Open AccessCommunication

Joint Constant-Modulus Waveform and RIS Phase Shift Design for Terahertz Dual-Function MIMO Radar and Communication System

Rui Yang

Hong Jiang

and

Liangdong Qu

College of Communication Engineering, Jilin University, Changchun 130012, China

Author to whom correspondence should be addressed.

Remote Sens. 2024, 16(16), 3083; https://doi.org/10.3390/rs16163083

Submission received: 22 July 2024 / Revised: 19 August 2024 / Accepted: 19 August 2024 / Published: 21 August 2024

(This article belongs to the Topic Emerging Terahertz Technologies for Integrated Sensing and Communication)

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Browse Figures

Figure 1
RIS-aided THz dual-function MIMO radar and communication system with multiple targets and UEs. "> Figure 2
Sum rate versus the transmit SNR for different algorithms. "> Figure 3
Beampatterns of the transmit waveforms achieved by different algorithms. "> Figure 4
Auto-correlation functions achieved by different algorithms. "> Figure 5
Sum rate versus the transmit SNR of our algorithm under different weighting coefficients and RIS conditions. "> Figure 6
Beampatterns of the transmit waveforms of our algorithm under different weighting coefficients and RIS conditions. "> Figure 7
Average detection probability versus the radar transmit SNR of our algorithm under different weighting coefficients and RIS conditions. "> Figure 8
Auto-correlation functions of the transmit waveforms of our algorithm under different weighting coefficients and RIS conditions. "> Figure 9
Sum rate versus the transmit SNR under different channel conditions of THz system. ">

Versions Notes

Abstract

This paper considers a terahertz (THz) dual-function multi-input multi-output (MIMO) radar and communication system with the assistance of a reconfigurable intelligent surface (RIS) and jointly designs the constant modulus (CM) waveform and RIS phase shifts. A weighted optimization scheme is presented, to minimize the weighted sum of three objectives, including communication multi-user interference (MUI) energy, the negative of multi-target illumination power and the MIMO radar waveform similarity error under a CM constraint. For the formulated non-convex problem, a novel alternating coordinate descent (ACD) algorithm is introduced, to transform it into two subproblems for waveform and phase shift design. Unlike the existing optimization algorithms that solve each subproblem by iteratively approximating the optimal solution with iteration stepsize selection, the ACD algorithm can alternately solve each subproblem by dividing it into multiple simpler problems, to achieve closed-form solutions. Our numerical simulations demonstrate the superiorities of the ACD algorithm over the existing methods. In addition, the impacts of the weighting coefficients, RIS and channel conditions on the radar communication performance of the THz system are analyzed.

Keywords:

terahertz (THz); dual-function MIMO radar and communication; reconfigurable intelligent surface (RIS); waveform and phase shift design; alternating coordinate descent (ACD)

1. Introduction

The boom in radar and communication technologies and the explosive growth of connecting equipments in the future sixth generation (6G) system have led to a serious scarcity of spectrum resources. The terahertz (THz) frequency band is recognized as a potential frequency band for 6G, due to its plentiful spectrum source, high transmission rate and sensing accuracy, and it has become a promising enabler for the integration of sensing and communication (ISAC) systems [1,2]. Waveform design is a key technique for the THz dual-function radar communication (DFRC) systems. Employing THz DFRC waveforms, the sensing of radar targets and communication by terminal devices can be implemented simultaneously, and spectrum resources can be effectively utilized.

The THz frequency band provides significant benefits to developing DFRC technology, along with several problems. The problems mainly include limited transmitting power and high path loss. Since THz systems operate on high frequencies, the output power of their power amplifiers rapidly decreases with the increase in the frequency. Therefore, the waveform design in THz DFRC systems has a stricter requirement for the peak-to-average power ratio (PAPR) of the transmitting signal. To avoid distortion of the nonlinear power amplifier and to maximize transmitting power and energy efficiency, the constant modulus (CM) waveform constraint is a solution when designing THz DFRC waveforms. Recently, some studies on waveform design of DFRC systems have considered CM constraints. In [3], multi-user interference (MUI) was minimized by similarity and CM waveform constraints. In [4], a dual-functional unimodular signal was designed by minimizing the ambiguity function sidelobes. In [5,6,7], symbol-level precoded transmitting signals were designed, to ensure radar functionality under communication and CM power constraints. A DFRC waveform with low integrated sidelobe levels was designed in [8] with CM waveform and information embedding constraints. With communication quality requirements in mind, CM waveforms were designed in [9,10,11] to improve radar performance.

