Decision Fusion in Centralized and Distributed Multiuser Millimeter-Wave Massive MIMO-OFDM Sensor Networks

Low-complexity fusion rules relying on hybrid combining are proposed for decision fusion in frequency selective millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) sensor networks (SNs). Both centralized (C-MIMO) and distributed (D-MIMO) antenna architectures are considered, where the error-prone local sensor decisions are transmitted over orthogonal subcarriers to a fusion center (FC) employing a large antenna array. Fusion rules are designed for the FC, followed by closed-form expressions of the false alarm and detection probabilities to comprehensively characterize the performance of distributed detection. Furthermore, efficient transmit signaling vectors are designed for optimizing the detection performance. Both the asymptotic performance analysis and the pertinent power reduction laws are presented for the large antenna regime considering both the C-MIMO and D-MIMO topologies, which potentially lead to a significant transmit power reduction. Low-complexity fusion rules and their analyses are also given for the realistic scenario of incorporating channel state information (CSI) uncertainty, where the sparse Bayesian learning (SBL) framework is utilized for the estimation of the sparse frequency selective mmWave massive MIMO channel. Finally, the performance of the proposed low-complexity detectors is characterized through extensive simulation results for different scenarios.


I. INTRODUCTION
N EXT-GENERATION (NG) sensor networks (SNs) are envisaged to support ultra-dense sensor connectivity to enable real-time monitoring of a large area for missioncritical tasks in applications such as surveillance, disaster management, and agriculture, among others [1], [2], [3].In these networks, battery-operated sensors are geographically distributed to monitor a specific event of interest.
The sensor measurements are generally compressed into single-bit decisions, due to power and bandwidth constraints, and are subsequently transmitted to the fusion center (FC), where the final decision related to the presence/absence of a signal of interest is arrived at using a suitable fusion rule.This paradigm is referred to as distributed detection [4].The rising interest in these applications both in current and NG networks, along with the drastic surge in the number of miniature sensors has led to a spectrum-crunch in the sub-6 GHz bands.This motivates one to look for alternative frequency bands that can provide an abundance for spectrum necessary for NG SNs.
In this respect, millimeter wave (mmWave) communication serves as a potential technology for supporting high data rates and bandwidth-intensive applications by leveraging the underutilized band ranging from 30 to 300 GHz [5].However, mmWave signals suffer from severe absorption and propagation losses, which renders the implementation of mmWave communication challenging [6].To overcome this challenge, one can employ massive multiple-input multipleoutput (MIMO) technology, wherein massive antenna arrays (AAs) are deployed at the base stations (BSs) within limited physical dimensions, thanks to the shorter wavelength of mmWave signals.This can play a pivotal role in compensating for the huge path losses at such higher frequencies, thus paving the way for the practical implementation of mmWave MIMO communication [7].Furthermore, orthogonal frequency division multiplexing (OFDM), which provides robustness to inter-symbol interference (ISI) and multipath distortion, is well suited for mmWave massive MIMO SNs.Thus, mmWave massive MIMO-OFDM SNs constitute an effective solution for the reliable detection of events of critical importance in high-speed networks.
The conventional fully digital precoding/combining architectures employed in such systems, where the signal processing (SP) is exclusively executed in the baseband, necessitates a substantially larger number of radio frequency (RF) chains since an independent RF chain is required for each antenna.This fact, coupled with the high sampling rate of the ADCs due to the larger bandwidth of the signals, makes the system power-hungry and costly, precluding the deployment of fully digital signal processing (DSP) schemes in NG SNs.This issue can be circumvented by implementing the popular hybrid SP architecture [8], wherein SP tasks like analog/digital transmit precoding (TPC) and receiver combining are executed in the RF and baseband domains, respectively.Such architectures require only a modest number of RF chains in contrast to conventional fully DSP.
In a conventional mmWave massive MIMO SN, also known as centralized (C-MIMO) architecture, hundreds of antennas are co-located at the FC situated at the cell center.Despite having low deployment costs, this architecture experiences high spatial channel correlation and eventually leads to loss of orthogonality for the mmWave channels.To overcome this concern, distributed (D-MIMO) antenna configuration, wherein a very large AA is distributed over multiple geographically dispersed FCs that are linked together by a high-speed optical fiber backhaul, has gained considerable research interest [9].Furthermore, the D-MIMO architecture potentially reduces the radio access distance between the sensors and the FCs, which facilitates significant power savings and improves the system performance over that of the C-MIMO architecture.The dense sensor deployment in NG networks naturally generates a tremendous amount of data that has to be processed to obtain reliable decisions regarding events of critical importance.Therefore, there is a crucial need to design efficient rules for decision fusion, which are investigated in this paper.A literature review is provided next.

