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Phase-only transmit beampattern design for large phased array antennas with multi-point nulling

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Abstract

In this paper, we investigate the problem of the phase-only transmit beampattern design for phased array system with large antennas. The goal of this paper is to maximize the mainlobe gain while nulling in multiple interference directions and keeping the shape of mainlobe similar to the quiescent beampattern. To this end, we formulate our optimization problem by maximizing mainlobe gain under the constraints of constant modulus, nulling levels as well as similarity between the designed beampattern and the quiescent one. To solve the resultant problem, an efficient iterative algorithm based on alternating direction method of multipliers (ADMM) framework is proposed. Concretely, two auxiliary variables are first introduced to modify the original problem, and then the primal variable in the ADMM framework is updated by utilizing the majorization–minimization and proximal algorithms. In addition, to facilitate the low-cost implementation of the transmit beampattern, we consider the problem of the one-bit transmit beampattern design, and propose an efficient ADMM algorithm to solve this problem. Numerical simulations are provided to demonstrate the effectiveness of the proposed algorithms.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 62001084 and 62031007.

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Correspondence to Jinyang He.

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Appendices

Proof of Proposition 1

Suppose the one-bit transmit beampattern points to \(\theta _0 \in [-90^{\circ }, 90^{\circ }]\), then the gain at \( \theta _0 \) is

$$\begin{aligned} \begin{aligned} G(\theta _0)&= \left| {{{\mathbf{w}}^T}{\mathbf{a}}({\theta _0})} \right| = \left| {\sum \limits _{n = 0}^{N - 1} {{w_n}{e^{j2\pi nd\sin {\theta _0}/\lambda }}} } \right| \\&= \left| {\sum \limits _{n = 0}^{N - 1} {{w_n}\cos (2\pi nd\sin {\theta _0}/\lambda ) + j\sum \limits _{n = 0}^{N - 1} {{w_n}\sin (2\pi nd\sin {\theta _0}/\lambda )} } } \right| \end{aligned} \end{aligned}$$
(39)

Since \( {w_n} \in \left\{ { - 1,1} \right\} ,n = 0, \cdots ,N - 1 \), (39) is equivalent to

$$\begin{aligned} \left| {\sum \limits _{n = 0}^{N - 1} {{w_n}\cos (2\pi nd\sin {\theta _0}/\lambda ) - j\sum \limits _{n = 0}^{N - 1} {{w_n}\sin (2\pi nd\sin {\theta _0}/\lambda )} } } \right| \end{aligned}$$
(40)

then we have

$$\begin{aligned} \begin{aligned} G(\theta _0) = \left| {\sum \limits _{n = 0}^{N - 1} {{w_n}{e^{j2\pi nd\sin ( - {\theta _0})/\lambda }}} } \right| = \left| {{{\mathbf{w}}^T}{\mathbf{a}}( - {\theta _0})} \right| = G(-\theta _0) \end{aligned} \end{aligned}$$
(41)

Therefore, this proof is completed.

Proof of Proposition 2

Let \( {\overline{{\mathbf{w}}} }_n \) being the vector \( {\mathbf{w}} \) whose n-th entry is zeroed and \( {{{\tilde{\mathbf{e}}}}_n} \) being an \(N\times 1\) vector, whose n-th entry is 1, and 0 for otherwise. Then the first term of the objective function in (31) can be expressed as

$$\begin{aligned} \begin{aligned} -{{{\mathbf {w}}}^{T}}{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}}){\mathbf {w}}&=-{{({{{\bar{\mathbf {w}}}}_{n}}+{{w}_{n}}{{{\tilde{\mathbf {e}}}}_{n}})}^{T}}{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}})\left( {{{\bar{\mathbf {w}}}}_{n}}+{{w}_{n}}{{{\tilde{\mathbf {e}}}}_{n}}\right) \\&=-\left[ \bar{\mathbf {w}}_{n}^{T}{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}}){{{\bar{\mathbf {w}}}}_{n}}+{{w}_{n}}\bar{\mathbf {w}}_{n}^{T}{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}}){{{\tilde{\mathbf {e}}}}_{n}} \right. \\&\quad \left. +{{w}_{n}}\tilde{\mathbf {e}}_{n}^{T}{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}}){{{\bar{\mathbf {w}}}}_{n}}+w_{n}^{2}\tilde{\mathbf {e}}_{n}^{T}{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}}){{{\tilde{\mathbf {e}}}}_{n}}\right] \\&={\tilde{\mathbf {e}}}_{n}^{\dag }{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}}){{{\tilde{\mathbf {e}}}}_{n}}\cdot w_{n}^{2}+2{\text {Re}}(\bar{\mathbf {w}}_{n}^{T}{\mathbf {a}}({{\theta }_{0}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{0}}){{{\tilde{\mathbf {e}}}}_{n}})\cdot {{w}_{n}}+{{\mathrm {c}}_{\mathrm {1}}} \end{aligned} \end{aligned}$$
(42)

