Using the MUSIC Method (Advanced Signal Processing Toolkit)

The multiple signal classification (MUSIC) method is a model-based spectral estimation method. The MUSIC method offers higher frequency resolution in the resulting power spectral density (PSD) than the fast Fourier transform (FFT)-based methods. However, the MUSIC method computes PSD magnitudes that generally are not proportional to the true PSD. Therefore, the PSD that the MUSIC method computes traditionally is known as the pseudo PSD.

Assuming that a time series can be represented by a series of complex sinusoids with additive white noise, the MUSIC method first computes the correlation matrix R_p of the time series and obtains the eigenvectors V₁, V₂, ..., V_p and eigenvalues λ₁, λ₂, ..., λ_p. The eigenvectors V₁, V₂, ..., V_M with large eigenvalues form the signal subspace. The remaining eigenvectors V_M+1, V_M+2, ..., V_p form the noise subspace. The signal subspace also can be represented by group vectors of complex sinusoids e(f₁), e(f₂), ..., e(f_M). The vector of a complex sinusoid is defined as follows:

If a time series contains a frequency component at f_i, the vector e(f_i) is uncorrelated with V_M+1, V_M+2, ..., V_p. Accordingly, the following equation generates a peak value at f_i:

Note  A power spectral density (PSD) normally gives a good indication of the attributes of spectral components contained in a time series. However, the peaks in the PSD computed with the MUSIC method just indicate the frequency locations of components in a time series. These peaks occur when the denominator of the previous equation reaches zero. Therefore, the magnitude of each peak does not indicate the spectral power at the corresponding frequency.

A modified MUSIC method uses the eigenvalues λ₁, λ₂, ..., λ_p as a weighting vector to compute the PSD as follows:

The modified MUSIC method, which also is called the Eigenvector method, can reduce the variance in the estimated PSD.

When using the MUSIC method, you must specify the size of the noise subspace, which is defined as p–M. In general, you can specify a rough percentage of the whole space for the size of the noise subspace. With a large size of the noise subspace, you can obtain a smooth PSD but the resulting PSD may miss weak spectral peaks in the signal. With a small size of the noise subspace, you can obtain a detailed PSD that reveals weak spectral peaks. However, a too small size for the noise subspace leads to spurious peaks in the resulting PSD. Refer to the Discrete Random Signals and Statistical Signal Processing book for more information about using the MUSIC method.

Use the TSA MUSIC VI to compute the PSD of a time series with the MUSIC method.