Blind Diagnosis for Millimeter-wave Large-scale Antenna Systems
11 Blind Diagnosis for Millimeter-waveMassive MIMO Systems
Rui Sun, Weidong Wang, Li Chen, Guo Wei, and Wenyi Zhang,
Senior Member, IEEE
Abstract —Millimeter-wave (mmWave) massive multiple-input-multiple-output (MIMO) systems rely on large-scale antennaarrays to combat large path-loss at mmWave band. Due tohardware characteristics and deploying environments, mmWavemassive MIMO systems are vulnerable to antenna elementblockages and failures, which necessitate diagnostic techniques tolocate faulty antenna elements for calibration purposes. Currentdiagnostic techniques require full or partial knowledge of channelstate information (CSI), which can be challenging to acquire inthe presence of antenna failures. In this letter, we propose ablind diagnostic technique to identify faulty antenna elements inmmWave massive MIMO systems, which does not require anyCSI knowledge. By jointly exploiting the sparsity of mmWavechannel and failure, we first formulate the diagnosis problemas a joint sparse recovery problem. Then, the atomic normis introduced to induce the sparsity of mmWave channel overcontinuous Fourier dictionary. An efficient algorithm based onalternating direction method of multipliers (ADMM) is proposedto solve the proposed problem. Finally, the performance ofproposed technique is evaluated through numerical simulations.
Index Terms —Array diagnosis, atomic norm, fault identifica-tion, massive MIMO, millimeter-wave communication
I. I
NTRODUCTION
Millimeter-wave (mmWave) massive multiple-input-multiple-output (MIMO) system is a key technology in currentand next-generation mobile communication systems, whichmay suffer from antenna element failures due to hardwarecharacteristics and deploying environments. MmWave activedevices like amplifiers and mixers are generally less reliablethan conventional sub-6G devices due to higher operatingfrequency and lower power efficiency [1], [2]. Besides,mmWave antenna elements are susceptible to blockages withcomparable sizes like dirt and precipitation owing to theshort wavelength at the mmWave band [3]. The existenceof antenna element failure will cause signal power reductionand radiation pattern distortion, which may lead to severedegradation in system performance. Therefore, diagnostictechniques for mmWave massive MIMO systems are of greatsignificance for system monitoring and maintenance.Extensive related works on the antenna array diagnosishave been proposed, including compressed sensing based ones[4]–[10] and deep learning based one [11]. The main ideaof compressed sensing based diagnostic techniques is to useknown channel state information (CSI) to generate fault-freereference signal and then subtract it from the received signal.
The authors are with the CAS Key Laboratory of Wireless-OpticalCommunications, University of Science and Technology of China, Hefei230027, China (e-mail: [email protected]; [email protected];[email protected]; [email protected]; [email protected]).
Therefore, the differential signal contains the informationof faulty antenna elements, which can be estimated fromcompressed measurements. The work [4] was the first tointroduce compressed sensing to the array diagnosis. Improve-ments over [4] include modifications in sampling methods[5]–[7] and refinements in sparse recovery algorithms [8]–[10]. In particular, the work in [12] extended the aboveantenna-only diagnosis to the joint diagnosis of antenna, phaseshifter, and RF chain for hybrid beamforming (HBF) systems.Deep-learning based diagnostic techniques include [11], whichutilizes a convolutional neural network (CNN) to detect theabnormality in the distribution of received signal and thenlocate faulty antenna elements.In order to distinguish antenna failure from channel fading,the above diagnostic techniques rely on perfect CSI to generatefault-free reference signal, which can be challenging to acquirein the presence of antenna failures. Recently, a diagnostictechnique proposed in [13] relaxes the CSI requirement, whichonly needs the angle-of-arrival (AOA) of each sub-path inthe channel, i.e., partial CSI. However, acquiring the AOA ofeach sub-path still can be a challenging task for a potentiallyfaulty system since faulty antenna elements will change arraygeometry and distort radiation pattern, which makes AOAestimation highly unreliable.Aiming at this limitation, in this letter, we propose a blinddiagnostic technique for mmWave massive MIMO systems.The term ‘blind’ represents that the proposed technique doesnot require any knowledge of the CSI, which is achieved byjointly exploiting the sparsity of the mmWave channel and thefailure. Specifically, we first formulate the diagnosis problemas a joint sparse recovery problem. Then, the atomic normis introduced to induce the sparsity of mmWave channel overthe continuous Fourier dictionary. An efficient algorithm basedon alternating direction method of multipliers (ADMM) isproposed to solve the joint sparse recovery problem. Finally,numerical simulations validate the proposed technique. To thebest of our knowledge, this work is the first to propose adiagnostic technique that does not require any knowledge ofthe CSI.
