Line-of-Sight MIMO for High Capacity Millimeter Wave Backhaul in FDD Systems
11 Line-of-Sight MIMO for High Capacity MillimeterWave Backhaul in FDD Systems
Ye Xue † , Xuanyu Zheng † , and Vincent Lau Abstract —Wireless backhaul is considered to be the key partof the future wireless network with dense small cell traffic andhigh capacity demand. In this paper, we focus on the designof a high spectral efficiency line-of-sight (LoS) multiple-inputmultiple-output (MIMO) system for millimeter wave (mmWave)backhaul using dual-polarized frequency division duplex (FDD).High spectral efficiency is very challenging to achieve for thesystem due to various physical impairments such as phase noise(PHN), timing offset (TO) as well as the poor condition number ofthe LoS MIMO. In this paper, we propose a holistic solution con-taining TO compensation, PHN estimation, precoder/decorrelatoroptimization of the LoS MIMO for wireless backhaul, and theinterleaving of each part. We show that the proposed solutionhas robust performance with end-to-end spectral efficiency of 60bits/s/Hz for 8x8 MIMO.
Index Terms —Line-of-Sight MIMO, Wireless backhaul, FDD,timing synchronization, phase noise
I. I
NTRODUCTION
Dense small cells and femtocells have been proposed toenhance the spatial reuse and boost the capacity of hot spotsin future wireless networks [1, 2]. As a result, future wirelessnetworks may comprise a substantial number of small base sta-tions, and the backhaul transmission will be a critical capacityand cost bottleneck. In this paper, we focus on designing avery high spectral efficiency mmWave backhaul transmissionusing MIMO technology for future wireless networks.MIMO has been widely used in 3GPP long term evolution(LTE) and 5G systems as a key technique to enable thecapacity requirement of the wireless access network. Leverag-ing the rich scattering in the non-line-of-sight (NLoS) fadingchannels, spatial multiplexing for MIMO systems has beenwidely studied in [3]. However, there are various technicalchallenges in adopting the MIMO technique for improvingthe spectral efficiency of the wireless backhaul. First, in thecontext of wireless backhaul, the propagation channel is pri-marily LoS due to the high carrier frequency together with thenarrow beamwidths being used. With limited scattering in thepropagation environment, the LoS MIMO channel responsescan be highly correlated, leading to a rank deficient channelmatrix. Nevertheless, by optimizing the antenna placements,the capacity and rank of the LoS MIMO channel can beimproved with a large separation between the antennas inthe arrays [4, 5]. However, it is costly to install arrayswith large antenna-separation, especially for MIMO systems.This poses the technical challenge in adopting the MIMO † These two authors contribute equally to the work.Y. Xue, X. Zheng and V. Lau are with the Hong Kong University of Scienceand Technology, Hong Kong, China technique for improving the spectral efficiency of the wirelessbackhaul since low antenna-separation arrays will lead to apoor condition number of the LoS MIMO channel and thespatial multiplexing benefit can be jeopardized.The second challenge is brought by the existence of physicalimpairments in the system, such as TO and PHN. The presenceof timing offsets among the transmit and receive antennawill bring severe inter-symbol interference (ISI), which willdegrade the performance of the wireless backhaul. As aresult, this poses a very stringent requirement for accuratetiming synchronization. Additionally, PHN in MIMO systemswill bring two penalties in the MIMO system, namely the demodulation penalty caused by phase distortion to signalconstellation and the multi-access interference (MAI) penalty induced by the coherency loss of the precoder and decorre-lator. These physical impairments usually exist in practicalwireless backhaul, the effect of which is more severe inmmWave systems [6] due to the high carrier frequency. Unlikewireless access applications, the target spectral efficiency ofwireless backhaul is very high. These physical impairmentscan be the performance bottleneck for wireless backhaul andpose a very stringent requirement for accurate estimation andcompensation of the TO and PHN. Unfortunately, the spatialmultiplexing in MIMO systems will cause severe MAI, whichis a huge hurdle for the compensation of these impairments.Furthermore, dual-polarization is usually adopted in wirelessMIMO backhaul [7] to overcome the space limitation, whichallows two orthogonal streams from