CN113660022B - Transceiver and IRS Optimization Design Method of Over-the-Air Computing System under Nonideal CSI - Google Patents

Abstract

The invention discloses a transceiver and an IRS optimization design method of an over-the-air computing system under non-ideal CSI, comprising the following steps: (1) According to the uncertainty domain of CSI, solve the error of CSI under the worst case of MSE at the receiving end

And calculate the MSE of the receiving end; (2) remove the constant modulus constraint of the IRS phase, take the minimum MSE of the receiving end as the goal, fix the transmit power t _k of each sensor and the combining coefficient m of the receiving end, and optimize the reflection phase vector v _k of the IRS; (3) Fix the reflection phase vector v _k of the IRS, and optimize the transmit power t _k of each sensor and the combining coefficient m of the receiving end; (4) Repeat steps (2)-(3) until the MSE iteratively converges at the receiving end; (5) pair The reflection phase vector v _k of the IRS carries out constant modulus constraint. The invention can reduce the receiving error of the system under the condition of non-ideal CSI under the condition that the total transmit power is limited.

Description

Transceiver and IRS Optimization Design Method for Over-the-Air Computing System under Nonideal CSI

technical field

The invention relates to the field of communication, and relates to a transceiver and an IRS optimization design method of an over-the-air computing system under non-ideal CSI.

Background technique

The Internet of Things (IoT) services are in full swing and touch every aspect of people's lives. As the most prominent feature of the current era of high-speed information, the astonishing growth of massive data is of vital significance to the Internet of Things. On the one hand, the growing number of IoT smart devices makes data collection more difficult. On the other hand, information fusion of massive data, or what we call data aggregation, is more challenging with the low spectral efficiency and high latency brought about by billions of sensors.

To address this problem, researchers propose a promising solution, Over-the-Air Computation (AirComp) that exploits the superposition properties of wireless multiple-access channels (MACs). Through the concurrent transmission of sensors and the weighted average function of distributed local computation in the system, the required data can be obtained directly at the receiving end. AirComp has attracted much attention in many aspects such as information theory, signal processing, transceiver design, etc. The study found that AirComp can improve communication efficiency and reduce the required bandwidth. However, the advantage of the above technique is premised that the MAC is not blocked.

To address this issue, intelligent reflective surfaces (IRS) are considered to improve signal propagation conditions by utilizing passive reflective elements to reflect incident signals at ideal angles, an emerging complementary technology. It can be easily attached to the exterior of a building and is inexpensive. In view of the real-time reconfiguration characteristics of IRS, its applications have been widely developed, such as secure communication in the Internet of Things, synchronous wireless information and power transmission system (SWIPT), edge computing (MEC), etc. Therefore, using the IRS to assist the AirComp system can reduce the MSE of AirComp by optimizing the transceiver and phase shift. However, all the above studies are highly dependent on perfect CSI, which is difficult to obtain in applications. Therefore, it is also necessary to introduce stochastic and deterministic robustness to mitigate performance degradation in terms of CSI inaccuracy. In addition, the wireless sensors of AirComp systems are often powered by power-limited beacons, and overall power constraints are also a challenge.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a transceiver and an IRS optimization design method of an air computing system under non-ideal CSI. Receive MSE at the receiving end.

In order to solve the complex non-convex problem, the present invention obtains the closed-form solution of the robust problem and the joint design of the transmitting power of each sensor, the combining coefficient of the receiving end and the IRS phase by using an efficient alternating optimization algorithm and KKT conditions.

In order to achieve the above object, the present invention provides the following technical solutions:

A transceiver and an IRS optimization design method for an over-the-air computing system under non-ideal CSI, comprising the following steps:

并计算接收端MSE，其中，k∈{1,2,…,K}，K表示空中计算系统中发射端传感器数量；(1) According to the uncertainty domain of CSI, solve the error of CSI in the worst case of MSE at the receiving end

And calculate the receiver MSE, where k∈{1,2,…,K}, K represents the number of transmitter sensors in the air computing system;

(2) remove the constant modulus constraint of the IRS phase, take minimizing the MSE of the receiving end as the goal, fix the transmit power t _k of each sensor and the combining coefficient m of the receiving end, and optimize the reflection phase vector v _k of the IRS;

(3) The reflection phase vector v _k of the IRS is fixed, and the transmit power t _k of each sensor and the combining coefficient m of the receiving end are optimized;

(4) Repeat steps (2)-(3) until the MSE iteratively converges at the receiving end;

(5) Constrain the constant modulus of the reflection phase vector v _k of the IRS.