A. LITERATURE REVIEW
Distributed detection for wireless sensor networks (WSNs) was first investigated by Li and Dai in their pioneering work in [10], wherein correlated observations were transmitted over a multiple access channel (MAC).The authors of [11] proposed modified amplify-and-forward (AF) and decodeand-forward (DF) based detection schemes for a MAC.A distributed detection algorithm was proposed and analyzed by Tepedelenlioglu and Dasarathan in [12] for a Gaussian MAC, wherein the sensor transmissions were constant modulus signals.Berger et al. [13] investigated the decentralized detection performance of MAC and parallel access channels (PAC) using noncoherent modulation.The optimality of the energy detector in the decision fusion was studied in [14] for a non-coherent Rayleigh fading MAC.In [15], Banavar et al. presented a novel AF-based distributed detection scheme for Ricean fading channels considering multiple antennas at the FC and characterized the performance of the system using an error exponent.This work was further extended in [16] to evaluate the optimal sensor transmit gains for different levels of channel state information (CSI) at the sensors.In [17], Ciuonzo et al. designed sub-optimal detection rules based on both the decode-and-fuse and decode-then-fuse principles for distributed MIMO SNs considering local decision transmission.The authors in [18] developed algorithms for distributed detection in WSNs considering both perfect and imperfect CSI both between the source and the sensors as well as between the sensors and the FC, followed by analyzing their performance in terms of the associated false alarm and detection probabilities.Novel DF relaying-based optimal and sub-optimal fusion rules were derived in [19] for cooperative WSNs.The authors of [20] successfully developed tests for multi-user (MU) cooperative MIMO spectrum sensing, incorporating realistic CSI uncertainty in the secondary user channels.The detection and estimation of a Gaussian signal in an AF WSN are investigated in [21] using both the energy and Neyman-Pearson (NP) detectors as well as the linear minimum mean-squared error estimator.In [22], Astaneh and Gazor studied the distributed detection performance of both the NP detector and of the generalized likelihood ratio test in an OFDM system.State-of-the-art detection rules were designed using the NP criterion in [23] for a massive MIMO-aided WSN, which also considers realistic imperfect CSI at the FC.In [24], a state of detection rules were developed for massive MIMO WSNs while considering both imperfect and perfect CSI availability, for both known/unknown parameter scenarios.A family of detection schemes were presented in [25] for mmWave massive C-MIMO WSNs for decision fusion, which were subsequently extended to the distributed antenna topology in [26].The sparse Bayesian learning (SBL) paradigm was proposed for sparse channel estimation in mmWave massive MIMO WSNs in [27], followed by the design of distributed detection schemes incorporating the associated CSI uncertainty in C-and D-MIMO architectures.Furthermore, the decision fusion in a mmWave massive MIMO WSN is investigated in [28], harnessing the SBL framework for realistic imperfect channel estimation.However, the analysis is restricted only to the centralized antenna architecture, where the sensors transmit their decisions to the FC over a flat fading channel.In [29], Gimenez et al. characterized the performance of a distributed hybrid precoding algorithm designed for a mmWave D-MIMO system in an indoor setting.Hybrid transmit/receive beamforming schemes were presented in [30] for heterogeneous systems to mitigate the inter-and intra-tier interferences in mmWave massive MIMO scenarios.In [9], the authors investigated the asymptotic spectral efficiencies of the MU mmWave massive MIMO downlink (DL) employing a hybrid precoding algorithm for C-and D-MIMO topologies.The authors of [31] proposed a cooperative MIMO-OFDM algorithm for enhancing energy efficiency and for interference mitigation in multi-hop 3D wireless camera SNs.Al-Jarrah et al. [32] studied parallel distributed detection in a WSN conceived for the local sensor decisions' transmission to the FC using a cooperative transmission strategy, where the FC receives each local decision as an OFDM block.The authors of [33] designed fusion rules for distributed detection by exploiting the probability density function (PDF) of the received signal, for an OFDM system.However, to the best of our knowledge, none of the existing treatises have proposed hybrid combining-based fusion rules for a WSN, where the sensors transmit their local decisions over frequency selective channels while leveraging the benefits of mmWave massive MIMO technologies.Hence, this paper explores the problem of distributed detection in an OFDM-based mmWave massive MIMO WSN, wherein the sensors transmit their decisions over orthogonal subcarriers.Table 1 contrasts the key features of the paper vis-à-vis those of the closely related works reviewed above.The detailed contributions of this work are further detailed below.

B. KEY CONTRIBUTIONS
• This study investigates an OFDM system wherein the sensors transmit their local decisions over single/multiple orthogonal subcarriers to the FC over a frequency selective mmWave massive MIMO channel.This scenario is in contrast to [15], [16], [24], [27] and [34], where analog observations are transmitted.Fusion rules are conceived for processing the wideband signal received at the FC, to determine a global decision.• Low-complexity detection rules based on hybrid receiver combining (RC) are designed for C-and D-MIMO antenna topologies for antipodal signaling and low signal-to-noise ratios (SNR).Closed-form expressions of the false alarm (P FA ) and detection (P D ) probabilities are for characterizing the system performance.
• Furthermore, a beneficial signaling vector design is conceived for the D-and C-MIMO topologies in order to improve the detection performance.• The asymptotic performance analysis and the associated power reduction laws are determined for a large number of antennas.The analysis clearly demonstrates the substantial sensor transmit power reduction attained without affecting the detection performance, thereby resulting in improved battery life.• The extension of the above framework to a realistic scenario incorporating CSI uncertainty is also investigated.The SBL approach [35], [36] is utilized for exploiting the sparsity of the mmWave channel to obtain improved estimates of the frequency selective mmWave massive MIMO-OFDM CSI.It is a promising sparse signal recovery method, which has also shown superior performance compared to various state-of-theart schemes in our recent work related to delay-Doppler domain sparse CSI estimation [37].Furthermore, suitable hybrid combining-based fusion rules are designed for both antenna topologies under realistic imperfect CSI.• Simulation results are provided for validating the theoretical findings for characterizing the performance of the proposed novel detectors.

C. ORGANIZATION
The organization of the paper is as follows.Section II discusses the system model of mmWave C-and D-massive MIMO-OFDM SNs.The fusion rules are determined in Section III and our signaling vector design procedure is described in Section IV.The asymptotic performance and the associated power reduction laws are analyzed in Section V. Furthermore, our novel SBL-based channel estimation scheme and the corresponding fusion rules are given in Section VI.Finally, our simulation results and conclusions are provided in Sections VII and VIII, respectively.Furthermore, we have provided a diagram, depicted in Fig. 1, to concisely outline the contributions in each section.

