# Multiple signal classification

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Multiple signal classification

MUltiple SIgnal Classification (MUSIC) is an algorithm used for frequency estimation[1] and emitter location.[2]

## MUSIC algorithm

In many practical signal processing problems, the objective is to estimate from measurements a set of constant parameters upon which the received signals depend. There have been several approaches to such problems including the so-called maximum likelihood (ML) method of Capon (1969) and Burg's maximum entropy (ME) method. Although often successful and widely used, these methods have certain fundamental limitations (especially bias and sensitivity in parameter estimates), largely because they use an incorrect model (e.g., AR rather than special ARMA) of the measurements. Pisarenko (1973) was one of the first to exploit the structure of the data model, doing so in the context of estimation of parameters of cisoids in additive noise using a covariance approach. Schmidt (1977), while working at ESL (now part of Northrop Grumman) and independently Bienvenu (1979) currently accepted high-resolution algorithms, MUSIC was the most promising and a leading candidate for further study and actual hardware implementation. However, although the performance advantages of MUSIC are substantial, they are achieved at a cost in computation (searching over parameter space) and storage (of array calibration data).

## Application to frequency estimation

MUSIC estimates the frequency content of a signal or autocorrelation matrix using an eigenspace method. This method assumes that a signal, x(n), consists of p complex exponentials in the presence of Gaussian white noise. Given an $M \times M$ autocorrelation matrix, $\mathbf{R}_x$, if the eigenvalues are sorted in decreasing order, the eigenvectors corresponding to the p largest eigenvalues span the signal subspace. Note that for M = p + 1, MUSIC is identical to Pisarenko's method. The general idea is to use averaging to improve the performance of the Pisarenko estimator.

The frequency estimation function for MUSIC is

$\hat P_{MU}(e^{j \omega}) = \frac{1}{\sum_{i=p+1}^{M} |\mathbf{e}^{H} \mathbf{v}_i|^2},$

where $\mathbf{v}_i$ are the noise eigenvectors and

$e = \begin{bmatrix}1 & e^{j \omega} & e^{j 2 \omega} & \cdots & e^{j (M-1) \omega}\end{bmatrix}^T.$

## History

MUSIC was originated by R. O. Schmidt in 1979 as an improvement to Pisarenko's method.

## References

1. ^ Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. ISBN 0-471-59431-8.
2. ^ Schmidt, R.O, "Multiple Emitter Location and Signal Parameter Estimation," IEEE Trans. Antennas Propagation, Vol. AP-34 (March 1986), pp.276-280.

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