Method for optimizing air computing system under constraint of non-ideal channel and total power

Abstract

The invention relates to the technical field of air computing, in particular to an air computing system optimization method under a non-ideal channel and total power constraint, which comprises the following steps: considering the optimal design problem of an air computing system under the total power constraint and imperfect statistical CSI (Channel State Information), and designing a scaling factor of an air computing receiving and transmitting end through joint optimization; and the MSE of system calculation distortion is minimized. Specifically, under the constraint of total power, a mean square error estimated by minimizing the sum of preprocessed signals of K sensors is taken as a target, a design problem of an optimal transmitter-receiver scaling factor is established, an original non-convex problem is converted into a convex problem by using an alternative optimization idea, and a closed-form solution of the optimization problem is obtained through a KKT condition. According to the method, the solving difficulty of the algorithm is reduced, and the calculation distortion is smaller in the full-power transmission range.

Description

Technical Field

[0001] The present invention relates to the technical field of air computing, and specifically relates to an optimization method for an air computing system under non-ideal channels and total power constraints. Background Art

[0002] Air computing has recently been used in many different disciplinary fields, such as machine learning over wireless networks, distributed sensing, distributed consensus, and distributed edge learning. These exciting applications have triggered research on air computing from different perspectives and have produced a variety of computing strategies. Currently, most research is mainly carried out based on perfect channel state information (CSI) of air computing.

[0003] In practical applications, due to quantization and feedback errors, calibration mismatches, and delay errors and noise caused by various factors, the central side can often only obtain imperfect CSI information. In addition to the channel conditions, the total power constraint of the system is also an issue worthy of consideration in the AirComp system. This is due to the presence of multiple sensors, and the wireless sensors of the AirComp system are usually powered by beacons with limited power, and the system must meet a certain total power constraint. However, the current methods mainly design the air computing system under the constraint of the peak power of a single sensor. Summary of the Invention

[0004] In order to solve the above technical problems, the purpose of the present invention is to provide an optimization method for an air computing system under non-ideal channels and total power constraints, and the specific technical solutions adopted are as follows:

[0005] Construct an air computing system model, including multiple sensors and a fusion center; at the sensor, after preprocessing the original signal, the signal is sent to the fusion center, and the fusion center aggregates the received signal according to the channel information, noise information between the sensor and the receiving end, and the scaling factor of the sensor to obtain an aggregated signal;

[0006] According to the error situation between the aggregated signal at the fusion center and the estimated signal after signal post-processing at the fusion center, combined with the characteristics of the normalized variance of the sensor preprocessed signal, obtain the mean square error of the air computing system model;

[0007] According to the total power of the air computing system model, the scaling factor of the sensor, and the minimization of the mean square error, construct an optimization problem, and use the Lagrange method and KKT conditions to solve the optimization problem to determine the optimal transceiver parameters.

[0008] Preferably, the aggregating the received signal according to the channel information, noise information between the sensor and the receiving end, and the scaling factor of the sensor to obtain an aggregated signal specifically includes:

[0009]

[0010] Among them, y represents the aggregated signal at the receiving end, h k represents the channel coefficient between the k-th sensor and the receiving end, K represents the total number of sensors, x k represents the signal after preprocessing of the k-th sensor, z represents the additive white Gaussian noise at the receiving end, and follows distribution, σ z 2 represents the variance of the additive white Gaussian noise at the receiving end, b k represents the scaling factor of the k-th sensor.

[0011] Preferably, the estimation function of the receiving end of the fusion center is specifically:

[0012]

[0013] Among them, represents the estimation function, m represents the scaling factor of the receiving end of the fusion center.

[0014] Preferably, according to the error situation between the aggregated signal at the fusion center and the estimated signal after signal post-processing at the fusion center, combined with the characteristics of the normalized variance of the sensor preprocessed signal, the mean square error of the air computing system model is obtained, specifically including:

[0015]

[0016] Among them, MES represents the mean square error of the air computing system model, u represents the objective function of the sensor, e k represents the CSI estimation error of the k-th sensor, represents the channel estimation vector of the k-th sensor, E[] represents the mathematical expectation.

