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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.

$\textstyle \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

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

and the log likelihood function is

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

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

as we saw before in (5.33).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University