each dipole antenna totravel in the same bandwidth at the same time (one verti-cally (V mode) and one horizontally (H mode)). In such adual-polarized system, high cross-polarization discrimination(XPD) of each dipole antenna is required to reduce theinterference between cross-polar link transmission. However,high XPD together with severe MAI will lead to inaccuratecross-polar link measurements for the TO and PHN, makingit more challenging to estimate and compensate these physicalimpairments. Thus, a novel and holistic solution consideringthe physical impairments for mmWave LoS MIMO backhaulwith dual-polarization is required. The precoder and decorrelator are used to mitigate interference betweenspatially multiplexed data streams. The precoder and decorrelator were de-signed based on the estimated CSI at the beginning of a frame and were fixedthroughout the frame. However, due to PHN, there is an increasing mismatchbetween the precoder/decorrelator and the effective channel (incorporating thePHN) and hence, the MAI increases. The cross-polar link represents the data link between the V mode (Hmode) at the transmitter and the H mode (V mode) at the receiver in a dual-polarized MIMO (See Fig.1), which is very weak in high XPD systems andwill be corrupted by the MAI caused by the signal from other links. a r X i v : . [ ee ss . SP ] J un There are many existing works that considered differentsubsets of the aforementioned physical impairments [6, 8–18].TO estimation has been well studied in [9–11]. In [9], theauthors propose a maximum likelihood (ML) estimator to theTO in the MIMO system with an impractical assumption thatthe TOs are identical across all the antennas. [10, 11] proposeestimation methods of different TOs over antennas. However,there are strict assumptions of TO in these works. Specifically,[10] assumes TOs are small and within one symbol time and[11] only considers TOs that are multiples of symbol time. Inaddition, these works only estimate the sum-offsets and thepreamble used in this work may fail due to the severe MAIinduced by the high XPD in the dual-polarized LoS MIMOchannel.PHN estimation algorithms in single-input single-output(SISO) systems are widely studied in [12–14, 19]. However,these methods cannot be applied to MIMO systems sincethe received signal at each antenna will be influenced bymultiple PHNs in the transmitters, which needs to be jointlyestimated. For MIMO systems, PHN estimation and compen-sation algorithms are proposed in [15–17] with data-aidedand decision-directed sum-PHNs estimators. However, [15]assumes that the channel is perfectly known at the receiverand [16, 17] have ignored the effect of TOs at the transceiver.In addition, the high XPD in dual-polarized MIMO is notconsidered, hence, the sum-PHNs extracted from the estimatedchannel will have large estimation errors for transceiver linkswith a small amplitude (e.g., the cross-polar links). In theseworks, after estimating the sum-PHNs, the authors designthe decorrelator to equalize the aggregated channel and sum-PHN matrix. These compensation schemes impose a heavycomputational burden at the receiver since the PHN is varyingover time and the decorrelator needs to be updated alongwith the variations of the PHN. In [6], the authors proposea method to estimate and compensate the per-antenna PHNfor MIMO, but they do not track the time-varying PHN andthey ignore the interplay with the precoder and decorrelator.Hence, existing works are not applicable when various impair-ments are considered in mmWave LoS MIMO. The precoderand decorrelator are key components in MIMO systems toachieve high spectral efficiency for wireless backhaul, sincethey enable stable multi-stream transmissions. There are alot of existing works on MIMO precoder/decorrelator design[20–22]. [20] proposes a unified linear transceiver designframework, in which the optimal decorrelator is fixed as theWiener filter and the design problem can be formulated asconvex optimization problems in terms of precoder underdifferent design criteria. In [21, 22], the authors consider theweighted MMSE optimization and propose low complexity Sum-offsets are the effective TOs in the transmitter-receiver links. Forexample, in an 8x8 MIMO, we have altogether 16 unknown TOs for all theantennas, but these works only estimate the 64 combinations of the sum-offsetsbetween the transmitter and receiver antennas. There are only 16 freedoms inthese 64 sum-offsets, but such information is not exploited in these works. Sum-PHNs are the