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Summary

Description
English: Developed according to [1].
Date
Source Own work
Author Kirlf
PNG development
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Source code
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Python code

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')

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
  1. Haykin, Simon S. Adaptive filter theory. Pearson Education India, 2008. - p. 108-142, 217-242

Captions

The example of the step size influence.

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25 February 2019

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Date/TimeThumbnailDimensionsUserComment
current07:59, 12 July 2019Thumbnail for version as of 07:59, 12 July 20191,500 × 1,200 (200 KB)KirlfAdditional step size. Grid.
11:43, 25 February 2019Thumbnail for version as of 11:43, 25 February 20191,500 × 1,200 (142 KB)KirlfUser created page with UploadWizard

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