Transmission Waveform Design of MIMO Dual-Functional Radar-Communication System

Abstract

Based on the multiple-input multiple-output (MIMO) dual-functional radar-communication (DFRC) system, a new DFRC transmission waveform is designed in this paper, The MIMO-DFRC system transmits radar detection waveform to targets and communication signals to downlink users simultaneously using orthogonal frequency division multiplexing (OFDM) chirp multi-channel orthogonal signals. Firstly, considering the key performance indicators of communication and sensing, an optimization model is established to minimize the joint function of downlink multi-user interference (MUI), waveform similarity and CRB under the constraint of total transmit power, and the tradeoff parameter is introduced to adjust the priority of communication and radar performance, and the penalty parameter is introduced to adjust the weight of reference waveform similarity and CRB. The constrained optimization problem can be further reduced to a non-convex quadratic constrained quadratic programming problem, which can be solved by semidefinite relaxation (SDR) technique. The global optimal solution can be obtained by transforming the constrained optimization problem into a semi-definite programming problem (SDP) by rank one approximation. The numerical results show that the designed DFRC transmission waveform can achieve better detection performance without sacrificing the communication performance in the real scenario, and the performance indicators of the new DFRC transmission waveform are significantly better than that of the traditional DFRC transmission waveform.

Appendices

Appendix 1

The constrained optimization problem of Eq. (17) is:

$$\begin{gathered} \mathop {\min }\limits_{{\mathbf{X}}} {\uprho }||{\mathbf{HX}} - {\mathbf{S}}||_{{\text{F}}}^{2} + (1 - {\uprho })\{ ||{\mathbf{X}} - {\mathbf{X}}_{0} ||_{{\text{F}}}^{2} - {\upmu }||{\dot{\mathbf{a}}}_{{\text{r}}} {\mathbf{a}}_{{\text{t}}}^{\text{H}} {\mathbf{X}}||{}_{{\text{F}}}^{2} \} \hfill \\ {\text{s.t.}}\frac{1}{{\text{L}}}||{\mathbf{X}}||_{{\text{F}}}^{{2}} = {\text{P}}_{{\text{t}}} \hfill \\ \end{gathered}$$

(21)

First, split the objective function in Eq. (21):

$${\uprho }||{\mathbf{HX}} - {\mathbf{S}}||_{{\text{F}}}^{2} + (1 - {\uprho })||{\mathbf{X}} - {\mathbf{X}}_{0} ||_{{\text{F}}}^{2} - {\upmu }(1 - {\uprho })||{\dot{\mathbf{a}}}_{{\text{r}}} {\mathbf{a}}_{{\text{t}}}^{\text{H}} {\mathbf{X}}||{}_{{\text{F}}}^{2}$$

(22)

It is not difficult to find that the objective function in Eq. (22) is composed of three F norms. First, we can integrate the first two terms of the objective function:

$$||\left[\sqrt {\uprho } {\mathbf{H}}^{\text{T}} ,\sqrt {1 - {\uprho }} {\mathbf{I}}_{{N_{t} }} \right]^{\text{T}} {\mathbf{X}} - \left[\sqrt {\uprho } {\mathbf{S}}^{\text{T}} ,\sqrt {1 - {\uprho }} {\mathbf{X}}_{{\mathbf{0}}}^{\text{T}} \right]^{\text{T}} ||_{{\text{F}}}^{2} - {\upmu }(1 - {\uprho })||{\dot{\mathbf{a}}}_{{\text{r}}} {\mathbf{a}}_{{\text{t}}}^{\text{H}} {\mathbf{X}}||{}_{{\text{F}}}^{2}$$

(23)

Then integrate the objective function of Eq. (23):

$$||\left[ {\left[\sqrt {\uprho } {\mathbf{H}}^{\text{T}} ,\sqrt {1 - {\uprho }} {\mathbf{I}}_{{N_{t} }} \right],{\text{j}} \sqrt {(1 - {\uprho })\mu } ({\dot{\mathbf{a}}}_{r} {\mathbf{a}}_{t}^{\text{H}} )^{\text{T}} } \right]^{\text{T}} {\mathbf{X}} - \left[ {\left[\sqrt {\uprho } {\mathbf{S}}^{\text{T}} ,\sqrt {1 - {\uprho }} {\mathbf{X}}_{{\mathbf{0}}}^{\text{T}} \right],{\mathbf{Q}}} \right]^{\text{T}} ||_{{\text{F}}}^{2}$$

