A Closed-Form ARMA-Based ML-Estimator of a Single-Tone Frequency

Abstract

The new synthesis methodology for single-tone frequency estimation is proposed on the basis of the maximum likelihood (ML) method and autoregressive moving average (ARMA) models. The novel frequency estimator for the scenario with a strongly correlated interference and an additive white Gaussian noise is synthesized in a closed form. The assumption about a constant value of that interference allows to simplify the mixture and transform it to ARMA (3,3) model. The synthesized estimator is invariant to the interference power when it is actually constant (has a unitary correlation coefficient). The particular and much more usual scenario without the interference and just with the noise is considered as well. For that case, the ML-estimator synthesized using the proposed methodology is equal to the existing modified least squares estimator. It is shown that the proposed ML-estimator has an advantage over the known one when the interference is present in the mixture.

Appendix A

Synthesis sequence for the ARMA estimator is given as follows. The residuals of linear prediction for the ARMA (2, 2) model are written as

$$\begin{aligned} \nu _n(x_n,x_{n-1},x_{n-2},\hat{\alpha })\equiv x_n-\hat{x}_n=x_n-\hat{\alpha }x_{n-1}+x_{n-2}, \; n= \overline{3,N} . \end{aligned}$$

(49)

Their joint probability density function (PDF) is

$$\begin{aligned} \begin{aligned}&f_{PDF}(\mathbf {\nu }| \alpha ,\sigma )=\frac{\sigma ^{-(N-2)}_{\nu }(\alpha ,\sigma )}{\sqrt{(2\pi )^{N-2}|\mathbf R|}} \\&\qquad \times \exp \left\{ -\frac{1}{2|\mathbf R|} \sum _{n=3}^N \sum _{k=3}^N \frac{d_{nk}\nu _n(x_n,x_{n-1},x_{n-2}|\alpha )\nu _k(x_k,x_{k-1},x_{k-2}|\alpha )}{\sigma ^2_\nu (\alpha ,\sigma )} \right\} , \end{aligned} \end{aligned}$$

(50)

where $\mathbf {\nu }=\{\nu _3,\dots , \nu _N\}$. We consider this function as a likelihood function and take a logarithm

$$\begin{aligned} \ln f(\alpha ,\sigma | \mathbf {\nu })= \ln \frac{\sigma ^{-(N-2)}_{\nu }(\alpha ,\sigma )}{\sqrt{(2\pi )^{N-2}|\mathbf R|}} -\frac{1}{2|\mathbf R| \sigma ^2_\nu (\alpha ,\sigma )} \sum _{n=3}^N \sum _{k=3}^N d_{nk} \cdot \nu _n(\alpha ) \cdot \nu _k(\alpha ) .\qquad \end{aligned}$$

(51)

After assumptions (30) the first likelihood equation by $\alpha$ can be written

$$\begin{aligned} \frac{\partial \ln f}{\partial \alpha }=-(N-2)\frac{\sigma _\nu '}{\sigma _\nu }+\frac{\sigma _\nu '}{\sigma ^3_\nu }\varSigma -\frac{1}{2\sigma ^2_\nu }\varSigma '=0, \end{aligned}$$

(52)

where the next values are used

$$\begin{aligned} \begin{aligned} \varSigma (\alpha )&= \sum _{n=3}^N \left[ \nu _n(\alpha ) \right] ^2 \equiv \sum _{n=3}^N \left[ x_n-\alpha x_{n-1}+x_{n-2} \right] ^2, \\ \varSigma '(\alpha )&= -\sum _{n=3}^N \left[ 2x_{n-1} (x_n-\alpha x_{n-1}+x_{n-2}) \right] , \\ \sigma _\nu&=\sigma \sqrt{1+\alpha ^2}, \qquad \qquad \sigma _\nu '=\sigma \alpha /\sqrt{2+\alpha }. \end{aligned} \end{aligned}$$

(53)

After taking to the common denominator and throwing away the cubic term the equation is set to its final form

$$\begin{aligned} \alpha ^2\varSigma '-2\alpha \varSigma +2\varSigma '=A\alpha ^2+B\alpha -2A=0, \end{aligned}$$

(54)

where coefficients are defined by equations

$$\begin{aligned} A= \sum _{n=3}^N \bigl [(x_n+x_{n-2})x_{n-1}\bigr ], \;B=\sum _{n=3}^N \bigl [2x^2_{n-1}- (x_n+x_{n-2} )^2 \bigr ]. \end{aligned}$$

(55)

The desired root is calculated as

$$\begin{aligned} \alpha = \left( -B+\sqrt{B^2+8A^2} \right) /2A. \end{aligned}$$

(56)

Choice of the correct root and the asymptotical unbiasedness can be shown in a way similar to the ARMA-I estimator.