Add files via upload

Author: abodedaniel

CRLB.m

@@ -0,0 +1,32 @@
+clc
+clear all
+close all
+%% y(t) = m + n(t) t= 1,....,N
+est_m = zeros(1,1000);
+var_est_m = zeros(1,10);
+i = 1;
+N = 1000;
+CRLB = zeros(1,10);
+for v = 1:1:10
+for mc = 1:1000
+m = 5; %%choose m = 5
+n = v * randn(1,N);  
+y = m + n;
+y = y';
+%plot(y)
+%% Creating a Maximum Likelihood Estimator
+est_m(mc) = (1/(N-1))*sum(y);
+end
+exp_est_m = sum(est_m)/mc;
+%%calculating the variance of 
+var_est_m(i) = (sum((est_m - exp_est_m).^2))/mc;
+CRLB(i) = var(y)/N;
+i = i+1;
+
+end
+plot(1:1:10,var_est_m);
+title('plot of var(mˆML), for N=1000 constant and sigma^2 changing')
+ylabel('var(mˆML)')
+xlabel('sigma^2')
+hold on
+plot(1:1:10,CRLB);

CRLB_vs_variance_plot.m

@@ -0,0 +1,34 @@
+clc
+clear all
+close all
+%% y(t) = m + n(t) t= 1,....,N
+est_m = zeros(1,1000);
+var_est_m = zeros(1,10);
+i = 1;
+N = 1000;
+CRLB = zeros(1,10);
+for v = 1:1:10
+for mc = 1:1000
+m = 5; %%choose m = 5
+n = v .* randn(1,N);  
+y = m + n;
+y = y';
+%plot(y)
+%% Creating a Maximum Likelihood Estimator
+est_m(mc) = (1/(N))*sum(y);
+end
+%% expectation of the Maximum Likelihood Estimator
+exp_est_m = sum(est_m)/mc;
+%%calculating the variance of 
+var_est_m(i) = (sum((est_m - exp_est_m).^2))/mc;
+CRLB(i) = v/N;
+i = i+1;
+
+end
+plot((1:1:10).^2,var_est_m);
+title('Comparing the CRLB to the variance')
+ylabel('var(mˆML)')
+xlabel('sigma^2')
+hold on
+plot((1:1:10).^2,CRLB,'r-');
+legend('var(mˆML)','CRLB');

Central_Limit_Theorem.m

@@ -0,0 +1,45 @@
+clc
+clear all
+close all
+%%Central Limit Theorem
+S = zeros(1,1000);   %%declaring vector S
+for j = 1:1000
+    N = 10;       %%value of N
+    a = -1;
+    b = 1;
+    ni = a+(b-a)*rand(1,N);  %%generating ni for size N
+    S(j) = sum(ni);          %%summing ni and storing in S
+end
+subplot(3,1,1)
+hist(S);
+title('Approximate pdf of Sj for size N = 10');
+ylabel('Frequency')
+xlabel('values of Sj')
+
+%% Repeated for N = 1000
+for j = 1:1000
+    N = 1000;       %%value of N
+    a = -1;
+    b = 1;
+    ni = a+(b-a)*rand(1,N);  %%generating ni for size N
+    S(j) = sum(ni);          %%summing ni and storing in S
+end
+subplot(3,1,2)
+hist(S);
+title('Approximate pdf of Sj for size N = 10000');
+ylabel('Frequency')
+xlabel('values of Sj')
+
+%% Repeated for size N = 100000
+for j = 1:1000
+    N = 100000;       %%value of N
+    a = -1;
+    b = 1;
+    ni = a+(b-a)*rand(1,N);  %%generating ni for size N
+    S(j) = sum(ni);          %%summing ni and storing in S
+end
+subplot(3,1,3)
+hist(S);
+title('Approximate pdf of Sj for size N = 100000');
+ylabel('Frequency')
+xlabel('values of Sj')

Least_square_criterion.m

@@ -0,0 +1,19 @@
+clc
+clear all
+close all
+%% y (t) = a1 sin(2?f0t) + a2 sin(2?f1t) + n(t), t= 1,....,N
+a1 = 1; 
+a2 = 1;
+f0 = 0.25;
+f1 = 0.4;
+standard_deviation = 1;
+m = 5; %%choose m = 5
+N = 10000;
+t = 1:1:N;
+n = standard_deviation .* randn(1,N);  
+y = (a1.*sin(2*pi*f0.*t)) + (a2.*sin(2*pi*f1.*t)) + n;
+y = y';
+J = (y - (a1.*sin(2*pi*f0.*t')) - (a2.*sin(2*pi*f1.*t')));
+J = J.^2;
+J1 = sum(J);
+disp(J1);