(24)

In Eq. (24), the matrix ${\mathbf{Q}} \in {{\mathbb{C}}}^{{L \times N_{r} }}$ is defined as an all zero matrix, ${\mathbf{B}} = \left[ {[\sqrt \rho {\mathbf{H}}^{\text{T}} ,\sqrt {1 - \rho } {\mathbf{I}}_{{N_{t} }} ],{\text{j}} \sqrt {(1 - \rho )\mu } ({\dot{\mathbf{a}}}_{r} {\mathbf{a}}_{t}^{\text{H}} )^{\text{T}} } \right]^{\text{T}} \in {{\mathbb{C}}}^{{(K + N_{t} + N_{r} ) \times N_{t} }}$,${\mathbf{C}} = \left[ {[\sqrt \rho {\mathbf{S}}^{\text{T}} ,\sqrt {1 - \rho } {\mathbf{X}}_{{\mathbf{0}}}^{\text{T}} ],{\mathbf{Q}}} \right]^{\text{T}} \in {{\mathbb{C}}}^{{(K + N_{t} + N_{r} ) \times L}}.$

Equation (21) can be further expressed as:

$$\begin{gathered} \mathop {\min }\limits_{{\mathbf{X}}} ||{\mathbf{BX}} - {\mathbf{C}}||_{{\text{F}}}^{2} \hfill \\ {\text{s.t.}}\frac{1}{{\text{L}}}||{\mathbf{X}}||_{{\text{F}}}^{2} = {\text{P}}_{{\text{t}}} \hfill \\ \end{gathered}$$

(25)

Appendix 2

Since the objective function and constraints of Eq. (18) are quadratic, it is a non-convex quadratic constrained quadratic programming (QCQP) problem, which can be further transformed into an SDR problem. First, the matrix is listed:

$$\begin{gathered} ||{\mathbf{BX}} - {\mathbf{C}}||_{F}^{2} = \hfill \\ ||vec({\mathbf{BX}} - {\mathbf{C}})||_{2}^{2} = \hfill \\ ||vec({\mathbf{BX}}) - vec({\mathbf{C}})||_{2}^{2} = \hfill \\ ||({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})vec({\mathbf{X}}) - vec({\mathbf{C}})||_{2}^{2} \hfill \\ \end{gathered}$$

(26)

Then, substitute the matrix in Eq. (26) with variables. Where, ${\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}} \in {{\mathbb{C}}}^{{(K + N_{t} + N_{r} )L \times N_{t} L}}$, ${\text{vec}} ({\mathbf{X}}) \in {{\mathbb{C}}}^{{N_{t} L \times 1}}$,Eq. (26) is rewritten as:

$$\begin{gathered} \mathop {\min }\limits_{{\mathbf{X}}} ||{\text{t}}{\text{vec}} ({\mathbf{C}}) - ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}}){\text{vec}} ({\mathbf{X}})||_{2}^{2} \hfill \\ {\text{s.t.}}\left\{ \begin{gathered} \frac{1}{{\text{L}}}||{\text{vec}} ({\mathbf{X}})||_{2}^{2} = {\text{P}}_{{\text{t}}} \hfill \\ {\text{t}}^{2} = 1 \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered}$$

(27)

where $t$ is an auxiliary variable. The purpose of adding ${\text{t}}^{2} = 1$ constraint is to homogenize the original constrained optimization problem. Split the objective function in Eq. (27) as:

$$\begin{gathered} ||{\text{t}}{\text{vec}} ({\mathbf{C}}) - ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}}){\text{vec}} ({\mathbf{X}})||_{2}^{2} = \hfill \\ \left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{H} \left[ \begin{gathered} \begin{array}{*{20}c} {({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})^{\text{H}} ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})} & {{\mathbf{ - }}({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})^{\text{H}} {\text{vec}} ({\mathbf{C}})} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {{\mathbf{ - }}{\text{vec}} ({\mathbf{C}})^{\text{H}} ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})} & {{\text{vec}} ({\mathbf{C}})^{\text{H}} } \\ \end{array} {\text{vec}} ({\mathbf{C}}) \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(28)