D. NOTATIONS
Boldface uppercase (resp.lowercase) letters denote matrices and vectors, where [X] i,j is the (i, j)th entry of X and x i is the ith entry of x, 0 K and I K are the K × 1 zero vector and K × K identity matrix, respectively, (.

II. SYSTEM MODEL
Consider K single-antenna sensors which are randomly distributed in a mmWave massive MIMO SN to sense the presence (H 1 ) (resp.absence (H 0 )) of a signal of interest.
The sensors subsequently send their local decisions to either a single FC or multiple FCs, depending on the antenna configuration, over a frequency selective fading channel.Let N s successive subcarriers be allocated to each sensor and the same signal be transmitted over all the N s subcarriers, so that beneficial frequency diversity is attained.Hence, the signal corresponding to the kth sensor, 0 ≤ k ≤ K − 1, becomes: where N = N s K is the total number of subcarriers.For antipodal signaling, the local decision x k corresponding to the kth sensor belongs to the set x k ∈ {u k , −u k }, so that its local false alarm and detection probabilities can be expressed as . Subsequently, the signals of the K sensors are processed at the FC to obtain a final decision.The frequency selective mmWave MIMO channel models for different antenna topologies are discussed next.

A. FREQUENCY SELECTIVE MMWAVE MIMO CHANNEL MODEL
In the C-MIMO configuration, the sensors transmit their local decisions to a single FC, located at the center of the cell.The FC is equipped with a massive antenna array (AA) containing M antennnas.The centralized antenna topology is depicted in Fig. 2a.Considering the D-delay channel model of [39] for the C-MIMO topology, the d-th delay tap g k,d ∈ C M×1 associated with the kth sensor and the FC is where β k characterizes the pathloss and lognormal shadowing [40], while h k,d is the small-scale fading vector [41] modeled as where τ k,i and αi k are the delay and complex channel gains of the kth sensor and the ith spatial multipath component, respectively.The parameters T s , L k , and p k (τ ) denote the sampling period, the number of multipath components, and the band-limited pulse shaping filter response at delay τ , respectively.The receive array response vector a r (θ , where θ i k is the angle of arrival (AoA) of the ith multipath and the kth sensor, while λ, and d denote the wavelength and receive antenna spacing, respectively.In the frequency domain, the lth subcarrier channel vector g l ∈ C M×1 is derived as where the variable k satisfies k = l N s , with a total of N s consecutive subcarriers allocated to the kth sensor and the channel assumed to be constant across the N s subcarriers.Using (3), the channel vector g l can be remodeled as denote the equivalent complex channel gain vector associated with the kth sensor and the lth subcarrier and the receive array response matrix of the kth sensor, respectively.
In the D-MIMO architecture, J FCs are uniformly distributed on a circle of radius ρ with R being the cell radius.All the J FCs are interconnected via the high-capacity and low-latency backhaul infrastructure constituted by optical fiber cables.Each FC is equipped with an independent N delement AA, hence the total number of antennas N tot across all FCs is N tot = N d J. Additionally, each FC utilizes a single RF chain connected to its N d -element AA.The RF combiner outputs from all the FCs are then directed to the baseband processing unit (BPU) for further signal processing, as illustrated in Fig. 2b.The circular layout is preferred because of its compatibility with the existing infrastructure and low optical backhaul installation requirements [9].To ensure a fair comparison with C-MIMO, it is assumed that N tot = M and J = K, similar to the model in [9].For D-MIMO, the channel vector related to the l k th subcarrier and the jth FC is formulated as where and L k,j represent the receive array response matrix, large-scale fading coefficient and the number of multipath components, respectively, for the kth sensor and the jth FC.Furthermore, is the equivalent complex channel gain vector of the jth FC, l k th subcarrier and the kth sensor.The decision rules for the C-and D-MIMO topologies, considering perfect CSI, are described in the subsequent section.

III. FUSION RULES FOR DIFFERENT ANTENNA ARCHITECTURES
In this section, fusion rules are developed for both antenna configurations, considering hybrid combining and perfect CSI at the FC.Next, the decision rule for the centralized architecture is discussed.

