Distributed Generative Adversarial Networks for mmWaveChannel Modeling in Wireless UAV Networks
DDistributed Generative Adversarial Networks for mmWaveChannel Modeling in Wireless UAV Networks
Qianqian Zhang, Aidin Ferdowsi, and Walid Saad
Bradley Department of Electrical and Computer Engineering, Virginia Tech, VA, USA, Emails: { qqz93,aidin,walids } @vt.edu. Abstract —In this paper, a novel framework is proposed toenable air-to-ground channel modeling over millimeter wave(mmWave) frequencies in an unmanned aerial vehicle (UAV)wireless network. First, an effective channel estimation approachis developed to collect mmWave channel information allowing eachUAV to train a local channel model via a generative adversarialnetwork (GAN). Next, in order to share the channel informationbetween UAVs in a privacy-preserving manner, a cooperativeframework, based on a distributed GAN architecture, is developedto enable each UAV to learn the mmWave channel distributionfrom the entire dataset in a fully distributed approach. The nec-essary and sufficient conditions for the optimal network structurethat maximizes the learning rate for information sharing in thedistributed network are derived. Simulation results show that thelearning rate of the proposed GAN approach will increase bysharing more generated channel samples at each learning iteration,but decrease given more UAVs in the network. The results alsoshow that the proposed GAN method yields a higher learningaccuracy, compared with a standalone GAN, and improves theaverage rate for UAV downlink communications by over ,compared with a baseline real-time channel estimation scheme.
I. I
NTRODUCTION
Millimeter wave (mmWave) frequencies are a pillar of next-generation communication systems, and they will support avariety of new applications, such as ultra-high-speed low-latency communications and airborne wireless networks. Inorder to overcome the fast attenuation of mmWave signals,multiple-input multiple-output (MIMO) technologies are oftenused so as to increase the cell throughput and reduce the multi-user interference. Compared with the sub-6 GHz spectrum, thehigher frequency of mmWave yields a shorter coherence timefor the wireless channels. Therefore, mmWave communicationlinks are more time-sensitive and require frequent channel mea-surements. However, real-time estimation of mmWave MIMOchannels can cause a heavy communication overhead [1]. Inorder to improve the transmission efficiency, it is essential tocharacterize the mmWave wireless link and accurately model itsunderlying MIMO channels, thus enabling realistic assessmentand effective deployment of mmWave wireless networks.Compared with a terrestrial communication network,mmWave channel modeling for an airborne, drone-based wire-less cellular system is more challenging, due to the mobilelocation of an aerial base station, and the limited studies onthe air-to-ground (A2G) channel characteristics [2]. Traditionalchannel modeling methods, such as ray-tracing, becomes verydifficult and time-consuming to measure A2G channels, andthe generated model cannot be flexibly generalized into othercommunication environments [3]. In order to address thischallenge, an unmanned aerial vehicle (UAV) base station can collect the A2G channel information during its wireless service,and, then, build a stochastic model to estimate the long-termchannel parameters. The A2G channel model enables the UAVbase station to estimate the mmWave link state, thus saving pilottraining time and transmit power for efficient communications.Therefore, wireless channel modeling is essential to support ascalable deployment of mmWave UAV wireless networks.In order to capture the stochastic characteristics of mmWavechannels from measurement results, a number of data-drivenmodeling approaches were developed in [1] and [4]–[8]. Tra-ditional methods, such as spatial-temporal correlation [4] andcompressed sensing [5], were investigated for characterizingmmWave MIMO transmissions. Recent works in [1], [7] and[6] developed machine learning methods to extract the prop-agation feature of mmWave-based communication links. Theauthors in [1] applied deep learning tools for estimating MIMOchannels over mmWave frequencies. A deep learning dataset isintroduced in [6] for the performance evaluation of mmWaveMIMO transmissions. However, all of the proposed modelingframeworks in [1] and [4]–[6] depend on the local dataset of asingle channel learner, and, thus, the generated channel model isconstrained by a limited amount of channel samples and a fewdedicated measurement environments. In order to extend thechannel model to large-scale application scenarios, a coopera-tive approach with distributed channel datasets was proposedin [7] to train the channel model using a federated learning(FL) framework. However, the centralized network structure