effective PHNs in the transmitter-receiver links. Forexample, in an 8x8 MIMO, we have altogether 16 unknown PHNs for all theantennas, but these works only estimate the 64 combinations of the sum-PHNsbetween the transmitter and receiver antennas. There are only 16 freedoms inthese 64 sum-PHNs, but such information is not exploited in these works. algorithms based on alternative optimization. However, in allthese works, the physical impairments such as TO and PHNhave been ignored. This may be justified for wireless accessapplications but these physical impairments cannot be ignoredfor wireless backhaul applications due to the very high targetspectral efficiency. Recently, many works have consideredthe physical impairments issue. Hardware impairments aware(HIA) MIMO transceivers are proposed in [23–25]. However,the effect of physical impairments is simply modeled asadditive Gaussian noise to each antenna, which is an over-simplification of the impairments due to the TO and the PHNin the MIMO systems. In summary, these existing solutionscannot be applied in our case because of very different targetspectral efficiency and the practical considerations of the LoSMIMO.In this paper, we adopt a holistic approach and propose apractical solution for a dual-polarized LoS MIMO in mmWavebackhaul addressing the aforementioned physical impairments.The solution is not a trivial combination of existing techniquesas each component is inter-related. Based on the proposedsolution, we can achieve very high spectral efficiency (e.g.,60 bits/s/Hz with 8x8 MIMO) and fully unleash the potentialof the LoS MIMO in mmWave backhaul applications. Thefollowing summarizes our contributions. • Decentralized Spatial Timing Estimation and Com-pensation:
We propose a low complexity spatial timingestimator that has a similar complexity to the cross-correlation approach [8, 11] but it is capable of utilizingspatial information across different antennas to estimatethe per-antenna TO. The proposed spatial TO estimatoronly requires local information and hence it can beimplemented separately at the transmitter and receiver. Toovercome the strong MAI induced by spatial multiplexingof dual-polarized LoS MIMO channels, we propose newpreamble sequences with improved auto-correlation andcross-correlation rejection. Based on this, we proposea decentralized timing compensation scheme where thetransmitter and receiver compensate for the TO basedonly on local information without any explicit signaling. • Decentralized Phase Noise Estimation and Compen-sation with Decision Feedback.
We propose a low com-plexity per-antenna PHN estimation and compensationscheme, which enables compensation for both the phasedistortion to the received symbols and the MAI caused bythe loss of coherence of the precoder and decorrelator dueto the drifting of PHN. To reduce the pilot overhead, weadopt the decision feedback and regression-based fusionto enhance the PHN estimation quality. The proposedPHN estimator has low complexity and requires local in-formation only. Based on this, we propose a decentralizedPHN compensation scheme, which compensates the per-antenna PHN locally at the transmitter and the receiver. • Robust MIMO Precoder and Decorrelator Design : Weexploit the MIMO precoder and decorrelator to suppress Note that such per-antenna compensation is not possible if one uses con-ventional PHN estimators, which only estimate and compensate the effectivesum-PHN. the inter-symbol interference (ISI) and MAI induced bythe physical impairments of the TO and the PHN. Weshow that the precoder and decorrelator can significantlyalleviate the requirement of the TO compensation andPHN compensation to achieve very high spectral effi-ciency for mmWave backhaul applications. The designis formulated as a nonconvex optimization problem. Byexploiting structures in the ISI and MAI, we transformthe problem into a tractable form and propose a low com-plexity solution using alternative optimization techniques.This paper is organized as follows. In Section II, we presentthe system model, including the mmWave dual-polarized LoSMIMO channel model, the TO model, and the PHN model,as well as the data path in the transmitter and receiver. InSection III, V and IV, the proposed timing synchronization,PHN estimation and compensation, as well as the procoderand decorrelator scheme, respectively, are illustrated. Thenumerical simulation results and the corresponding discussionsare provided in Section VI. Finally, Section VII summarizesthe whole work.
