MUSIC: MUSIC

MUltiple SIgnal Classification (MUSIC) is an algorithm used for frequency estimation and emitter location.
In many practical signal processing problems, the objective is to estimate from measurements a set of constant parameters upon which the received signals depend. For example, high-resolution direction-of-arrival (DOA) estimation is important in many sensor systems such as radar, sonar, electronic surveillance, and seismic exploration. High-resolution frequency estimation is important in numerous applications, recent examples of which include the design and control of robots and large flexible space structures. In such problems, the functional form of the underlying signals can often be assumed to be known (e.g., narrow-band plane waves, cisoids). The quantities to be estimated are parameters (e.g., frequencies and DOA's of plane waves, cisoid frequencies) upon which the sensor outputs depend, and these parameters are assumed to be constant.
There have been several approaches to such problems including the so-called maximum likelihood (ML) method of Capon (1969) and Burg's (1967) 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) were the first to correctly exploit the measurement model in the case of sensor arrays of arbitrary form. Schmidt, in particular, accomplished this by first deriving a complete geometric solution in the absence of noise, then cleverly extending the geometric concepts to obtain a reasonable approximate solution in the presence of noise. The resulting algorithm was called MUSIC (Multiple SIgnal Classification) and has been widely studied. In a detailed evaluation based on thousands of simulations, M.I.T.'s Lincoln Laboratory concluded that, among 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).

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

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