MUSIC (MUltiple SIgnal Classification) is an algorithm used for frequency estimation and radio direction finding.
History 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 complex sinusoids in additive noise using a covariance approach. Schmidt (1977), while working at Northrop Grumman and independently Bienvenu and Kopp (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, the Massachusetts Institute of Technology's Lincoln Laboratory concluded in 1998 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).
Theory MUSIC method assumes that a signal vector, x {\displaystyle \mathbf {x} } , consists of p {\displaystyle p} complex exponentials, whose frequencies ω {\displaystyle \omega } are unknown, in the presence of Gaussian white noise, n {\displaystyle \mathbf {n} } , as given by the linear model
x = A s + n . {\displaystyle \mathbf {x} =\mathbf {A} \mathbf {s} +\mathbf {n} .}
Here A = [ a ( ω 1 ) , ⋯ , a ( ω p ) ] {\displaystyle \mathbf {A} =[\mathbf {a} (\omega _{1}),\cdots ,\mathbf {a} (\omega _{p})]} is an M × p {\displaystyle M\times p} Vandermonde matrix of steering vectors a ( ω ) = [ 1 , e j ω , e j 2 ω , … , e j ( M − 1 ) ω ] T {\displaystyle \mathbf {a} (\omega )=[1,e^{j\omega },e^{j2\omega },\ldots ,e^{j(M-1)\omega }]^{T}} and s = [ s 1 , … , s p ] T {\displaystyle \mathbf {s} =[s_{1},\ldots ,s_{p}]^{T}} is the amplitude vector. Note that that the formulation of the steering vector matrix as Vandermonde assumes the signals are incident on a uniform, linear array. Another crucial assumption is that number of sources, p {\displaystyle p} , is less than the number of elements in the measurement vector, M {\displaystyle M} , i.e. p < M {\displaystyle p<M} . The M × M {\displaystyle M\times M} autocorrelation matrix of x {\displaystyle \mathbf {x} } is then given by
R x = A R s A H + σ 2 I , {\displaystyle \mathbf {R} _{x}=\mathbf {A} \mathbf {R} _{s}\mathbf {A} ^{H}+\sigma ^{2}\mathbf {I} ,}
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