Fourth-order cumulants-based sparse representation approach for DOA estimation in MIMO radar with unknown mutual coupling

Joint Smoothed l0-Norm DOA Estimation Algorithm for Multiple Measurement Vectors in MIMO Radar

“…In the far field, there are P uncorrelated targets regarded as point scatterers at the same range. In addition, it is assumed that $P\le M+N-2$ [22,25,26]. Thus, at the receive array, the N antennas are impinged by the echo signals reflected by the P targets.…”

“…Then, by matched filtering operation, the $N\times 1$ dimensional complex envelop of the output of the m th carrier matched filter is expressed as [14,27] $\begin{array}{c}\hfill {\mathrm{boldx}}_m\left(t\right)=\sum _{p=1}^P{\mathbf{a}}_r\left({\theta }_p\right){\mathbf{a}}_{tm}^{\mathtt{T}}\left({\theta }_p\right)s_p\left(t\right)+{\mathrm{boldn}}_m\left(t\right),\end{array}$ where ${\mathbf{a}}_r\left({\theta }_p\right)={\left[1,e^{j\pi sin\left({\theta }_p\right)},e^{j\pi 2sin\left({\theta }_p\right)},\dots ,e^{j\pi (N-1)sin\left({\theta }_p\right)}\right]}^{\mathtt{T}}$ is the receive steering vector, ${\mathrm{bolda}}_{tm}$ is the m th element of the transmit steering vector ${\mathbf{a}}_t\left({\theta }_p\right)={\left[1,e^{j\pi sin\left({\theta }_p\right)},e^{j\pi 2sin\left({\theta }_p\right)},\dots ,e^{j\pi (M-1)sin\left({\theta }_p\right)}\right]}^{\mathtt{T}}$, $s_p\left(t\right)$ contains the target reflection coefficient and the transmitted baseband signal such as non-circular signal, and ${\mathbf{n}}_m\left(t\right)$ is the noise vector after the m th matched filter. After all the matched filters, the received data vector, i.e., the vector composed of the outputs of the M matched filters, is given by [2,25,28] …”

“…It is written as $\mathrm{boldb}\left({\theta }_p\right)={false[1,e^{j\pi sin\left({\theta }_p\right)},e^{j\pi 2sin\left({\theta }_p\right)},\dots ,e^{j\pi (M+N-2)sin\left({\theta }_p\right)}false]}^{\mathtt{T}},$ where $\mathrm{boldb}\left({\theta }_p\right)\in {\mathrm{double-struckC}}^{(M+N-1)\times 1}$. According to the element structures of $\mathrm{bolda}\left({\theta }_p\right)$ and $\mathrm{boldb}\left({\theta }_p\right)$, their relationship can be expressed as [25] $\mathrm{bolda}\left({\theta }_p\right)=\mathrm{boldGb}\left({\theta }_p\right),$ where $\mathrm{boldG}={[{\mathbf{L}}_0^{\mathrm{T}},{\mathbf{L}}_1^{\mathrm{T}},\cdots ,{\mathbf{L}}_{M-1}^{\mathrm{T}}]}^{\mathrm{T}}\in {\mathrm{double-struckC}}^{MN\times (M+N-1)}$ and ${\mathbf{L}}_m=[{\mathrm{bold0}}_{N\times m},{\mathrm{boldI}}_N,{\mathrm{bold0}}_{N\times (M-m-1)}]\in {\mathrm{double-struckC}}^{N\times (M+N-1)}$ for $m=0,1,...,M-1$. Based on the relationship in Equation (9), a reduced-dimensional matrix can be defined as [25] $\mathrm{boldR}={\left({\mathbf{G}}^{\mathrm{H}}\mathrm{boldG}\right)}^{false(-\frac{1}{2}false)}$…”

