Manifold optimization for hybrid beamforming in dual-function radar-communication system

Abstract

We study hybrid beamforming design for millimeter wave dual-function radar-communication system, which simultaneously performs downlink communication and target detection. With two typical analog beamformer structures, we consider the hybrid beamforming design to minimize a weighted summation of radar and communication performance. Leveraging Riemannian optimization theory, a manifold alternating direction method of multipliers is developed for the fully-connected structure. While for the partially-connected structure, a low-complexity Riemannian product manifold trust region algorithm is proposed to approach a near-optimal solution. Numerical simulations are provided to demonstrate the effectiveness of the proposed methods.

Appendices

Appendix A: Derivation of Riemannian gradient

Recalling the Riemannian gradient is calculated as

$$\begin{aligned} {\begin{aligned} {\text {grad}}J\left( {{{{\mathbf {F}}}_{PS}}{\mathbf {,}}{{{\mathbf {F}}}_{BB}}} \right) = \left( {{\text {gra}}{{\text {d}}_{{{{\mathbf {F}}}_{PS}}}}J\left( {{{{\mathbf {F}}}_{PS}}{\mathbf {,}}{{{\mathbf {F}}}_{BB}}} \right) ,{\text {gra}}{{\text {d}}_{{{{\mathbf {F}}}_{BB}}}}J\left( {{{{\mathbf {F}}}_{PS}}{\mathbf {,}}{{{\mathbf {F}}}_{BB}}} \right) } \right) \\ \end{aligned} } \end{aligned}$$

(52)

where the Riemannian gradients ${\text {gra}}{{\text {d}}_{{{{\mathbf {F}}}_{PS}}}}J\left( {{{{\mathbf {F}}}_{PS}},{{{\mathbf {F}}}_{BB}}} \right)$ and ${\text {gra}}{{\text {d}}_{{{{\mathbf {F}}}_{BB}}}}J\left( {{{{\mathbf {F}}}_{PS}},{{{\mathbf {F}}}_{BB}}} \right)$ are computed by orthogonal projection from Euclidean space to the tangent space (Absil et al., 2009) as

$$\begin{aligned} {\text {gra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}} J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) {\text { }}&= {\text {Pro}}{{\text {j}}_{{\mathbf{{F}}_{PS}}}}\left( {{\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) } \right) \nonumber \\&= {\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) \end{aligned}$$

(53a)

$$\begin{aligned}&\quad - \mathfrak {Re} \{ {\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) \circ \mathbf{{F}}_{PS}^*\} \circ {\mathbf{{F}}_{PS}} \nonumber \\ {\text {gra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) {\text { }}&= {\text {Pro}}{{\text {j}}_{{\mathbf{{F}}_{BB}}}}\left( {{\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) } \right) \nonumber \\&= {\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) \end{aligned}$$

(53b)

$$\begin{aligned}&\quad - \mathfrak {Re} \left\{ {{\text {tr}}\left( {\mathbf{{F}}_{BB}^H{\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}g\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) } \right) } \right\} {\mathbf{{F}}_{BB}} \end{aligned}$$

(53c)

where ${{\text {Gra}}{{\text {d}}_{{{{\mathbf {F}}}_{PS}}}}J\left( {{{{\mathbf {F}}}_{PS}}{\mathbf {,}}{{{\mathbf {F}}}_{BB}}} \right) }$ and ${{\text {Gra}}{{\text {d}}_{{{{\mathbf {F}}}_{BB}}}}J\left( {{{{\mathbf {F}}}_{PS}}{\mathbf {,}}{{{\mathbf {F}}}_{BB}}} \right) }$ denote, respectively, the Euclidean gradients with respect to ${{{{\mathbf {F}}}_{PS}}}$ and ${{{{\mathbf {F}}}_{BB}}}$, and they are calculated as

$$\begin{aligned} {\text {Gra}}{{\text {d}}_{{{{\mathbf {F}}}_{PS}}}}J\left( {{{{\mathbf {F}}}_{PS}},{{{\mathbf {F}}}_{BB}}} \right) =&2\varphi \left\{ {\left( {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Com}}} \right) \mathbf{{F}}_{BB}^H} \right\} \circ {\mathbf{{F}}_D} \nonumber \\&+ 2\left( {1 - \varphi } \right) \left\{ {\left( {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Rad}}} \right) \mathbf{{F}}_{BB}^H} \right\} \circ {\mathbf{{F}}_D} \end{aligned}$$

(54a)

$$\begin{aligned} {\text {Gra}}{{\text {d}}_{{{{\mathbf {F}}}_{BB}}}}J\left( {{{{\mathbf {F}}}_{PS}},{{{\mathbf {F}}}_{BB}}} \right) =&2\varphi {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) ^H}\left\{ {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Com}}} \right\} \nonumber \\&+ 2(1 - \varphi ){\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) ^H}\left\{ {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Rad}}} \right\} \end{aligned}$$

(54b)

Appendix B: Derivation of Riemannian hessian

The Riemannian Hessian of $J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right)$ at $\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \in {\mathcal {M}}$ is a linear operator ${\text {Hes}}{{\text {s}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}J:{T_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}{{\mathcal {M}}}{\text { }} \mapsto {T_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}{{\mathcal {M}}}$ (Boumal, 2020; Absil et al., 2009; Udriste, 2013) defined by

$$\begin{aligned} \begin{aligned} {\text {Hes}}&{{\text {s}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] = {\nabla _{{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}}}{\text {grad}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \end{aligned} \end{aligned}$$