即CSI的不确定域被限制在以ε为半径的区域内。Further, in the step (1), the uncertainty domain of CSI is

That is, the uncertainty region of CSI is limited to the region with ε as the radius.

表示信道反馈的CSI误差，

表示第k个传感器经由第k个IRS阵面反射到达接收基站的真实的等效级联信道，

表示第k个传感器经由第k个IRS阵面反射到达接收基站的估计的等效级联信道。其中，N表示IRS反射单元数目，k∈{1,2,…,K}，K表示空中计算系统中发射端传感器数量，‖‖₂表示向量Frobenius范数，()^H表示矩阵共轭转置；in,

represents the CSI error of the channel feedback,

represents the real equivalent concatenated channel of the kth sensor reaching the receiving base station through the kth IRS front reflection,

represents the estimated equivalent concatenated channel of the kth sensor arriving at the receiving base station via the kth IRS front reflection. Among them, N represents the number of IRS reflection units, k∈{1,2,…,K}, K represents the number of transmitter sensors in the air computing system, _‖‖2 represents the vector Frobenius norm, () ^H represents the matrix conjugate transpose ;

The worst-case MSE problem at the receiver is formulated as

其中，m为接收端的合并系数，v_k为第k个IRS的反射向量，t_k表示第k个传感器的发射功率，σ²为加性高斯白噪声功率，||²表示复数的模的平方；Constraints are

Among them, m is the combining coefficient of the receiving end, v _k is the reflection vector of the kth IRS, _tk is the transmit power of the kth sensor, σ ² is the additive white Gaussian noise power, and || ² is the square of the modulus of the complex number ;

Since the cascaded equivalent channels of each sensor are independent of each other, the worst-case MSE problem is transformed into

Further, in the step (1), the error of the CSI in the worst case of the receiving end MSE is:

为中间变量，λ_k表示对于第k个传感器的等效级联信道误差优化时的KKT(Karush-Kuhn-Tucker)乘子，且where () ^* denotes the conjugate of complex numbers,

is an intermediate variable, λ _k represents the KKT (Karush-Kuhn-Tucker) multiplier when optimizing the equivalent cascaded channel error for the kth sensor, and

At this time, it is the worst case of CSI, and the receiving MSE of the receiving end is

Further, in the step (2), by fixing the transmit power t _k of each sensor and the combining coefficient m of the receiving end, the following optimization problem is constructed to optimize the reflection phase vector v _k of the IRS:

The optimization objective is:

The constraints are: |v _k (n)| ² = 1, n∈{1,…,N}

Among them, v _k (n) represents the n-th element in the reflection vector of the k-th IRS; since each sensor is independent of each other, the summation optimization problem can be decomposed into K sub-optimization problems, k∈{1,2,…, K}, K represents the number of transmitter sensors in the air computing system; to make the problem solvable, first temporarily remove the constant modulus constraint of the IRS phase, and the optimization objective becomes

为中间变量，

表示空中计算系统里K个传感器的总功率限制。in,

is an intermediate variable,

Represents the total power limit of the K sensors in the air computing system.

Further, in the step (2), the reflection phase vector of the optimal IRS is

表示计算矩阵的Moore-Penrose逆。in,

Represents computing the Moore-Penrose inverse of a matrix.

Further, in the step (3), by fixing the reflection phase vector v _k of the IRS, the following optimization problem is constructed to optimize the transmission power t _k of each sensor and the combining coefficient m of the receiving end:

The optimization objective is:

Obtain the optimal intermediate variable by derivation

and

表示计算矩阵的Moore-Penrose逆。in,

Represents computing the Moore-Penrose inverse of a matrix.

Further, in the step (5), the constant modulus constraint is performed on the reflection phase vector v _k of the IRS as

where ‖‖ ₂ represents the Frobenius norm of the vector.

Beneficial effects: The present invention minimizes the receiving MSE at the receiving end by designing the transmit power of each sensor at the transmitting end of the IRS-assisted AirComp system, the combining coefficient at the receiving end and the IRS phase. To solve the complex non-convex problem, we obtain the closed-form solution of the robust problem and the joint design of the transmit power of each sensor, the combining coefficient at the receiver and the IRS phase by utilizing an efficient alternating optimization algorithm and KKT conditions. The alternating optimization algorithm is used to solve the original non-convex and highly coupled complex optimization problem, and the non-convex problem is transformed into a convex problem through intermediate variables, and a closed-form solution is obtained, which has absolute advantages compared with other invented non-closed-form solutions. . In addition, the present invention can effectively reduce the reception error of the system under the condition of non-ideal CSI under the condition that the total transmit power is limited.