Notations:
We use a lowercase and an uppercase bold letterto represent a vector and a matrix, respectively. (cid:107) a (cid:107) and (cid:107) a (cid:107) represent the l and l norm of the vector a , respectively. Inparticular, (cid:107) a (cid:107) A denotes the atomic norm of a . For a matrix A , A T , A H denote the transpose and conjugate transpose of A ,respectively. (cid:107) A (cid:107) F and tr( A ) represent the Frobenius normand the trace of A , respectively. I N represents an N × N iden-tity matrix. CN ( µ, σ ) denotes circularly symmetric complexGaussian distribution with mean µ and variance σ . U ( a, b ) a r X i v : . [ ee ss . S Y ] J a n denotes uniform distribution over the interval [ a, b ] .II. S YSTEM M ODEL
We consider an analog beamforming (ABF) mmWave mas-sive MIMO system equipped with an N -element uniformlinear array (ULA) , which is the array-under-test (AUT).A single-antenna diagnostic transmitter (TX) is adopted totransmit test symbols. The AUT receives the test symbol andconducts the diagnosis. In the absence of antenna failure, thefault-free received symbol can be expressed as y (cid:48) = f T h x + w, (1)where f ∈ C N is the combing vector, h ∈ C N is the channelvector between the TX and the AUT, x is the transmit symbol, w ∼ CN (0 , / SNR) is the noise and
SNR is the signal-to-noise ratio.The channel h is assumed as the block-fading mmWaveclustered channel [14], [15], which can be expressed as h = L (cid:88) l =1 α l a ( θ l ) , (2)where a ( θ l ) = [1 , e j πd sin θ l , · · · , e j πd ( N −
1) sin θ l ] T is thearray response vector, L is the number of sub-paths, α l ∼CN (0 , /L ) and θ l ∼ U ( − π/ , π/ are the complex gain andthe angle-of-arrival (AOA) of the l -th sub-path, respectively,and d = 1 / is the element spacing relative to the wavelength.In the presence of faulty antenna elements, the actualchannel vector will deviate from the ideal one. Therefore, thereceived symbol under antenna faults can be expressed as y = f T ( h + h f ) x + w, (3)where h f ∈ C N is the fault-induced channel deviation. Due tothe failure sparsity that usually only a few antenna elementsare faulty [4], [5], h f is assumed as a sparse vector, in whichnon-zero entries indicate faulty antenna elements.To detect faults, the TX transmit test symbol x = 1 throughout the diagnosis. Within the channel coherence time,the AUT uses K random combining vectors to receive thesignal, yielding the observation model y = F ( h + h f ) + w , (4)where y = [ y , · · · , y K ] T ∈ C K , F = [ f , · · · , f K ] T ∈ C K × N , w = [ w , · · · , w K ] T ∈ C K , (5)in which y k , f k , and w k are the received symbol, the combingvector, and the noise corresponded to the k -th measurement,respectively.The goal of diagnosis is to recover the sparse fault-inducedchannel deviation h f , in which non-zero entries indicate faultyantenna elements. From (4), it can be observed that h f iscoupled with the fault-free channel vector h . Since the system The proposed diagnostic technique can be easily extended to other systemarchitectures (like digital beamforming (DBF) and hybrid beamforming (HBF)systems) and other array structures (like uniform planer array (UPA)). contains potentially faulty antenna elements, the channel vec-tor h can not be estimated by conventional channel estimationtechniques like pilot-based training, and thus recovering h f under unknown h can be challenging.To cope with this issue, prior works require full or partialCSI knowledge to perform the diagnosis. The diagnostictechnique proposed in [5] requires full CSI, which assumesa free-space wireless