A. DECISION RULE FOR A C-MIMO BASED WSN
After the N-point fast Fourier transform (FFT) of OFDM, the signal y(l) ∈ C M×1 received at the FC corresponding to the lth subcarrier becomes where the additive white Gaussian noise (AWGN) w(l) ∈ C M×1 is distributed as w(l) ∼ CN (0, σ 2 w I M ).After stacking N s outputs of the kth sensor, one obtains the equivalent received signal vector where is the equivalent channel vector and the equivalent noise vector , where C w = σ 2 w I MN s .Furthermore, the above signal y k is distributed as y k ∼ CN (μ y k , C w ), where the mean μ y k is defined as μ y k = √ p u g k x k .Upon concatenating the output vectors of all K sensors, one obtains where Y = [y 0 , . . ., denote the equivalent received signal, channel, transmitted signal and noise matrices, respectively.Upon using the NP criterion, the log-likelihood ratio (LLR) test [42] for the scenario considering the C-MIMO configuration and perfect CSI can be formulated as Note that the expression in ( 8) is obtained by exploiting the independence of y k across different sensors.For antipodal signaling, the LLR test reduces to Notice that the simplification of the above expression requires the addition of a large number of exponential terms.Hence, the above test statistic is computationally complex and unsuitable for light-weight practical implementation.To mitigate this concern, a hybrid combining-based two-step procedure is utilized for approximating the test in ( 8) [25], as illustrated in Fig. 3.In the first step, the received signal is processed by an analog RC f RF,l ∈ C M×1 , followed by the digital RC f BB,k ∈ C N s ×1 in the next step.After that, the hybrid RC outputs are processed to form a global decision.
The RF combiner for the kth sensor is chosen as f RF,l = a r (θ i k k ), which is the receive array response vector of the path having the maximum gain and i k is the corresponding index.After RF combining, the processed signal ỹ(l) is given by ỹ(l) = √ p u f H RF,l g l x k + w(l), where the equivalent noise is w(l) = f H RF,l w(l) ∼ CN (0, σ 2 w ).Upon exploiting the asymptotic orthogonality of the mmWave massive MIMO channels [43] formulated as: where are the equivalent channel gain and noise vectors, respectively.Furthermore, the baseband RC is selected as f BB,k = Mβ k L k α k , which aims for maximizing the signal's SNR.Hence, the baseband RC output z k is obtained as where For the centralized mmWave massive MIMO-OFDM WSN, the LLR test based on the hybrid combiner output z can be simplified to Under the low SNR assumption, the test in ( 13) reduces to where a k P D,k − P F,k .The above expression follows from the approximations e −s ≈ 1 − s and ln(1 + s) ≈ s.Observe that the detector in ( 14) has low complexity in contrast to the fusion rule in (9), which requires the addition of a large number of exponential terms.Moreover, the proposed detector is suitable for practical scenarios, since WSNs are typically resource-constrained. Furthermore, T CP (z) in ( 14) is a complex Gaussian random variable since it is a weighted sum of K complex Gaussian random variables.For a scenario wherein the sensors have the same local false alarm and detection probabilities, i.e., P F,k = P f and The performance of the test in ( 14) is presented next.Theorem 1: Considering the C-MIMO architecture of our mmWave massive MIMO-OFDM WSN, the detection (P D ) and false alarm (P FA ) probabilities of the test T CP (z) in ( 14) are where the means μ T CP |H 0 , μ T CP |H 1 and the variances σ 2 T CP |H 1 under the H 0 and H 1 hypotheses, respectively, are where The fusion rule for the distributed antenna architecture is discussed next.

B. DECISION RULE FOR A D-MIMO BASED WSN
In the D-MIMO architecture, the task of assigning sensors to the operating FCs is crucial, hence diverse FC selection algorithms have been developed in [9].The distance (D)based algorithm, where an sensor is allocated to its nearest FC according to the minimum distance criterion, is widely used in the literature [9].Hence, we also employ the D-based algorithm to design detectors for the distributed antenna architecture.To implement the D-based algorithm, a distance matrix D ∈ R K×J is determined so that its (k, j)th entry d k,j = [D] k,j denotes the distance of the kth sensor from the jth FC.Let S ∈ R K×J be the binary correction matrix, where [S] k,j = 1, when the jth FC is assigned to the kth sensor and [S] k,j = 0, otherwise.Furthermore, S H S = I K since only a single sensor is allocated to each FC and J = K.
For the D-MIMO topology, the signal received at the jth FC y j (l j ) ∈ C N d ×1 corresponding to the l j th subcarrier is y j (l j ) = √ p u g l j ,j x k j + w j (l j ), where is distributed as w j (l j ) ∼ CN (0, σ 2 w I N d ) and the channel vector g l j ,j is defined as g l j ,j = N d β k j ,j L k j ,j A r,k j ,j α l j ,j , ∀l j ∈ A k j .Similarly, a two-step architecture is utilized for the D-MIMO based WSN to reduce the computational complexity of the LRT, as demonstrated in Fig. 4. For the jth FC and the k j th sensor, the RF combiner f RF,D,j ∈ C N d ×1 is chosen as f RF,D,j = a r (θ i k j k j ,j ), ∀l j , which represents the receive array response vector with maximum path gain.Hence, the output of the RF combiner can be expressed as where the noise wj (l j ) = f H RF,D,j w j (l j ) is distributed as wj (l j ) ∼ CN (0, σ 2 w ).Furthermore, observe that the quantity f H RF,D,j g l j ,j reduces to L k j ,j α i k j l j ,j .After stacking the RF combiner outputs corresponding to the k j th sensor, one obtains where ỹj ∈ C N s ×1 and α j ∈ C N s ×1 denote the equivalent RF combiner output vector and the channel gain vector associated with the jth FC, respectively.The equivalent noise vector wj ∈ C N s ×1 follows the Gaussian normal distribution, which is given as wj ∼ CN (0, σ 2 w I N s ).Furthermore, to maximize the SNR at the jth FC, the baseband combiner is chosen as f BB,D,j = N d β k j ,j L k j ,j α j .Therefore, the hybrid combiner output z k j at the jth FC is where the quantity d k j ,j is defined as d k j ,j = L k j ,j α j 2 and the equivalent noise w k j is distributed as The hybrid combiner outputs gleaned from K FCs are processed using the correction matrix S at the baseband processing unit (BPU) to derive the soft decisions of all sensors.The kth sensor soft decision is which is distributed as Considering the D-MIMO topology, the LLR test of mmWave massive MIMO-OFDM based WSNs can be expressed as Furthermore, for the low SNR regime, the above test reduces to where γ is the threshold.Since, the test in ( 24) is a weighted linear combination of Gaussian random variables, the test statistic T DP (z) is complex Gaussian distributed.The performance of the test in ( 24) is presented next.Theorem 2: For the test T DP (z) in ( 24), the detection (P D ) and false alarm (P FA ) probabilities used for decision fusion at the mmWave distributed massive MIMO-OFDM based WSNs, are where the means μ T DP |H 0 , μ T DP |H 1 and the variances under hypotheses H 0 and H 1 , respectively, are given as Proof: Similar to the proof of Theorem 1.
In the subsequent section, the signaling vector design for the perfect CSI scenario aimed at improving the detection performance is presented.