[0017] Preferably, the optimization problem is specifically:

[0018]

[0019] Among them, PW represents the total power of the air computing system model, P represents the given maximum power.

[0020] Preferably, the Lagrangian expression is specifically:

[0021]

[0022] Among them, represents the Lagrangian function, σ e represents the standard deviation of the channel estimation noise, σ zdenotes the standard deviation of the additive white Gaussian noise at the receiver, and λ denotes the KKT multiplier.

[0023] Preferably, the KKT conditions are specifically as follows:

[0024]

[0025] where b k denotes the scaling factor of the k-th sensor, and {b k} denotes the set of scaling factors of all sensors.

[0026] The embodiments of the present invention have at least the following beneficial effects:

[0027] The present invention comprehensively considers the constraint conditions closer to the actual situation, gives the optimal system transceiver strategy under the conditions of statistical channel state information error and total power constraint, and minimizes the computational distortion of the system. By using the idea of alternating optimization and KKT conditions to transform and solve non-convex problems, the solution difficulty of the algorithm is reduced, and a closed-form solution to the optimization problem is given. Compared with the traditional optimization design scheme, it has smaller computational distortion within the full-power transmission range. Description of the Drawings

[0028] In order to more clearly illustrate the technical solutions and advantages in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0029] Figure 1 is a flowchart of an optimization method for an air computing system under non-ideal channels and total power constraints provided by the present invention;

[0030] Figure 2 is a schematic diagram of an air computing system model provided by the present invention;

[0031] Figure 3 is a schematic diagram of the relationship between MSE and total power constraint provided by the present invention;

[0032] Figure 4 is a schematic diagram of the relationship between the average computational MSE and the total number of sensors provided by the present invention;

[0033] Figure 5 is a schematic diagram of the relationship between MSE and signal-to-noise ratio provided by the present invention. Detailed Embodiments

[0034] To further elaborate on the technical means and effects adopted by the present invention to achieve the intended invention purpose, the following combines the accompanying drawings and preferred embodiments to detail the specific implementation manner, structure, features, and effects of an optimization method for an air computing system under non-ideal channels and total power constraints proposed according to the present invention. In the following description, different "one embodiment" or "another embodiment" do not necessarily refer to the same embodiment. In addition, the specific features, structures, or characteristics in one or more embodiments can be combined in any suitable form.

[0035] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs.

[0036] The following specifically describes the specific solution of an optimization method for an air computing system under non-ideal channels and total power constraints provided by the present invention with reference to the accompanying drawings.

[0037] The main objective of the present invention is to consider the optimal design problem of an air computing system under total power constraints and imperfect statistical CSI. By jointly optimizing the scaling factors of the air computing transceiver, the system computing distortion MSE is minimized. Specifically, under the total power constraint, aiming at minimizing the mean square error estimated by the sum of the preprocessed signals of K sensors, the design problem of the optimal transmitter-receiver scaling factors is established, and using the idea of alternating optimization, the original non-convex problem is transformed into a convex problem, and the closed-form solution of the optimization problem is obtained through the KKT conditions. In addition, the impact of channel estimation error on system performance is also studied.

[0038] Please refer to Figure 1 , which shows the flowchart of an optimization method for an air computing system under non-ideal channels and total power constraints provided by an embodiment of the present invention. The method includes the following steps:

[0039] Step 1, construct an air computing system model, including multiple sensors and a fusion center; at the sensor, after preprocessing the original signal, the signal is sent to the fusion center, and the fusion center aggregates the received signal according to the channel information, noise information, and the scaling factor of the sensor to obtain an aggregated signal.

[0040] In this embodiment, as Figure 2 shown, a single-antenna air computing system model is composed of K sensors and a fusion center (FC). Taking any one sensor as an example, the original signal of the k-th sensor is s k , the k-th

[0041] sensor's preprocessed signal is where xk denotes the signal of the k-th sensor after preprocessing, denotes the preprocessing function of the k-th sensor. The objective function received by the fusion center in the air computing system model is where ψ represents the postprocessing function of the receiving end of the fusion center, and K represents the total number of sensors.