In addition, with high path loss, the THz frequency band is susceptible to direct line of sight (LOS). The reconfigurable intelligent surface (RIS) design is an effective solution for mitigating the problem [12,13]. RIS is a metasurface containing many reflective elements, in which graphene material can be used in the THz frequency with high directivity. RIS creates an additional link to improve the propagation quality of the THz DFRC system. More recently, the investigation of the RIS-aided DFRC system has received much attention [14,15,16,17,18,19]. In [14,15], joint waveform and phase shift design in RIS-assisted DFRC systems were studied by minimizing the MUI. Refs. [16,17] also considered the above joint design, to maximize the radar output signal-to-interference-plus-noise ratio (SINR) or to minimize the weighted mean square cross-correlation pattern. Ref. [18] addressed the above joint design, to minimize MUI energy while maximizing radar SINR. In [19], a CM waveform, receive adaptive filter and RIS phase shift were jointly designed for a DFRC system by maximizing the radar SINR.

As we know, communication MUI has an impact on communication data rates. Meanwhile, radar target illumination power is closely related to the probability of detection, and radar waveform similarity is extremely sensitive to pulse compression and ambiguity properties. In this paper, to obtain a more flexible compromise between the performance of radar and communication, we present a multi-objective weighted scheme to jointly design a CM waveform and RIS phase shifts for a THz dual-function multi-input multi-output (MIMO) radar and communication system. We focus on minimizing the weighted sum of three objectives, including the communication MUI energy, the negative of multi-target illumination power and the MIMO radar waveform similarity error under a CM constraint. The same optimization problem has not been addressed.

For the formulated non-convex problem, we present a novel alternating coordinate descent (ACD) algorithm, to transform it into two subproblems and to alternately solve each subproblem. Unlike some existing optimization algorithms, it can avoid iteratively approximating the optimal solution with the chosen iteration stepsize. Specifically, the proposed ACD algorithm can derive closed-form solutions for weighted optimization problems. Our numerical simulations demonstrate the superiority of our ACD algorithm. In addition, the impacts of the weighting coefficients, RIS and channel conditions on the radar communication performance of the THz system are analyzed.

Notation:

I_{N}

represents the N-dimensional identity matrix;

v e c (\cdot)

represents vectorization;

t r (\cdot)

represents the trace;

ℜ \{\cdot\}

represents taking the real part; ⊗ represents the Kronecker product;

E

represents the expectation;

C

represents complex number domain;

C N (\cdot)

represents the circularly symmetric complex Gaussian distribution;

D i a g (a)

represents the diagonal matrix with vector

a

along the diagonal and

d i a g (A)

represents the vector formed by the diagonal of matrix

A

;

{(\cdot)}^{T}

{(\cdot)}^{*}

and

{(\cdot)}^{H}

represent the transpose, conjugate and conjugate transpose, respectively;

|\cdot|

represents the modulus;

{∥\cdot∥}_{2}

and

{∥\cdot∥}_{F}

represent the

l_{2}

and Frobenius norm, respectively.

2. System Model

Figure 1 shows an RIS-aided THz dual-function MIMO radar and communication system with multiple targets and user equipments (UEs). We assume that the DFRC base station (BS) has M antennas, to form a uniform linear array. It detects T point targets and communicates with K downlink UEs simultaneously. The RIS with R elements is used to assist downlink communication, which is assumed to be far from the targets, such that its impact on radar detection can be ignored. Let

X \in C^{M \times L}

denote the DFRC transmit signal with code length L.

2.1. MIMO Radar Model

We assume that the T point targets of interest are located at directions

{\{ϕ_{t}\}}_{t = 1}^{T}

. Then, the illumination power at the tth target is

P (ϕ_{t}) = {∥a^{H} (ϕ_{t}) X∥}_{2}^{2} = {|I_{L} \otimes a^{H} (ϕ_{t}) x|}^{2} = x^{H} A_{t} x

(1)

where

a (ϕ_{t}) = {[1, e^{- j 2 π f d sin ϕ_{t} / c}, \dots, e^{- j 2 π f d (M - 1) sin ϕ_{t} / c}]}^{T} \in C^{M \times 1}

denotes the transmit steering vector of the MIMO radar [20] at

ϕ_{t}

, with f, c and d being the center frequency at THz band, the velocity of the electromagnetic wave and the interval between the adjacent antennas of the DFRC BS, respectively;

x = v e c (X) \in C^{M L \times 1}

and

A_{t} = I_{L} \otimes [a (ϕ_{t}) a^{H} (ϕ_{t})] \in C^{M L \times M L}

. The multi-target average illumination power is

P = \frac{1}{T} \sum_{t = 1}^{T} P (ϕ_{t}) = \frac{1}{T} \sum_{t = 1}^{T} x^{H} A_{t} x = x^{H} Ax

(2)

where

A = \frac{1}{T} \sum_{t = 1}^{T} A_{t}

. From the aspect of radar performance, we wish to maximize P or minimize

- P

, to obtain a high probability of detection.

