File:Visualization of the matrix of the discrete Fourier transform for n=8.svg

English: Depiction of the matrix of the DFT for n=8. Each element is represented by a picture of its location in the complex plane in relation to the unit circle. See also https://commons.wikimedia.org/wiki/File:Visualization_of_the_matrix_of_the_discrete_Fourier_transform_for_n%3D4.svg. Generated by the following Python code.

from numpy import linspace,ones_like,real,imag,cos,sin,pi,exp,arange
import matplotlib.pyplot as plt  
from matplotlib import cm

def fft_matrix_viz(n):
	 L = 1.
	 vsep = -1.
	 markersize = 75./float(n)
	 i = complex(0.,1.)
	 t = linspace(0,L,501,endpoint=True)
	 kvals = range(0,n)#-n//2,n)
	 colormap = cm.get_cmap('hsv') # a cyclic colormap
	 nc = len(kvals)
	 colors = colormap( linspace(0,1,nc,endpoint=False) )  # get nc colors from the colormap
	 # pastelize colors
	 p = 0.5
	 colors = (1-p)*colors + p*ones_like(colors)
	 # darken colors
	 d = 0.9
	 colors *= [d,d,d,1]
	 fontsize = 20*4/n
	 plt.figure(figsize=(12, 12))

	 theta = linspace(0,2*pi,100)
	 c = cos(theta)
	 s = sin(theta)
	 def plotZ(x,y,Z):
	     radius = 0.3
	     plt.plot(x+radius*c, y+radius*s,'k',alpha=0.35)  # draw unit circle
	     plt.plot([x,x+radius*real(Z)],[y,y+radius*imag(Z)],'k',alpha=0.35) # draw radius
	     plt.plot(x+radius*real(Z),y+radius*imag(Z),'o',markersize=markersize,color=colors[(j*k)%len(kvals)]) # mark Z

	 plt.subplot(1,1,1,aspect=1,frameon=False)
	 for k in kvals:
	     zd = exp(-2*pi*i*k*arange(n)/float(n))
	     for j,Z in enumerate(zd): plotZ(j,vsep*k,Z)
	     plt.text(-0.5,vsep*k,str(k),va='center',ha='right',fontsize=fontsize)

	 for j in range(n):
	     plt.text(j,vsep*(min(kvals)-0.45),str(j),ha='center',fontsize=fontsize)
	 plt.xlim(-.5,n+.5)
	 plt.ylim(vsep*(max(kvals)+1),vsep*(min(kvals)-1))
	 plt.xticks([])
	 plt.yticks([])

	 plt.savefig('dft_visualization_'+'n'+str(n).zfill(4)+'.svg',bbox_inches='tight')

fft_matrix_viz(8)