Likelihood Function

The likelihood function $l_x(\underline{\theta})$ is defined as the probability density function of $x$ given $\underline{\theta}= [A,\phi,\omega_0 ,\sigma_v^2]^T$ , evaluated at a particular $x$ , with $\underline{\theta}$ regarded as a variable.

In other words, the likelihood function $l_x(\underline{\theta})$ is just the PDF of $x$ with a particular value of $x$ plugged in, and any parameters in the PDF (mean and variance in this case) are treated as variables.

$\parbox{0.8\textwidth}{The \emph{maximum likelihood estimate} of the parameter vector $\underline{\theta}$\ is defined as the value of $\underline{\theta}$\ which maximizes the likelihood function $l_x(\underline{\theta})$\ given a particular set of observations $x$.}$

For the sinusoidal parameter estimation problem, given a set of observed data samples $x(n)$ , for $n=0,1,2,\ldots,N-1$ , the likelihood function is

$$l_x(\underline{\theta}) = \frac{1}{\pi^N \sigma_v^{2N}} e^{-\frac{1}{\sigma_v^2}\sum_{n=0}^{N-1} \left\vert x(n) - {\cal A}e^{j\omega_0 n}\right\vert^2}$$ (6.50)

and the log likelihood function is

$$\log l_x(\underline{\theta}) = -N\log(\pi \sigma_v^2) -\frac{1}{\sigma_v^2}\sum_{n=0}^{N-1}\left\vert x(n) - {\cal A}e^{j\omega_0 n}\right\vert^2.$$ (6.51)

We see that the maximum likelihood estimate for the parameters of a sinusoid in Gaussian white noise is the same as the least squares estimate. That is, given $\sigma_v$ , we must find parameters ${\cal A}$ , $\phi$ , and $\omega_0$ which minimize

$$J(\underline{\theta}) = \sum_{n=0}^{N-1} \left\vert x(n) - {\cal A}e^{j\omega_0 n}\right\vert^2$$ (6.52)

as we saw before in (5.33).