of sines and cosines in the series. Each additional wave added would "correct" error in the sum of the previous waves.

For engineers and scientists, there is usually a point where you eventually have enough sines and cosines summed up to give a "good enough" approximation to the curve you want to describe, and you simply drop the rest of the terms in the sine and cosine series.

So that's the basic idea behind Fourier series. They're just sums of sines and cosines to approximate any given function.

The *frequencies* of the sines and cosines used in Fourier series are defined in terms of the interval over which you are approximating the function you are interested in.

the frequency w of the n-th term in the series is k=(2*pi*n)/N, where N is the interval length.

So n would range over the values from zero to infinity, if you used an infinite number of waves to approximate the function.

So what have we done really?

If the function you are approximating using the sines and cosines was a function over an interval of *time* you have just found a way to convert it into a function of *frequency*.

So your original function f(n) has been converted to a new function, F(k).

So what is F(k)? What good is it? It is a plot showing the relative amplitudes of the different waves of different frequencies in the system. For different types of systems, it will have different meanings. In the case of audio systems, F(k) shows whether the sound waves are carring mostly deep bass (low frequency) or high treble (high frequency) sounds.

Of course, with the usual fourier transform, you need more terms to be more accurate... an infinate number of terms for exact precision. But we have restricted ourselves to an interval N, so we have N terms in the series.

You might not be surprised to hear that it is slow to calculate.

So most computer programmers use the FFT to do the same thing that the Fourier transform does.

Now you should note that in the frequency k, "n" will have even and odd values.

An FFT separates the "infinite" sum of the Fourier Transform into two sums, a sum of even-n terms, and a sum of odd-n terms.

In complex variable notation, the even terms are of the form
f(n)*e^i[(2 pi)(n*k)/N ]
and the odd terms are of the form
f(n+1)*e^i[(2 pi)(k*(n+1))/N]

You can factor out the e^i(2 pi)/N from the odd terms, giving
F(k)=Feven + e^i (2 pi)/N Fodd where Feven and Fodd are the even- and odd-termed infinite sums that together make up the Fourier Transform F(k).

Now you treat Feven as being a sum of sines and cosines, essentially, you are taking the Fourier Transform of a part of a Fourier Transform. We can then say

Feven = Feveneven + e^i(8 pi)/N Foddodd

Notice the factor in the exponential went from 2 to 8. We cut the interval in half when we went from F(k) to just Feven, and we cut it in half again going from Feven to Feveneven, so that's just N/2, then N/4.

So it is recursive. You keep chopping into even and odd halves, down to the point where you just have one single value, and you end up with N/N=1, and your equation will look like

Fevenoddeveneven.....oddeveneven = f(n)

Which is kinda weird, because f(n) was an *input* you stuffed into the FFT in the first place!

BUT.... you don't plot f(n) at the same x-coordinate, that would just give you your original starting function that you wanted to transform. It gets shuffled to a *new* x-coordinate. You need to know at what new x-coordinate, k, to plot the value f(n).

You do this by reversing the pattern of even's and odd's in the subscript of

Fevenoddeveneven.....oddeveneven = f(n)

Then setting even=0, odd=1, then sorting the Feveneven....evenevenoddeven's, so that the subscripts are in increasing order in binary.

I'm drawing this from "Numerical Recipes." I've glossed through a lot of the fine details, but the important facts are that you are separating the Fourier Transform terms by evens and odds and performing a shuffle to reorder the input values of f(n) along the x-axis to obtain F(k).

And I may have missed some key steps, but this roughs out the big picture of what an FFT does with the raw f(n) data.

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Answer 10

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