ofthe FL framework requires a global controller for informationaggregation, and, thus, it cannot operate in a fully distributednetwork. Meanwhile, the FL-based discriminative approach tochannel modeling in [7] requires pilot signals to model the accu-rate channel state information (CSI). Furthermore, the work in[8] characterizes time-varying channel models, by continuouslyexchanging data in a distributed wireless system. However,sharing the raw cellular data in a real-time manner yields aheavy communication overhead, and violates the privacy ofmobile users by revealing their location-time information toother unauthorized entities.The main contribution of this paper is a novel framework thatcan perform data collection and channel modeling for mmWavecommunications in a distributed UAV network. First, an effec-tive channel measurement approach is developed to collect thereal-time channel information allowing each UAV to train alocal model via a generative adversarial network (GAN). Next,to expand the application scenarios of the trained mmWavechannel model into a broader spatial-temporal domain, a coop-erative learning framework, based on the distributed framework a r X i v : . [ c s . I T ] F e b f brainstorming GANs [9], is developed for each UAV to learnthe channel distribution from other agents in a fully distributedmanner. This generative approach allows us to characterize anunderlying distribution of the mmWave channels based on theentire spatial-temporal domain of measured channel dataset.Meanwhile, to avoid revealing the real measured data or thetrained channel model to other agents, each UAV shares syn-thetic channel samples that are generated from its local channelmodel in each iteration. We derive the necessary and sufficientconditions for the optimal network structure that maximizes thelearning rate for information sharing in the distributed network.Simulation results show that the learning rate of the proposedGAN approach will increase by sharing more generated channelsamples in each iteration, but it decreases for larger networks.The results also show that the proposed GAN approach yieldsa higher learning accuracy, compared with a standalone GANwithout information sharing, and improves the average data rateof UAV downlink communications by over , compared witha baseline real-time channel estimation scheme.The rest of this paper is organized as follows. Section IIpresents the communication model and data collection. TheUAV network, learning framework, and problem formulationare presented in Section III. The optimal network structure andlearning solutions are derived in Section IV. Simulation resultsare shown in Section V. Conclusions are drawn in Section VI.II. C OMMUNICATION M ODEL AND D ATA C OLLECTION
A. Millimeter Wave Channel Model
Consider an aerial cellular network, in which a set of UAVsprovide mmWave downlink communications to ground userequipment (UE). Each UAV and each UE will be equipped with M and N antennas, respectively. The MIMO channel matrix H ∈ C N × M can be given by H = (cid:80) Kk =1 α k a r ( φ rk ) a Ht ( φ tk ) ,where ( · ) H is conjugate transpose, K is the number of paths, α k is the complex gain of path k , and a t ( φ tk ) ∈ C M × and a r ( φ rk ) ∈ C N × are the transmit steering vector of angle of departure φ tk and receive vector of angle of arrival φ rk , respectively. Weassume uniform linear antenna arrays [5], with the steering vec-tors given by a t ( φ t ) = [1 , e j πλ sin( φ t ) , · · · , e j ( M − πλ sin( φ t ) ] T and a r ( φ r ) = [1 , e j πλ sin( φ r ) , · · · , e j ( N − πλ sin( φ r ) ] T , where λ is the carrier wavelength.Given the fact that the A2G channel via mmWave frequencieshas very few scattering paths, we assume that K = 1 fora line-of-sight (LOS) scenario, i.e. each LOS A2G channelconsists of a single path that directly connects the UAV andthe UE, while in an non-line-of-sight (NLOS) state scenario,the number of paths is zero. Then, for a UAV located atcoordinates x and a UE located at coordinates y , the A2Gchannel model at the service time t can be rewritten as H ( x , y , t ) = α ( x , y , t ) a r ( x , y ) a Ht ( x , y ) , where | α NLOS | = 0 and | α LOS | > . Here, the values of a r ( x , y ) and a t ( x , y ) willbe uniquely determined by the locations of the UAV-UE pair.Then, the estimation of the channel matrix can be obtained bydetermining the parameter α , in terms of the transmitter’s andreceiver’s locations, as well as the service time. B. Channel Estimation and Data Collection