Least_square_criterion_plot_against_fo.m

@@ -0,0 +1,22 @@
+clc
+close all
+%% y (t) = a1 sin(2?f0t) + a2 sin(2?f1t) + n(t), t= 1,....,N
+%% plotting the Least square criterion as a function of fo with fixed variance
+a1 = 1;
+a2 = 1;
+i = 1;
+J1 = zeros(length(0:0.01:0.5),1);
+for f0 = 0:0.01:0.5
+J = (y - (a1.*sin(2*pi*f0.*t')) - (a2.*sin(2*pi*f1.*t'))).^2;
+J = sum(J);
+J1(i) = J;
+i = i+1;
+end
+plot(0:0.01:0.5,J1);
+title('Plot of J as a function of fo for a1 = 1 and a2 = 1')
+ylabel('magnitude of J')
+xlabel('values of fo')
+[f0_value,J_value] = ginput(1); %picking the minimum coordinates
+disp('minimum coordinate of J as picked by manually by ginput() is')
+disp(['value for f0 = ' num2str(f0_value)])
+disp(['at J = ' num2str(J_value)])

Least_square_criterion_plot_against_fo_f1.m

@@ -0,0 +1,24 @@
+%% a1 and a2 must be set to one in Least_square_criterion.m
+%% a1=a2=1 and variance constant, this code plot J(fo,f1) vs fo, f1 in a 3D plot
+clc
+close all
+%% y (t) = a1 sin(2?f0t) + a2 sin(2?f1t) + n(t), t= 1,....,N
+i = 1;
+k = 1;
+J1 = zeros(length(0:0.01:0.5),length(0:0.01:0.5));
+for f1 = 0:0.01:0.5
+    for f0 = 0:0.01:0.5
+        J = (y - (a1.*sin(2*pi*f0.*t')) - (a2.*sin(2*pi*f1.*t'))).^2;
+        J1(i,k) = sum(J);
+        k = k+1;
+    end
+    i = i+1;
+    k=1;
+end
+surf(0:0.01:0.5,0:0.01:0.5,J1);
+title('Plot of J(fo,f1) vs fo, fi')
+xlabel('values of fo')
+ylabel('values of f1')
+zlabel('Amplitude of J(fo,f1)')
+%[f0_value,f1_value,J_value] = ginput(1) %picking the minimum coordinates
+%plot3(0:0.01:0.5,0:0.01:0.5,J1);

Least_square_criterion_plot_against_var.m

@@ -0,0 +1,25 @@
+clc
+close all
+%% y (t) = a1 sin(2?f0t) + a2 sin(2?f1t) + n(t), t= 1,....,N
+%% plotting J vs variance with fixed fo
+f0 = 0.25;
+a1 = 1;
+a2 = 1;
+N = 10000;
+t = 1:1:N;
+i=1;
+var_ = zeros(length(0.1:0.1:1),1);
+J1 = zeros(length(0.1:0.1:1),1);
+for s_d = 0.1:0.1:1
+    n1 = s_d .* n; 
+    y = (a1.*sin(2*pi*f0.*t)) + (a2.*sin(2*pi*f1.*t)) + n1;
+    var_(i) = var(y);
+    y = y';
+    J = (y - (a1.*sin(2*pi*f0.*t')) - (a2.*sin(2*pi*f1.*t'))).^2;
+    J1(i) = sum(J);
+    i = i + 1;
+end
+plot(var_,J1)
+title('Plot of J vs variance with fo and f1constant and a1 = a2 = 1')
+ylabel('Amplitude of J')
+xlabel('Value of variance')

Least_square_criterion_plot_against_var_with_f1_f0_constant.m

@@ -0,0 +1,21 @@
+clc
+close all
+%% y (t) = a1 sin(2?f0t) + a2 sin(2?f1t) + n(t), t= 1,....,N
+f0 = 0.25;
+f1 = 0.4;
+m = 5; %%choose m = 5
+N = 10000;
+t = 1:1:N;
+i=1;
+var_ = zeros(length(1:0.2:10),1);
+J1 = zeros(length(1:0.2:10),1);
+for s_d = 1:0.2:10
+    n1 = s_d .* n; 
+    y = (a1.*sin(2*pi*f0.*t)) + (a2.*sin(2*pi*f1.*t)) + n1;
+    var_(i) = var(y);
+    y = y';
+    J = (y - (a1.*sin(2*pi*f0.*t')) - (a2.*sin(2*pi*f1.*t'))).^2;
+    J1(i) = sum(J);
+    i = i + 1;
+end
+plot(var_,J1)