[0042] It should be noted that the preprocessing function of the sensor is the same as the postprocessing function of the receiving end. The implementer can specify the preprocessing function of a specific sensor and the postprocessing function of the receiving end. For example, it can be a function for calculating the arithmetic mean, weighted sum, maximum value, and minimum value, etc. No more introduction will be given here.

[0043] In this embodiment, it is assumed that the signal x after preprocessing by the sensor k has a normalized variance, and the transmission synchronization of the sensors is good. By parallelizing the preprocessing operations of each sensor and the postprocessing operations of the receiving end, the fusion center at the receiving end can directly obtain all the required signals, without the need for one-to-one transmission and calculation as in the traditional scheme. The sensor measurement scenario with normalized variance in the air computing framework of this embodiment can be applied to the general scenario with non-zero mean and non-normalized variance.

[0044] More specifically, assume that the sensor sends the signal x k with a mean of μ and a variance of σ 2 , The normalized transmitted signal is Sent from the sensor to the fusion center through the air computing system model, the fusion center can obtain Since both and are constants, The optimal estimate of can be obtained through the optimal estimate of.

[0045] Furthermore, the sensor linearly scales the signal by a transmission factor and then simultaneously sends the scaled signal through the multiple access channel. The aggregated signal received by the receiving end is:

[0046]

[0047] where y represents the aggregated signal at the receiving end, h k represents the channel coefficient between the k-th sensor and the receiving end, K represents the total number of sensors, x k represents the signal of the k-th sensor after preprocessing, z represents the additive Gaussian white noise at the receiving end, and follows distribution, σ z 2 represents the variance of the additive Gaussian white noise at the receiving end, b kDenotes the scaling factor of the k-th sensor.

[0048] The estimation function at the receiving end of the fusion center is specifically:

[0049]

[0050] Where, Denotes the estimation function, m denotes the scaling factor at the receiving end of the fusion center. The Rx scaling factor applied to the signal and noise is for estimating the calculation output. It can be seen that there is distortion in the estimation function compared with the objective function of the fusion center.

[0051] Step 2: According to the error situation between the aggregated signal at the fusion center and the estimated signal after signal post-processing at the fusion center, combined with the characteristics of the normalized variance of the preprocessed sensor signal, obtain the mean square error of the air computing system model.

[0052] In Step 1, the calculations of the transceiver are all based on the fusion center's acquisition of perfect channel state information (CSI). This embodiment also considers the actual channel estimation error. Generally, the receiver can perform channel estimation through pilots or training sequences during channel estimation to obtain CSI. However, in most practical applications, due to quantization and feedback errors, calibration mismatches, and delay errors and noise caused by various factors, the central end can often only obtain imperfect CSI information. In this embodiment, an additive error is used to describe the imperfect CSI of the estimation.

[0053] It should be noted that in the problem of imperfect CSI, the error can be modeled as a statistical model or a deterministic model. Among them, the statistical model requires estimating the statistics of the error, such as the mean and variance, etc. The deterministic model requires obtaining the range of values of the error. The set of all values within this range is also called the uncertainty set, and the elements in this set are the possible values of the imperfect CSI. In this embodiment, the statistical model is used to model the imperfect CSI.

[0054] In this embodiment, the imperfect CSI based on the statistical model is defined as e k Denotes the CSI estimation error of the k-th sensor, which is a CSCG vector with a mean of 0 and a variance of σ e 2 ; h k Denotes the channel coefficient between the k-th sensor and the receiving end, Denotes the channel estimation vector of the k-th sensor.

[0055] Then, combined with the CSI error of the air computing system model, the CSI error can be substituted into the description formula of the received signal, and then the estimation function considering the CSI error can be obtained, specifically:

[0056]

[0057] Among them, represents the estimation function considering the CSI error at the receiving end of the fusion center.