In order to estimate the A2G mmWave channel, each UAVtransmits a pilot symbol with signal power P . Let w and q bethe beamforming and combining vectors for channel estimation,respectively. Then, the received pilot signal at the UE is r = √ P q H Hw + q H n , (1)where n ∼ CN ( , σ UE I N ) is the noise vector. Let ⊗ be theKronecker product, and vec ( · ) be the vectorization of a matrix.Then, the received pilot signal in (1) can be rewritten as r = √ P ( w T ⊗ q H ) vec ( H ) + q H n , = √ P ( w T ⊗ q H )( a ∗ t ⊗ a r ) α ( x , y , t ) + q H n , = βα ( x , y , t ) + q H n , (2)where ( · ) T is transpose, ( · ) ∗ is complex conjugate, and β = √ P ( w T ⊗ q H )( a ∗ t ⊗ a r ) ∈ C . After receiving r , each UE willsend the pilot training information to the UAV via a sub- GHzuplink [10]. Note that, the beamforming and combining vectorsare known by the BS for training purpose. Therefore, based onthe pilot signal and location information, the UAV located at x can estimate the downlink channel gain towards a UE located y at time t via ˜ α ( x , y , t ) = rβ − = α ( x , y , t ) + ˜ n, (3)where ˜ n = q H n β − is the uncorrelated estimation error.During the aerial cellular service, the channel gain ˜ α can bemeasured and collected by each UAV over a spatial-temporaldomain. We denote the channel dataset of a given UAV i as S i = { s n } n =1 , ··· ,S i = { x n , y n , t n , ˜ α n } n =1 , ··· ,S i , where S i = |S i | is the number of data samples. Based on S i ,each UAV i can build its own model for estimating A2GmmWave channels in its dedicated measurement area. However,over a large spatial-temporal domain, it is very challenging todevelop a stochastic model that properly captures the amplitudeand phase coefficients of the MIMO channel response, dueto distinct communication environments and a large span ofchannel parameter values. To address this challenge, we willintroduce a deep learning approach to enable accurate A2Gchannel modeling over mmWave spectrum.III. C HANNEL M ODELING VIA D ISTRIBUTED
GAN S Given the channel dataset S i , each UAV i can train itsown channel model, based on a deep neural network (DNN),to characterize the underlying distribution ( x , y , t, α ) ∼ f i .This channel distribution enables each UAV to estimate itsmmWave link gain α , while identifying the spatial-temporalrange of ( x , y , t ) that defines the applicable domain of thetrained channel model. However, in practice, each UAV onlyhas a limited amount of channel data samples. Thus, a mmWavechannel model that is trained based on a local dataset, can bebiased and only feasible for a limited spatial-temporal domain.Once the UAV moves to an unvisited area, pilot measurementwill again be necessary in order to acquire the propagationfeature of the new environment and update the channel model.However, both data collection and model update processes areig. 1: An illustration of the distributed GAN framework with fourUAVs, where O = { } , O = { , } , O = { } , and O = { } . time-demanding and energy-consuming for a UAV platform.Therefore, to avoid repeated channel estimations within thesame area, a UAV can learn the channel information from otherUAVs that operated in this region. However, raw data exchangein a distributed manner can yield a heavy communicationoverhead, and it may raise privacy concerns by sharing thelocation-time information of mobile UEs to an unauthorizedUAV, especially when each UAV belongs to a different networkoperator. A. Distributed GAN Framework: Preliminaries
In order to share channel data in a communication-efficientand privacy-preserving approach within the UAV network, adistributed GAN framework is proposed to cooperatively modelthe A2G mmWave channel. The GAN framework trains a modelto generate channel samples from an underlying distributionof its dataset, without explicitly revealing the data distributionor showing real data samples. Given a set I of I UAVs, weconsider that the available data in S = S ∪ · · · ∪ S I follows adistribution f . The local dataset S i of each UAV i is collectedfrom different geographic areas or at different service times.Hence, each local dataset S i follows a distribution f i that doesnot span the entire space of the real channel distribution.In a GAN framework, each UAV i has a generator G i , a dis-criminator D i and a local dataset S i . The generator G i ( z , θ gi ) is a DNN with a parameter vector θ gi , which maps a randominput z to the channel sample space S , and the discriminator D i ( s , θ di ) is another DNN with a parameter vector θ di that takesa channel sample s as an input and outputs a value between and . If the output of D i is close to , then the input sample s is similar to the real data sample in S i ; otherwise, a zerooutput of D i means that the input data is fake. Therefore, thegenerator of each UAV i aims to generate channel samples closeto the real measurement data, while the discriminator tries todistinguish the fake samples from the real channel samples.The goal is to train the generator distribution f gi of eachUAV i to find the entire channel distribution f , under theconstraint that no UAV i sends its real dataset S i or its DNNparameters