# Efficient Support Vector Regression for Wideband DOA Estimation Using a Genetic Algorithm
## 2 Wideband Array Signal Model  
Consider a uniform linear array (ULA) with \(M\) sensors and \({}_{K}\) wideband signals \(s(t)=\left[s_{1}(t),\cdots,s_{K}(t)\right]^{\text{T}}\) in the far-field from direction \(\theta=\left[\theta_{1},\cdots,\theta_{K}\right]^{\text{T}}\). Then, the received signal at the \({}_{m}\)-th sensor of the ULA can be expressed as  
\[x_{m}(t)=\sum_{k=1}^{K}s_{k}(t-\tau_{m,k})+n_{m}(t),m=1,2,\cdots,M \tag{1}\]  
where \((\cdot)^{\text{T}}\) is the transpose operation, \(\tau_{m,k}=(m-1)d\sin\theta_{k}/c\) denotes the time delay of the \(k\)-th target at the \({}_{m}\)-th sensor relative to the reference sensor, \(d\) is the spacing between sensors, \({}_{c}\) is the speed of electromagnetic wave propagation, and \(n_{m}(t)\) denotes the noise observed at the \({}_{m}\)-th sensor. For the whole array, variations in the envelope and phase of the wideband signal across different sensors cannot be ignored. Therefore, it is typically necessary to divide the array output for frequency division processing to establish a frequency-domain model. We transform the temporally sampled data of the \(M\) sensors into the frequency domain using a discrete Fourier transform (DFT), which divides it into \(J\) sub-bands. Then, the mathematical model of the \(j\)-th frequency point is [43]:  
\[\mathbf{x}(f_{j})=\mathbf{A}\big{(}f_{j},\theta\big{)}\mathbf{s}(f_{j})+ \mathbf{n}\big{(}f_{j}),\;j=1,2,\cdots,J \tag{2}\]where \(\mathbf{x}(f_{j})=\left[x_{1}(f_{j}),\cdots,x_{M}(f_{j})\right]^{\mathrm{T}}\) and \(\mathbf{n}(f_{j})=\left[n_{1}(f_{j}),\cdots,n_{M}(f_{j})\right]^{\mathrm{T}}\) denote the Fourier coefficient vector of the array's received data and noise at the frequency point \(f_{j}\), respectively; \(x_{m}(f_{j})\) is the Fourier coefficient of the signal \(x_{m}(t)\) received at the \(m\)-th sensor. \(\mathbf{s}(f_{j})=\left[s_{1}(f_{j}),\cdots,s_{K}(f_{j})\right]^{\mathrm{T}}\) is the \(K\times 1\) signal vector. \(\mathbf{A}\big{(}f_{j},\theta\big{)}=\left[\mathbf{a}\big{(}f_{j},\theta_{1} \big{)},\mathbf{a}\big{(}f_{j},\theta_{2}\big{)},\cdots,\mathbf{a}\big{(}f_{j },\theta_{K}\big{)}\right]\) is the array manifold matrix at the frequency point \(f_{j}\), and \(\mathbf{a}\big{(}f_{j},\theta_{k}\big{)}\in\mathbf{C}^{M\times 1}\) is the steering vector corresponding to the \(k\)-th signal, \(\mathbf{a}\big{(}f_{j},\theta_{k}\big{)}=\left[1,e^{-2\pi f_{j}d\sin\theta_{k} /c},\cdots,e^{-2\pi f_{j}(M-1)d\sin\theta_{k}/c}\right]^{\mathrm{T}}\). Equation (2) represents the mathematical model of the wideband array signal for a single snapshot. In practice, a whole period of time can be divided into \({}_{L}\) equally spaced segments, provided that the duration of each segment exceeds the correlation time between signal and noise. This ensures the acquisition of multiple snapshots in the frequency domain. The mathematical model for the \(l\)-th snapshot can be expressed as  
\[\mathbf{x}_{l}\big{(}f_{j}\big{)}=\mathbf{A}\big{(}f_{j},\theta\big{)}\mathbf{ s}_{l}\big{(}f_{j}\big{)}+\mathbf{n}_{l}\big{(}f_{j}\big{)},\;l=1,2,\cdots,L \tag{3}\]  
Given that the signal is uncorrelated with the noise, and different channel noises are uncorrelated, the covariance matrix for the \(j\)-th frequency point is  
\[\mathbf{R}_{j}=E\Big{[}\mathbf{x}_{l}\big{(}f_{j}\big{)}\mathbf{x}_{l}^{ \mathrm{H}}\big{(}f_{j}\big{)}\Big{]}=\mathbf{A}\big{(}f_{j},\theta\big{)} \mathbf{R}_{s}\mathbf{A}^{\mathrm{H}}\big{(}f_{j},\theta\big{)}+\sigma_{n}^{2 }\mathbf{I} \tag{4}\]  
where \(E[\cdot]\) represents the expectation operation, \(\mathbf{R}_{s}=E\big{[}\mathbf{s}_{l}\big{(}f_{j}\big{)}\mathbf{s}_{l}^{ \mathrm{H}}\big{(}f_{j}\big{)}\big{]}\) is the signal covariance matrix, \(\sigma_{n}^{2}\) denotes the noise power, and \({}_{\mathrm{I}}\) is the identity matrix. The superscript \(\left(\cdot\right)^{\mathrm{H}}\) indicates the conjugate transpose operation. From Equation (4), we observe that \(\mathbf{A}\big{(}f_{j},\theta\big{)}\) depends on both the signal incidence angle \(\theta\) and the frequency \(f_{j}\).