Demonstration of the discrete cosine transform, a variation of the fast fourier transform, often used in image and audio compression

dct.py

import numpy as np
import matplotlib.pyplot as plt 

N = 100 
x = np.zeros(N)
x[50] = 1 

# compute the dct
X = np.zeros(N)
for k in range(N):
    X[k] = np.sum([x[n]*np.cos(np.pi/N*(n+0.5)*k) for n in range(N)])

# invert the dct
X_ = np.zeros(N)
for k in range(N):
    X_[k] = X[0]/2 + np.sum([X[n]*np.cos(np.pi/N*n*(k+0.5)) for n in range(N)])
X_ *= 2/N 

fig, ax = plt.subplots(3)
ax[0].plot(x)
ax[0].set_title('Original Signal')
ax[0].set_xticks([])
ax[1].plot(X)
ax[1].set_title('Discrete Cosine Transform (DCT)')
ax[1].set_xticks([])
ax[2].plot(X_)
ax[2].set_title('Inverted DCT')
plt.show()

Generates the following figure ...