θ gi and θ di to other UAVs. Instead, as shown in[9], each UAV i only shares the generated samples from G i ineach training iteration. Fig. 1 illustrated the proposed distributedGAN framework, where the input of the discriminator for each UAV i consists of the real samples from the local dataset S i , and the generated samples from the local generator G i and the generators of its neighboring UAVs. In the distributedGAN framework [9], the generators collaboratively generatechannel samples to fool all of the discriminators while thediscriminators try to distinguish between the generated and realchannel samples. Let N i be the set of UAVs from whom UAV i receives generated samples, and let O i be the UAV set towhom UAV i sends its generated samples. Then, for each UAV i , the interaction between its generator and discriminator can bemodeled by a game-theoretic framework with a value function: V i ( D i , G i , { G j } j ∈N i ) = E s ∼ f bi [log D i ( s )]+ E z ∼ f zi [log(1 − D i ( G i ( z )))] , (4)where f bi is a mixture distribution of UAV i ’s local dataset S i and received data from all neighboring UAVs in N i , and f zi isthe sampling distribution of the random input z . Here, we define f bi = π i f i + (cid:80) j ∈N i π gij f gj , where π i = S i / ( S i + η (cid:80) j ∈N i S j ) , π gij = ηS j / ( S i + η (cid:80) j ∈N i S j ) , and ηS j is the number ofgenerated samples that UAV j transmits to UAV i in eachiteration, with η > . Thus, the first term in (4) forces thediscriminator to output one for local real data and channelsamples from other UAVs, and the second term penalizes gener-ated data samples created by the local generator. Therefore, thegenerator of each UAV aims to minimizing the value function,while the discriminator tries to maximize this value. Thus,the local training within each UAV between its generator anddiscriminator forms a zero-sum game, and the total utilityfunction of the distributed GAN network is [9] V ( { D i } Ii =1 , { G i } Ii =1 ) = I (cid:88) i =1 V i ( D i , G i , { G j } j ∈N i ) , (5)where all generators aim at minimizing the total utility functiondefined in (5), while all discriminators try to maximize thisvalue. Therefore, based on [9], the optimal discriminators andgenerators can be derived as a min-max problem as follows: { D ∗ i } Ii =1 , { G ∗ i } Ii =1 = arg min G , ··· ,G I arg max D , ··· ,D I V. (6)Note that, the optimal discriminators and generators in (6) forthe distributed GAN learning depend on the structure of theUAV communication system, which is defined as next. B. UAV Communication Network
The communication structure of the UAV network is denotedby a directed graph G = ( I , E ) , where I is the set of UAVs,and E is the set of edges. Each edge e ij is an ordered UAVpair that corresponds to an air-to-air (A2A) communication link.For example, for any i, j ∈ I , if e ij ∈ E , then in each learningiteration, UAV i will send its generated data to the discriminatorof UAV j . Meanwhile, for any u, v ∈ I , if we can start from u ,follow a set of connected non-repeated edges in E , and finallyreach v , then we say that a path E u,v exists from u to v , andthe length l u,v equals to the number of edges on E u,v .In order to efficiently share the generated channel samples,orthogonal frequency-division multiple access (OFDMA) tech-niques with I available resource blocks (RBs) are used toupport the A2A communication over sub 6-GHz frequencies[10]. In order to avoid communication interference, we requirethe number of communication links not to exceed the numberof RBs, i.e., |E| ≤ I , which is reasonable for UAV networks.Meanwhile, assuming a fixed hovering location for each UAVduring the learning stage, the A2A communication rate fromUAV i to j using RB b is given as R ij = w b log (1 + P ij h ij /σ ) , where w b is the A2A communication bandwidth, P ij and h ij are the transmit power and path loss from UAV i to j , and σ is the noise power. A signal-to-noise ratio (SNR)threshold τ is introduced, such that for any UAV pair ( i, j ) , ifthe received SNR at UAV j is lower than τ , i.e. P ij h ij /σ < τ ,then, no RB will be assigned to this A2A communication link,i.e., e ij / ∈ E . In each iteration, each UAV i sends ηS i generatedsamples to its neighbors O i , and the transmission time for thegenerated channel samples should not exceed t τ .Here, we define the convergence time C of the distributedGAN approach as the expected number of iterations that isrequired for the learning process to converge, multiplied by theduration of each learning iteration. To facilitate the analysis, weconsider a fixed size for each UAV’s dataset, i.e. S = · · · = S I , and a homogeneous UAV communication network, where N = · · · = N I = N . Then, the probability that the learningprocess converges after iteration T can be derived as follows. Theorem 1.