channel and proposes to calculate thechannel vector using known sub-path AOA and gain (i.e., θ l and α l ). Hence, one can generate the fault-free channelvector h using (2) and subtract it from the received signal, andthe impact of channel can be eliminated. A recent work [13]relaxes the CSI requirement in the sense that it only requiresthe AOA of each sub-path (i.e., θ l ), whose main idea is toproject the received signal onto the null space of AOAs andthus the impact of channel can also be eliminated.It can be found that the above diagnostic techniques followthe routine of “cancel-then-recover” in the sense that theyseek to cancel the impact of channel from the received signalfirst and then recover fault-induced channel deviation, whichrequires full or partial CSI knowledge. Such CSI assumptionsometimes can be challenging to satisfy, especially for theoutdoor online diagnosis in a complex multipath scatteringenvironment. In the following, we propose a diagnostic tech-nique that does not require any knowledge of the CSI (i.e.,blind diagnosis). This is achieved by jointly exploiting themmWave channel sparsity and the failure sparsity. Besides,the fault-free channel vector h can also be recovered, whichprovides an approach to perform channel estimation underantenna element failures.III. F AULT D IAGNOSIS
A. Problem Formulation
Recall that our goal is to recover h and h f simultaneouslyunder the observation model y = F ( h + h f ) + w . (6)Intuitively speaking, it seems impossible to distinguish channelfading from antenna failures, and hence recovering h and h f simultaneously can be a challenging task. To decouple the fail-ure deviation h f from the channel h , we need to exploit theirstructural characteristics. The fault-induced channel deviation h f is sparse itself due to the failure sparsity that only a fewantenna elements are faulty. The ideal channel vector h is alsosparse in the Fourier dictionary since the mmWave channel issparse in the angle domain (i.e., L (cid:28) N ) [14]. Therefore, h and h f are sparse under different dictionaries, which allowsthem to be recovered simultaneously [16].To jointly recover h and h f , we need to introduce a-prior information on them. The fault-induced channel deviation h f issparse itself. We adopt the well-known l norm as the sparsity-inducing norm for h f : (cid:107) h f (cid:107) = N (cid:88) n =1 | h f ,n | , (7)where h f ,n is the n -th entry in h f . The ideal channel vector h is sparse in the Fourier dictio-nary. To be more specific, h is sparse in the continuous Fourierdictionary A = { a ( θ l ) | θ l ∈ [ − π/ , π/ } . (8)We adopt the atomic norm as the sparsity-inducing norm for h . The atomic norm of h over the dictionary A is defined as[17] (cid:107) h (cid:107) A = inf (cid:40)(cid:88) l | α l | (cid:12)(cid:12)(cid:12)(cid:12) h = (cid:88) l α l a l , a l ∈ A (cid:41) , (9)which can be regarded as the l norm over the continuousdictionary A .Using the l norm and the atomic norm as sparsity-inducingnorm for h f and h , respectively, the optimization problem forjoint recovery can be expressed as { ˆ h , ˆ h f } = arg min h , h f (cid:107) y − F ( h + h f ) (cid:107) + τ (cid:107) h (cid:107) A + λ (cid:107) h f (cid:107) , (10)where τ and λ are regularization parameters controlling thesparsity penalty on h and h f , respectively. The optimizationproblem (10) tends to find a sparse vector h f and a vector h that is sparse under the dictionary A to fit the observationmodel.Although the optimization problem (10) is clear in itsform, it can not be directly solved since it involves infinite-dimensional variable optimization due to the continuity ofthe dictionary A . To cope with this issue, a conventionalapproach is to discretize the angle domain into a grid, whichforms a discrete Fourier transform (DFT) dictionary [18].This approach, however, induces the off-grid error since theAOAs will not lie exactly on the grid, which may significantlydegrades the recovery performance [19]. In the following, wedevelop an efficient algorithm for solving the optimizationproblem (10). B. An Efficient Diagnostic Algorithm