[0058] Furthermore, the mean square error is used to quantify the computational distortion existing between the estimated signal and the true signal at the receiving end. The computational distortion can be expressed as, that is, the mean square error of the air computing system model can be expressed as:

[0059]

[0060] Among them, MES represents the mean square error of the air computing system model, u represents the objective function of the sensor, and e k represents the CSI estimation error of the k-th sensor, represents the channel estimation vector of the k-th sensor, and E[] represents taking the mathematical expectation.

[0061] Since the expectation is expanded according to the distribution of the transmitted signal x k and the sensed sensor data x k is statistically independent on different devices then the mean square error is simplified to:

[0062]

[0063] Among them, σ e represents the standard deviation of the channel estimation noise, σ z represents the standard deviation of the additive Gaussian white noise at the receiving end, σ e 2 represents the variance of the channel estimation noise, and σ z 2 represents the variance of the additive Gaussian white noise at the receiving end.

[0064] Step 3: According to the total power of the air computing system model, the scaling factor of the sensor, and the minimization of the mean square error, construct an optimization problem, and use the Lagrangian method and KKT conditions to solve the optimization problem to determine the optimal transceiver parameters.

[0065] It should be noted that since the preprocessed signal of the sensor considered in this embodiment has a normalization method, taking any one sensor as an example, the average transmission power of the k-th sensor can be expressed as E[|b k x k | 2 = |b k | 2 , and then the total power consumption of the air computing system model can be expressed as

[0066] In this embodiment, the optimization problem is to minimize the mean square error of the air computing system model through the optimal transceiver scaling factor design problem under the constraint of the total power of the air computing system model. The optimization problem can be expressed by the formula as follows:

[0067]

[0068] where PW represents the total power of the air computing system model, P represents the given maximum power, {b k} represents the set of scaling factors of all sensors, and subject to means "under the condition of...", represents finding the minimum value of the mean square error of the air computing system model. It can be understood that the optimization problem in this embodiment includes signal misalignment error and CSI-related error.

[0069] Thus, due to the coupling between {b k} and m in the constraint conditions, this problem is non-convex and thus difficult to optimize and solve. In this embodiment, considering the optimal transceiver factor design, the total power is minimized and optimized under error constraints and imperfect CSI. The idea of alternating optimization is adopted, that is, while optimizing other variables, some variables are alternately fixed.

[0070] Specifically, by fixing m and {b k}, the problem can be conveniently solved and finally a closed-form solution of the optimal strategy can be obtained.

[0071] According to 's function definition, given the complex scaling factor m and channel coefficient h k at the receiver, the k-th sensor can always adjust the phase of its transmitter's scaling factor b k for phase compensation without changing its amplitude, so that the result of the mh k b k term is real and non-negative, thereby minimizing |mh in k b k - 1|. It can be seen that only the magnitudes of m, {h k}, and {b k} affect the minimization of the mean square error in the optimization problem. Therefore, without loss of generality, it is set that where represents the set of real numbers,

[0072] Based on this, the Lagrangian expression of the optimization problem is specifically constructed as follows:

[0073] ​

[0074] Among them, represents the Lagrangian function, and σ e 2 represents the variance of the channel estimation noise, and σ z 2 represents the variance of the additive white Gaussian noise at the receiver, and λ represents the KKT multiplier.

[0075] Furthermore, the necessary conditions for the optimal solution, that is, the KKT conditions, can be specifically expressed as:

[0076]

[0077] Among them, b k represents the scaling factor of the k-th sensor, and {b k} represents the set of scaling factors of all sensors.

[0078] By analysis, it can be seen that in the above conditions is obviously established. According to the condition it can be further solved to obtain Furthermore, in order to find the solution that satisfies the KKT conditions, two cases of the Lagrange multiplier are considered, namely λ = 0 and λ > 0. The following explains the two cases separately.

[0079] On the one hand, when λ = 0, at this time always holds, and the sensor can process and transmit signals without using up the system power budget. The solution of b k can be obtained as However, at this time, the constraint condition is an ineffective constraint. Furthermore, minimizing the mean square error can be updated as And the optimal solution m = 0 obtained at this time does not meet the conditions, so λ = 0 does not hold.