where \(c_1\) is a constant. In addition, the second term of (31) can be written as

$$\begin{aligned} \begin{aligned} {{\rho }_{t}}\left( N-{{{\mathbf {w}}}^{T}}{{{\mathbf {t}}}^{(k)}}\right)&={{\rho }_{t}}\left( N-{{\left( {{{\bar{\mathbf {w}}}}_{n}}+{{w}_{n}}{{{\tilde{\mathbf {e}}}}_{n}}\right) }^{T}}{{{\mathbf {t}}}^{(k)}}\right) \\&=-{{\rho }_{t}}\tilde{\mathbf {e}}_{n}^{T}{{{\mathbf {t}}}^{(k)}}\cdot {{w}_{n}}+{{\mathrm {c}}_{\mathrm {2}}} \end{aligned} \end{aligned}$$
(43)

where \(c_2\) is a constant. Similarly, the last term of the objective function in (31) can be written as transformed as

$$\begin{aligned} \begin{aligned}&\sum \limits _{q=1}^{Q}{\frac{{{\rho }_{q}}}{2}{{\left\| x_{q}^{(k)}-{{{\mathbf {w}}}^{T}}{\mathbf {a}}({{\theta }_{q}})+h_{q}^{(k)} \right\| }^{2}}}=\text { }\sum \limits _{q=1}^{Q}{\frac{{{\rho }_{q}}}{2}{{\left\| x_{q}^{(k)}-{{\left( {{{\bar{\mathbf {w}}}}_{n}}+{{w}_{n}}{{{\tilde{\mathbf {e}}}}_{n}}\right) }^{T}}{\mathbf {a}}({{\theta }_{q}})+h_{q}^{(k)} \right\| }^{2}}} \\&\quad = \sum \limits _{q=1}^{Q}{\frac{{{\rho }_{q}}}{2}\left[ \tilde{\mathbf {e}}_{n}^{T}{\mathbf {a}}({{\theta }_{q}}){{{\mathbf {a}}}^{\dag }}({{\theta }_{q}}){{{\tilde{\mathbf {e}}}}_{n}}\cdot w_{n}^{2}-2{\text {Re}}\left( {{{\mathbf {a}}}^{\dag }}({{\theta }_{q}}){{{\tilde{\mathbf {e}}}}_{n}}(x_{q}^{(k)}+h_{q}^{(k)}-\bar{\mathbf {w}}_{n}^{T}{\mathbf {a}}({{\theta }_{q}})) \right) {{w}_{n}}\right] }+{{{\text {c}}}_{3}} \end{aligned} \end{aligned}$$
(44)

where \(c_3\) is a constant. After ignoring the constant, the optimization problem (31) can be reformulated as a function with respect to \(w_n\) as

$$\begin{aligned} \begin{aligned}&\mathop {\min }\limits _{{w_n}} \,\kappa ({w_n}) = {d_1} \cdot w_n^2 + {d_2} \cdot {w_n} \\&\quad s.t. - 1 \le {w_n} \le 1 \end{aligned} \end{aligned}$$
(45)

where \( d_1 \) and \( d_2 \) are defined as

$$\begin{aligned} {d_1} =&{{\tilde{\mathbf{e}}}}_n^{T}{\mathbf{a}}({\theta _0}){{\mathbf{a}}^{\dag }}({\theta _0}){{{\tilde{\mathbf{e}}}}}_n + \sum \limits _{q = 1}^Q {\frac{{{\rho _q}}}{2}{{\tilde{\mathbf{e}}}}_n^{T}{\mathbf{a}}({\theta _q}){{\mathbf{a}}^{\dag }}({\theta _q}){{{{\tilde{\mathbf{e}}}}}_n}} \end{aligned}$$
(46a)
$$\begin{aligned} {d_2} =&2{\mathop {\Re }\nolimits } \left( {{\bar{\mathbf{w}}}}_n^\mathrm{{T}}{\mathbf{a}}({\theta _0}){{\mathbf{a}}^{\dag }}({\theta _0}){{{{\tilde{\mathbf{e}}}}}_n}\right) - {\rho _t}{\tilde{\mathbf{e}}}_n^{T}{{\mathbf{t}}^{(k)}} \nonumber \\&-\sum \limits _{q = 1}^Q {{\rho _q}{\mathop {\Re }\nolimits } \left( {{{\mathbf{a}}^{\dag }}({\theta _q}){{{\tilde{\mathbf{e}}}}}_n}(x_q^{(k)} + u_q^{(k)} - {{\bar{\mathbf{w}}}}_n^{T}{\mathbf{a}}({\theta _q})) \right) } \end{aligned}$$
(46b)

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He, J., Cheng, Z., He, Z. et al. Phase-only transmit beampattern design for large phased array antennas with multi-point nulling. Multidim Syst Sign Process 33, 597–619 (2022). https://doi.org/10.1007/s11045-021-00815-7

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