The main difficulty of solving (10) arises from theatomic norm (cid:107) h (cid:107) A over the continuous Fourier dictio-nary A . Fortunately, the minimization of (cid:107) h (cid:107) A admitsthe following semidefinite program (SDP) thanks to theCarath´eodory–Fej´er–Pisarenko decomposition [17], [20] arg min u ∈ C N v ∈ R (cid:18) N tr (cid:0) T ( u ) (cid:1) + v (cid:19) s . t . (cid:20) T ( u ) hh H v (cid:21) (cid:23) , (11)where tr( · ) denotes the trace of a matrix, and T ( u ) is theHermitian Toeplitz matrix with vector u as its first column.Therefore, the optimization problem (10) has the equivalentSDP { ˆ h , ˆ h f } = arg min h , h f , u ,v (cid:107) y − F ( h + h f ) (cid:107) + τ (cid:18) N tr (cid:0) T ( u ) (cid:1) + v (cid:19) + λ (cid:107) h f (cid:107) s . t . (cid:20) T ( u ) hh H v (cid:21) (cid:23) . (12) Related works on the atomic norm suggest using the CVXtoolbox to solve the SDP (12) [17], which can be time-consuming for large-scale problems. To solve (12) in anefficient way, we develop an algorithm based on the alternatingdirection method of multipliers (ADMM) algorithm [21].The ADMM algorithm integrates the augmented Lagrangianmethod with the dual ascent method, which shows excellentefficiency in solving large-scale problems.We first rewrite (12) into the ADMM form: { ˆ h , ˆ h f } = arg min h , h f , u ,v (cid:107) y − F ( h + h f ) (cid:107) + τ (cid:18) N tr (cid:0) T ( u ) (cid:1) + v (cid:19) + λ (cid:107) h f (cid:107) , s . t . Z = (cid:20) T ( u ) hh H v (cid:21) (cid:23) , (13)where Z ∈ C ( N +1) × ( N +1) is an auxiliary matrix.The augmented Lagrangian function for (13) can be ex-pressed as L ρ ( v, u , h , h f , Z , Λ )= 12 (cid:107) y − F ( h + h f ) (cid:107) + τ (cid:18) N tr (cid:0) T ( u ) (cid:1) + v (cid:19) + λ (cid:107) h f (cid:107) + (cid:28) Λ , Z − (cid:20) T ( u ) hh H v (cid:21)(cid:29) + ρ (cid:13)(cid:13)(cid:13)(cid:13) Z − (cid:20) T ( u ) hh H v (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) , (14)where Λ ∈ C ( N +1) × ( N +1) is the Lagrangian multiplier, ρ isa penalty parameter, and (cid:104)· , ·(cid:105) denotes the real inner product.The ADMM algorithm minimizes the augmented La-grangian function by iteratively updating the following vari-ables: v ( l +1) = arg min v L ρ ( v, u ( l ) , h ( l ) , h ( l )f , Z ( l ) , Λ ( l ) ) , u ( l +1) = arg min u L ρ ( v ( l +1) , u , h ( l ) , h ( l )f , Z ( l ) , Λ ( l ) ) , h ( l +1) = arg min h L ρ ( v ( l +1) , u ( l +1) , h , h ( l )f , Z ( l ) , Λ ( l ) ) , h ( l +1)f = arg min h f L ρ ( v ( l +1) , u ( l +1) , h ( l +1) , h f , Z ( l ) , Λ ( l ) ) , Z ( l +1) = arg min Z L ρ ( v ( l +1) , u ( l +1) , h ( l +1) , h ( l +1)f , Z , Λ ( l ) ) , Λ ( l +1) = Λ ( l ) + ρ (cid:32) Z ( l +1) − (cid:34) T (cid:0) u ( l +1) (cid:1) h ( l +1) h ( l +1)H v ( l +1) (cid:35)(cid:33) , (15)where the superscript ( l ) denotes the l -th iteration.To perform the above updates in an explicit way, we firstintroduce the partitions Z ( l ) = (cid:34) Z ( l )0 z ( l )1 z ( l )1 H Z ( l ) N +1 ,N +1 (cid:35) , Λ ( l ) = (cid:34) Λ ( l )0 λ ( l )1 λ ( l )1 H Λ ( l ) N +1 ,N +1 (cid:35) . (16)The augmented Lagrangian function is convex and differ-entiable with respect to the variables v , u , and h . Setting their gradient to zero, these updates have closed-form expressions v ( l +1) = Z ( l ) N +1 ,N +1 + 1 ρ (cid:16) Λ ( l ) N +1 ,N +1 − τ (cid:17) , u ( l +1) = Ψ − (cid:18) T ∗ (cid:18) Z ( l )0 + 1 ρ Λ ( l )0 (cid:19)(cid:19) − τ ρ e , h ( l +1) = ( F H F + 2 ρ I N ) − (cid:16) F H ( y − Fh ( l )f ) + 2 λ ( l )1 + 2 ρ z ( l )1 (cid:17) , (17)where Ψ is a diagonal matrix and Ψ i,j = N − j + 1 , j =1 , · · · , N , e is a zero vector with the first entry being one,and T ∗ ( · ) generates a vector whose i -th element is the traceof the ( i − -th subdiagonal of the input matrix.The update of h f can be written as h ( l +1)f = arg min h f (cid:107) ( y − Fh ( l +1) ) − Fh f (cid:107) + λ (cid:107) h f (cid:107) , (18)which is the well-known LASSO problem and can be solvedefficiently by the algorithm proposed in [21].Finally, the update of Z can be expressed as Z ( l +1) = arg min Z (cid:23) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z − (cid:34) T (cid:0) u ( l +1) (cid:1) h ( l +1) h ( l +1)H v ( l +1) (cid:35) − ρ Λ ( l ) (cid:124) (cid:123)(cid:122) (cid:125) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (19)which amounts to projecting the Hermitian matrix G onto thepositive semidefinite cone and can be performed by computingthe eigenvalue decomposition of G and setting all negativeeigenvalues to zero.The overall algorithm for solving (12) is summarized inAlgorithm 1. The computational complexity mainly arisesfrom the updates of h , h f , and Z . The updates of h and h f involve matrix inversion, which has the complexity of O ( N ) .The update of Z requires eigenvalue decomposition and hasthe complexity of O (( N + 1) ) . Therefore, the update of Z dominates the overall computational complexity and we have O (( N + 1) ) for Algorithm 1.IV. N UMERICAL S IMULATIONS
In this section, we conduct numerical simulations to eval-uate the performance of proposed diagnostic technique. Thenumber of antenna elements is set to N = 64 . The number offaulty antenna elements is set to 3, whose locations are chosenuniformly at random. The amplitude and phase of the entriesin the fault-induced channel deviation h f follow U (0 . , and U (0 , π ) , respectively. We adopt the success probability as theperformance metric, which is defined as the probability thatthe status of all antenna elements is correctly identified. Forcomparison, we adopt the diagnostic techniques proposed in[5] (denoted as Eltayeb18) and [13] (denoted as Medina20)as benchmark techniques, which require full and partial CSI,respectively.First, we evaluate the effect of the number of measurements.Fig. 1 shows the success probability versus the number ofmeasurements under different SNRs, where the number of sub-paths is set to L = 4 . It can be observed that all techniquesperform better with a larger number of measurements and/or Algorithm 1:
An Efficient Algorithm for Solving (12)Initialization: Z (0) = , Λ (0) = , h (0)f = ; while not converged do v ( l +1) = Z ( l ) N +1 ,N +1 + ρ (cid:16) Λ ( l ) N +1 ,N +1 − τ (cid:17) ; u ( l +1) = Ψ − (cid:16) T ∗ (cid:16) Z ( l )0 + ρ Λ ( l )0 (cid:17)(cid:17) − τ ρ e ; h ( l +1) = ( F H F +2 ρ I N ) − (cid:16) F H ( y − Fh ( l )f ) + 2 λ ( l )1 + 2 ρ z ( l )1 (cid:17) ; h ( l +1)f =arg min h f (cid:107) ( y − Fh ( l +1) ) − Fh f (cid:107) + λ (cid:107) h f (cid:107) ,which is the LASSO problem and can be solvedby the algorithm proposed in [21]; Perform eigenvalue decomposition on (cid:34) T (cid:0) u ( l +1) (cid:1) h ( l +1) h ( l +1)H v ( l +1) (cid:35) − ρ Λ ( l ) and set allnegative eigenvalues to zero, yielding Z ( l +1) ; Λ ( l +1) = Λ ( l ) + ρ (cid:32) Z ( l +1) − (cid:34) T (cid:0) u ( l +1) (cid:1) h ( l +1) h ( l +1)H v ( l +1) (cid:35)(cid:33) ; end