[0080] On the other hand, when λ > 0, at this time the error constraint will tighten in the optimal case. The constraint conditions in the optimization problem can be rewritten as Furthermore, the optimization problem can be updated as:

[0081]

[0082] Among them, represents the scaling factor of the k-th sensor. Substituting the constraint condition into minimizing the mean square error, it can be expressed as: Reducing the number of optimization variables, the optimal solution for minimizing the mean square error can be obtained as:

[0083]

[0084] Furthermore, the optimal solution of the scaling factor finally obtained by optimizing the problem can be expressed as:

[0085]

[0086] In summary, the present invention mainly relates to a system model based on air computing, aiming to minimize the estimation error at the receiver by optimally designing the sensor scaling factor under the total power constraint. By introducing an imperfect CSI model, mean square error quantization, and the Lagrangian optimization method, an optimization problem is finally constructed, and the solution idea and optimal strategy are obtained through alternating optimization and KKT conditions.

[0087] It should be noted that when the noise power approaches 0, that is, σ z 2 →0, at this time, the air computing system is in a high signal-to-noise ratio state, and the mean square error MSE of the air computing system model under imperfect CSI is dominated by the signal misalignment error and the CSI error. In this case, instead of approaching 0. Due to the existence of channel estimation error σ e 2 , even if the total transmit power of the system becomes infinite, it is inevitable that the calculated MSE is non-zero. And in practice, high transmit power will also amplify the channel estimation error, which is different from the case of perfect CSI (i.e., σ e 2 = 0). In the case of perfect CSI, at high signal-to-noise ratio, the mean square error approaches 0.

[0088] When the noise power approaches infinity, that is, σ z 2 →∞, at this time, the air computing system is in a low signal-to-noise ratio state, and the mean square error MSE of the air computing system model is mainly affected by noise. The error caused by noise can be compressed by reducing the scaling factor at the receiver. At this time, the mean square error value of imperfect CSI is close to the mean square error value of perfect CSI, that is, the influence of the channel estimation error at this time can be ignored.

[0089] Finally, this embodiment provides the optimal MSE strategy performance of the AirComp system under imperfect CSI, considering the actual situation of the CSI error model and the total power constraint. The wireless channel from each transmitter to the fusion center is modeled as an independent and identically distributed Rayleigh fading model, then h k can be modeled as an independent and identically distributed circularly symmetric complex Gaussian (CSCG) random variable with zero mean and unit variance. The receiver noise σ z 2= 1. Meanwhile, in this embodiment, the Monte Carlo method is adopted, and 10 4 random realizations are performed for each experiment, and then the experimental results are averaged to obtain the final simulation results.

[0090] In Figure 3 , the AirComp performance achieved by the proposed algorithm and the benchmark algorithm is compared under different total power constraints P and the number of sensors K. Among them, σ e 2 = 0.3. It can be seen that within the entire transmit power range, the proposed optimization strategy is superior to the benchmark scheme. Under low power conditions (such as P ≤ 5W), the performance of the traditional scheme is close to that of the strategy proposed in this embodiment. Because in this case, the error caused by noise dominates in the MSE, and the CSI error has little impact on the system performance. Under high power conditions (such as P ≥ 30W), the MSE of the system is dominated by the signal misalignment error and the CSI error, and the performance gap between the traditional scheme and this embodiment gradually becomes larger. This is because the CSI error is amplified by the high transmit power, thus reducing the MSE performance.

[0091] As Figure 4 shows, it reflects the relationship between the mean square error MSE and the number of sensors K, and studies the performance of the optimal calculation MSE strategy of the AirComp system with different numbers of sensors K. Among them, the result of the normalized average MSE is represented by E[MSE] / K. The upper limit of the total power is P = 10K, σ e 2 = 0.3. Obviously, as the total number of sensors K and the signal-to-noise ratio increase, the average calculation MSE decreases.

