# Online Sparse DOA Estimation Based on Sub-Aperture Recursive LASSO for TDM-MIMO Radar
## 2 Materials and Methods
### Signal Model  
Suppose that there are \(K\) impinging signals with angular parameter, \(\theta_{k}\), \(k=1,\ldots,K\). According to Figure 1, the steering expression of the \(m\)-th virtual array element is written as  
\[\begin{split} a(\theta_{k})&=\exp(j(m-1)d_{r}\sin \theta_{k})\\ &=\exp(j(m_{t}-1)d_{t}\sin\theta_{k})\cdot\exp(j(m_{r}-1)d_{r}\sin \theta_{k})\end{split} \tag{1}\]  
where \(m_{t}=1,2,\ldots,M_{t}\) and \(m_{r}=1,2,\ldots,M_{r}\) represent the sequence number of transmitting and receiving array elements, respectively. In addition, \(m\), \(m_{t}\) and \(m_{r}\) satisfy the relationship \(m=(m_{t}-1)M_{r}+m_{r}\). For all virtual elements, the steering vector can be denoted as  
\[\mathbf{a}(\theta_{k})=[a_{1}(\theta_{k}),a_{2}(\theta_{k}),\ldots,a_{M}( \theta_{k})]^{T} \tag{2}\]  
where \(M=M_{t}\cdot M_{r}\). We assume that there are \(L\) TDM periods and each period contains \(N\) pulses. Hence, the \(\ell\)-th complex baseband receiver signal \(\mathbf{y}(\ell)\in\mathbb{C}^{M}\) can be denoted as [19]  
\[\mathbf{y}(\ell)=\exp(-j\cdot\boldsymbol{\gamma}\omega_{d})\odot\mathbf{a}( \theta_{k})\frac{1}{\sqrt{N}}s(\ell)+\mathbf{e}(\ell) \tag{3}\]  
where \(\odot\) represents the Hadamard product, \(s(\ell)\) the unknown and deterministic target signals, \(\mathbf{e}(\ell)\) the additive zero mean Gaussian noise, and \(\sigma^{2}\) the variance. \(\omega_{d}\) denotes the Doppler frequency of the target. \(\boldsymbol{\gamma}=\mathbf{t}\otimes\mathbf{1}_{M_{r}}\) represents the TDM-induced Doppler frequency shift (DFS), where \(\mathbf{t}=[t_{1},t_{2},t_{3},t_{4}]\) represents the time-division scheme of the signal transmission, and \(\mathbf{1}_{M_{t}}\) denotes the full one-column vector with a length of \(M_{r}\).  
To simplify the model, it is assumed that \(\omega_{d}=0\), and \(N=1\). For the \(\ell\)-th snapshot of a given TDM period, model (3) can be simplified to  
\[\mathbf{y}(\ell)=\mathbf{A}\mathbf{s}(\ell)+\mathbf{e}(\ell) \tag{4}\]  
Figure 1: Array element distribution of MIMO radar (\(M_{t}=4\), \(M_{r}=4\)).  
where the steering matrix is given by  
\[\mathbf{A}=[\mathbf{a}(\theta_{1}),\mathbf{a}(\theta_{2}),\dots,\mathbf{a}(\theta _{K})]=\left[\begin{array}{c}\mathbf{b}_{1}^{*}\\ \mathbf{b}_{2}^{*}\\ \vdots\\ \mathbf{b}_{M}^{*}\end{array}\right]\in\mathbb{C}^{M\times K} \tag{5}\]  
The receiver signals \(\mathbf{y}\) of the \(M\) channels are arranged as \(1,2,\cdots,M_{t}\) according to the sub-aperture order.  
The TDM-MIMO radar system transmits the signals in a time-division manner per the transmitting scheme, as shown by Figure 2a. Only one transmitting antenna is available for each time slot. In the case of the four receivers, the received signals are shown in Figure 2b. The blocks with order 1-4 denote the sub-aperture data received in the first period, and so on. Any set of sub-apertures is equivalent to a SIMO subarray with \(M_{r}\) receivers. In the traditional batch processing method, all sub-aperture data must be received completely to formulate the full-aperture data before processing. When dealing with a massive TDM-MIMO data set, the dimension of aperture data will become quite high. As a result, the high dimensionality of the full-channel data increases the temporal and spatial complexity. Moreover, the period of reception will introduce additional delay. Both of these reasons lead to poor real-time performance of batch processing. To solve this problem, we propose an online processing method that processes sub-aperture data along receiving. The proposed method allows beamforming to begin directly after the first sub-aperture data are received. Doing so introduces two benefits, one is that it can save the time cost of measurement collection, the other is that online processing allows for much less iterations, avoiding high-dimensional matrix inversion and multiplication, resulting in lower temporal and spatial complexity.  
Figure 2: (a) Time–division scheme of the TDM transmit signal, (b) sub–aperture receiver data.