Description of drawings

1 is a schematic diagram of a practical application scenario of the present invention;

Fig. 2 is the flow chart of the present invention;

FIG. 3 is a graph of the received NMSE versus the number N of IRS reflection units using the optimization method of the present invention.

Detailed ways

The technical solutions provided by the present invention will be described in detail below with reference to specific embodiments, and it should be understood that the following specific embodiments are only used to illustrate the present invention and not to limit the scope of the present invention.

The technical terms involved in the present invention are explained as follows:

CSI: channel state information;

AirComp: air computing system;

IRS: Intelligent Reflector;

MSE: Minimum Mean Squared Error.

A transceiver and intelligent reflector (IRS) optimization design method of an air computing system (AirComp) under non-ideal channel state information (CSI) of the present invention, wherein, the transmitting end is K single-antenna sensors to transmit the required information of the IoT system. The environmental information reaches the receiving end through the transmission of K IRSs with N reflecting units. The receiving end is a single antenna and the required amount of reception is the sum of the information sent by the K single-antenna sensors.

并联合优化系统发射端各个传感器的发射功率t_k、接收端的合并系数m和IRS相位v_k，从而最小化接收端的接收最小均方误差(MSE)。The invention takes the limitation of the total transmission power as the constraint condition, and studies the error of the MSE and CSI of the receiving end under the worst CSI condition of the receiving end MSE

And jointly optimize the transmit power t _k of each sensor at the transmitter end, the combining coefficient m and the IRS phase v _k at the receiver end, so as to minimize the minimum mean square error (MSE) of the receiver end.

In order to reduce the receiving MSE at the receiving end, the present invention minimizes the receiving MSE at the receiving end by designing the transmit power of each sensor at the transmitting end of the IRS-assisted AirComp system, the combining coefficient and the IRS phase at the receiving end. In order to solve the complex non-convex problem, the present invention obtains the closed-form solution of the robust problem and the joint design of the transmitting power of each sensor, the combining coefficient of the receiving end and the IRS phase by using an efficient alternating optimization algorithm and KKT conditions. The alternating optimization algorithm is used to solve the original non-convex and highly coupled complex optimization problem, and the non-convex problem is transformed into a convex problem through intermediate variables, and a closed-form solution is obtained, which has absolute advantages compared with other invented non-closed-form solutions. .

As shown in Figure 1, the transmitter sends the environmental information required by the IoT system for K single-antenna sensors. In order to enhance the received signal at the receiver, K IRSs are used to reconstruct the propagation environment. The main optimization design idea is to first analyze the error of the receiving MSE and the CSI in the worst case of the receiver MSE according to the uncertainty field of CSI; then, under the worst case MSE, first remove the constant modulus constraint of the IRS phase, and fix The transmit power of each sensor and the combining coefficient of the receiving end, optimize the reflection phase vector of the IRS, then fix the reflected phase vector of the IRS, optimize and fix the transmitting power of each sensor and the combining coefficient of the receiving end, repeat the reflected phase vector of the IRS and the transmitting and receiving end of the system Alternate optimization steps of the coefficients until the received MSE of the system converges; the optimization result of the reflected phase vector of the IRS is limited by constant modulus. The invention can effectively reduce the reception error of the system under the condition of non-ideal CSI under the condition that the total transmit power is limited.

The optimal design method of the transceiver of the AirComp system and the IRS under the non-ideal CSI of the present invention comprises the following steps:

(1) According to the uncertainty domain of CSI, solve the error of CSI in the worst case of MSE at the receiving end

即CSI的不确定域被限制在以ε为半径的区域内；Among them, the uncertainty domain of CSI is

That is, the uncertainty domain of CSI is limited to the area with ε as the radius;