Meanwhile, given that the total transmitting power by M antennas of MIMO radar is

P_{t}

, the CM constraint needs to be considered, which is written as

|X (m, l)| = |x_{n}| = \sqrt{\frac{P_{t}}{M}}

(3)

where

X (m, l)

is the

(m, l)

th entry of

X

for

m = 1, \dots, M

l = 1, \dots, L

and

x_{n}

denotes the nth entry of

x

for

n = 1, \dots, M L

In addition, the DFRC waveform should be designed to be as similar as possible to the desired MIMO radar waveform with good pulse compression and ambiguity properties. The similarity error is defined as

S E = {∥X - X_{0}∥}_{F}^{2} = {∥x - x_{0}∥}_{2}^{2}

(4)

where the reference waveform matrix

X_{0} \in C^{M \times L}

is composed of orthogonal linear frequency modulation (LFM) waveforms with the

(m, l)

th element:

X_{0} (m, l) = \sqrt{\frac{P_{t}}{M}} exp \{\frac{j 2 π m (l - 1)}{L}\} exp \{\frac{j π {(l - 1)}^{2}}{L}\}

(5)

and

x_{0} = v e c (X_{0}) \in C^{M L \times 1}

. From this aspect, we aim to minimize

S E

to improve the radar resolution capability.

2.2. Communication Model

We assume that the THz channels from DFRC BS to UEs, from DFRC BS to RIS and from RIS to UEs are, respectively, represented as

H_{b u} \in C^{K \times M}

H_{b r} \in C^{R \times M}

and

H_{r u} \in C^{K \times R}

. Then, the UEs’ received signal

Y_{c} \in C^{K \times L}

Y_{c} = (H_{b u} + H_{r u} Θ H_{b r}) X + W_{c} = HX + W_{c}

(6)

where

Θ = D i a g (θ) \in C^{R \times R}

is the RIS phase shift matrix with

θ = {[θ_{1}, \dots, θ_{R}]}^{T} \in C^{R \times 1}

and

|θ_{r}| = 1, \forall r \in {1, 2, \dots, R}

;

W_{c} = [{w_{c}}_{1}, {w_{c}}_{2}, \dots, {w_{c}}_{L}] \in c^{K \times L}

is the white Gaussian noise with

{w_{c}}_{l} \sim C N (0, N_{0} I_{K}), \forall l \in {1, 2, \dots, L}

; and

H = (H_{b u} + H_{r u} Θ H_{b r}) \in C^{K \times M}

is the equivalent communication channel.

Suppose the communication symbol to be sent is

S \in C^{K \times L}

; then,

Y_{c}

can be further represented as

Y_{c} = S + \underset{MUI}{\underset{⏟}{H X - S}} + W_{c}

(7)

The middle part in (7) is defined as the MUI; thereby, the communication downlink MUI energy is

E_{MUI} = {∥H X - S∥}_{F}^{2} = {∥\tilde{H} x - s∥}_{2}^{2}

(8)

where

\tilde{H} = I_{L} \otimes H \in C^{K L \times M L}

and

s = v e c (S) \in C^{K L \times 1}

To evaluate

E_{MUI}

, the sum rate of a multi-UE communication system is given by [21]

Γ = \sum_{i = 1}^{K} {log}_{2} (1 + γ_{i})

(9)

where the ith UE’s SINR

γ_{i}

γ_{i} = \frac{E ({|S (i, j)|}^{2})}{\underset{MUI energy}{\underset{⏟}{E ({|H (i, :) X (:, j) - S (i, j)|}^{2})}} + N_{0}}

(10)

where

S (i, j)

is the

(i, j)

th entry of

S

, and where

H (i, :)

and

X (:, j)

are the ith row and jth column of

H

and

X

, respectively. Due to the constant energy of

E ({|S (i, j)|}^{2})

, given the communication symbol

S

, we increase the sum rate

Γ

by minimizing the MUI energy

E_{MUI}

3. Problem Formulation and Algorithm Proposal

We jointly design the CM DFRC waveform

X

and the RIS phase shifts

Θ

by minimizing the weighted sum of three quantities, i.e.,

E_{MUI}

- P

and

S E

. Then, we cast a non-convex problem and develop a novel alternating coordinate descent (ACD) algorithm.

3.1. Optimization Problem Formulation

Utilizing the weighted optimization scheme, the proposed objective function is

F (x, θ) = ρ_{1} {∥\tilde{H} x - s∥}_{2}^{2} - ρ_{2} x^{H} Ax + ρ_{3} {∥x - x_{0}∥}_{2}^{2}

(11)

where the weighting coefficients

ρ_{1}, ρ_{2}

and

ρ_{3}

, respectively, indicate the priorities of

E_{MUI}

- P

and

S E

with

ρ_{1} + ρ_{2} + ρ_{3} = 1

. Combining the objective function in (11) with the constraints, we obtain the problem as

\begin{matrix} min_{x, θ} & ρ_{1} {∥\tilde{H} x - s∥}_{2}^{2} - ρ_{2} x^{H} Ax + ρ_{3} {∥x - x_{0}∥}_{2}^{2} \\ s . t . & |x_{n}| = \sqrt{\frac{P_{t}}{M}}, n = 1, \dots, M L \\ |θ_{r}| = 1, r = 1, \dots, R \end{matrix}

(12)

We observe that with the CM constraint, the problem in (12) is non-convex and non-homogeneous, and there is coupling between the optimized variables

x

and

θ

. Therefore, it is a challenge to solve the problem and obtain the closed-form solutions. In this paper, we present a novel ACD algorithm.

