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Spectral Audio Signal Processing
    Spectrum Analysis of Sinusoids
       Optimal Peak-Finding in the Spectrum
          Least Squares Sinusoidal Parameter Estimation

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Least Squares Sinusoidal Parameter Estimation

There are many ways to define ``optimal'' in signal modeling. Perhaps the most elementary case is least squares estimation. Every estimator tries to measure one or more parameters of some underlying signal model. In the case of sinusoidal parameter estimation, the simplest model consists of a single complex sinusoidal component in additive white noise:

$\displaystyle x(n) \isdef {\cal A}e^{j\omega_0 n} + v(n) \protect$ (5.9)

where $ {\cal A}= A e^{j\phi}$ is the complex amplitude of the sinusoid, and $ v(n)$ is white noise (defined in §C.3). Given measurements of $ x(n)$, $ n=0,1,2,\ldots,N-1$, we wish to estimate the parameters $ (A,\phi,\omega_0 )$ of this sinusoid. In the method of least squares, we minimize the sum of squared errors between the data and our model. That is, we minimize

$\displaystyle J(\underline{\theta}) \isdef \sum_{n=0}^{N-1}\left\vert x(n)-{\hat x}(n)\right\vert^2 \protect$ (5.10)

with respect to the parameter vector

$\displaystyle \underline{\theta}\isdef \left[\begin{array}{c} A \\ [2pt] \phi \\ [2pt] \omega_0 \end{array}\right], $

where $ {\hat x}(n)$ denotes our signal model:

$\displaystyle {\hat x}(n)\isdef \hat{{\cal A}}e^{j\hat{\omega}_0n} $

Note that the error signal $ x(n)-\hat{{\cal A}}e^{j\hat{\omega}_0n}$ is linear in $ \hat{{\cal A}}$ but nonlinear in the parameter $ \hat{\omega}_0$. More significantly, $ J(\underline{\theta})$ is non-convex with respect to variations in $ \hat{\omega}_0$. Non-convexity can make an optimization based on gradient descent very difficult, while convex optimization problems can generally be solved quite efficiently [21,84].


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Next: Sinusoidal Amplitude Estimation

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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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