“…Therefore, the dimensional reduction for the received data is carried out as follows [25]:$\begin{array}{cc}\hfill \stackrel{false˜}{\mathrm{boldx}}\left(t\right)& =\mathbf{R}\mathbf{G}\mathbf{B}\mathbf{s}\left(t\right)+\mathbf{R}\mathbf{n}\left(t\right)\hfill \\ & ={\left({\mathrm{boldG}}^{\mathrm{normalH}}\mathbf{G}\right)}^{(-\frac{1}{2})}{\mathrm{boldG}}^{\mathrm{normalH}}\mathbf{G}\mathbf{B}\mathbf{s}\left(t\right)+\stackrel{false˜}{\mathrm{boldn}}\left(t\right)\hfill \\ & =\mathbf{F}\mathbf{B}\mathbf{s}\left(t\right)+\stackrel{false˜}{\mathrm{boldn}}\left(t\right),\hfill \end{array}$ where $\tilde{\mathbf{x}}\left(t\right)=\mathrm{boldR}\mathrm{boldx}\left(t\right)\in {\mathrm{double-struckC}}^{(M+N-1)\times 1}$, $\mathrm{boldB}=[\mathbf{b}\left({\theta }_1\right),\dots ,\mathbf{b}\left({\theta }_P\right)]\in {\mathrm{double-struckC}}^{(M+N-1)\times P}$ and $\tilde{\mathbf{n}}\left(t\right)\in {\mathrm{double-struckC}}^{(M+N-1)\times 1}$ are the new low-dimensional received data vector, transmit–receive steering matrix and noise vector, respectively. In addition, $\mathrm{boldF}={\left({\mathbf{G}}^{\mathrm{H}}\mathrm{boldG}\right)}^{\frac{1}{2}}$ is a known diagonal matrix, and its $(i,i)$th element can be expressed as …”

“…To eliminate the Gaussian white noise or colored noise in $\tilde{\mathbf{x}}\left(t\right)$, fourth-order cumulant is exploited under the circumstance of collecting J snapshots. The definition of the FOC is given as follows [12,24,25] $\begin{array}{cc}\hfill C_{4\stackrel{false˜}{x}}(k_1,k_2,k_3,k_4)& =\mathrm{cum}\left\{{\tilde{\mathbf{x}}}_{k_1},{\tilde{\mathbf{x}}}_{k_2}^{*},{\tilde{\mathbf{x}}}_{k_3},{\tilde{\mathbf{x}}}_{k_4}^{*}\right\}=\mathrm{E}\left({\stackrel{false˜}{\mathrm{boldx}}}_{k_1}{\stackrel{false˜}{\mathrm{boldx}}}_{k_2}^{*}{\stackrel{false˜}{\mathrm{boldx}}}_{k_3}{\stackrel{false˜}{\mathrm{boldx}}}_{k_4}^{*}\right)-\mathrm{E}\left({\stackrel{false˜}{\mathrm{boldx}}}_{k_1}{\stackrel{false˜}{\mathrm{boldx}}}_{k_2}^{*}\right)\mathrm{E}\left({\stackrel{false˜}{\mathrm{boldx}}}_{k_3}{\stackrel{false˜}{\mathrm{boldx}}}_{k_4}^{*}\right)\hfill \\ & -\mathrm{E}\left({\stackrel{false˜}{\mathrm{boldx}}}_{k_1}{\stackrel{false˜}{\mathrm{boldx}}}_{k_3}\right)\mathrm{E}\left({\stackrel{false˜}{\mathrm{boldx}}}_{k_2}^{*}{\stackrel{false˜}{\mathrm{boldx}}}_{k_4}^{*}\right)-\mathrm{E}\left({\stackrel{false˜}{\mathrm{boldx}}}_{k_1}{\stackrel{false˜}{\mathrm{boldx}}}_{k_4}^{*}\right)\mathrm{E}\left({\stackrel{false˜}{\mathrm{boldx}}}_{k_2}^{*}{\stackrel{false˜}{\mathrm{boldx}}}_{k_3}\right),\hfill \end{array}$ where …”