3.2. Joint Design with ACD Algorithm

The idea of the ACD algorithm is to fix

θ

to optimize

x

, and then to fix

x

to optimize

θ

. The two subproblems are alternately optimized until the objective function converges. In each subproblem, the coordinate descent is utilized to divide the subproblem into multiple simpler problems and obtain the closed-form solutions.

3.2.1. Constant-Modulus Waveform Design

By fixing

θ

, the subproblem with respect to

x

can be expressed as

\begin{matrix} min_{x} & ρ_{1} {∥\tilde{H} x - s∥}_{2}^{2} - ρ_{2} x^{H} Ax + ρ_{3} {∥x - x_{0}∥}_{2}^{2} \\ s . t . & |x_{n}| = \sqrt{\frac{P_{t}}{M}}, n = 1, \dots, M L \end{matrix}

(13)

Firstly, we expand the objective function in (13) as

\begin{matrix} F (x) = x^{H} (ρ_{1} {\tilde{H}}^{H} \tilde{H} - ρ_{2} A + ρ_{3} I) x - x^{H} (ρ_{1} {\tilde{H}}^{H} s + ρ_{3} x_{0}) - (ρ_{1} s^{H} \tilde{H} + ρ_{3} x_{0}^{H}) x \\ + ρ_{1} s^{H} s + ρ_{3} x_{0}^{H} x_{0} \\ = x^{H} Qx - 2 ℜ (e^{H} x) + c o n s t_{1} \end{matrix}

(14)

where

Q = ρ_{1} {\tilde{H}}^{H} \tilde{H} - ρ_{2} A + ρ_{3} I \in C^{M L \times M L}

e = ρ_{1} {\tilde{H}}^{H} s + ρ_{3} x_{0} \in C^{M L \times 1}

and

c o n s t_{1} = ρ_{1} s^{H} s + ρ_{3} x_{0}^{H} x_{0}

is a constant independent of x. Ignoring the constant term, the subproblem in (13) is rewritten as

\begin{matrix} min_{x} & x^{H} Qx - 2 ℜ (e^{H} x) \\ s . t . & |x_{n}| = \sqrt{\frac{P_{t}}{M}}, n = 1, \dots, M L \end{matrix}

(15)

The new objective function in (15) can be unfolded as

\begin{matrix} x^{H} Qx - 2 ℜ (e^{H} x) = Q_{i, i} {|x_{i}|}^{2} + 2 ℜ (\sum_{n = 1, n \neq i}^{M L} x_{i} Q_{n, i} x_{n}^{*}) + \sum_{k = 1, k \neq i}^{M L} \sum_{n = 1, n \neq i}^{M L} x_{k}^{*} Q_{k, n} x_{n} \\ - 2 ℜ (x_{i} e_{i}^{*}) - 2 ℜ (\sum_{n = 1, n \neq i}^{M L} x_{n} e_{n}^{*}) \\ = ℜ (x_{i} f_{i}) + c_{i} \end{matrix}

(16)

where

Q_{p, q}

is the

(p, q)

th element of

Q

e_{p}

is the pth element of

e

f_{i} = 2 (\sum_{n = 1, n \neq i}^{M L} Q_{n, i} x_{n}^{*} - e_{i}^{*})

and

c_{i} = Q_{i, i} {|x_{i}|}^{2} + \sum_{k = 1, k \neq i}^{M L} \sum_{n = 1, n \neq i}^{M L} x_{k}^{*} Q_{k, n} x_{n} - 2 ℜ (\sum_{n = 1, n \neq i}^{M L} x_{n} e_{n}^{*})

Then, the optimization problem in (15) can be divided into

M L

subproblems, and the ith subproblem is

\begin{matrix} min_{x_{i}} & ℜ (x_{i} f_{i}) + c_{i} \\ s . t . & |x_{i}| = \sqrt{\frac{P_{t}}{M}} \end{matrix}

(17)

At each iteration, the ith subproblem only optimizes over one variable (i.e.,

x_{i}

) in

x

while substituting the remaining elements as the values obtained in the previous iteration. This means that

f_{i}

and

c_{i}

are constants independent of the optimization variable

x_{i}

. Therefore, (17) can be further simplified as

min_{φ_{x_{i}}} \sqrt{\frac{P_{t}}{M}} |f_{i}| cos (φ_{x_{i}} + φ_{f_{i}})

(18)

where

φ_{x_{i}}

and

φ_{f_{i}}

denote the phases of

x_{i}

and

f_{i}

, respectively. So, we obtain

φ_{x_{i}} = π - φ_{f_{i}}

(19)

