Multiple Signal Classification (MUSIC)

An observation signal vector can be expressed as

\begin{align*} \mathbf{x} = \mathbf{A} \mathbf{s} + \mathbf{n}, \end{align*}

where

• \(\mathbf{A} = \begin{bmatrix} \mathbf{a}(\omega_1) & \mathbf{a}(\omega_2) & \ldots & \mathbf{a}(\omega_r) \end{bmatrix}\) is a \(N \times r\) Vandermonde matrix, \(N > r\); Each column is a steering vector, e.g., \(\mathbf{a}(\omega) \triangleq \begin{bmatrix} 1 \\ e^{j\omega} \\ e^{j2\omega} \\ \vdots \\ e^{j(N-1)\omega} \end{bmatrix}\).
• \(\mathbf{s} = \begin{bmatrix} s_1 \\ s_2 \\ \vdots \\ s_r \end{bmatrix}\) is a signal vector.
• \(\mathbf{n} \sim \mathcal{CN}(0, \sigma^2\mathbf{I}_N)\) is a complex additive Gaussian noise vector.

Then autocorrelation matrix of observation can be obtained by

\begin{align*} \mathbf{R}_{\mathbf{x}} &= \mathbf{E}[ \mathbf{x} \mathbf{x}^H] \\ &= \mathbf{A} \mathbf{R}_{\mathbf{s}} \mathbf{A}^H + \sigma^2 \mathbf{I}_N, \end{align*}

where \(\mathbf{R}_{\mathbf{s}} \triangleq \mathbf{E}[ \mathbf{s} \mathbf{s}^H]\) is the autocorrelation of signal vector \(\mathbf{s}\). In practice, the autocorrelation of observation is usually calculated based on a series of observation samples, i.e., \(\hat{\mathbf{R}}_{\mathbf{x}} = \dfrac{1}{M} \sum_{i=1}^M \mathbf{x}_i \mathbf{x}_i^H\), where \(M\) is the number of observation samples available.

The autocorrelation matrix of observation can be eigen value decomposed (EVD) as

\begin{align*} \mathbf{R}_{\mathbf{x}} &= \begin{bmatrix} \mathbf{U}_{\mathbf{s}} & \mathbf{U}_{\mathbf{n}} \end{bmatrix} \begin{bmatrix} \mathbf{\Lambda}_{\mathbf{s}} & \mathbf{0} \\ \mathbf{0} & \sigma^2 \mathbf{I}_{N-r} \end{bmatrix} \begin{bmatrix} \mathbf{U}_{\mathbf{s}}^H \\ \mathbf{U}_{\mathbf{n}}^H \end{bmatrix}, \end{align*}

where

• The eigen vectors in \(\mathbf{U}_{\mathbf{s}}\) corresponding to \(p\) largest eigen values span the signal subspace.
• The eigen vectors in \(\mathbf{U}_{\mathbf{n}}\) corresponding to the remaining \(N - r\) eigen values (equal to \(\sigma^2\)) span the noise subspace.

Considering the fact that the signal subspace and noise subspace are orthogonal and complementary to each other, i.e. \(\text{span}(\mathbf{U}_{\mathbf{s}}) \perp \text{span}(\mathbf{U}_{\mathbf{n}})\), \(\forall \mathbf{v} \in \text{span}(\mathbf{U}_{\mathbf{s}})\), \(\mathbf{U}_{\mathbf{n}}^H \mathbf{v} = \mathbf{0}\). In order to depict the orthogonality quantitatively, a squared norm is introduced as follows.

\begin{align*} d^2(\mathbf{v}) &= \| \mathbf{U}_{\mathbf{n}}^H \mathbf{v}\|^2 \\ &= \mathbf{v}^H \mathbf{U}_{\mathbf{n}} \mathbf{U}_{\mathbf{n}}^H \mathbf{v} \\ &= \mathbf{v}^H (\mathbf{I}_N - \mathbf{U}_{\mathbf{s}} \mathbf{U}_{\mathbf{s}}^H) \mathbf{v} \end{align*}

MUSIC uses the reciprocal of the squared norm \(d^{-2}\) as the metric, in which case \(r\) sharp peaks appear at the signal frequencies. The MUSIC spectrum can be expressed as

\begin{align*} P_{\text{music}}(\omega) &= \frac{1}{d^2(\mathbf{a}(\omega))} \\ &= \frac{1}{\mathbf{a}^H(\omega) \mathbf{U}_{\mathbf{n}} \mathbf{U}_{\mathbf{n}}^H \mathbf{a}(\omega)} \\ & = \frac{1}{\mathbf{a}^H(\omega) (\mathbf{I}_N - \mathbf{U}_{\mathbf{s}} \mathbf{U}_{\mathbf{s}}^H) \mathbf{a}(\omega)}, \end{align*}

where \(\omega\) is the candidate frequency, and \(\mathbf{a}(\omega)\) is the candidate steering vector.

So long as the number of component (e.g., \(r\) above) is known in advance (as its drawback), MUSIC can outperforms DFT based schemes, since it can achieve superresolutional estimation for the spectrum, rather than the discrete DFT bins.

1. http://en.wikipedia.org/w/index.php?title=MUSIC_(algorithm)
2. 张贤达，矩阵分析与应用，清华大学出版社。