The Mathematical Estimation Problem

Since our study group has planned to design a poly tuner, by utilizing multi pitch estimation, this is a brief investigation into the basics of estimation theory.

The \(N\)-point dataset is \(\texttt{x} = x(n) = \{x[0], x[1], ... ,x[N-1] \}\) is dependent on an unknown parameter \(\theta\), which is to be determined based on the data. Therefore, an estimator \(\hat{\theta}\) is defined as

When determining good estimators the first step is to mathematically model the data. Because the data are random, it is described by its probability density function (PDF).

Therefore, it is clear that the signal is gaussian distributed with mean \(\theta\). The next figure shows the dependency between these by an example where \(N=1\) for \(x[0]\).

import pymatbridge as mpb
ip = get_ipython()
mpb.load_ipython_extension(ip)

Starting MATLAB on ZMQ socket ipc:///tmp/pymatbridge
Send 'exit' command to kill the server
............................MATLAB started and connected!

%%matlab
sigma = 1;
theta1 =-18; theta2 = theta1+10; theta3 = theta2+10;
        
x0 = linspace(theta1-5,theta3+5,1000);

p1 = 1./sqrt(2*pi*sigma.^2) .* exp( -(1)./(2*sigma.^2).*(x0-theta1).^2 );
p2 = 1./sqrt(2*pi*sigma.^2) .* exp( -(1)./(2*sigma.^2).*(x0-theta2).^2 );
p3 = 1./sqrt(2*pi*sigma.^2) .* exp( -(1)./(2*sigma.^2).*(x0-theta3).^2 );
plot(x0,p1,'k',x0,p2,'.k',x0,p3,'--k','linewidth',2)

title('Dependence of PDF on unknown parameter \theta','fontsize',18)
xlabel('x[0]','fontsize',16)
ylabel('p(x[0];\theta)','fontsize',16); ylim([0 1])
legend(sprintf('\\theta1 = %1.0f',theta1),sprintf('\\theta2 = %1.0f',theta2),sprintf('\\theta3 = %3.0f',theta3))

From this figure it can be seen that the value of \(\theta\) affects the probaility of x[0], and we should be able to infer the value of \(\theta\) from the observed value of \(x[0]\). i.e. If we masure x[0] to be positive, it is most likely that \(\theta = \theta3\).

In order to design a good estimator it is required to specify a PDF which is consistent with the problem constraints and any prior knowledge.

This will be continued later...