Given the UAV network structure G , the probability p G ( T ) that the generator distribution f gi of each UAV i coversthe entire data distribution f after the T -th iteration can begiven, based on the maximum shortest-path l max in G , as p G ( T ) = T ≥ l max η l max (1 + N η ) l max − + T >l max T (cid:88) i = l max +1 i − (cid:89) j = l max (cid:18) − η l max (1 + N η ) j − (cid:19) η l max (1 + N η ) i − . (7) Proof.
Proof is available in [11].Theorem 1 shows that the convergence iteration is greaterthan or equal to the maximum shortest-path length l max . Thisimplies that to optimize the convergence rate for data sharingand channel modeling in the UAV network, it is necessaryto minimize the maximum length of shortest paths among allUAVs. Then, based on Theorem 1, the convergence iteration T G ∈ N + with a confidence level p τ ∈ (0 , is given by p G ( T G − < p τ ≤ p G ( T G ) . (8)That is, after T G iterations, the generator distribution of eachUAV is guaranteed to cover the entire channel distribution witha probability p τ . Meanwhile, we assume the local adversarialtraining between the generator and discriminator within eachUAV to be perfect with a constant time cost t c . Then, giventhe network structure G , the overall convergence time of thedistributed GAN learning is C ( G ) = ( t τ + t c ) · T G .Consequently, in the distributed UAV network with limitedcommunication resources, the objective for the cooperative mmWave channel modeling is to form an optimal A2A com-munication network G , such that the expected convergence timeof the distributed GAN learning is minimized, i.e., min G C ( G ) (9a)s. t. (cid:88) e ij ∈E P ij ≤ P max , ∀ i ∈ I , (9b) P ij h ij /σ ≥ τ, ∀ e ij ∈ E , (9c) ηS i /R ij ≤ t τ , ∀ e ij ∈ E , (9d) ∃ E i,j ⊂ E , ∀ i, j ∈ I , (9e) |E| ≤ I. (9f)Here, (9b) limits the maximum transmit power P max for eachUAV, (9c) and (9d) set thresholds for the received SNR andthe transmission time of each A2A communication link, (9e)requires a strongly connected network in G such that each localchannel dataset can be learned by all the other UAVs via thedistributed GAN framework, and (9f) avoids the interferenceover A2A communication links. Note that, in order to solve (9),a central controller is required to optimize the communicationstructure based on the path loss information between eachUAV pair. However, in the distributed UAV network, such acentralized entity is often not available, which makes (9) verychallenging to solve.IV. O PTIMAL LEARNING FOR DISTRIBUTED
GAN S A. Optimal network structure for A2A UAV communications
In order to optimally solve (9) in a distributed mannerwithout a central controller, we derive the graphic property forthe UAV network structure, based on (9e) and (9f), as follows.
Theorem 2.
Under the constraint that the number of commu-nication edges is smaller than or equal to the number of UAVs,the strongly connected network must have a ring structure, i.e., N i = O i = 1 , N i ∩ N j = ∅ , and O i ∩ O j = ∅ , ∀ i, j ∈ I .Proof. Proof is available in [11].Theorem 2 shows that, given constraints (9e) and (9f), thenetwork structure of the UAV communication system must bea ring, where each UAV receives the channel sample from oneUAV, and sends its generated data to another UAV.Based on Theorems 1 and 2, we can equivalently reformulate(9) into a set of distributed optimization problems, such that theobjective of each UAV i is to choose the optimal single UAV O i = { o i } to whom UAV i sends its generated channel samples,so as to minimize the convergence time over its maximumshortest-path while satisfying constraints (9b)-(9d), i.e., min o i ∈I − i l max i ( G + e i,o i ) (10a)s. t. P i,o i ≤ P max , (10b) P i,o i h i,o i /σ ≥ τ, (10c) ηS i /R i,o i ≤ t τ , (10d)where I − i is the set of UAVs except for i , G + e i,o i is the graphstructure generated by adding an edge e i,o i to G , and l max i ishe maximum shortest-path from UAV i to any other UAVs. Wedefine the set of feasible UAVs to whom UAV i can send its gen-erated channel samples while satisfying constraints (10b)-(10d)as J i = { j ∈ I − i | P ij ≤ P max , P ij h ij /σ ≥ τ, ηS i /R ij ≤ t τ } .Then, the necessary condition for a feasible solution to (10) isprovided next. Proposition 1 (Necessary condition) . A feasible topology so-lution to (10) exists, only if (cid:83) Ii =1 J i = I and ∀ i, J i (cid:54) = ∅ hold.Proof. Proof is available in [11].Proposition 1 shows that if the union of feasible sets does notcover all UAVs, then the UAV network cannot form a stronglyconnected graph and a feasible solution to (10) does not exist.Based on Proposition 1 and Theorem 2, we derive the sufficientcondition for the optimal network structure that maximizes theconvergence rate for the distributed GAN learning as follows.