表示信道反馈的CSI误差，

表示第k个传感器经由第k个IRS阵面反射到达接收基站的真实的等效级联信道，

表示第k个传感器经由第k个IRS阵面反射到达接收基站的估计的等效级联信道。其中，N表示IRS反射单元数目，k∈{1,2,…,K}，K表示AirComp系统中发射端传感器数量，‖‖₂表示向量Frobenius范数，()^H表示矩阵共轭转置；in,

represents the CSI error of the channel feedback,

represents the real equivalent concatenated channel of the kth sensor reaching the receiving base station through the kth IRS front reflection,

represents the estimated equivalent concatenated channel of the kth sensor arriving at the receiving base station via the kth IRS front reflection. Among them, N represents the number of IRS reflection units, k∈{1,2,…,K}, K represents the number of transmitter sensors in the AirComp system, _‖‖2 represents the vector Frobenius norm, () ^H represents the matrix conjugate transpose;

At this point, the worst-case MSE problem at the receiver is formulated as

其中，m为接收端的合并系数，v_k为第k个IRS的反射向量，t_k表示第k个传感器的发射功率，σ²为加性高斯白噪声功率，||²表示复数的模的平方；Constraints are

Among them, m is the combining coefficient of the receiving end, v _k is the reflection vector of the kth IRS, _tk is the transmit power of the kth sensor, σ ² is the additive white Gaussian noise power, and || ² is the square of the modulus of the complex number ;

Since the cascaded equivalent channels of each sensor are independent of each other, the above worst-case MSE problem transforms into

Among them, k∈{1,2,…,K}, K represents the number of transmitter sensors in the AirComp system;

The error of the CSI in the worst case above is

为中间变量，λ_k表示对于第k个传感器的等效级联信道误差优化时的KKT(Karush-Kuhn-Tucker)乘子，且where () ^* denotes the conjugate of complex numbers,

is an intermediate variable, λ _k represents the KKT (Karush-Kuhn-Tucker) multiplier when optimizing the equivalent cascaded channel error for the kth sensor, and

At this time, it is the worst case of CSI, and the receiving MSE of the receiving end is

(2) Removing the constant modulus constraint of the IRS phase, aiming at minimizing the MSE of the receiving end, fixing the transmit power t _k of each sensor and the combining coefficient m of the receiving end, and optimizing the reflection phase vector v _k of the IRS. By fixing the transmit power t _k of each sensor and the combining coefficient m of the receiving end, the following optimization problem is constructed to optimize the reflection phase vector v _k of the IRS:

The optimization objective is:

The constraints are: |v _k (n)| ² = 1, n∈{1,…,N}

Among them, v _k (n) represents the n-th element in the reflection vector of the k-th IRS; since each sensor is independent of each other, the summation optimization problem is decomposed into K sub-optimization problems, k∈{1,2,…,K }, K represents the number of transmitter sensors in the air computing system; to make the problem solvable, first temporarily remove the constant modulus constraint of the IRS phase, and the optimization objective becomes

为中间变量，

表示AirComp系统里K个传感器的总功率限制；in,

is an intermediate variable,

Indicates the total power limit of the K sensors in the AirComp system;

Among them, the reflection phase vector of the optimal IRS is

表示计算矩阵的Moore-Penrose逆。in,

Represents computing the Moore-Penrose inverse of a matrix.

(3) Fix the reflection phase vector v _k of the IRS, and optimize the transmission power t _k of each sensor and the combining coefficient m of the receiving end; by fixing the reflection phase vector v _k of the IRS, the following optimization problem is constructed to optimize the transmission power t _k and the receiving end of each sensor. Merging factor m at the end:

The optimization objective is:

Obtain the optimal intermediate variable by derivation

and

表示计算矩阵的Moore-Penrose逆。in,

Represents computing the Moore-Penrose inverse of a matrix.

(4) Repeat steps (2)-(3) until the MSE at the receiving end is iteratively converged.

(5) Constrain the constant modulus of the reflection phase vector v _k of the IRS, namely

where ‖‖ ₂ represents the Frobenius norm of the vector.

As shown in Figure 2, the main process of the present invention is to first analyze the error between the receiving MSE and the CSI in the worst case of the MSE at the receiving end according to the uncertainty field of the CSI; then, in the worst case of the MSE, first remove the IRS The constant modulus constraint of the phase, fix the transmit power of each sensor and the combining coefficient of the receiving end, optimize the reflection phase vector of the IRS, then fix the reflection phase vector of the IRS, optimize and fix the transmitting power of each sensor and the combining coefficient of the receiving end, repeat the reflection of the IRS Alternate optimization steps of the phase vector and the coefficients of the transceiver end of the system until the receiving MSE of the system converges; the optimization result of the reflected phase vector of the IRS is limited by constant modulus.