Finally, the closed-form solution of

x_{i}

for the ith subproblem is given by

x_{i} = - \sqrt{\frac{P_{t}}{M}} e^{- j φ_{f_{i}}}

(20)

for

i = 1, \dots, M L

3.2.2. RIS Phase Shift Design

We note that only the MUI energy in the objective function (12) is related to

θ

; then, the subproblem with respect to

θ

with fixed

x

\begin{matrix} min_{θ} & {∥\tilde{H} x - s∥}_{2}^{2} \\ s . t . & |θ_{r}| = 1, r = 1, \dots, R \end{matrix}

(21)

Similarly, we expand the above objective function as

\begin{matrix} {∥\tilde{H} x - s∥}_{2}^{2} = {∥HX - S∥}_{F}^{2} \\ = {∥(H_{b u} + H_{r u} Θ H_{b r}) X - S∥}_{F}^{2} \\ = t r (Θ^{H} H_{r u}^{H} H_{r u} Θ H_{b r} X X^{H} H_{b r}^{H}) + t r (Θ^{H} H_{r u}^{H} (H_{b u} X - S) X^{H} H_{b r}^{H}) \\ + t r (H_{b r} X {(H_{b u} X - S)}^{H} H_{r u} Θ) + t r ({(H_{b u} X - S)}^{H} (H_{b u} X - S)) \\ = θ^{H} B θ + 2 ℜ (d^{T} θ) + c o n s t_{2} \end{matrix}

(22)

where

B = H_{r u}^{H} H_{r u} ⊙ {(H_{b r} X X^{H} H_{b r}^{H})}^{T} \in C^{R \times R}

d = d i a g (H_{b r} X {(H_{b u} X - S)}^{H} H_{r u}) \in C^{R \times 1}

and

c o n s t_{2} = t r ({(H_{b u} X - S)}^{H} (H_{b u} X - S))

is a constant independent of

θ

Ignoring the constant term, the subproblem in (21) is rewritten as

\begin{matrix} min_{θ} & θ^{H} B θ + 2 ℜ (d^{T} θ) \\ s . t . & |θ_{r}| = 1, r = 1, \dots, R \end{matrix}

(23)

In (23), the objective function can be further unfolded as

\begin{matrix} θ^{H} B θ + 2 ℜ (d^{T} θ) = B_{j, j} {|θ_{j}|}^{2} + 2 ℜ (\sum_{r = 1, r \neq j}^{R} θ_{j} B_{r, j} θ_{r}^{*}) + \sum_{s = 1, s \neq j}^{R} \sum_{r = 1, r \neq j}^{R} θ_{s}^{*} B_{s, r} θ_{r} \\ + 2 ℜ (θ_{j} d_{j}) + 2 ℜ (\sum_{r = 1, r \neq j}^{R} θ_{r} d_{r}) \\ = ℜ (θ_{j} g_{j}) + c_{j} \end{matrix}

(24)

where

g_{j} = 2 (\sum_{r = 1, r \neq j}^{R} B_{r, j} θ_{r}^{*} + d_{j})

and where

c_{j} = B_{j, j} {|θ_{j}|}^{2} + \sum_{s = 1, s \neq j}^{R} \sum_{r = 1, r \neq j}^{R} θ_{s}^{*} B_{s, r} θ_{r} + 2 ℜ (\sum_{r = 1, n \neq j}^{R} θ_{r} d_{r})

Thus, the optimization problem in (23) can be divided into R subproblems, and the jth subproblem is

\begin{matrix} min_{θ_{j}} & ℜ (θ_{j} g_{j}) + c_{j} \\ s . t . & |θ_{j}| = 1 \end{matrix}

(25)

Finally, the closed-form solution of

θ_{j}

for the jth subproblem is

θ_{j} = - e^{- j φ_{g_{j}}}

(26)

for

j = 1, \dots, R

, where

φ_{g_{j}}

denotes the phases of

g_{j}

We summarize the procedure of the proposed ACD algorithm in Algorithm 1. The coordinate descent algorithm in each subproblem can monotonically decrease the objective function, as the objective function has a lower bound, which ensures that the ACD algorithm can achieve convergence. In addition, the computational complexities of the two subproblems are

O (M^{2} L^{2})

and

O (R^{2})

, respectively. Hence, the total computational complexity of each iteration for the proposed ACD algorithm is

O (M^{2} L^{2} + R^{2})

. In practical THz dual-function MIMO radar and communication systems, due to the low output power of the THz transmitter it is usually necessary to increase the number of antennas M to improve the transmitting power, which increases the computational complexity of the ACD algorithm. To avoid using a large number of antennas, some hardware requirements need to be considered, such as a high-purity THz source, a high-gain and high-power amplifier, and a high-sensitivity THz receiver.

Algorithm 1 Proposed ACD algorithm for joint CM waveform and RIS phase shift design

Input:

t = 0

, initial variables

x^{(0)}

θ^{(0)}

and convergence tolerance

ϵ

Output:

x

θ

1 Repeat

t = t + 1

3 Update

x^{(t)}

initialize

x^{(t)} = x^{(t - 1)}

for

i = 1, \dots, M L

calculate

f_{i}

via (16) and obtain its phase

φ_{f_{i}}

Update

x_{i}^{(t)}

via (20).

