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add the example2
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‎example2/M_sequence_and_inverse_M_sequence.py‎

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import os
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import numpy as np
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import matplotlib.pyplot as plt
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L = 30 # the len of sequence
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x1 = 1; x2 = 1; x3 = 1; x4 = 0 # the init_valce of register
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#print("the x4 :" , x4)
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S = 1
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U = [] # Inverse_M_sequence
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M = [] # M_sequence
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for i in range(1, L + 1, 1):
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IM = S ^ x4
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if IM == 0:
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U.append(-1) # Inverse_M_sequence
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else:
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U.append(1) # Inverse_M_sequence
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S = not(S)
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m = x3 ^ x4
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M.append( m ) # M_sequence
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x4 = x3; x3 = x2; x2 = x1 ; x1 = m
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print("the M_sequence:", M)
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print("the_inverse_M_sequence:", U)
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#figure
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plt.figure(1)
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ax = plt.subplot(2, 1, 1)
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ax1 = plt.subplot(2, 1, 2)
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plt.sca(ax)
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ax.plot(M, 'Red', label = 'M_sequence')
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plt.title('M_sequence')
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#plt.xlabel("k")
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plt.ylabel("amplitude")
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ax.legend()
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plt.sca(ax1)
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ax1.plot(U, 'Blue', label = 'Inverse_M_sequence')
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plt.title('Inverse_M_sequence')
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plt.xlabel("k")
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plt.ylabel("amplitude")
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plt.legend()
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plt.show()
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exit()
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‎example2/Noise_Singal_ratio_SISO.py‎

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import os
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import numpy as np
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import matplotlib.pyplot as plt
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a = np.array([1, -0.4]); b = np.array([1]); c = np.array([1, -0.4])
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na = len(a) -1; nb = len(b) - 1; nc = len(c) - 1
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#print("the na: ", na)
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n = max(na, nb, nc)
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print("the max of (na, nb, nc):", n)
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a0 = a; b0 = b; c0 = c
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for i in range(na, n, 1):
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a0 = np.append(a0, 0)
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for i in range(nb, n, 1):
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b0 = np.append(b0, 0)
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for i in range(nc, n, 1):
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c0 = np.append(c0, 0)
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print("the a0: ", b0)
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p = []; qg = []; qh = []
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deltau2 = 1; deltav2 = 1
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for i in range(n):
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p.append(a0[i])
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qg.append(b0[i])
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qh.append(c0[i])
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for i in range(n - 1, 0 , -1):
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for j in range(0, 2, 1):
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p_2[j, i] = p[0] * p[i]
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‎example2/White_Noise_And_Colored_Noise.py‎

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import os
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import numpy as np
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import matplotlib.pyplot as plt
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L = 500 # the len of noise
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# e(k) = c/d * f(k) f(k) is white_noise and e(k) is colored_noise
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d = np.array([1, -1.5, 0.7, 0.1]) # the c
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c = np.array([1, 0.5, 0.2]) # the d
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nd = len(d) - 1 #
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nc = len(c) - 1 #
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white_noise_init = np.zeros(nc) # the init_value of white_noise
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colored_noise_init = np.zeros(nd) # the init_valuw of colored_noise
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white_noise = np.random.normal(0, 1, L) ## the normal distribution
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colored_noise = [] # colored_noise
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random_walk = [0.0]
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for i in range(L):
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old_total_noise = random_walk[i]
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new_noise = white_noise[i]
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random_walk.append(old_total_noise + new_noise)
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noise_1 = np.dot(-d[1 : nd + 1], colored_noise_init.transpose()) ##d * colored_noise_init
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white_noise_init_array = np.insert(white_noise_init, 0, white_noise[i])
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noise_2 = np.dot(white_noise_init_array, c.transpose()) # c * white_noise
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colored_noise.append(noise_1 + noise_2) ## save the colored_noise
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#update the data
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for j in range(nd - 1, 0, -1):
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colored_noise_init[j] = colored_noise_init[j - 1]
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colored_noise_init[0] = noise_2 + noise_1
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for k in range(nc-1, 0, -1):
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white_noise_init[k] = white_noise_init[k - 1]
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white_noise_init[0] = white_noise[i]
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random_walk = np.array(random_walk)
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#figure
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plt.figure(1)
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ax = plt.subplot(2, 1, 1)
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ax1 = plt.subplot(2, 1, 2)
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plt.sca(ax)
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ax.plot(white_noise, 'Red', label = 'white_noise')
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plt.title('white_noise')
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#plt.xlabel("k")
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plt.ylabel("amplitude")
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ax.legend()
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plt.sca(ax1)
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ax1.plot(colored_noise, 'Blue', label = 'colored_noise')
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plt.title('colored_noise')
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plt.xlabel("k")
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plt.ylabel("amplitude")
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plt.legend()
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plt.figure(2)
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ax3 = plt.subplot(1, 1, 1)
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ax3.plot(random_walk, 'Green' ,label = "random_walk")
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plt.title('random_walk')
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plt.xlabel("k")
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plt.ylabel("amplitude")
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plt.legend()
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plt.show()
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exit()
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‎example2/forgetting_factor_recursive_least_square.py‎