Proposition 2 (Sufficient condition) . Given that (cid:83) Ii =1 J i = I and J i (cid:54) = ∅ hold for all i ∈ I , the optimal UAV networkstructure is G ∗ = ( I , E ) , where E ⊆ { e ij | i ∈ I , j ∈ J i } and l max i ( G ∗ ) = I − , ∀ i ∈ I .Proof. Proof is available in [11].Consequently, the optimal UAV network G ∗ that minimizesthe convergence time C ( G ∗ ) has a ring structure with a com-munication link set E ⊆ { e ij | i ∈ I , j ∈ J i } . B. Optimal learning for distributed GANs
Based on the optimal network structure G ∗ , according to [9,Proposition 1 and Theorem 1], the optimal generator of eachUAV i for the distributed GAN learning is G ∗ i ∼ f gi ∗ = f bi =( f i + f gj ) / for e ji ∈ E , and the optimal discriminator is D ∗ i = f bi / ( f bi + f gi ∗ ) = 0 . . That is, for each UAV i , its generator’sdistribution f gi ∗ equals to the mixture of the channel distribution f i from its local dataset S i and the generator’s distribution f gj from neighboring UAV j , and, thus, the discriminator cannotdistinguish the generated channel samples from the real data. Inthis case, the learning process in the UAV network convergesto a Nash equilibrium (NE), and the generator of each UAV i learns the entire distribution of mmWave channels, i.e., G ∗ i ∼ f .The formation approach of the optimal UAV network, as well asthe distributed GAN learning algorithm for mmWave channelmodeling, is summarized in Algorithm 1.V. S IMULATION R ESULTS AND A NALYSIS
For our simulations, we consider an airborne network withfour UAVs that provide wireless service within a geographicarea of × m . Each UAV has a mmWave channel dataset[6] that covers one of four regions in the area without overlap,i.e., business blocks, residential areas, rural region and a citypark. For simulation parameters, we set M = 256 , N = 64 , f = 30 GHz, w b = 2 MHz, P max = 40 dBm, σ = − dBm/Hz, τ = 10 dB, t τ = 0 . second, η = 1 . , and S i = 1000 for each UAV i . We implement a neural network (NN) with twoconvolutional layers for the GAN discriminator, and anotherNN with two transposed convolutional layers for the generator. Algorithm 1
UAV network formation with distributed GANlearning for mmWave channel modeling
UAV Network Formation:
1. Each UAV i uses its own RB to measure channel h ij for j ∈ I − i ,and broadcasts the feasible UAV set J i ;2. If (cid:83) Ii =1 J i = I and (cid:84) Ii =1 J i (cid:54) = ∅ , go to step 3; otherwise, theUAVs need to adjust their locations, and then, go back to step 1;3. Start with the network graph where E = { e ij | i ∈ I , j ∈ J i } ;4. For each UAV i with |J i | > ,Remove one edge e ij from E where j ∈ J i , while guaranteeing ( (cid:83) k ∈I − i J k ) ∪ ( J i − j ) = I ; Until |J i | = 1 for all i ∈ I . Distributed GAN learning:
A. Initialize G i and D i for each UAV i ∈ I ;B. Repeat:
Parallel for all i ∈ I :a. Sample π i u real channel samples: s (1) i , · · · , s ( π i u ) i ∼ S i ;b. Generate u channel samples G i ( z (1) ) , · · · , G i ( z ( u ) ) from f Gi and f zi ;c. Send π oi u generated data to each UAV o ∈ O i , and receive π ij u data samples s (1) j , · · · , s ( π ij u ) j from each UAV j ∈ N i ;d. Update θ di via gradient ascent: ∇ θ di V ( D i ( θ di )) = u ∇ θ di [ (cid:80) π i uk =1 log( D i ( s ( k ) i ))+ (cid:80) j ∈N i (cid:80) π ij uk =1 log( D i ( s ( k ) j ))+ (cid:80) uk =1 log(1 − D i ( G i ( z ( k ) )))] ;e. Update θ gi via gradient descent: ∇ θ gi V ( G i ( θ gi )) = u ∇ θ gi (cid:80) uk =1 log(1 − D i ( G i ( z ( k ) ))) ; Until convergence to the NE.
