## MUSIC (Multiple Signal Classification) Algorithm

MUSIC is a spatial spectrum estimation algorithm based on second order statistics.  It attracted intensive studies due to the following perceived advantages.

• Capability to handle multiple simultaneous sources
• High precision measurement
• High spatial resolution
• Adjustable to small observable data cases
• Possible for real time implementation.

This short paper the basic idea and implementation are described. Its use for resolving multiple sound sources with antenna array is discussed.

Generalizing Figure 1 to N sources to M microphones, we have the microphone captured sound vector, $y_m\left(t\right)=\sum_{n=1}^{N}{s_n\ \left(t\right)e^{-j2\pi f\left(m-1\right)dsin\left(\theta_n\right)/v}}+noise_m\left(t\right)$ $=\sum_{n=1}^{N}{s_n\left(t\right)a_m\left(\theta_n\right)}+noise_m\left(t\right)$

We can further write this into the following matrix form, $Y=AS+N$

where $Y = [y1(t), y2(t), \dots , yM(t)]$, $S = [s1(t), s2(t), \dots , sN(t)]$, and $A=\left[a_1\left(\theta_1\right),a_2\left(\theta_2\right),\dots,a_M\left(\theta_N\right)\right]$.

The covariance matrix, $R_y=E\left[YY^H\right]=AR_sA^H+R_n$, is Hermitian and positive definite since $R_n$ is always positive definite.
By eigenvalue decomposition, $vi$ and $\lambda i$ are the eigenvector and corresponding eigenvalue, we have $R_yv_i=\lambda_iv_i.$

The noise dimension has smaller eigenvalue, which is the noise floor, while the dimensions that contain signals will have larger eigenvalue. Therefore, the noise subspace can be constructed by $E_N=\left[v_{N+1},v_{N+1},\dots,v_M\right]$.

The signal dimension will have smaller value if it is projected into the noise subspace. Therefore, the following formula will have larger value and appear to be a peak, $\frac{1}{\left|E_n^Ha\left(\theta\right)\right|^2}\ =\ \frac{1}{a^H\left(\theta\right)E_nE_n^Ha\left(\theta\right)}$

The above formula defines the MUSIC algorithm. The number of peaks indicates the number of independent sound sources and the corresponding direction $\theta$ defines the incoming direction of each sound source.