IV. SIGNALING VECTOR DESIGN
In this section, the signaling vector x = [x 0 , x 1 , . . ., x K−1 ] T is designed for maximizing the performance of the proposed C-and D-MIMO detectors.Next, the signaling vector design for the centralized architecture is presented.

A. SIGNALING VECTOR DESIGN FOR A C-MIMO BASED FC
For the signaling vector u = [u 0 , u 1 , . . ., u K−1 ] T , the performance can be enhanced by the deflection coefficient maximization procedure of [42], given by Substituting the expressions of σ 2 T CP |H 0 , μ T CP |H 1 and μ T CP |H 0 from Theorem 1, (29) can be simplified as Consider the diagonal matrices ∈ C K×K , ∈ C K×K and ∈ C K×K , where their diagonal entries are Upon utilizing the above matrices, the expression in (30) can be simplified as Observe that the above expression is non-convex, therefore, it is challenging to directly maximize (32).For finding a tractable solution, the expression in (32) can be rewritten as where = uu H H and = uu H H + .Based on the Rayleigh quotient standard form, the modified objective function is max.
The optimization problem in (34) can now be solved in an iterative fashion and the solution during the ith iteration is described next.
Theorem 3: To improve the detection performance of the proposed test in (14) for the centralized antenna topology, the signaling vector u (i) of the ith iteration can be defined as , where the ith iteration vector v (i) can be determined from max .
where (i) .During the (i−1)st iteration, (i−1) and (i−1) are derived by substituting u (i) in place of u in (33), where u is initialized to u (0) = 1 K during the 0th iteration.
Upon solving the optimization problem in (35), one obtains v (i) = κν (i−1) , where the power scaling factor is κ and the eigenvector ν (i−1) of (i−1) is associated with the maximum eigenvalue.Therefore, the signaling vector of the ith iteration is u 1) .The overall computational complexity of obtaining the signaling vector after convergence is O(nK 3 ), with K denoting the dimension of the matrix and n representing the total number of iterations.The signaling vector design for the distributed antenna topology is discussed next.

B. TRANSMIT SIGNALING VECTOR DESIGN FOR A D-MIMO BASED FC
Similarly, the ith iteration's signaling vector u (i) D of the D-MIMO topology is obtained as with ν (i−1) D as the eigenvector for the maximum eigenvalue of (i−1) D and α denoting the power scaling factor.The matrices D and D are obtained upon replacing d k and M by d k,j k and N d , respectively, in (30).The asymptotic performance analysis and power scaling laws for C-and D-MIMO architectures are presented in the subsequent section.

V. ASYMPTOTIC ANALYSIS AND POWER SCALING LAWS
This section demonstrates the advantage of using a massive MIMO in distributed and centralized antenna architectures.The asymptotic performance analysis provides insights regarding the rate at which the reduction in the sensor transmit power is achieved.The asymptotic performance analysis for the centralized topology is discussed next.

A. ASYMPTOTIC PERFORMANCE OF THE C-MIMO SYSTEM
Consider the power scaling p u = pu M , with the average sensor transmit power as pu .For this scenario, the asymptotic detection performance of the C-MIMO test is discussed below.
Theorem 4: The asymptotic false alarm (P a FA ) and detection (P a D ) probabilities of the test in ( 14) for a frequency selective mmWave C-MIMO based SN are where γ is the normalized detection threshold.The normalized asymptotic means for the alternative and null hypotheses, denoted by μ a T CP |H 1 and μ a T CP |H 0 , respectively, are expressed as The asymptotic performance analysis for the distributed architecture is presented in the subsequent subsection.

B. ASYMPTOTIC PERFORMANCE OF THE D-MIMO SYSTEM
Considering the D-MIMO topology, the asymptotic detection (P a D ) and false alarm (P a FA ) probabilities for the test in (24), when N d → ∞ and power scaling p u = pu N d , are expressed as where γ is the normalized detection threshold.The normalized asymptotic means μ a T DP |H 1 and μ a T DP |H 0 for the alternative and null hypotheses, respectively, are The above expressions can be derived using similar steps to those of Theorem 4. Through the analytical expressions derived above, it is evident that one can scale the sensor transmit power as 1 M and 1 N d , for C-and D-MIMO based architectures, respectively while considering a large AA at the FC.Therefore, one can improve the sensor battery life, which is crucial for the reliable operation of next-generation SNs.The Bayesian learning (BL) based channel estimation framework for the imperfect CSI scenario is discussed in the next section.

VI. FRAMEWORK FOR SBL-BASED CSI ESTIMATION
This section discusses the frequency selective mmWave MIMO channel estimation framework using the SBL-based approach [35], [36].Subsequently, the fusion rules are determined for both antenna topologies in the face of realistic CSI uncertainty.The sparse beamspace channel representation for the centralized architecture is discussed next.

A. SPARSE BEAMSPACE CHANNEL REPRESENTATION
To model the frequency selective mmWave MIMO channel for the C-MIMO based FC, a M size angular grid is chosen, where the AoAs are defined as R = {ψ n : ψ n ∈ [0, π], ∀1 ≤ n ≤ M}.Furthermore, the angles satisfy the following condition [44] The corresponding receive array response dictionary matrix A R,C ∈ C M×M can be expressed as Owing to the selection of AoAs, the above matrix follows the [44].Upon using the quantities defined above, the lth subcarrier's channel vector g l ∈ C M×1 can be characterized as where h b,l ∈ C M×1 denotes its equivalent sparse beamspace channel vector.Similarly, the channel vector g l j ,j ∈ C N d ×1 , ∀l j ∈ A k j , associated with the jth FC for D-MIMO is modeled as where the equivalent sparse beamspace channel associated with the jth FC is h b,l j ,j ∈ C N d ×1 , ∀l j ∈ A k j and the receive array response dictionary matrix A R,D ∈ C N d ×N d is determined by using the AoAs from the set R = {ψ v : The BL algorithm harnessed for estimating the sparse frequency selective mmWave massive MIMO channel is presented next.