4 Update

θ^{(t)}

initialize

θ^{(t)} = θ^{(t - 1)}

for

j = 1, \dots, R

calculate

g_{j}

via (24) and obtain its phase

φ_{g_{j}}

Update

θ_{j}^{(t)}

via (26).

5 Until

|F (x^{(t)}, θ^{(t)}) - F (x^{(t - 1)}, θ^{(t - 1)})| / |F (x^{(t)}, θ^{(t)})| \leq ε

4. Numerical Simulations

By simulations, we verified the effectiveness of the proposed joint design algorithm for the THz DFRC system, and we analyzed the effects of weighting coefficients and RIS on the DFRC performance. We assumed that the transmitting power was

P_{t} = 10

dBm and that the transmitting signal-to-noise ratio (SNR) was

P_{t} / N_{0}

. The numbers of BS antennas, UEs and RIS elements were

M = 8

K = 4

and

R = 40

, respectively, and the code length was

L = 20

. The center frequency was

f = 0.3

THz and the interval between adjacent antennas of BS was

d = 0.5

mm. There were

T = 3

targets at

ϕ_{1} = - 20 °

ϕ_{2} = 0 °

and

ϕ_{3} = 35 °

, respectively. The channels

H_{b u}

H_{b r}

and

H_{r u}

were generated according to the geometric-based THz channel model in [22], and each entity of the communication symbol

S

was the quadrature phase shift keying (QPSK) signal with the power of 10 mW. The MATLAB R2016b software is used.

4.1. Comparison of ACD Algorithm and Two Existing Optimization Algorithms

Firstly, we compared the ACD algorithm with the existing gradient projection (GP) [23] and Riemannian conjugate gradient (RCG) [24] algorithms, which were applied to the problem (12). All three algorithms require alternating optimization of

x

and

θ

; the difference lies in the procedure for solving each subproblem. In the following numerical simulation experiments, we investigated the DFRC performance in terms of the sum rate, beampattern and auto-correlation function (ACF) of the transmit waveforms. ‘GP’ and ‘RCG’ were used to denote the gradient projection and Riemannian conjugate gradient algorithms, respectively.

Figure 2 shows the sum rate versus the transmit SNR achieved by the different algorithms. The weighting coefficients were

ρ_{1} = 0.9

and

ρ_{2} = ρ_{3} = 0.05

, in which the larger weight of the communication performance was set to obtain the higher sum rate. ‘Zero MUI’ refers to the perfect communication scenario without MUI energy, which is shown as a benchmark. We can see that under the same SNR, the sum rate of our algorithm was always higher than the ‘GP’ and ‘RCG’ algorithms and that the difference gradually increased as the SNR increased. The reason was that the ‘GP’ and ‘RCG’ algorithms needed to iteratively approximate the optimal solution with appropriate iteration stepsizes. Also, the choice of the optimal iteration stepsize is an unresolved issue in these two algorithms, since too small an iteration stepsize will lead to slow convergence speed while too large an iteration stepsize will cause divergence of the objective function. Fortunately, our algorithm could obtain the closed-form solutions for the weighted optimization problem. Therefore, the waveform designed by the ACD algorithm had more outstanding communication performance than the ‘GP’ and ‘RCG’ algorithms.

Figure 3 shows the beampatterns of the transmit waveforms achieved by the different algorithms. The weighting coefficients were

ρ_{2} = 0.9

and

ρ_{1} = ρ_{3} = 0.05

, in which the larger weight of the radar detection performance was set to achieve the higher beampattern mainlobe-to-sidelobe ratio (MSR). The black dashed lines are the true angles of the targets. We see that the waveforms designed by the three algorithms all obtained high target illumination power in the directions of the targets, while the proposed algorithm could achieve higher MSR than the ‘GP’ and ‘RCG’ algorithms, which means that the waveform designed by the ACD algorithm owned less energy leakage, resulting in more outstanding performance in radar detection.

Figure 4 shows the ACFs of the transmit waveforms achieved by the different algorithms. The weighting coefficients were

ρ_{3} = 0.9

and

ρ_{1} = ρ_{2} = 0.05

, in which the larger weight of the range resolution performance was set to obtain better ACF. The waveform designed by our proposed algorithm had lower ACF sidelobes and higher similarity to the desired radar waveform

X_{0}

. Thus, the transmit waveform designed by our ACD algorithm had more outstanding radar pulse compression and range resolution properties.