Fig. 2 shows the convergence rates of the distributed GANlearning, for different sizes of shared data samples η and fordifferent numbers of UAVs, respectively. Note that, in eachiteration, each UAV i sends ηS i generated channel samplesto its neighboring UAVs in O i . As shown in the upper plotof Fig. 2, when η becomes larger, the convergence rate ofour distributed GAN approach becomes faster. Given that themaximum path length l max for a four-UAVs network equals tothree, the convergence probabilities remains to be zero, until T is equal to or greater than three. Meanwhile, our proposedalgorithm shows a rapid convergence property when η = 1 . ,and the distributed GAN learning converges with a probabilityof over after six iterations. Next, we show the relationshipbetween the convergence rate and the number of UAVs inthe lower plot of Fig. 2, for a fixed generated sample size η = 1 . . For larger network sizes, the convergence rate of thechannel modeling process decreases, due to a longer path lengthin the distributed learning system. Therefore, in a large UAVnetwork, the size η of the generated channel samples needs tobe adaptively adjusted to guarantee an efficient learning.In Fig. 3, we evaluate the channel modeling accuracy andthe communication performance of the proposed distributedGAN algorithm. First, in the upper plot of Fig. 3, we intro-duce a baseline scheme that performs local channel modelingwithout information sharing and a distributed learning schemethat shares raw channel data between each UAV, and Jensen-Shannon divergence (JSD) is used as the performance metric,where a lower value of JSD indicates a higher learning accuracy.Fig. 3 first shows that the modeling accuracy of the pro-ig. 2: Convergence rate for different sizes of shared samples η anddifferent network sizes. Fig. 3:
Average JSD( f , f gi ) of channel modeling and average data rateof UAV downlink communications for different network sizes. posed distributed GAN approach outperforms the local learningscheme. Given more UAVs in the system, each UAV coversa smaller service area, and the local generator distributiononly applies to a limited spatial domain. Thus, the modelingaccuracy of the local learning scheme decreases for a largernetwork size. However, using the distributed learning approach,each UAV can learn the A2G channel property over a largerlocation domain from the generated samples of other UAVs.Thus, the modeling accuracy for the proposed approach staysthe same for different network sizes. Moreover, due to a limitedtraining time and the inevitable training error at each UAV,the overall distributed GAN training of the UAV network mayconverge to a local optimum. This explains the performancegap between the proposed learning scheme and the raw datasharing scheme. In the lower plot of Fig. 3, we evaluate thetime-average data rate of the UAV A2G communications witha MHz bandwidth, using the proposed channel modelingapproach and two other schemes: A baseline scheme thatrequires a constant pilot training on mmWave channels, andan upper-bound scheme that assumes a known CSI. Fig. 3shows that, given more UAVs in the network, the averagedata rates of all three schemes increase, due to an averagelysmaller service area for each UAV. Our proposed method applies the trained channel model for downlink transmissions,thus avoiding constant channel estimation. Therefore, comparedwith the real-time measurement scheme, the proposed methodimproves the time-average data rate by over . However, dueto the inevitable training error in the proposed channel model,our distributed GAN method yields a lower data rate, comparedwith a perfect CSI scheme.VI. C
ONCLUSION
In this paper, we have proposed a novel framework formmWave channel modeling in a UAV cellular network. Basedon the distributed GANs, a cooperative learning framework hasbeen developed for each UAV to learn the mmWave channeldistribution from other agents in the privacy-preserving and dis-tributed manner. We have derived the necessary and sufficientconditions for the optimal network structure of informationsharing that maximizes the learning rate. Simulation resultshave shown that the learning rate will increase by sharing moregenerated samples, but decrease given a larger UAV networksize. The results also show that the proposed distributed GANapproach yields a higher learning accuracy, compared with astandalone GAN, and it improves the average data rate of UAVdownlink communications by over , compared with thebaseline real-time channel estimation scheme.R
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