Notations:
In this paper, lowercase and upper bold faceletters stand for column vectors and matrices, respectively. Theoperations ( · ) T and ( · ) H are respectively, the operations oftranspose and conjugate transpose. The entry in the i -th rowand j -th column of matrix A is [ A ] i,j while the n -th elementof vector a is a n . a ∗ represents the complex conjugate of a . I N is the N × N identity matrix and N denotes the all one vectorof dimension N . The norms ||·|| , |·| and ||·|| F are respectively,the L , L and Frobenius norm. (cid:12) and ⊗ represents theHadamard product and Kronecker product, respectively. e j ( · ) isan elementwise operator for vector or matrix input. CN ( µ, σ ) denotes the Complex Gaussian distribution with mean, µ and variance σ . diag ( a ) denotes a diagonal matrix whosediagonal elements are filled with elements of vector a , and vec ( · ) is the vectorization operator. Finally, g ( · ) ∗ f ( · ) denotesthe convolution of g ( · ) and f ( · ) .II. S YSTEM M ODEL
In this paper, we consider mmWave MIMO wireless back-haul in a dual-polarized FDD system. Both the transmitterand the receiver comprise an antenna array mounted on a polewith a primarily LoS channel in between. Each of the transmit(and receive) antennas has a local oscillator that is looselysynchronized to a master control unit. The illustration of thesystem is shown in Fig.1. We shall elaborate on each part ofthis system below.
A. LoS MIMO Channel Model
In mmWave backhaul, a terrestrial link usually exists.Therefore, we consider the Rummler model [26] in this work,which is an LoS propagation model with a single NLoS pathcaused by the terrestrial reflection between two fixed antennatowers. For a single-input-single-output (SISO) system, theimpulse response of the Rummler model can be specified as h ( t ) = δ ( t ) + βe j πf τ d δ (cid:0) t − τ d (cid:1) , (1) d1 LoS path
NLoS path
V H D Tx Rx
Master
Control
Unit Master Control UnitFlat panel dual-polarized MIMO antennae array
Fig. 1. Illustration of antenna arrays and geometry of an × LoS dual-polarized MIMO system. Flat-panel dual-polarized MIMO antenna arrays areapplied with antenna spacing d , d , where d √ d . The distancebetween the transmitter and the receiver is D . where the first term represents the LoS path and the secondterm represents the NLoS path, τ d is the propagation delayassociated with the difference in the propagation time betweenthe LoS path and the NLoS path and is assumed to be withinone symbol time. f denotes the notch frequency, whichwill be anywhere in the spectral efficiency. We consider theminimum-phase case of the Rummler model ( β ≤ ), wherewe set the channel gains β = 1 − − ρ , with ρ denoting thenotch depth, which relates the power of the LoS path to thatof the NLoS paths.Generalizing the SISO Rummler model to a dual-polarized N × M MIMO system, the channel model is given by ˜ H ( t ) = H LoS ( t ) + H NLoS ( t ) , = H LoS ( t ) + β (cid:12) R (cid:12) H LoS ( t − τ d ) (2)where β and R is the random magnitude attenuation and therandom phase rotation matrix, respectively, caused by reflec-tion.Since β and R are multiplied element-wisely to the LoSchannel response, the expression for the NLoS propagationin (2) can represent any NLoS propagation. The LoS channel H LoS ( t ) is highly deterministic [27] and can be modeled as H LoS = H xp ⊗ J N × M (cid:12) H A , (3)where H xp ∈ C × contains the cross-polar gains, J N × M isthe N × M all ones matrix and H A ∈ C N × M is the arrayresponse matrix between transmitter and receiver with N and M flat-panel dual-polarized MIMO antenna arrays. Note thatin this paper, we assume spherical curvature of the propagatingwaves, thus the array response H A is given by [ H A ] i,j = e − j πλ d i,j , where d i,j is the distance between the i -th receiver antenna andthe j -th transmit antenna. A similar propagation assumptionand the array response matrix can be found in [6], in whichthe dual-polarized array is not considered. In this paper, we consider the H mode and V mode of one dipole antennaas two different antennas. For example, in Fig.1, there are 4 dipole antennasat each side of the transmitter and the receiver, and the system is consideredto be a × MIMO system. c o rr e l a ti on m e t r i c f o r ti m i ng o ff s e t correct timing offset Fig. 2. Correlation metric for TO estimation in a cross-polar link for differentnumbers of antennas. The traditional ZC sequence is used as the preamblesequence for TO estimation.
