Wideband MIMO Radar Transmit Beampattern Synthesis via Majorization–Minimization

Abstract

We consider the design of constant-modulus waveforms for wideband multiple-input multiple-output radar. The aim is to match a desired transmit beampattern. To tackle the non-convex design problem, we develop an iterative algorithm, which is based on cyclic optimization and majorization–minimization. We prove that the sequence of the objective values of the proposed algorithm has guaranteed convergence. Moreover, we can obtain closed-form solutions for the associated optimization problems at every iteration. Furthermore, the proposed method can be implemented without matrix inversion and hence is computationally efficient. Simulation results demonstrate that the performance of the proposed algorithm is better than existing methods.

APPENDIX

Note that

$$\begin{aligned} \mathbf{{F}}_{p_1}^H\mathbf{{A}}_{p_1}^H{{\mathbf{{A}}}_{p_1}}{\mathbf{{F}}_{p_1}}&= \left( {\sum \limits _{k_1 = 1}^{K _1} {{\mathbf{{a}}_{k_1p_1}}{} \mathbf{{a}}_{k_1p_1}^H} \otimes \mathbf{{e}}_{p_1}^*} \right) \left( {{\mathbf{{I}}_M} \otimes {\mathbf{{e}}_{p_1}^T}} \right) \nonumber \\&= \sum \limits _{k_1 = 1}^{K_1} {{\mathbf{{a}}_{k_1p_1}}{} \mathbf{{a}}_{k_1p_1}^H} \otimes \left( {\mathbf{{e}}_{p_1}^*{\mathbf{{e}}_{p_1}^T}} \right) . \end{aligned}$$

(40)

From (40), we can see that $\sum \limits _{k _1= 1}^{K_1} {{\mathbf{{a}}_{k_1p_1}}{} \mathbf{{a}}_{k_1p_1}^H}$ and $\mathbf{{e}}_{p_1}^*{\mathbf{{e}}_{p_1}^T}$ are both Toeplitz matrices. In addition, since ${\varvec{P}} = \sum \limits _{p_1= 1}^{N_1} {\varvec{F}}_{p_1}^H{\varvec{A}}_{p_1}^H{\varvec{A}}_{p_1}{\varvec{F}}_{p_1} + \sum \limits _{p_2= 1}^{N_2} {\varvec{F}}_{p_2}^H{\varvec{A}}_{p_2}^H{\varvec{\varGamma }}_{p_2}^H{\varvec{\varGamma }}_{p_2}{\varvec{A}}_{p_2}{\varvec{F}}_{p_2}$, it can be checked that $\mathbf{{P}}$ is a Toeplitz-block Toeplitz matrix, and any block in $\mathbf{{P}}$ (i.e., ${\mathbf{{P}}_{ij}}$ ) is a Toeplitz matrix.