4.2. Impact of Weighting Coefficients and RIS on Radar and Communication Performance

Figure 5 shows the sum rate versus the transmit SNR of our algorithm under different weighting coefficients and RIS conditions. The ‘No RIS’ refers to the downlink communication without RIS assistance. The weighting coefficients were chosen as

ρ_{1} = 0.3

ρ_{2} = ρ_{3} = 0.35

(approximately same weight) and

ρ_{1} = 0.9

ρ_{2} = ρ_{3} = 0.05

(larger weight of communication performance), respectively. The simulation shows that under the same SNR, the larger the weighting coefficient

ρ_{1}

the smaller the MUI energy and the higher the sum rate. The difference became more obvious as the SNR increased. Nevertheless, the sum rate was always lower than that of the ideal communication with ‘Zero MUI’. Therefore, increasing the weighting coefficient

ρ_{1}

was conducive to improving the sum rate, and vice versa. In addition, due to the creation of an additional communication link, the RIS-aided downlink communication achieved a higher sum rate under the same condition. Therefore, deploying RIS in a downlink communication is beneficial for improving communication performance.

Figure 6 shows the beampatterns of the transmit waveforms of our algorithm under different weighting coefficients and RIS conditions. The weighting coefficients were chosen as

ρ_{2} = 0.3

ρ_{1} = ρ_{3} = 0.35

(approximately same weight) and

ρ_{2} = 0.9

ρ_{1} = ρ_{3} = 0.05

(larger weight of radar detection performance), respectively. The black dashed lines are the true angles of the targets. We found that the larger the weighting coefficient

ρ_{2}

the greater the multi-target illumination power and the higher the beampattern MSR. Therefore, increasing the weighting coefficient

ρ_{2}

is in favor of increasing the illumination power in the directions of the targets, thereby improving the radar detection performance, and vice versa. In addition, unlike the impact of RIS on communication performance, the radar beampattern performance of an RIS-aided system is not better than that of a ‘No RIS’ system under the same condition, which is because although RIS does not directly affect the illumination power at the targets, the additional RIS phase shift constraint reduces the freedom of the waveform design, resulting in the degradation of the radar detection performance.

To further validate the conclusion obtained in Figure 6, Figure 7 shows the average detection probability of multiple targets versus the radar transmit SNR of our algorithm under different weighting coefficients and RIS conditions. The weighting coefficients were still chosen as

ρ_{2} = 0.3

ρ_{1} = ρ_{3} = 0.35

and

ρ_{2} = 0.9

ρ_{1} = ρ_{3} = 0.05

, respectively. The false alarm probability of the MIMO radar was set to

P_{FA} = 10^{- 3}

. According to [25], the detection probability was calculated by the noncentrality parameter of the noncentral Chi-squared distribution function, which was influenced by the covariance of the designed waveform. We can see from Figure 7 that under the same SNR, the larger the weighting coefficient

ρ_{2}

the greater the multi-target illumination power and the higher the radar detection probability. Thus, increasing the weighting coefficient

ρ_{2}

can indeed improve the radar detection performance. In addition, the detection probability of the RIS-aided system was lower than that of the ‘No RIS’ system, especially when

ρ_{2}

was small. Thus, deploying RIS in a downlink communication will deteriorate the radar detection performance.

Figure 8 shows the ACFs of the transmit waveforms of our algorithm under different weighting coefficients and RIS conditions. The weighting coefficients were chosen as

ρ_{3} = 0.3

ρ_{1} = ρ_{2} = 0.35

(approximately same weight) and

ρ_{3} = 0.9

ρ_{1} = ρ_{2} = 0.05

(larger weight of range resolution performance), respectively. We found that the larger the weighting coefficient

ρ_{3}

the lower the ACF sidelobes and the smaller the radar waveform similarity error; thus, the ACF of the transmit waveform was closer to that of the desired waveform

X_{0}

. Therefore, increasing the weighting coefficient

ρ_{3}

is beneficial for improving the radar pulse compression and ambiguity properties, and vice versa. Nevertheless, under the same condition, the RIS-aided system had higher ACF sidelobes than the ‘No RIS’ system. The reason was that though the RIS had no direct effect on the radar waveform similarity error, the RIS phase shift constraint limited the freedom of the waveform design and degraded the range resolution ability of the MIMO radar.

In summary, the communication data rate, radar detection probability and range resolution performance a were mainly affected by the weighting coefficients

ρ_{1}

ρ_{2}

and

ρ_{3}

, respectively, and they were also impacted by the RIS. By increasing one of the weighting coefficients, the corresponding performance would be improved but the other aspects of the performance would be lost. In practice, the optimal weighting coefficients are not fixed, and they can be flexibly adjusted according to the degree of emphasis on different performances. Specifically, the more the emphasis is placed on a certain performance, the larger the corresponding weighting coefficient, and vice versa. Despite the fact that the radar detection probability and range resolution performance will be affected by deploying RIS in downlink, it can improve the multi-UE communication performance. Therefore, it is necessary to achieve a flexible performance trade-off for the THz DFRC system by adjusting the weighting coefficients as well as RIS.