XPD is set to a typical value of dB, and theTO is set to be 4 times the symbol time. With dual-polarized MIMO antennas, XPD of the antennashas a significant influence on the system performance. Ingeneral, high antenna XPD is desired to leverage the benefitsof dual-polarization [28, 29]. However, high XPD also causesvery weak cross-polar transmission links, i.e., the link betweenthe horizontal mode to the vertical mode of each dipoleantenna. This will cause a great challenge for the estimationof the TO and PHN since the physical impairments aredifferent on each antenna and the estimation for these physicalimpairments requires measurements from all the transceiverlinks. Moreover, the intensity of the MAI will increase as thenumber of antennas of the MIMO system increases, which willfurther distort the cross-polar measurements. For example, inan × MIMO with a typical
XPD = 20 dB, the power ofthe MAI will be dB of the desired signal for a cross-polarlink. We show this influence on TO estimation in Fig. 2 withthe timing correlation metric in a cross-polar link using thetraditional Zadoff–Chu (ZC) sequence. It is observed that asthe number of antennas increases, the severe MAI will induceseveral peaks of similar intensity in the timing correlationmetric. As a result, there might be large timing estimationerrors due to the false peaks in the metric. B. Timing offset Model
In mmWave MIMO wireless backhaul, the transmit antennasare not collocated and hence the clock of each transmitantenna/receive antenna is only roughly synchronized to amaster clock. Let τ txj = T | offset txj − offset tx | and τ rxi = T | offset rxi − offset tx | denote the normalized TO of the clock atthe j -th transmit antenna and i -th receive antenna, respectively,w.r.t. the symbol duration T . Without loss of generality, weassume τ rx = 0 and TOs τ txj and τ rxi are independent quasi-static random processes uniformly distributed in (0 , τ max ) ,where τ max > is the maximum timing delay that canoccur in the system. We denote τ tx = [ τ tx , ..., τ txM ] T and In practice, the drift in the oscillators is a slowly varying process withcoherence time in the order of minutes or hours [30].
Preamble (Density depends on how fast delay drifts)
Pilot (Density depends on how fast PHN drifts)
Data L t … … L d Subframe 0 Subframe 1 … … Subframe N sf -1 L p U T U P S S U P S Fig. 3. Frame structure for the transmission of preamble, pilot, and datasymbols from one transmitter. τ rx = [ τ rx , ..., τ rxN ] T as the TO vector for the transmitterand receiver, respectively, and let τ = (cid:104) ( τ rx ) T , ( τ tx ) T (cid:105) T . C. Phase Noise Model
In this paper, we consider the independent phase noise θ txj ( n ) on each transmit antenna and θ rxi ( n ) on each receiveantenna. For free-running oscillators, the discrete time PHNs θ txj ( n ) and θ rxi ( n ) for j = 1 , ..., M and i = 1 , ..., N can bemodeled as a Wiener process [15, 16, 31], which are given by θ txj ( n ) = θ txj ( n −
1) + ∆ txj ( n ) ,θ rxi ( n ) = θ rxi ( n −
1) + ∆ rxi ( n ) . (4)The terms ∆ txj ( n ) and ∆ rxi ( n ) are random phase innovationsfor the oscillators at each sample, assumed to be whitereal Gaussian processes with ∆ txj ( n ) ∼ N (cid:16) , σ txj (cid:17) and ∆ rxi ( n ) ∼ N (cid:16) , σ rxi (cid:17) , respectively. σ txj and σ rxi standsfor the variance of the innovations at the j -th and i -th transmitand receive antennas, respectively, which are given by [16, 32] σ txj = 2 πc txj T s ,σ rxi = 2 πc rxi T s , (5)where c txj and c rxi denote the one-sided 3 dB bandwidth ofthe Lorentzian spectrum of the oscillators at the j -th and i -th transmit and receive antennas, respectively, and T s is thesampling time. We assume that σ txj and σ rxi are knownat the receiver since they are dependent on the oscillatorproperties. D. Transmit and Receive Data Path