[0092] As Figure 5 shows, it reflects the influence of imperfect channel estimation on the received mean square error MSE. Generally speaking, the channel estimation noise variance σ e 2 depends on the channel SNR and the estimation method. In the simulation, four groups of different noise power ratios σ e 2 / σ z 2 are set to evaluate the influence of channel estimation error on the achieved MSE. In the low signal-to-noise ratio region, due to the small power, the change of MSE is not significant. However, in the high signal-to-noise ratio region, the power is large, and a small change in σ e 2 / σ z 2 (when the signal-to-noise ratio = 20dB) will also have a greater impact on the system performance. Therefore, in the high signal-to-noise ratio region, the accuracy of channel estimation should be improved as much as possible to improve the system performance.

[0093] The embodiments described above are only used to illustrate the technical solutions of the present application, rather than to limit it; although the present application has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements on some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application, and should all be included within the protection scope of the present application.

Claims

1. A method for optimizing an air computing system under non-ideal channels and total power constraints, characterized in that: The method comprises the following steps: Constructing an air computing system model, including multiple sensors and a fusion center; the sensor includes pre-processing the original signal and sending the signal to the fusion center, and the fusion center aggregates the received signal to obtain an aggregated signal according to the channel information between the sensor and the receiving end, the noise information and the scaling factor of the sensor; According to the error between the aggregated signal at the fusion center and the estimated signal after signal post-processing at the fusion center, combined with the characteristics of the normalized variance of the sensor pre-processing signal, the mean square error of the air computing system model is obtained; According to the total power of the air computing system model, the scaling factor of the sensor and the minimization of the mean square error, an optimization problem is constructed, and the Lagrangian method and KKT conditions are used to solve the optimization problem and determine the optimal transceiver parameters.

2. The method for optimizing an air computing system under non-ideal channel and total power constraints according to claim 1, characterized in that: The step of aggregating the received signal to obtain an aggregate signal according to the channel information between the sensor and the receiving end, the noise information and the scaling factor of the sensor specifically includes: Where y represents the aggregate signal at the receiving end, h k represents the channel coefficient between the kth sensor and the receiver, K represents the total number of sensors, and x k represents the signal after preprocessing of the kth sensor, z represents the additive Gaussian white noise at the receiving end, and obeys Distribution, σ z 2 represents the variance of the additive white Gaussian noise at the receiving end, b k represents the scaling factor of the kth sensor.

3. The method for optimizing an air computing system under non-ideal channel and total power constraints according to claim 2, characterized in that: The estimation function of the receiving end of the fusion center is specifically: in, represents the estimation function, and m represents the scaling factor of the receiver at the fusion center.

4. The method for optimizing an air computing system under non-ideal channel and total power constraints according to claim 3, characterized in that: The method of obtaining the mean square error of the airborne computing system model based on the error between the aggregated signal at the fusion center and the estimated signal after signal post-processing at the fusion center and the characteristics of the normalized variance of the sensor pre-processed signal specifically includes: Where MES represents the mean square error of the air computing system model, u represents the objective function of the sensor, and e k represents the CSI estimation error of the kth sensor, represents the channel estimation vector of the kth sensor, and E[] represents the mathematical expectation.

5. The method for optimizing an air computing system under non-ideal channel and total power constraints according to claim 4, characterized in that: The optimization problem is specifically: subject to PW≤P Where PW represents the total power of the air computing system model, and P represents the given maximum power.

6. The method for optimizing an air computing system under non-ideal channels and total power constraints according to claim 5, characterized in that: The Lagrangian expression is as follows: in, represents the Lagrangian function, σ e represents the standard deviation of the channel estimation noise, σ z represents the standard deviation of the additive white Gaussian noise at the receiving end, and λ represents the KKT multiplier.

7. The method for optimizing an air computing system under non-ideal channels and total power constraints according to claim 6, characterized in that: The KKT conditions are as follows: Among them, b k represents the scaling factor of the kth sensor, {b k } represents the set of scaling factors for all sensors.

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