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import os
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import numpy as np
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import matplotlib.pyplot as plt
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import numpy.matlib
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#the least_square
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a = np.array([1, -1.5, 0.7]); b = np.array([1, 0.5]); d = 3 #
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len_a = len(a) -1; len_b = len(b) - 1 ; L = 1500
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Input_init = np.zeros(d + len_b) #the init_value of input
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Output_init = np.zeros(len_a) # the init_value of ouput
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white_noise = np.random.normal(0, 1, L) # the white_noise
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white_noise_1 = np.sqrt(0.1) * np.random.normal(0, 1, L)
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#theta = np.append(a[1: len_a + 1], b) #the permeter of object
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theta_1 = np.zeros(len_a + len_b + 1)
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P = np.eye(len_a + len_b + 1, len_a + len_b + 1)
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forgetting_factor = 0.98 #the forgetting_factor
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phi = np.array([]) # the
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thetae_1 = []
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thetae_2 = []
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thetae_3 = []
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thetae_4 = []
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theta_real_1 = []
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theta_real_2 = []
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theta_real_3 = []
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theta_real_4 = []
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for i in range(L):
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if i == 500:
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a = [1, -1, 0.4]; b = [1.5, 0.2]
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theta = np.append(a[1: len_a + 1], b) #the permeter of object
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#print("the theta: ", theta)
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theta_real_1.append(theta[0])
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theta_real_2.append(theta[1])
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theta_real_3.append(theta[2])
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theta_real_4.append(theta[3])
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temp = np.append(-Output_init, Input_init[d - 1: d + len_b]) # X
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temp_y = np.dot(temp, theta.transpose()) + white_noise_1[i] # Y
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phi = np.concatenate((phi, temp))
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temp_matrix = np.matrix(temp)
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P_matrix = np.matrix(P)
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K = (P_matrix * temp_matrix.transpose()) /(forgetting_factor + temp_matrix * P_matrix *temp_matrix.transpose())
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thetae_temp = theta_1 + K * (temp_y - temp * theta_1)
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thetae_1.append(thetae_temp[0,0])
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thetae_2.append(thetae_temp[1,1])
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thetae_3.append(thetae_temp[2,2])
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thetae_4.append(thetae_temp[3,3])
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P = (np.eye(len_a + len_b + 1) - K * temp) * P /forgetting_factor
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#update the data
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theta_1 = thetae_temp
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for k in range(d + len_b - 1, 0, -1):
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Input_init[k] = Input_init[k - 1]
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Input_init[0] = white_noise[i]
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for j in range(len_a - 1, 0, -1):
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Output_init[j] = Output_init[j - 1]
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Output_init[0] = temp_y
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plt.figure(1)
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ax = plt.subplot(1, 1, 1)
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plt.sca(ax)
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ax.plot(thetae_1, 'Red', label = 'thetae_1')
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ax.plot(thetae_2, 'Blue', label = 'thetae_2')
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ax.plot(thetae_3, 'Green', label = 'thetae_3')
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ax.plot(thetae_4, 'Yellow', label = 'thetae_4')
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ax.plot(theta_real_1, 'brown', label = 'theta_real_1', linestyle ="-" )
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ax.plot(theta_real_2, 'cyan', label = 'theta_real_2', linestyle="-" )
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ax.plot(theta_real_3, 'burlywood', label = 'theta_real_3', linestyle="-" )
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ax.plot(theta_real_4, 'coral', label = 'theta_real_4', linestyle="-" )
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plt.title('thetae')
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plt.xlabel("k")
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plt.ylabel("value")
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ax.legend()
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plt.show()

‎example2/least_square.py‎

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import os
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import numpy as np
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import matplotlib.pyplot as plt
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#the least_square
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a = np.array([1, -1.5, 0.7]); b = np.array([1, 0.5]); d = 3 #
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len_a = len(a) -1; len_b = len(b) - 1 ; L =500
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Input_init = np.zeros(d + len_b) #the init_value of input
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Output_init = np.zeros(len_a) # the init_value of ouput
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x1 = 1; x2 = 1; x3 = 1; x4 = 0; S = 1 # the init_value of move_register and square_wave
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white_noise = np.random.normal(0, 1, L) # the white_noise
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phi = np.array([]) # the
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y = np.array([])
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U = np.array([]) #the inverse_M_sequence
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theta = np.append(a[1: len_a + 1], b) #the permeter of object
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for i in range(L):
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# Y = AX + xi(noise)
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temp = np.append(-Output_init, Input_init[d - 1: d + len_b]) # X
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phi = np.concatenate((phi, temp))
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#print("the phi: ", phi)
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temp_y = np.dot(temp, theta.transpose()) + white_noise[i] # Y
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#save the y
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y = np.append(y, temp_y)
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#the inverse_M_sequence
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IM = S ^ x4
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if IM == 0:
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U = -1
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else:
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U = 1
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S = not(S); M = x3 ^ x4
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#update the data
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x4 = x3; x3 = x2; x2 = x1; x1 = M
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for k in range(d + len_b - 1, 0, -1):
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Input_init[k] = Input_init[k - 1]
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Input_init[0] = U
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for j in range(len_a - 1, 0, -1):
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Output_init[j] = Output_init[j - 1]
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Output_init[0] = temp_y
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# the np.narray to np.matrix
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y_matrix = np.matrix(y.reshape(len(y), 1))
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phi_matrix = np.matrix(phi.reshape((L, 4)))
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print("the phi_matrix_transpose*phi_matrix:", np.matmul(phi_matrix.transpose(), phi_matrix))
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temp_phi_phi = np.linalg.inv(np.matmul(phi_matrix.transpose(), phi_matrix))
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temp_phi_phi_phi = np.dot(temp_phi_phi, phi_matrix.transpose())
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theta_estimate = np.dot(temp_phi_phi_phi, y)
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print("the theta_estimate: ", theta_estimate)
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