B. SBL-BASED CSI ESTIMATION
This subsection discusses the channel estimation procedure of the C-MIMO configuration, given in (41).The channel vector of the D-MIMO topology defined in ( 42) can be estimated using a similar approach.Let x p ∈ C K×1 be the transmitted pilot vector obeying |x p,k | 2 = 1, ∀k, and W C ∈ C M×M be the training RF combining matrix, which is chosen as the normalized DFT matrix of size M.For a block of N f training frames, the channel is assumed to be constant.After combining, the output corresponding to the lth subcarrier y (n) (l) ∈ C M×1 at the nth frame, becomes [41] y (n) where the AWGN during the nth frame, denoted by n (n) (l) ∈ C M×1 , follows the distribution n (n) (l) ∼ CN (0, σ 2 n I M ).Upon using (41), the above expression can be simplified to where Stacking all RF combiner outputs associated with the lth subcarrier y (n) where y(l) = [(y (1) (l)) T , . . ., (y Furthermore, the columns of the sensing matrix are orthogonal, i.e., Q H Q = N f I M .Thus, using the received signal in (45) and the SBL framework, the lth subcarrier beamspace channel estimate ĥb,l can be obtained as follows.The proposed SBL-based framework utilizes a parameterized Gaussian prior formulated as p(h b,l ; ) = CN (0, ) [45], where = diag(γ 1 , γ 2 , . . ., γ M ) and the associated hyperparameter vector is defined as γ = [γ 1 , γ 2 , . . ., γ M ] T .Note that here the hyperparameter γ i ∈ R + signifies the prior variance of the ith element.These hyperparameters are iteratively estimated employing the well-known expectation-maximization (EM) algorithm.The main steps in the EM procedure are summarized below.
Let ˆ (q) (l) be the qth iteration estimate of the hyperparameter matrix (l).In the qth iteration, the expectation step (E-step) determines the average log-likelihood L( (l)| ˆ (q) (l)) of the data set {y(l), h b,l }, given as The first quantity in the above expression can be simplified as ).It can be observed that the simplified term is independent of the hyperparameter vector γ , thus, it is ignored in the maximization step (M-step).The second term is evaluated using the a posteriori probability density of h b,l [45], which is p(h b,l |y(l); ˆ (q) (l)) ∼ CN (μ (q) (l), (q) (l)) with a mean of μ (q) Consequently, the second term reduces to

Algorithm 1: SBL-Based Frequency Selective mmWave Massive MIMO Channel Estimation
Input : Pilot output y(l), stopping parameter , sensing matrix Q and pilot power p p . 1 Initialization: (0) (l) = I M 2 Set q = 0 and (−1) (l) = 0 M×M 3 while ˆ (q) (l) − ˆ (q−1) (l) F > do 4 E-step: Determine a posteriori covariance and mean 5 (q) M-step: Evaluate hyperparameter estimates end for In the M-step, the hyperparameter vector estimate γ (q+1) is evaluated by maximizing the above cost-function with respect to γ , as Observe from (46) that the estimation of the hyperparameter vector γ is decoupled with respect to each γ i .Hence, it can be solved to obtain the estimates γi (q+1) during the (q + 1)st iteration of the EM algorithm as where [ (q) (l)] i,i and [μ (q) (l)] i denote the (i, i)th and ith elements of the a posteriori covariance matrix (q) (l) and mean vector μ (q) (l), respectively.On convergence, the SBL-based estimate is determined by substituting the hyperparameter matrix estimate into the a posteriori mean.The sparse CSI estimation technique is conveniently summarized in Algorithm 1.The fusion rule and its detection performance for the centralized topology considering imperfect CSI scenario are discussed below.

C. DECISION RULES FOR C-MIMO BASED FC WITH CSI UNCERTAINTY
Employing the technique formulated in Algorithm 1, the beamspace channel estimate of the lth subcarrier, is obtained upon convergence, as ĥb,l = μ (q) (l), ( where μ (q) (l) represents the a posteriori mean.Observe that the a posteriori covariance matrix l = (q) (l) is diagonal, since the matrix (q) (l) is diagonal.The mmWave massive MIMO estimate of the lth subcarrier is expressed as and the beamspace estimation error is e b,l = ( ĥb,l − h b,l ), which is distributed as e b,l ∼ CN (0, l ).Hence, the corresponding estimation error e l can be expressed as e l = ĝl − g l = A R,C e b,l , which is distributed as . Therefore, using (5), the equivalent system model for the lth subcarrier, considering the imperfect CSI, can be remodeled as where the noise w(l) w(l) − √ p u e l x k is distributed as ) and the variables l and k are related as k = l N s .Now, a two-step procedure is conceived for reducing the computational complexity of the conventional detectors, wherein the received signal is processed using a hybrid combiner in the first step.A global decision is arrived at in the second step by fusing the hybrid combiner outputs.
The RF combiner is selected for the lth subcarrier as w RF,l = a R,C (ψ i l ), where i l denotes the maximum estimated path gain index.The RF combiner output obtained by exploiting the asymptotic orthogonality property of the mmWave MIMO channel in (10) is where ĥi l ,l = [ ĥb,l ] i l and w(l) = w H RF,l w(l) is distributed as w(l) ∼ CN (0, σ 2 l ) with σ 2 l = σ 2 w + p u w H RF,l C l w RF,l |u k | 2 .Using (52), the stacked RF combiner output ỹk of the kth sensor, similar to (11), is where wk ∼ CN (0, Ck ) and the ith diagonal entry of Ck is . The baseband combiner for the kth sensor is chosen as w BB,k = ĥk ∈ C N s ×1 for maximizing the output SNR of the signal in (53).Thus, the digital combiner output z k is formulated as: where the equivalent noise Using the received signal in (54), the LLR test can be expressed as where we have zk = √ p u ĥk 2 (z * k u k ).Upon employing the low SNR approximations of e −s ≈ 1 − s and ln(1 + s) ≈ s, the expression in (56) can be simplified to: where γ is the detection threshold.The above test T CIP (z) represents a weighted linear combination of Gaussian random variables (z * k u k ) and it is distributed as where we have H ∈ {H 0 , H 1 }.The expressions of the means and variances under both hypotheses can be formulated as Based on the above quantities, the detection (P D ) and false alarm (P FA ) probabilities are All the above expressions can be obtained along similar lines to those of Theorem 1.The fusion rule for the D-MIMO topology considering CSI uncertainty and its detection performance are considered next.

