fminunc not converging objective function
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I'm trying to minimize the following from observations data (which now is synthetic):
Observation generation:
Fs = 80E6;
lags = (-10:10)';
tau = lags/Fs;
bw = 0.1;
[obs, obs_A, obs_C] = Raa_parabola(lags, bw);
function [y, A, C] = Raa_parabola(lags,bw)
%RAA_PARABOLA generates a parabola function given a lag vector.
% tau: time vector
% bw: bandwidth at RF
x2 = 1/bw;
x1 = -x2;
A = 1/x1/x2;
B = 0;
C = A*x1*x2;
y = A*lags.^2 + B*lags + C;
end
Which generates a parabola given a tau vector and a bandwidth bw (Fig. 1)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1694741/image.jpeg)
Adding phase to this observations:
f = 420E3;
ph = exp(1j*2*pi*f*tau);
obs = obs.*ph;
Thus the objective function will have 2 parabola parameters and 1 last parameter to obtain the phase, defined as:
function F = myfunc1(x, o, lags, tau)
m_mag = x(1)*lags.^2 + x(2); % magnitude
m_phase = exp(1j*2*pi*x(3)*tau); % phase
m = m_mag.*m_phase; % model
e = m - o; % error = model - observations
F = e'*e; % mean square error
end
With the idea to generate a kinda least squares minimization and use F as the mean square error.
x0 = [0,0,0];
fun = @(x) myfunc1(x, obs, lags, tau);
options = optimoptions('fminunc', 'Display', 'iter', 'StepTolerance', 1e-20, 'FunctionTolerance', 1e-9, ...
'MaxFunctionEvaluations', 300, 'DiffMinChange', 1e-5);
[x,fopt] = fminunc(fun, x0,options);
fprintf("Observation coefficients: A = %.2f, C = %.2f\n",obs_A,obs_C)
disp(x);
y = x(1)*lags.^2 + x(2);
y = y.*exp(1j*2*pi*x(3)*tau);
figure(1); clf;
subplot(4,1,1); plot(lags,real(obs),'LineWidth',2);hold on;
subplot(4,1,2); plot(lags,imag(obs),'LineWidth',2);hold on;
subplot(4,1,3); plot(lags,abs(obs),'LineWidth',2);hold on;
subplot(4,1,4); plot(lags,angle(obs)*180/pi,'LineWidth',2);hold on;
subplot(4,1,1); plot(lags,real(y),'LineWidth',1.5); legend('Obs','Model');
subplot(4,1,2); plot(lags,imag(y),'LineWidth',1.5); legend('Obs','Model');
subplot(4,1,3); plot(lags,abs(y),'LineWidth',2);hold on;
subplot(4,1,4); plot(lags,angle(y)*180/pi,'LineWidth',2);hold on;
Notice that values x(1) and x(2) converge to a valid point (for me) but parameter 3 should be 420E3.
Where is my misconception?
Thank you very much.
1 Comment
Torsten
on 14 May 2024
Edited: Torsten
on 14 May 2024
Fs = 80E6;
lags = (-10:10)';
tau = lags/Fs;
bw = 0.1;
[obs, obs_A, obs_C] = Raa_parabola(lags, bw);
f = 420E3;
ph = exp(1j*2*pi*f*tau);
obs = obs.*ph;
x0 = [0,0,0];
fun = @(x) myfunc1(x, obs, lags, tau);
options = optimoptions('fminunc', 'Display', 'iter', 'StepTolerance', 1e-20, 'FunctionTolerance', 1e-9, ...
'MaxFunctionEvaluations', 300, 'DiffMinChange', 1e-5);
[x,fopt] = fminunc(fun, x0,options)
fprintf("Observation coefficients: A = %.2f, C = %.2f\n",obs_A,obs_C)
y = x(1)*lags.^2 + x(2);
y = y.*exp(1j*2*pi*x(3)*tau);
figure(1); clf;
subplot(4,1,1); plot(lags,real(obs),'LineWidth',2);hold on;
subplot(4,1,2); plot(lags,imag(obs),'LineWidth',2);hold on;
subplot(4,1,3); plot(lags,abs(obs),'LineWidth',2);hold on;
subplot(4,1,4); plot(lags,angle(obs)*180/pi,'LineWidth',2);hold on;
subplot(4,1,1); plot(lags,real(y),'LineWidth',1.5); legend('Obs','Model');
subplot(4,1,2); plot(lags,imag(y),'LineWidth',1.5); legend('Obs','Model');
subplot(4,1,3); plot(lags,abs(y),'LineWidth',2);hold on;
subplot(4,1,4); plot(lags,angle(y)*180/pi,'LineWidth',2);hold on;
function [y, A, C] = Raa_parabola(lags,bw)
%RAA_PARABOLA generates a parabola function given a lag vector.
% tau: time vector
% bw: bandwidth at RF
x2 = 1/bw;
x1 = -x2;
A = 1/x1/x2;
B = 0;
C = A*x1*x2;
y = A*lags.^2 + B*lags + C;
end
function F = myfunc1(x, o, lags, tau)
m_mag = x(1)*lags.^2 + x(2); % magnitude
m_phase = exp(1j*2*pi*x(3)*tau); % phase
m = m_mag.*m_phase; % model
e = m - o; % error = model - observations
F = e'*e; % mean square error
end
Accepted Answer
Matt J
on 14 May 2024
Edited: Matt J
on 14 May 2024
You need to a better choice of units for x(3), at least for the optimization step. Below, I modify the objective function so that x(3) is measured in MHz instead of Hz.
Fs = 80E6;
lags = (-10:10)';
tau = lags/Fs;
bw = 0.1;
[obs, obs_A, obs_C] = Raa_parabola(lags, bw);
f = 420E3;
ph = exp(1j*2*pi*f*tau);
obs = obs.*ph;
s=[1,1,1e6]; %unit scaling
x0 = [0,0,0];
fun = @(x) myfunc1(x.*s, obs, lags, tau);
options = optimoptions('fminunc', 'Display', 'iter', 'StepTolerance', 1e-20, ...
'FunctionTolerance', 1e-9);
[x,fopt] = fminunc(fun, x0,options);
x=x.*s;
x1=x(1),x2=x(2),x3=x(3)
fopt
function [y, A, C] = Raa_parabola(lags,bw)
%RAA_PARABOLA generates a parabola function given a lag vector.
% tau: time vector
% bw: bandwidth at RF
x2 = 1/bw;
x1 = -x2;
A = 1/x1/x2;
B = 0;
C = A*x1*x2;
y = A*lags.^2 + B*lags + C;
end
function F = myfunc1(x, o, lags, tau)
m_mag = x(1)*lags.^2 + x(2); % magnitude
m_phase = exp(1j*2*pi*x(3)*tau); % phase
m = m_mag.*m_phase; % model
e = m - o; % error = model - observations
F = e'*e; % mean square error
end
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