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See Also

Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Spectrum Analysis of Sinusoids
       Optimal Peak-Finding in the Spectrum
          Likelihood Function

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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}\index{... ...unction $l_x(\underline{\theta})$\ 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... ...a_v^2}\sum_{n=0}^{N-1} \left\vert x(n) - {\cal A}e^{j\omega_0 n}\right\vert^2} $

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. $

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 $

as we saw before in (4.10).


Subsections
Previous: Maximum Likelihood Sinusoid Estimation
Next: Multiple Sinusoids in Additive Gaussian White Noise

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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