D. DECISION RULES FOR A D-MIMO BASED FC WITH CSI UNCERTAINTY
For the D-MIMO topology, the beamspace channel vector h b,l j ,j , ∀l j ∈ A k j , is estimated at each FC using the SBL-based approach.Thus, the estimated channel vector corresponding to the l j th subcarrier at the jth FC can be expressed as The estimated beamspace channel vector can be determined as ĥb,l j ,j = μ (q) j (l j ), where μ (q) j (l j ) denotes the a posteriori mean.For the l j th subcarrier and the jth FC, the estimation error vector can be expressed as e l j ,j = ĝl j ,j − g l j ,j = A R,D e b,l j ,j , which is distributed as e l j ,j ∼ CN (0, C l j ), where C l j = A R,D l j A H R,D and l j is the a posteriori covariance matrix.Using the above expressions, the received signal y j (l j ) ∈ C N d ×1 , ∀l j ∈ A k j , at the jth for the l j th subcarrier, can be remodeled as y j (l j ) = √ p u (ĝ l j ,j − e l j ,j )x k j + w j (l j ) = √ p u ĝl j ,j x k j + wj (l j ), (65) where the noise vector wj (l j ) = w j (l j ) − √ p u e l j ,j x k j is distributed as wj (l j ) ∼ CN (0, σ 2 w I N d + p u C l j |u k j | 2 ).Employing the two-step architecture, the RF combiner w RF,j is selected as w RF,j = a R,D (ψ i j ), ∀l j , which represents the receive array response vector corresponding to the maximum estimated path gain of ĥb,l j ,j .The RF combiner output of the l j th subcarrier and the jth FC obtained by leveraging the asymptotic orthogonality property of the mmWave MIMO channel, can be simplified as where the quantities wj (l j ) = w H RF,j wj (l j ) ∼ CN (0, σ 2 w + p u w H RF,j C l j w RF,j |u k j | 2 ) and ĥi j ,l j ,j = [ ĥb,l j ,j ] i j .After stacking all the RF combiner outputs corresponding to the k j th sensor, one obtains where wk j ∼ CN (0, Čk j ) and the covariance matrix Čk j is diagonal with nth diagonal entry as Selecting the baseband combiner vector as w BB,j = ĥj ∈ C N s ×1 , the hybrid combiner output of the k j th sensor can be determined as where the quantities w k j = ĥH j wk j ∼ CN (0, σ 2 k j ) and σ 2 k j = ĥH j Čk j ĥj .The hybrid combiner outputs are then transmitted to the BPU for final processing.At the BPU, the correction matrix S is used to combine the received signals to obtain ordered soft sensor decisions.Thus, the hybrid combined output signal of the kth sensor can be given as where the quantities w k ∼ CN (0, σ 2 k ) and 2 = ĥH j k Čk ĥj k .Further, z k in (69) is distributed as Using (69), the LLR test statistic for z, considering the D-MIMO topology, can be modeled as where γ is the detection threshold.Furthermore, the expressions of P D and P FA for T DIP (z) in (71) can be derived as where the means μ T DIP |H 0 , μ T DIP |H 1 and the variances σ 2 T DIP |H 0 , σ 2 T DIP |H 1 under null and alternative hypotheses, respectively, are determined as The proof of the above expressions follows along similar lines to Theorem 1.The simulation results to validate the proposed analyses are discussed in the subsequent section.Observe that the simulated plots coincide with the analytical findings derived, thus validating the accuracy of our analysis.Moreover, the performance improves upon increasing the values of N s .Thus, one can fine-tune the number of subcarriers according to the detection performance desired.Fig. 5(b) compares the detection performance obtained using the signaling vector designed in Section IV to that of a uniform transmit signaling vector design, wherein all the sensors transmit at equal power, i.e., u = [u 0 , u 1 , . . ., u K−1 ] T , where u k ∈ {−1, 1}, for p u = −15 dB.Observe from the figure that one can significantly enhance the detection performance by employing the transmit signaling vector, which demonstrates the value of our optimal signaling paradigm conceived for practical mmWave massive MIMO-OFDM systems.Fig. 5(c) verifies the power scaling laws determined in Section V for a centralized mmWave massive MIMO-OFDM system.As shown therein, the figure supports our conclusion that the sensor transmit power can be reduced according to 1/M, without affecting the detection performance.This key result pertaining to the asymptotic performance analysis helps us in improving the battery life of the sensors, which is one of the critical resources in a WSN.Moreover, the detection performance improves with an increase in the value of pu .