We consider a frame structure, illustrated in Fig. 3, in thispaper. There is a preamble sequence U T = [ a , ..., a M ] T ∈ C M × L t with L t symbols at the beginning of each frame fortiming synchronization and channel estimation. Specifically,since timing offset is a slowly varying process, we use thepreamble of the first frame (initial frame) to perform timingsynchronization, while preambles of the rest frames are usedfor channel estimation. Each frame consists of N sf subframes,and within each subframe, there is a data section with L d symbols, denoted by S = [ s , ..., s M ] T ∈ C M × L d , where s j ∈ C L d × is the transmitted data symbols at j -th transmit antenna.In addition, there is a pilot sequence U P = [ b , ..., b M ] T ∈ C M × L p with length L p at the beginning of each subframe forthe estimation and tracking of the PHN over the entire frame.To summarize, the first subframe of each frame has the lengthof L t + L d symbols, while the other subframes are of length L p + L d symbols. As a result, the overall pilot and preambleoverhead is given by L t +( N sf − L p N sf L d . In practice, the periodof the preamble is comparable to the coherence time of thechannel (which is ˜ ms for fixed-point wireless backhaulapplications [33]). The period of the pilot is more frequent as ithas to track the variation of the PHN θ txj ( n ) and θ rxi ( n ) . Fora typical oscillator at f c = 28 GHz, the effective bandwidth ofthe power spectral density (PSD) of the PHN is Hz [34]and hence, the coherence time of the PHN is around . ms.Consequently, a pilot density of pilots per frame would besufficient. Hence, for a system with a symbol rate of M/s,the pilot and preamble overhead is around 5% for L t = 256 , L p = 64 , L d = 1280 and N sf = 100 , which is quite low.Figure.4 illustrates the uplink and downlink datapath of thetransmitter and receiver for the wireless backhaul betweentwo base stations (BSs) A and B. We consider an FDDsystem, and the transmitter and the receiver of a site sharea common oscillator. Hence, the PHN and the TO duringdownlink for BS A (BS B) as a receiver (transmitter) areidentical to the PHN and the TO during uplink for BS A(BS B) as a transmitter (receiver), i.e., in Fig. 4 we have θ ULA,j ( n ) = θ DLA,j ( n ) , τ ULA,j = τ DLA,j , ∀ j and θ ULB,i ( n ) = θ DLB,i ( n ) , τ ULB,i = τ DLB,i , ∀ i . To avoid the abuse of notation, we assume θ txj ( n ) = θ ULA,j ( n ) , τ txj = τ ULA,j , θ rxi ( n ) = θ ULB,i ( n ) , τ rxi = τ ULB,i in the following illustrations.Let g ( t ) be the response of the pulse-shaping filters evalu-ated at t . The equivalent channel for the j -th transmit antennaand the i -th receive antenna is given by h τ i,j ( t ) = (cid:104) ˜ H ( t ) (cid:105) i,j ∗ g (cid:0) t − (cid:0) τ txj + τ rxi (cid:1) T (cid:1) , (6)where the superscript τ means the variable is parameterizedby τ . Let u j = [ u j (0) , u j (1) , ..., u j ( L − T denote thecomplex-valued symbol sequence of length L transmitted bythe j -th transmitter, which can be a preamble sequence, pilotsequence or data. Suppose the received waveform is sampledat a rate of Q samples per symbol (i.e., T s = TQ ), the receivedsignal of i -th receive antenna at the n -th sample is given by y i ( n ) (7) = M (cid:88) j =1 L − (cid:88) k =0 e j [ θ txj ( n )+ θ rxi ( n ) ] h τi,j ( nT s − kT ) u j ( k ) + v i ( n ) , where k is the symbol index, and v i ( n ) is the complexadditive white Gaussian noise (AWGN) with zero mean andvariance σ , i.e., v i ( n ) ∼ CN (0 , σ ). Specifically, to show asymbol level signal model, the received signal at the n = kQ - th sample for symbol index k = 0 , ..., L − is given by: y i ( kQ ) = e j [ θ txi ( kQ )+ θ rxi ( kQ ) ] (cid:124) (cid:123)(cid:122) (cid:125) PHN distortion h τi,i (0) u i ( k ) (cid:124) (cid:123)(cid:122) (cid:125) desired signal + L − (cid:88) k (cid:48) =1; k (cid:48) (cid:54) = k h τi,i (( k − k (cid:48) ) T ) u i ( k (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) ISI + M (cid:88) j =1; j (cid:54) = i L − (cid:88) k (cid:48) =0 h τi,j (( k − k (cid:48) ) T ) u j ( k (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) MAI + v i ( kQ ) (cid:124) (cid:123)(cid:122) (cid:125) AW GN . (8)From (8), we have the following observations. First, the PHNwill introduce a random phase distortion on the received signalconstellation during the demodulation process. Furthermore,the MAI and ISI of the spatially multiplexed streams willbe mitigated by the precoder and the decorrelator, which aredesigned based on the estimated channel at the beginning ofeach frame and remain constant throughout the frame. Whilethe channel will be quasi-static within a frame, the PHNprocess will be time-varying and hence the PHN distortionwill induce loss of coherency of the precoder and decorrelatorfor symbols within a frame. Second, the impacts of the TOappear in both the ISI and the MAI terms, as illustrated in(8), which will increase both types of interference. Thesephysical impairments may not be a significant performancebottleneck in regular wireless access applications. However,due to the very high spectral efficiency target in wirelessbackhaul applications, they can be a significant bottleneck.III. T IMING O FFSET E STIMATION AND C OMPENSATION
We consider the TO issue in this section. Conventionalcorrelation-based TO estimators correlate the preamble se-quences with delayed replicas of the received samples at eachreceive antenna, and find the maximum correlation peak in thetiming metric to estimate the sum-offsets in each transmitter-receiver link. For an N × M MIMO system, the correlatorcan estimate
N M sum-offsets for all the transmitter-receiverlinks, but there are only N + M degrees of freedom inthese sum-offsets as we only have one unknown TO for eachindividual antenna to be estimated. Hence, there are correla-tions among the N M sum-offsets, and this correlation can befurther exploited to enable a more accurate TO estimation andcompensation for each individual antenna.Conventionally, the ZC sequence is used as preamble in LTEsystems, serving as the primary synchronization signal (PSS)to extract timing information. The cross-correlation of the ZCsequence is low for traditional wireless MIMO applicationsdue to the moderate MAI, thus the accurate estimations of thesum-offsets can be guaranteed. However, the cross-correlationof the ZC sequence is not sufficiently low to mitigate theMAI for a dual-polarized LoS MIMO channel with high XPD,especially for the cross-polar links, which experience severe
PHNTiming Offset
DuplexingDACADC
Timing Sync PHNTiming estimationChannel EstimationPrecoder(feedback)/Decorrelator(local) PHN trackingPHN Estimation Timing Offset
Duplexing
DAC
ADC
Timing Sync .........
PHN CompensationPHN Compensation De mod Mod De mod Mod U T U P S Detection
Detection U T Control unit
PHN Timing Offset
Duplexing
DAC
ADC
Timing SyncPHN Timing estimation Channel EstimationPrecoder(feedback)/
Decorrelator(local)
PHN trackingPHN EstimationTiming Offset
Duplexing DACADC
Timing Sync ... ... ...
PHN CompensationPHN Compensation De mod Mod
Demod
Mod U T U P S Detection
Detection U T Control unit
A B
Data pathControl path
Timing Estimation and CompensationPhase Noise Estimation and Compensation
Precoder/DecorrelatorControl module
Data module
ULDL θ A,1 (n) τ A,1 θ A,M (n) τ A,M θ B,1 (n) θ B,N (n) τ B,1 τ B,N
TxTx RxRx
Fig. 4. Datapath for the proposed mmWave MIMO backhaul in FDD system.
