File:Steepest descent.png

import numpy as np 
import matplotlib.pyplot as plt
from scipy.linalg import toeplitz

def convmtx(h,n):
    return toeplitz(np.hstack([h, np.zeros(n-1)]), np.hstack([h[0], np.zeros(n-1)]))

def MSE_calc(sigmaS, R, p, w):
    w = w.reshape(w.shape[0], 1)
    wH = np.conj(w).reshape(1, w.shape[0])
    p = p.reshape(p.shape[0], 1)
    pH = np.conj(p).reshape(1, p.shape[0])
    MSE = sigmaS - np.dot(wH, p) - np.dot(pH, w) + np.dot(np.dot(wH, R), w)
    return MSE[0, 0]

def mu_opt_calc(gamma, R):
    gamma = gamma.reshape(gamma.shape[0], 1)
    gammaH = np.conj(gamma).reshape(1, gamma.shape[0])
    mu_opt = np.dot(gammaH, gamma) / np.dot(np.dot(gammaH, R), gamma)
    return mu_opt[0, 0]

M = 5 # number of sensors

h = np.array([0.722-1j*0.779, -0.257-1j*0.722, -0.789-1j*1.862])
L = len(h)-1 # number of signal sources
H = convmtx(h,M-L)

sigmaS = 1 # the desired signal's (s(n)) power
sigmaN = 0.01 # the noise's (n(n)) power

# The correlation matrix of the received signal:
# Rxx = E\{x(n)x(n)^{H}\}, where ^\{H\} means Hermitian

Rxx = (sigmaS)*(np.dot(H,np.matrix(H).H))+(sigmaN)*np.identity(M)

# The cross-correlation vector between the tap-input vector x(n) and the desired response s(n):
p = (sigmaS)*H[:,0]
p = p.reshape((len(p), 1))

# Solution of the Wiener-Hopf equation:
wopt = np.dot(np.linalg.inv(Rxx), p)
MSEopt = MSE_calc(sigmaS, Rxx, p, wopt)

# Steepest descent algorithm testing:
coeff = np.array([1, 0.9, 0.5, 0.2, 0.1]) 
lamda_max = max(np.linalg.eigvals(Rxx))
mus = 2/lamda_max*coeff # different step sizes

N_steps = 100
MSE = np.empty((len(mus), N_steps), dtype=complex)

for mu_idx, mu in enumerate(mus):
    w = np.zeros((M,1), dtype=complex)
    for N_i in range(N_steps):
        w = w - mu*(np.dot(Rxx, w) - p)
        MSE[mu_idx, N_i] = MSE_calc(sigmaS, Rxx, p, w)

MSEoptmu = np.empty((1, N_steps), dtype=complex)
w = np.zeros((M,1), dtype=complex)
for N_i in range(N_steps):
    gamma = p - np.dot(Rxx,w)
    mu_opt = mu_opt_calc(gamma, Rxx)
    w = w - mu_opt*(np.dot(Rxx,w) - p)
    MSEoptmu[:, N_i] = MSE_calc(sigmaS, Rxx, p, w)

x = [i for i in range(1, N_steps+1)]

plt.figure(figsize=(5, 4), dpi=300)

for idx, item in enumerate(coeff):
    if item == 1:
        item = ''
    plt.loglog(x, np.abs(MSE[idx, :]), label='$\mu = '+str(item)+'\mu_{max}$')

plt.loglog(x, np.abs(MSEoptmu[0, :]), label='$\mu = \mu_{opt}$')
    
plt.loglog(x, np.abs(MSEopt*np.ones((len(x), 1), dtype=complex)), label = 'Wiener solution')
plt.grid(True)
plt.xlabel('Number of steps')
plt.ylabel('Mean-Square Error')
plt.title('Steepest descent')
plt.legend(loc='best')
plt.minorticks_on()
plt.grid(which='major')
plt.grid(which='minor', linestyle=':')
plt.savefig('SD.png')