Because the product of a matrix's conjugate transpose and the matrix itself is a semi positive definite matrix, and $\left[ \begin{gathered} \begin{array}{*{20}c} {({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})^{\text{H}} ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})} & {{\mathbf{ - }}({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})^{\text{H}} {\text{vec}} ({\mathbf{C}})} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {{\mathbf{ - }}{\text{vec}} ({\mathbf{C}})^{\text{H}} ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})} & {{\text{vec}} ({\mathbf{C}})^{\text{H}} } \\ \end{array} {\text{vec}} ({\mathbf{C}}) \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}}} \\ { - {\text{vec}} ({\mathbf{C}})} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}}} \\ { - {\text{vec}} ({\mathbf{C}})} \\ \end{array} } \right]^{\text{H}}$ Therefore, the matrix is a positive semi definite matrix. At the same time, the constraints in Eq. (27) can be expressed as:

$$\begin{gathered} ||vec({\mathbf{X}})||_{2}^{2} = \left[ \begin{gathered} vec({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{\text{H}} \left[ \begin{gathered} \begin{array}{*{20}c} {{\mathbf{\rm I}}_{{{\text{N}}_{{\text{t}}} {\text{L}}}} } & 0 \\ \end{array} \hfill \\ \begin{array}{*{20}c} 0 & { \, 0} \\ \end{array} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} vec({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right] = {\text{LP}}_{{\text{t}}} \hfill \\ {\text{t}}^{2} = \left[ \begin{gathered} vec({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{\text{H}} \left[ \begin{gathered} \begin{array}{*{20}c} {{\mathbf{0}}_{{{\text{N}}_{{\text{t}}} {\text{L}}}} } & 0 \\ \end{array} \hfill \\ \begin{array}{*{20}c} 0 & { \, 1} \\ \end{array} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} vec({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right] = 1 \hfill \\ \end{gathered}$$

(29)

The constrained optimization problem of Eq. (26) can be finally expressed as:

$$\begin{gathered} \mathop {\min }\limits_{{\mathbf{X}}} {\text{tr}} \left( {\left[ \begin{gathered} \begin{array}{*{20}c} {({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})^{\text{H}} ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})} & {{\mathbf{ - }}({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})^{\text{H}} {\text{vec}} ({\mathbf{C}})} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {{\mathbf{ - }}{\text{vec}} ({\mathbf{C}})^{\text{H}} ({\mathbf{I}}_{{\mathbf{L}}} \otimes {\mathbf{B}})} & {{\text{vec}} ({\mathbf{C}})^{\text{H}} } \\ \end{array} {\text{vec}} ({\mathbf{C}}) \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{\text{H}} } \right) \hfill \\ s.t.\left\{ \begin{gathered} {\text{tr}} \left( {\left[ \begin{gathered} \begin{array}{*{20}c} {{\mathbf{\rm I}}_{{{\text{N}}_{{\text{t}}} {\text{L}}}} } & 0 \\ \end{array} \hfill \\ \begin{array}{*{20}c} 0 & { 0} \\ \end{array} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{\text{H}} } \right) = {\text{LP}}_{{\text{t}}} \hfill \\ {\text{tr}} \left( {\left[ \begin{gathered} \begin{array}{*{20}c} {{\mathbf{0}}_{{{\text{N}}_{{\text{t}}} {\text{L}}}} } & 0 \\ \end{array} \hfill \\ \begin{array}{*{20}c} 0 & { \, 1} \\ \end{array} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{\text{H}} } \right) = 1 \hfill \\ {\text{rank}} \left( {\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{\text{H}} } \right) = 1,\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]\left[ \begin{gathered} {\text{vec}} ({\mathbf{X}}) \hfill \\ {\text{ t}} \hfill \\ \end{gathered} \right]^{\text{H}} \ge 0 \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered}$$

(30)