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. The values are also coded in the shading of the unit disk. Generated by the following Python code.
from numpy import linspace,ones_like,real,imag,cos,sin,pi,exp,arange,mod
import matplotlib.pyplot as plt
from matplotlib import cm
def fft_matrix_viz(n):
L = 1.
vsep = -1.
lwfac = 4
markersize = 75./float(n)/2 *lwfac
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)
cmrotation = 0.95 #0.8 #0.05
colors = colormap( mod( linspace(0,1,nc,endpoint=False) + cmrotation, 1 ) ) # get nc colors from the colormap
# pastelize colors
p = 0.55
colors = (1-p)*colors + p*ones_like(colors)
# darken colors
d = 1 #0.9
colors *= [d,d,d,1]
clockcolor = '#888888'
fontsize = 20*4/n*1.5*2
figsize = 20
plt.figure(figsize=(figsize,figsize))
theta = linspace(0,2*pi,100)
c = cos(theta)
s = sin(theta)
def plotZ(x,y,Z):
radius = 0.35
tickfraci = 0.15
tickfraco = 0.05
color = colors[(j*k)%len(kvals)]
plt.fill(x+radius*c, y+radius*s,color=color,alpha=.75) # fill unit circle
plt.plot(x+radius*c, y+radius*s,color=clockcolor,lw=lwfac,alpha=1)#0.35) # draw unit circle
for ticktheta in linspace(0,2*pi,n,endpoint=False): # draw ticks
tc,ts = cos(ticktheta),sin(ticktheta)
plt.plot([x+(1-tickfraci)*radius*tc,x+(1-tickfraco)*radius*tc],[y+(1-tickfraci)*radius*ts,y+(1-tickfraco)*radius*ts],color=clockcolor,lw=lwfac,alpha=1)#0.35)
plt.plot([x,x+radius*real(Z)],[y,y+radius*imag(Z)],color=clockcolor,lw=lwfac,alpha=1)#0.35) # draw radius
plt.plot(x+radius*real(Z),y+radius*imag(Z),'o',markersize=markersize,color='#505050',alpha=1) # 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)-0.50),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(f'dft_visualization_rev2_n{str(n).zfill(4)}.svg',bbox_inches='tight')
fft_matrix_viz(8)