(55)

for the point ${{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }} \in {T_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}{{\mathcal {M}}}$. $\nabla$ denotes the Levi-Civita connection of ${\mathcal {M}}$. Specifically, for the RPM, the Riemannian Hessian of $J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right)$ holds that (Absil et al., 2009)

$$\begin{aligned} \begin{aligned}&{\text {Hes}}{{\text {s}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] \\&\quad = \left( {{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] ,} \right. \left. {{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] } \right) \end{aligned} \end{aligned}$$

(56)

where ${{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}$ is tangent vector on the tangent space, i.e. ${{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }} \in {T_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}{{\mathcal {M}}}$, ${{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) }$ and ${{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) }$ denote, respectively, the Riemannian Hessian of $J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right)$ with respect to ${{\mathbf{{F}}_{PS}}}$ and ${{\mathbf{{F}}_{PS}}}$. Based on the classical expression of the Levi-Civita connection on a Riemannian submanifold of a Euclidean space (Boumal, 2020; Absil et al., 2009; Udriste, 2013), the Riemannian Hessian can be calculated via direction derivative in the embedding space followed by an orthogonal projection to the tangent space. Thus, the Riemannian Hessian ${{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) }$ and ${{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) }$ are computed as Li et al. (2020); Absil et al. (2013)

$$\begin{aligned}&{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] \nonumber \\&= {\nabla _{{{{\varvec{\zeta }}}_{{\mathbf{{F}}_{PS}}}}}}{\text {grad}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \nonumber \\&= {\text {Pro}}{{\text {j}}_{{\mathbf{{F}}_{PS}}}}\left( {{\text {Dgra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] } \right) \nonumber \\&= {\text {Pro}}{{\text {j}}_{{\mathbf{{F}}_{PS}}}}\left( {{\text {eHes}}{{\text {s}}_{{\mathbf{{F}}_{PS}}}}J - \mathfrak {Re} \left( {{\mathbf{{F}}_{PS}} \circ {{\left( {{\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}}J} \right) }^*}} \right) \circ {{{\varvec{\zeta }}}_{{\mathbf{{F}}_{PS}}}}} \right) \end{aligned}$$

(57a)

$$\begin{aligned}&{\text {Hes}}{{\text {s}}_{{\mathbf{{F}}_{BB}}}}{J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) }\left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] \nonumber \\&= {\nabla _{{\mathbf{{\zeta }}_{{\mathbf{{F}}_{BB}}}}}}{\text {grad}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \nonumber \\&{\text { = Pro}}{{\text {j}}_{{\mathbf{{F}}_{BB}}}}\left( {{\text {Dgra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{{{\varvec{\zeta }}}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] } \right) \nonumber \\&= {\text {Pro}}{{\text {j}}_{{\mathbf{{F}}_{BB}}}}\left( {{\text {eHes}}{{\text {s}}_{{\mathbf{{F}}_{BB}}}}J} \right) + \mathfrak {Re} \left\{ {{\text {tr}}\left( {\mathbf{{F}}_{BB}^H{\text {Gra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}J} \right) } \right\} {{{\varvec{\zeta }}}_{{\mathbf{{F}}_{BB}}}} \end{aligned}$$

(57b)

where ${{\text {Dgra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{\mathbf{{\zeta }}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] }$ and ${{\text {Dgra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) \left[ {{\mathbf{{\zeta }}_{\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{BB}}} \right) }}} \right] }$ are the directional derivative of the Riemannian gradient ${{\text {gra}}{{\text {d}}_{{\mathbf{{F}}_{PS}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right) }$ and ${\text {gra}}{{\text {d}}_{{\mathbf{{F}}_{BB}}}}J\left( {{\mathbf{{F}}_{PS}},{\mathbf{{F}}_{PS}}} \right)$, the Euclidean Hessian ${\text {eHes}}{{\text {s}}_{{\mathbf{{F}}_{PS}}}}J$ and ${\text {eHes}}{{\text {s}}_{{\mathbf{{F}}_{BB}}}}J$ denote, respectively, the directional derivative of the Euclidean gradient (54a) and (54b), which are computed as

$$\begin{aligned} {\text {eHes}}{{\text {s}}_{{\mathbf{{F}}_{PS}}}}J&= 2\varphi \left\{ {\left( {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Com}}} \right) {{\varvec{\zeta }}}_{BB}^H} \right\} \circ {\mathbf{{F}}_D} \\&\quad + 2\left( {1 - \varphi } \right) \left\{ {\left( {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Rad}}} \right) {{\varvec{\zeta }}}_{BB}^H} \right\} \circ {\mathbf{{F}}_D}\\&\quad + 2\left\{ {\left( {\left( {{\mathbf{{F}}_D} \circ {{{\varvec{\zeta }}}_{PS}}} \right) {\mathbf{{F}}_{BB}} + \left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {{{\varvec{\zeta }}}_{BB}}} \right) \mathbf{{F}}_{BB}^H} \right\} \circ {\mathbf{{F}}_D} \\ {\text {eHes}}{{\text {s}}_{{\mathbf{{F}}_{BB}}}}J&= 2\varphi {\left( {{\mathbf{{F}}_D} \circ {{{\varvec{\zeta }}}_{PS}}} \right) ^H}\left\{ {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Com}}} \right\} \\&\quad + 2(1 - \varphi ){\left( {{\mathbf{{F}}_D} \circ {{{\varvec{\zeta }}}_{PS}}} \right) ^H}\left\{ {\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {\mathbf{{F}}_{BB}} - {\mathbf{{F}}_{Rad}}} \right\} \\&\quad + 2{\left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) ^H}\left\{ {\left( {{\mathbf{{F}}_D} \circ {{{\varvec{\zeta }}}_{PS}}} \right) {\mathbf{{F}}_{BB}} + \left( {{\mathbf{{F}}_D} \circ {\mathbf{{F}}_{PS}}} \right) {{{\varvec{\zeta }}}_{BB}}} \right\} \end{aligned}$$