As shown in FIG. 3 , the optimized design scheme proposed by the present invention can effectively reduce the reception error of the system under the condition of non-ideal CSI under the condition that the total transmit power is limited. Compared with the traditional scheme without considering the robustness design, it has performance advantages.

The technical means disclosed in the solution of the present invention are not limited to the technical means disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features. It should be pointed out that for those skilled in the art, without departing from the principle of the present invention, several improvements and modifications can be made, and these improvements and modifications are also regarded as the protection scope of the present invention.

Claims (2)

1. A transceiver and IRS optimization design method of an air computing system under non-ideal CSI is characterized in that: the method comprises the following steps:

(1) according to the uncertain domain of the CSI, solving the error of the CSI under the condition that MSE of the receiving end is the worst

And computing MSE at a receiving end, wherein K belongs to {1,2, …, K }, and K represents an aerial meterCalculating the number of transmitting end sensors in the system;

the uncertain domain of CSI is

Namely, the uncertain domain of the CSI is limited in a region with epsilon as a radius;

wherein,

a CSI error representing the feedback of the channel,

representing the true equivalent concatenated channel that the kth sensor reflected via the kth IRS front to the receiving base station,

an estimated equivalent concatenated channel representing the arrival of the kth sensor at the receiving base station via the kth IRS front reflection; where N represents the number of IRS reflecting units, K ∈ {1,2, …, K }, and K represents the number of transmitting-end sensors in the airborne computing system, |₂Represents the vector Frobenius norm, ()^HRepresenting a matrix conjugate transpose;

the MSE problem of the worst case at the receiving end is expressed as

The constraint condition is

Where m is the combining coefficient of the receiving end, v_kIs the reflection vector of the kth IRS, t_kRepresenting the transmission power, σ, of the kth sensor²Is additive white Gaussian noise power, | non-dominant²A square of a modulus representing a complex number;

since the cascade equivalent channels of each sensor are independent of each other, the MSE problem in the worst case translates into

The error of CSI under the condition of the MSE (mean square error) of the receiving end is

Wherein (C)^*Which represents the conjugate of the complex number,

is an intermediate variable, λ_kRepresents the KKT multiplier in the optimization of the equivalent cascade channel error for the kth sensor, and

at this time, the CSI is the worst case, and the receiving MSE of the receiving end is

(2) Removing constant modulus constraint of IRS phase, fixing transmitting power t of each sensor by taking minimum receiving end MSE as target_kAnd the combination coefficient m of the receiving end, optimizing the reflection phase vector v of the IRS_k；

By fixing the transmission power t of the individual sensors_kAnd a combination coefficient m of a receiving end, constructing a reflection phase vector v for optimizing IRS (interference rejection ratio) by the following optimization problem_k：

The optimization target is as follows:

the constraint conditions are as follows: | v_k(n)|²＝1，n∈{1,…,N}

Wherein v is_k(n) denotes the nth element in the reflection vector of the kth IRS; because the sensors are independent, the summation optimization problem can be decomposed into K sub-optimization problems, K belongs to {1,2, …, K }, and K represents the number of transmitting end sensors in the aerial computing system; to make the problem solvable, the constant modulus constraint of the IRS phase is temporarily removed first and the optimization objective becomes

Wherein,

is the intermediate variable(s) of the variable,

representing the total power limit of K sensors in the airborne computing system;

the reflection phase vector of the optimal IRS is

Wherein,

Moore-Penrose inverse representing the computational matrix;

(3) reflection phase vector v of fixed IRS_kOptimizing the individual sensor transmission power t_kAnd a merging coefficient m of the receiving end;

reflection phase vector v by fixed IRS_kThe following optimization problem is constructed to optimize the transmission power t of each sensor_kAnd a combining coefficient m of the receiving end:

the optimization target is as follows:

obtaining the optimal intermediate variable by derivation

And is provided with

Wherein,

Moore-Penrose inverse representing the computational matrix;

(4) repeating the steps (2) - (3) until MSE iteration of the receiving end converges;

(5) reflection phase vector v to IRS_kAnd carrying out constant modulus constraint.

2. The method of claim 1, wherein the method comprises: in the step (5), the reflection phase vector v of IRS_kIs subjected to constant modulus confinement of

Wherein |₂Representing the Frobenius norm of the vector.

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