Max-min secure energy efficiency optimization for simultaneous transmitting and reflecting (STAR)-reflective intelligent surface (RIS)-assisted near-field wideband terahertz (THz) communication

1 . A method for maximizing a minimum secure energy efficiency in a near-field, wideband terahertz (THz) communication system, the method comprising:

generating, at a base station, a plurality of wideband THz signals using a hybrid analog-digital beamforming architecture that includes a plurality of radio-frequency (RF) chains, phase shifters (PS), and a plurality of true time delay (TTD) units;

applying, by the plurality of TTD units, respective time delays to compensate for beam-splitting effects across a plurality of subcarriers;

configuring, by a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) comprising N elements, a set of phase-shifts and amplitude coefficients for each element of the STAR-RIS to simultaneously transmit and reflect the THz signals toward a plurality of user equipment (UEs) on a transmission side and a reflection side of the STAR-RIS;

determining, by a processor, a secrecy rate for each UE as a difference between an achievable rate of the UE and an intercept rate of an eavesdropper (EV); and

optimizing, by the processor, a plurality of optimization variables including:

digital beamforming

(

W

k

D

)

,

PS-based analog beamforming (W PS ), matrices of the TTD units (D k ), and transmission and reflection coefficients of the STAR-RIS (θ t and θ r ) to maximize a minimum secure energy efficiency among all UEs.

2 . The method of claim 1 , wherein the optimizing is performed subject to constraints including (i) a transmit-power budget, (ii) a unit-modulus constraint of the analog beamforming, (iii) an energy-conservation constraint at the STAR-RIS where a square of a transmission amplitude coefficient

(

ρ

t

,

n

2

)

of plus a square of a reflection amplitude coefficient

(

ρ

r

,

n

2

)

for each element equals one, and (iv) a non-negative time-delay constraint for the true-time-delay units.

3 . The method of claim 2 , wherein the optimizing is performed using a penalty dual decomposition (PDD) algorithm that constructs an augmented Lagrangian including dual variables and a penalty factor to enforce a plurality of equality constraints.

4 . The method of claim 3 , wherein the optimizing further comprises an inner loop that optimizes the augmented Lagrangian by block coordinate descent (BCD) and an outer loop that updates the dual variables and the penalty factor based on an equality constraint threshold.

5 . The method of claim 4 , wherein the BCD comprises iteratively updating the plurality of optimization variables, wherein, during each iteration, the processor fixes all but one optimization variable and updates the remaining optimization variables to minimize the augmented Lagrangian until convergence.

6 . The method of claim 1 , further comprising:

maximizing the minimum secure energy efficiency among all UEs by defining the secure energy-efficiency value for a UE is a ratio of a secrecy-rate of the UE to a total power consumption used to serve the UE.

7 . The method of claim 1 , wherein the secrecy rate is a non-negative difference between the achievable rate of the UE and a maximum achievable rate among a plurality of EVs when intercepting the user equipment.

8 . The method of claim 1 , further comprising:

operating the base station and the STAR-RIS in a wideband THz frequency range under near-field conditions.

9 . The method of claim 1 , further comprising:

applying the optimization variables obtained from the optimizing in real-time to control the base station and the STAR-RIS during wideband THz data transmission.

10 . A system for maximizing a minimum secure energy efficiency in a near-field, wideband terahertz (THz) communication system, the system comprising:

a base station;

a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) comprising N elements, and,

a processor configured to:

generate, at a base station, a plurality of wideband THz signals using a hybrid analog-digital beamforming architecture that includes a plurality of radio-frequency (RF) chains, phase shifters (PS), and a plurality of the true time delay (TTD) units;

apply, via the plurality of TTD units, respective time delays to compensate for beam-splitting effects across a plurality of subcarriers;

configure, via the STAR-RIS, a set of phase-shifts and amplitude coefficients for each element of the STAR-RIS to simultaneously transmit and reflect the THz signals toward a plurality of user equipment (UEs) on a transmission side and a reflection side both-sides of the STAR-RIS;

determine, a secrecy rate for each UE as a difference between an achievable rate of the UE and an intercept rate of an eavesdropper (EV); and

optimize a plurality of optimization variables including: digital beamforming

(

W

k

D

)

,

PS-based analog beamforming (W PS ), matrices of the TTD units (D k ), and transmission and reflection coefficients of the STAR-RIS (θ t and θ r ) to maximize a minimum secure energy efficiency among all UEs.

11 . The system of claim 10 , wherein each RF chain of the plurality of RF chains is coupled to one or more TTD units, wherein outputs of the one or more TTD units are distributed via analog phase-shifters to a linear antenna array.

12 . The system of claim 10 , wherein the STAR-RIS comprises a uniform planar array (UPA) structure.

13 . The system of claim 10 , wherein the processor is further configured to optimize the plurality of optimization variables subject to constraints including (i) a transmit-power budget, (ii) a unit-modulus of the analog beamforming, (iii) an energy-conservation constraint at the STAR-RIS where a square of a transmission amplitude coefficient

(

ρ

t

,

n

2

)

of plus a square of a reflection amplitude coefficient

(

ρ

r

,

n

2

)

for each element equals one, and (iv) a non-negative time-delay constraint for the true-time-delay units.

14 . The system of claim 13 , wherein the processor is further configured to optimize the plurality of optimization variables using a penalty dual decomposition (PDD) algorithm that constructs an augmented Lagrangian including dual variables and a penalty factor to enforce a plurality of equality constraints.

15 . The system of claim 14 , wherein the optimizing further comprises an inner loop that optimizes the augmented Lagrangian by block coordinate descent (BCD) and an outer loop that updates the dual variables and the penalty factor based on an equality constraint violation threshold.

16 . The system of claim 15 , wherein the BCD comprises iteratively updating the plurality of optimization variables, wherein, during each iteration, the processor fixes all but one optimization variable and updates the remaining optimization variables to minimize the augmented Lagrangian until convergence.

17 . The system of claim 10 , wherein the processor is configured to maximize the minimum secure energy efficiency among all UEs by defining the secure energy-efficiency value for a UE is a ratio of a secrecy-rate of the UE to a total power consumption used to serve the UE.

18 . The system of claim 10 , wherein the secrecy rate is a non-negative difference between the achievable rate of the UE and a maximum achievable rate among a plurality of EVs when intercepting the user equipment.

19 . The system of claim 10 , wherein the processor is further configured to operate the base station and the STAR-RIS in a wideband THz frequency range under near-field conditions.

20 . The system of claim 10 , wherein the processor is further configured to apply the optimization variables obtained from the optimizing in real-time to control the base station and the STAR-RIS during wideband THz data transmission.