Maximum likelihood (ML) method
The method, which was created by Capon in 1961 [16], estimates the coefficients of a digital rejector filter that is tailored to the frequency being looked at. This produces a power that shows the PSD's value for that frequency. To achieve this, we reduce the output signal's power.
\(E\left\{y^2(k)\right\}=E\left\{\left(\sum_{i=0}^{M-1}{h(i)x(k-i}\right)^2\right\}=H^t{\ R}_x\ H\) (43)
The property expresses the constraint that the eigenfunctions are sinusoidal functions.
\(\sum_{i=0}^{M-1}{h(k)e^{-j\omega k}}=H^t\ E=1\) (44)
with:
\(E=\left[1e^{-j\omega k}e^{-j\omega2k}....e^{-j\omega(M-1)}\right]^t\)
\(H=\left[h(0)h(1)....h(M-1)\right]^t\)
\({\ R}_x\) : an autocorrelation matrix of the signal \(x(k)\).
Fixing the frequency power equates to reducing the power of other frequencies in the signal.
Let \({x(k)=ae}^{(j(\omega k+\varphi))}\) be the sinusoidal signal of frequency \(\omega\) at the input of filter H. The corresponding output is then equal to:
\(y(k)=H(\omega)x(k)=aH(\omega)e^{(j(\omega k+\varphi))}\) (45)
If we adjust this filter to the frequency, it means that:
\(y(k)=x(k)\)
We will have:
\(H(\omega)=1=\sum_{k=0}^{M-1}{h(k)e^{-j\omega k}}\) (46)
Condition (46) is verified if:
\(h(k)=\frac{1}{M}e^{j\omega k}, 0\le k\le M-1\) (47)
The output at time k is as follows:
\(y(k)=1=\sum_{i=0}^{M-1}{h(i)x(k-i)=H^t\left[\begin{matrix}ae^{(j(\omega k+\varphi))}\\ae^{(j(\omega(k-1)+\varphi))}\\\begin{matrix}\vdots\\ae^{(j(\omega(k-(M-1))+\varphi))}\\\end{matrix}\\\end{matrix}\right]}\)
\({=H}^tEae^{(j(\omega k+\varphi))}{=H}^tE.y(k)\) (48)
We obtain the final condition:
\(H^tE=1\)
We determine the optimal filter by minimising the criterion with the constraint (44) in consideration.
\({C=H}^t{\ R}_x\ H-(\left(H^tE-1\right)\) (49)
With \(\lambda\) a Lagrange factor. This filter is equal to:
\({\ H=\sigma^2{{\ R}_x}^{-1}\ E\ \ }\) (50)
With,
\(\sigma^2=(=\frac{1}{E^{\ast\ t}\ {{\ R}_x}^{-1}\ E}\) (51)
The PSD value represents the maximum likelihood for the frequency \(\omega\) .