LinearEstimation.m

@@ -0,0 +1,20 @@
+clc
+clear all
+close all
+%% y(t) = m + n(t) t= 1,....,N
+v1 = zeros(5,5);
+for j = 0:4
+for i = 1:5
+m = 5; %%choose m = 5
+N = 1000;
+var = 1+(j*10);    %%value for variance
+v2 = 1+j;
+n = var * randn(1,N);  
+y = m + n;
+y = y';
+%plot(y)
+%% Creating a Maximum Likelihood Estimator
+est_m = (1/N)*sum(y);
+v1(v2,i) = est_m;
+end
+end

LinearEstimation_0.m

@@ -0,0 +1,23 @@
+clear all
+close all
+clc
+%% y(t) = m + n(t),  t = 1.....N
+m = 5;
+N= 1000;
+t = 1:1:N
+y = zeros(N,5);
+n = zeros(N,1);
+std_d = 1;
+i = 1;   %%counter for filling y for different values of variance
+for std_d = 0.1:0.2:1
+    n = std_d * randn(N,1);
+    y(:,i) = m + n;
+    subplot(5,1,i);
+    stem(t(1:100),y(1:100,i));
+    title(['Plot of y for variance = ' num2str(var(y(:,i)))])
+    xlabel('time(1:100)')
+    ylabel('Amplitude')
+    i = i+1;
+end
+
+

LinearEstimation_1.m

@@ -0,0 +1,19 @@
+clear all
+close all
+clc
+%% y(t) = m + n(t),  t = 1.....N
+m = 5;
+y = [];
+n = [];
+est_m = [];
+std_d = 1;
+i = 1; %%counter for filling y for different values of variance
+for N = [10, 100, 1000, 10000, 100000]
+    n = std_d * randn(N,1);
+    y = m + n;
+    est_m(i) = sum(y)/N;
+    disp(['Estimated value for N = ' num2str(N) ' is ' num2str(est_m(i))])
+    i = i+1;
+end
+
+

LinearEstimation_1_1.m

@@ -0,0 +1,20 @@
+clear all
+close all
+clc
+%% realization of the performance of the estimator for various value of variance
+%% y(t) = m + n(t),  t = 1.....N
+m = 5;
+y = [];
+n = [];
+est_m = [];
+N = 1000;
+i = 1; %%counter for filling y for different values of variance
+for std_d = [2, 4, 6, 8, 10]
+    n = std_d * randn(N,1);
+    y = m + n;
+    est_m(i) = sum(y)/N;
+    disp(['Estimated value for variance = ' num2str(var(y)) ' is ' num2str(est_m(i))])
+    i = i+1;
+end
+
+

LinearEstimation_3.m

@@ -0,0 +1,28 @@
+clc
+clear all
+close all
+%% Realizing the bias of the estimator for different values of N
+%% y(t) = m + n(t) t= 1,....,N
+est_m = zeros(1,1000);
+b = zeros(1,10);
+i = 1;
+for N = 10:10000:100000
+for mc = 1:1000
+m = 5; %%choose m = 5
+n = randn(1,N);  
+y = m + n;
+y = y';
+%plot(y)
+%% Creating a Maximum Likelihood Estimator
+est_m(mc) = (1/N)*sum(y);
+end
+exp_est_m = sum(est_m)/mc;
+%%calculating the bias
+b(i) = exp_est_m - m;
+b(i) = abs(b(i));
+i = i+1;
+end
+plot(10:10000:100000,b);
+title('plot for the bias for different values of N')
+xlabel('values of N')
+ylabel('value of bias')

LinearEstimator_4_0.m

@@ -0,0 +1,24 @@
+clc
+clear all
+close all
+%% y(t) = m + n(t) t= 1,....,N
+est_m = zeros(1,1000);
+b = zeros(1,100);
+i = 1;
+for N = 10
+for mc = 1:10
+m = 100; %%choose m = 5
+n = randn(1,N);  
+y = m + n;
+y = y';
+%plot(y)
+%% Creating a Maximum Likelihood Estimator
+est_m(mc) = (1/(N-1))*sum(y);
+end
+exp_est_m = sum(est_m)/mc;
+%%calculating the bias
+b(i) = exp_est_m - m;
+b(i) = abs(b(i));
+i = i+1;
+end
+plot(10:1000:100000,b);