10 20 30 40 50 60
Number of measurements S u cc e ss p r obab ili t y Eltayeb18, SNR=10dBEltayeb18, SNR=30dBMedina20, SNR=10dBMedina20, SNR=30dBproposed, SNR=10dBproposed, SNR=30dB
Fig. 1. Success probability versus the number of measurements under differentSNRs. larger SNR. Among all diagnostic techniques, the diagnosiswith full CSI (Eltayeb18) outperforms others since the impactof channel can be completely eliminated using known CSI.When the SNR is sufficiently large, the proposed techniquehas slightly better performance than the diagnosis with partialCSI (Medina20).Next, we assume that the estimated sub-path gain ˆ α l con-tains estimation error, which is defined as ˆ α l = α l + δ α α e , (20)where α e ∼ CN (0 , represents gain estimation error and δ α is the error intensity. The performance of different techniquesunder sub-path gain estimation error is shown in Fig. 2,where SNR = 30 dB and the number of measurements isset to K = N to ensure sufficient measurements for alltechniques. We can observe that the performance of diagnosiswith full CSI (Eltayeb18) degrades significantly under gainestimation errors, while other techniques are not affected bythe estimation error since they do not require the knowledge ofsub-path gain. Besides, the more sub-paths in the channel, the Gain error intensity S u cc e ss p r obab ili t y Eltayeb18, L=1Eltayeb18, L=4Medina20, L=1Medina20, L=4proposed, L=1proposed, L=4
Fig. 2. Success probability versus sub-path gain error intensity under differentnumber of sub-paths.
AOA error intensity S u cc e ss p r obab ili t y Eltayeb18, L=1Eltayeb18, L=4Medina20, L=1Medina20, L=4proposed, L=1proposed, L=4
Fig. 3. Success probability versus sub-path AOA error intensity underdifferent number of sub-paths. worse the performance since more errors will be introduced.Finally, we assume that the estimated AOA ˆ θ l containsestimation error, which is defined as ˆ θ l = θ l + δ θ θ e π, (21)where θ e ∼ N (0 , represents the AOA estimation error and δ θ is the error intensity. The success probabilities of differenttechniques under AOA estimation error are shown in Fig. 3,where SNR = 30 dB and the number of measurements is setto K = N . It can be observed that benchmark diagnostictechniques are highly sensitive to AOA estimation errors. Onthe contrary, the performance of proposed diagnostic techniqueis not affected under all intensities of AOA estimation errorsince it does not require any CSI knowledge, which showsstrong robustness against channel estimation errors.V. C ONCLUSIONS
In this letter, we have proposed a blind diagnostic techniquefor mmWave massive MIMO systems to locate faulty antennaelements. By jointly exploiting the sparsity of the mmWavechannel and the failure, the location of faulty antenna elementscan be identified without any knowledge of the CSI. A novelatomic norm has been introduced as the sparsity-inducingnorm of the mmWave channel, and the diagnosis problem hasbeen formulated as a joint sparse recovery problem. An effi-cient ADMM-based diagnostic algorithm has been proposedto solve the joint sparse recovery problem. Numerical resultshave shown that the proposed technique has strong robustnessagainst channel estimation errors compared with prior works,which provides a practical approach for the outdoor onlinediagnosis. R
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