# Source localization

Direction-of-arrival (DOA) estimation is a fundamental problem in array signal processing, which has been widely applied in wireless communications and other areas. In 5G wireless communications, massive MIMO (multiple-input multiple-output) is a key enabling technology to increase spectral and energy efficiency by employing a very large number of antennas at the base station. It is well known that the more the number of antennas, the more the degrees-of-freedom (DOFs), and the larger the array aperture, the better the angular resolution. Obviously, massive MIMO has a much smaller angular resolution than conventional MIMO, which implies a better estimation accuracy.

Among various DOA estimation algorithms, MUSIC (MUltiple Signal Classification) is the most popular one as a super-resolution method. The MUSIC spatial pseudo-spectrum can be defined as $\hat{p}_{\tiny{\mbox{MUSIC}}}(\theta) = \frac{1}{\boldsymbol{a}^{\mathrm{H}}(\theta) \hat{\boldsymbol{E}}_{n} \hat{\boldsymbol{E}}_{n}^{\mathrm{H}} \boldsymbol{a}(\theta)}, \nonumber$

where $\boldsymbol{a}(\theta)$ is the steering vector, and $\hat{\boldsymbol{E}}_{n}$ is composed of the noise subspace eigenvectors which can be estimated from the sample covariance matrix $\hat{\boldsymbol{R}}_{xx} = \frac{1}{K} \sum_{k=1}^{K} \boldsymbol{x}(k) \boldsymbol{x}^{\mathrm{H}}(k)$. Here, $\boldsymbol{x}(k)$ is the array received vector at the $k$-th snapshot and $K$ is the number of snapshots. By searching the peaks of the MUSIC spectrum, the DOAs of sources can be estimated. It is noted that to eigen-decompose the sample covariance matrix $\hat{\boldsymbol{R}}_{xx}$, the number of snapshots $K$ should be larger than number of antennas, which is very large in 5G base station.

On the other hand, when the number of snapshots is limited (i.e., $K$ is less than the number of antennas at the base station), the spatial spectrum can be estimated by solving the following compressive sensing optimization problem $\begin{array}{rcl}&\min\limits_{\boldsymbol{p}, \sigma_{n}^{2}}& \left\vert \hat{\boldsymbol{R}}_{xx} - \boldsymbol{A} \boldsymbol{P} \boldsymbol{A}^{\mathrm{H}} - \sigma_{\boldsymbol{n}}^{2}\boldsymbol{I} \right\vert_{F} \nonumber \\& subject\ to & \boldsymbol{P} \geq \boldsymbol{0}, \sigma_{n}^{2} > 0 \nonumber \end{array}$

where $\boldsymbol{p}$ is the spatial spectrum distribution on the sample grids of the observed spatial domain and $\boldsymbol{P} = \mbox{diag}{\boldsymbol{\{p\}}}$ is the corresponding diagonal matrix, $\boldsymbol{A} = [\boldsymbol{a}(\theta_{1}), \boldsymbol{a}(\theta_{2}), \cdots, \boldsymbol{a}(\theta_{L})]$ is the array manifold matrix, $\boldsymbol{I}$ is an identity matrix, and $\gamma$ is a regularization parameter controlling the tradeoff between the sparsity of the spatial spectrum and the residual norm of covariance matrix fitting.

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VOCAL Technologies offers custom designed source localization solutions in 5G from array configuration to DOA estimation. Our custom implementations of such systems are meant to deliver optimum performance for your specific signal processing task. Contact us today to discuss your solution.