This notebook introduces the Fast Fourier Transform (FFT). We will use it to analyze a WAV file.

Mathematically, the FFT evaluates a polynomial in all complex n-th roots of unity simultaneously. If n is a power of 2 (or has at least not many prime large prime factors) the algorithm is very fast.

To demonstrate this, we set xi equal to all 8-th roots of unity.

>xi=exp(I*2*Pi/8); z=xi^(0:7); ... plot2d(re(z),im(z),points=1):

These are the complex values

for n=8, k=0,...8.

For a test, we generate a complex polynomial of degree 7. We take a random polynomial.

>p=random(1,8)+1i*random(1,8);

We evaluate this polynomial at all points of z. Note that a polynomial is represented in EMT in a vector starting with the lowest coefficient.

>longestformat; evalpoly(z,p)

[ 3.61513775646999+4.027250420449434i, -0.06067706070439005+0.8797662696327804i, 0.1325483481709888-0.1451728061492038i, -0.0005680160112071508-0.9346386176109833i, 1.422171838062166-1.867849911920374i, 0.3965191106395581+0.1482400937388588i, -0.4481910174229353-0.0126947295288784i, 0.1863898275842544+0.1666181435935085i ]

This is exactly the same as FFT of p.

>max(abs(fft(p)-evalpoly(z,p)))

1.799837261104273e-015

The inverse of an evaluation is interpolation. Thus the inverse operation to fft() interpolates values in the roots of unity with a polynomial of degree n. The result contains the coefficients of the interpolating polynomial.

The inverse of fft() is ifft() in EMT. We can go back and forth.

>short ifft(fft(p))-p

[ 0+0i , 0+0i , 0+0i , 0+0i , 0+0i , 0+0i , 0+0i , 0+0i ]

The inverse FFT is almost the same as the original FFT. The operation involves two conjugations and the factor n.

>max(abs(8*conj(ifft(conj(p)))-fft(p)))

0

For those interested in the mathematics, we can write b=fft(a) as

From this,

because the last sums are 0, if j is not equal to k.

FFT can also be used to evaluate a trigonometric series at equidistant points.

Let us take a cosine series

and evaluate it directly. Our series has only three terms.

>t=(0:127)*2*Pi/128; s=1+2*cos(t)+cos(2*t); plot2d(t,s):

Now we evaluate it with FFT.

This is very much faster. For the FFT, we must set the rest of the coefficients to 0.

>a=zeros(1,128); a[1:3]=[1,2,1]; totalmax(abs(re(fft(a))-s))

1.776356839400251e-015

For another example, we evaluate

for n=1024 in n equidistant points.

>n=1024; d=2*pi/n; plot2d((0:n-1)*d,im(fft(0|1/(1:n-1)))):

FFT and its inverse operation can be used to determine frequencies in a signal. The reason for this is that the functions

are orthogonal, and the interpolation does in fact select each coefficient so that the distance

is minimal.

The following signal is made of the frequencies 20 and 55.

>t=0:Pi/128:2*Pi-Pi/128; s=2*cos(20*t)+cos(55*t);

We can see these frequencies with a Fourier transformation.

>f=abs(ifft(s)); plot2d(0:127,f[1:128]):

Adding randomly distributed noise does still shows the frequencies clearly.

>s=s+normal(size(s)); f=abs(ifft(s)); plot2d(0:127,f[1:128],title="FFT"):

Euler can generate sound using its matrix language. The sound can be saved in WAV format and played back by the system.

At first we create 2 seconds of sound parameters.

>t=soundsec(2);

Then we generate the sound.

>s=sin(440*t)+sin(220*t)/2+sin(880*t)/2;

We can easily make a Fourier analysis of it.

>analyzesound(s):

If you have speakers on, you can hear the sound file.

>playwave(s);

Now we add a lot of noise and listen to it.

>sa=s+normal(size(s)); playwave(sa);

We can also make a picture of the sound frequencies versus the length of the sound.

>mapsound(sa):

The Fourier transformation has a very interesting property. The folding of two sequences is defined as

E.g., if we fold a sequence a with b=[1/2,1/2], we take the average of two neighboring elements, with a[0]=a[n], a[-1]=a[n-1] etc. This can be computed by a simple multiplication of the Fourier transformation.

>a=1:8; b=[0.5,0.5,0,0,0,0,0,0]; real(ifft(fft(a)*fft(b)))

[4.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5]

There is a function for this in EMT.

>fftfold(a,[0.5,0.5])

[4.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5]

There is also fold(), which uses a loop to compute the fold, but does not go modulo n. Consequently, the resulting vector is shorter.

>fold(a,[0.5,0.5])

[1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5]

Finally, we are going to demonstrate Gibbs phenomenon. Summing up sin(nt)/n for odd n approaches a triangle wave form but with an error at 0 and pi.

>t=linspace(0,pi,500)';

To compute the sum sin(k*t)/k of k=1,...,100, we use the matrix language of Euler.

>s=sum(sin(t*(1:100))/(1:100))*2;

The plot shows that the Fourier sum converges to pi-x.

>plot2d(t',s'); plot2d("pi-x",color=3,add=1):

But there is an overshoot close to 0, which is called Gibbs phenomenon. Let us measure this overshoot for a very large n.

>t=linspace(0,pi,5000)'; max(sum(sin(t*(1:1000))/(1:1000))'*2-(pi-t'))

0.5622809267763276

Using a Riemann sum, one can see that the limit of this value for n to infinity is the following.

>integrate("sinc(x)",0,pi)*2-pi

0.5622814503751381

Let us plot some the truncated series for n=1,3,...,61 using a 3D plot.

>n=1:2:61; ... t=linspace(0,pi,500)'; ... S=cumsum(sin(t*n)/n); ... plot3d(n/10,t,S,hue=1,zoom=5):