This
The ROC plot of the proposed D-MIMO test in (24) is depicted in Fig. 6(a) for N s ∈ {1, 2, 4}, considering N tot = 1024 and p u = −5 dB.The analytical results derived in Theorem 2 are also verified using the simulation results.The performance improves as N s increases and the capability of the scheme can be further enhanced by employing the deflection coefficient maximization-based signaling vector, as demonstrated in Fig. 6(b).When contrasting Fig. 5(a) and Fig. 6(a), the performance degradation in D-MIMO against C-MIMO is attributed to the sensor distribution range.The asymptotic performance of the test in (24) with power scaling is explored in Fig. 6(c).Observe that one can scale the sensor transmit power as 1/N d , for the D-MIMO topology.massive MIMO SNs, employing D-and C-MIMO antenna topologies, where the local sensor decisions are transmitted to the FC over orthogonal subcarriers.For this framework, the decision rules were designed for the perfect CSI scenario and their performance was characterized via explicit analytical expressions for the resultant false alarm and detection probabilities.Furthermore, signaling vectors were obtained based on the deflection coefficient maximization criterion for both antenna topologies to improve the detection performance.The asymptotic performance and the power reduction laws were also characterized for demonstrating the rate at which the sensors can reduce their transmit power when the number of antennas at the FC becomes very large.Additionally, an SBL-based sparse mmWave massive MIMO CSI estimation procedure was developed followed by the associated fusion rules conceived for distributed detection under CSI uncertainty, for both antenna configurations.A comprehensive set of simulation results was presented to validate the performance of the proposed algorithms.Future research will prioritize the exploration of distributed detection in scenarios that take into account the allocation of multiple sensors per FC within the distributed antenna topology.Additionally, there is potential for further investigations into multi-cell scenarios.

APPENDIX A PROOF OF THEOREM 1
Substituting the expression of z k from ( 12) in (14), one obtains The mean of T CP (z) for hypothesis H 0 is T CP |H 0 can be expressed as Similarly, the mean and variance can be obtained for T CP (z) in (72) under hypothesis H 1 .

APPENDIX B PROOF OF THEOREM 4
Consider the modified test statistic TCP (z) = T CP (z) denotes the normalized mean under the null hypothesis.For the power scaling p u = pu M , with the average sensor transmit power as pu , the normalized asymptotic mean for hypothesis H 0 can be determined as On substituting the expressions of μ T CP |H 0 and σ T CP |H 0 from ( 15) and ( 17) in (75), the above expression reduces to On similar lines, one can obtain the normalized asymptotic mean under hypothesis H 1 .

FIGURE 1 .
FIGURE 1. Flow of the paper.

FIGURE 3 .
FIGURE 3. Hybrid combining based system model for decision fusion in the C-MIMO configuration.

FIGURE 4 .
FIGURE 4. Hybrid combining based system model for decision fusion in the D-MIMO topology.
section demonstrates the simulation results to corroborate the analytical performance of the C-and D-MIMO detectors and to contrast their performance to those of other detectors.For the C-MIMO configuration, K = 16 sensors are uniformly distributed in the range spanning from r min to R, where r min = 1 m and R = 200 m.In the D-MIMO architecture, J = 16 FCs are located on a circle of radius ρ = 0.6R and K = 16 sensors are uniformly distributed in the region [0, ρ − r min ] ∪ [ρ + r min , R].The system under consideration is operating at f c = 28 GHz and the antenna spacing is chosen as d = λ/2.The local false alarm and detection probabilities are taken as P F,k ∼ U [0.01, 0.16] and P D,k ∼ U [0.2, 0.95], respectively.The frequency selective mmWave massive MIMO channel is considered to be spatially sparse with L k = L k,j = 4 paths and D = 4 delay taps.The classic raised cosine pulse shaping filter is utilized with a roll-off factor of 0.6 and the delays τ k,i are chosen uniformly in the range [0, D − 1].The large-scale fading coefficients of the C-and D-MIMO topologies are modeled similar to[40], asβ k = v k (ρ k /r min ) ν and β k,j = v k,j(d k,j /r min ) ν , respectively.The quantities v k and v k,j are generated as log-normal random variables with a mean of μ v = 4 dB and standard deviation of σ v = 2 dB.Furthermore, ρ k and d k,j denote the distance from the kth sensor to the centralized FC and the jth FC for the C-and D-MIMO configurations, respectively, and ν = 2.The noise variances are set to σ 2 w = σ 2 n = 1.Fig.5(a) demonstrates the effect of varying the number of subcarriers assigned to each sensor, i.e., N s ∈ {1, 2, 4}, on the receiver operating characteristic (ROC) plot of the C-MIMO detector in(14) for M = 256 and p u = −10 dB.

Fig. 7 (
a) portrays the performance of the C-and D-MIMO based tests of (14) and (24), respectively, considering N s ∈ {1, 2}, M = N tot = 1024, P FA = 0.1, p u = −10 dB and sensor position range [0.5R, R].Notice that the performance is enhanced as the density of sensors increases.Moreover, the D-MIMO detector outperforms the C-MIMO detector owing to an increase in the number of FCs deployed upon increasing the sensor density.Considering M = N tot = 1024, P FA = 0.1 and p u = −10 dB, it becomes clear from Fig. 7(b) that the D-MIMO detector's performance is maximized at ρ/R = 0.6.This happens because for ρ/R = 0.6, the sensors