In this text, I talk about Fast Fourier Transform (FFT) which I used for arbitrary precision multiplication. I will write about FFT multiplication for arbitrary precision arithmetic, later. Before that, I would like to touch on Karatsuba and Toom-Cook multiplications, too. But, I start with only FFT now.
At first, I would like to start with “Fourier Series” which makes Joseph Fourier popular. “Fourier Series” is sum of sines and cosines which is expansion of a periodic function. Briefly, sum of sines and cosines represents a function or approximate.
a0, ai ve bi constants are coefficients of Fn(x).
Square Wave approximation with Fourier series,
Periodic functions are represented with sum of complex exponentials. When we look at exponential functions, we see famous Euler’s number (e)
Recall; e constant
At this point, we see Euler’s formula. He discovered the relation between exponential function and trigonometric function.
Recall; complex number
As is known, π radians equals to 180 degrees. So, if we put π, we see that .
cos(π)=-1 and sin(π)=0
Complex numbers are represented with x, y on 2D coordinate system and π = 180 radians, cos(π) + isin(π) = -1
Now, if we focus on FFT, we see that integrable aperiodic functions are transformable with general Fourier Series formula, too. Time - frequency transformation is done at this point.
The FFT transformation is done in that way.
FFT Inversion is similar. (Be careful about the sign.)
But, these functions are continuous and we must use discrete version of FFT in the digital world.
When we focus on Discrete FFT, we encounter Riemann sum approximation which is a familiar method to be used for integration in digital world. As you see, if we draw more rectangles, we get more accurate results.
and Discrete FFT is seen in this form,
, n complex number array is transformed by DFT.
As an output, we get N-periodic complex number array.
Also, inverse of this operation is similar.
There is something that needs to be added. DFT algorithm's complexity is O(n²) and this means; the number of steps to do this operation, grows exponentially. So, we have to optimize it as much as possible(if we can). We may spend hours/days for an operation which can be done in minutes.
Fast Fourier Transform is developed for this reason and its complexity is O(nlogn). It brings significant time savings.
Divide and conquer philosophy lies behind it. The problem is divided into sub problems, solved recursively.
In this part, I shall mention about Cooley-Tukey FFT (Radix-2 Decimation-in-Time ) algorithm.
At first, we calculate odd and even indexed elements, separately and then bring together them again.
As you see, 2m represents even and 2m+1 represents odd index.
And we don’t do the same operations thanks to symmetry property of DFT.
In detailed form,
As you see, there is symmetry property. So, there is a common factor.
Taking into account that the DFT is periodic,
Finally, happy ending is reached...
Image sources; Wolfram Alpha, Wikipedia.