4.3. Impact of Channel Conditions on the Performance of THz System

Finally, the impact of channel conditions on the performance of the THz dual-function MIMO radar and communication system was considered under imperfect channel state information (CSI), which is crucial for practical applications. We assumed that we only knew the estimated THz channels

H_{b u}

H_{b r}

and

H_{r u}

, as well as the distribution of the estimation errors

Δ H_{b u}

Δ H_{b r}

and

Δ H_{r u}

. Each entity of the estimation errors followed a complex Gaussian distribution with a mean of zero and a variance of 0.1. Under perfect CSI, the estimated THz channels were the true channels. Under imperfect CSI, the true channels were

H_{b u} + Δ H_{b u}

H_{b r} + Δ H_{b r}

and

H_{r u} + Δ H_{r u}

, respectively. Figure 9 shows the sum rate versus the transmit SNR under different channel conditions when the weighting coefficients were

ρ_{1} = 0.9

and

ρ_{2} = ρ_{3} = 0.05

. Compared to the perfect CSI, the sum rate only slightly decreased under imperfect CSI. Therefore, the THz dual-function MIMO radar and communication system is robust against imperfect CSI.

5. Conclusions

This paper formulated a new joint CM waveform and phase shift design problem for the RIS-aided THz dual-function MIMO radar and communication system with multiple targets and users. The weighted sum of the communication MUI energy, the negative of the multi-target illumination power and the MIMO radar waveform similarity error were minimized under the CM constraint, which is a non-convex problem. A novel ACD algorithm was introduced, to handle the optimization problem and obtain closed-form solutions. The simulations proved that our proposed ACD algorithm is superior to several existing algorithms. In addition, the proposed weighted optimization scheme for joint CM waveform and RIS phase shift design can implement a flexible radar and communication performance trade-off, and it is robust against imperfect CSI. In future research, the issue of MIMO DFRC and wideband radar [26,27] could be further investigated. Due to the large bandwidth of the THz frequency band, we will consider designing wideband transmit waveforms for THz dual-function MIMO radar and communication systems.

Author Contributions

Conceptualization, R.Y.; methodology, R.Y.; software, R.Y.; validation, R.Y.; formal analysis, R.Y.; investigation, R.Y.; resources, H.J. and L.Q.; data curation, R.Y.; writing—original draft preparation, R.Y.; writing—review and editing, R.Y. and H.J.; visualization, R.Y.; supervision, H.J. and L.Q.; project administration, H.J. and L.Q.; funding acquisition, H.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Natural Science Foundation of Jilin Province under Grants 20220101100JC and 20180101329JC and by the National Natural Science Foundation of China under Grants 61371158 and 61771217.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. RIS-aided THz dual-function MIMO radar and communication system with multiple targets and UEs.

Figure 2. Sum rate versus the transmit SNR for different algorithms.

Figure 3. Beampatterns of the transmit waveforms achieved by different algorithms.

Figure 4. Auto-correlation functions achieved by different algorithms.

Figure 5. Sum rate versus the transmit SNR of our algorithm under different weighting coefficients and RIS conditions.

Figure 6. Beampatterns of the transmit waveforms of our algorithm under different weighting coefficients and RIS conditions.

Figure 7. Average detection probability versus the radar transmit SNR of our algorithm under different weighting coefficients and RIS conditions.

Figure 8. Auto-correlation functions of the transmit waveforms of our algorithm under different weighting coefficients and RIS conditions.

Figure 9. Sum rate versus the transmit SNR under different channel conditions of THz system.

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Yang, R.; Jiang, H.; Qu, L. Joint Constant-Modulus Waveform and RIS Phase Shift Design for Terahertz Dual-Function MIMO Radar and Communication System. Remote Sens. 2024, 16, 3083. https://doi.org/10.3390/rs16163083

AMA Style

Yang R, Jiang H, Qu L. Joint Constant-Modulus Waveform and RIS Phase Shift Design for Terahertz Dual-Function MIMO Radar and Communication System. Remote Sensing. 2024; 16(16):3083. https://doi.org/10.3390/rs16163083

Chicago/Turabian Style

Yang, Rui, Hong Jiang, and Liangdong Qu. 2024. "Joint Constant-Modulus Waveform and RIS Phase Shift Design for Terahertz Dual-Function MIMO Radar and Communication System" Remote Sensing 16, no. 16: 3083. https://doi.org/10.3390/rs16163083

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Joint Constant-Modulus Waveform and RIS Phase Shift Design for Terahertz Dual-Function MIMO Radar and Communication System

Abstract

1. Introduction

2. System Model

2.1. MIMO Radar Model

2.2. Communication Model

3. Problem Formulation and Algorithm Proposal

3.1. Optimization Problem Formulation

3.2. Joint Design with ACD Algorithm

3.2.1. Constant-Modulus Waveform Design

3.2.2. RIS Phase Shift Design

4. Numerical Simulations

4.1. Comparison of ACD Algorithm and Two Existing Optimization Algorithms

4.2. Impact of Weighting Coefficients and RIS on Radar and Communication Performance

4.3. Impact of Channel Conditions on the Performance of THz System

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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