MAI. Thus, the traditional ZC preamble is not enough toisolate MAI in cross-correlation, which hinders correlator-based timing synchronization from fully utilizing the spatialinformation. In addition, the high spectral efficiency of wire-less backhaul poses a very stringent requirement for accuratetiming synchronization, thus preamble sequences with a goodauto-correlation property that is robust to ISI are also required.This motivates us to design preamble sequences with superiorauto/cross-correlation properties for the TO estimation.
A. Preamble Sequence Design
Let a j = [ a j (0) , ..., a j ( L t − T denote the preamblesequence transmitted by the j -th transmit antenna. The cross-correlation of two preamble sequences a j and a j (cid:48) is definedas η j,j (cid:48) ( l ) = L t − − l (cid:88) k =0 a j ( k + l ) a ∗ j (cid:48) ( k ) = η ∗ j (cid:48) ,j ( − l ) , ∀ l = 0 , ..., L t − , (9)where l represents the l -th lag. Note that (9) reduces to theauto-correlation of a j when j = j (cid:48) . A satisfying set ofpreamble sequences should have a very low cross-correlationfor all possible lags and a high auto-correlation only when l = 0 , thus, the correlation peak occurs only when the twosequences are from the same antenna and are perfectly aligned.However, [35] shows that it is impossible to design such a set.Fortunately, it is feasible to achieve the required properties in aspecific lag interval. Since the TO will not exceed a maximumvalue of τ max in each transmit and receive antenna, we canseek preamble sequences with the required properties at thelag interval ≤ l ≤ (cid:100) τ max (cid:101) . Such a set of sequences can beobtained by solving the following optimization problems [36]: min U T M (cid:88) j =1 M (cid:88) j (cid:48) =1 L t − (cid:88) l =1 − L t ω l f | η j,j (cid:48) ( l ) | − ω L t M s . t . | a j ( k ) | = 1 , k = 0 , ..., L t − , j = 1 , ..., M, (10) where ω l = ω − l ≥ , l = 0 , ..., L t − are non-negative weightsassigned to different time lags and are defined in our problemas ω ± l = (cid:26) , ≤ l ≤ (cid:100) τ max (cid:101) , , otherwise . The unimodular constraint in (10) ensures the symbols inpreamble sequences have a constant amplitude. The optimiza-tion problem (10) is solved via the majorization-minimization(MM) algorithm and can be implemented efficiently for verylong sequences via fast Fourier transform (FFT) [6]. Fig. 5shows the auto/cross-correlation of the proposed preamblesequence, compared with the ZC and Walsh sequence. It showsthat the proposed sequences provide -70 dB isolation fromMAI in cross-correlation and -70 dB isolation from ISI inauto-correlation at desired lags, which is sufficient for TOestimation for our channel model with high XPD. -300 -200 -100 0 100 200 300 lags -100-80-60-40-200 l og10 ( | , ( k ) | / D ) Proposed SeqWalsh SeqZC Seq (a) Auto-correlation of a . -300 -200 -100 0 100 200 300 lags -120-100-80-60-40-200 l og10 ( | , ( k ) | / D ) Proposed SeqWalsh SeqZC Seq (b) Cross-correlation of a and a .Fig. 5. Auto/cross correlation of the proposed sequence, the ZC sequence,and the Walsh sequence, with M = 8 , L t = 256 and τ max = 5 . B. Per-Antenna Timing Offset Estimation
To obtain the per-antenna TO, we first estimate the
N M sum-offsets via finding the maximum correlation peak in thetiming correlation metric. For a high resolution TO estimation,we consider upsampling at the received signal. Correlatingthe Q -fold upsampled version of the received signal at the i -th receive antenna and the preamble from the j -th transmit antenna, the timing correlation metric at the s f -th sample shift is given by Λ i,j ( s f ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L t − (cid:88) k =0 a ∗ j ( k ) · y i ( kQ + s f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The sum-offset for the j -th transmit antenna and the i -threceive antenna, τ i,j = τ txj + τ rxi , is obtained by selectingthe highest peak in the correlation metric, as given by ˆ τ i,j = T s T · (cid:18) arg max s f =0 ,..., (cid:100) Qτ max (cid:101) Λ i,j ( s f ) (cid:19) . Then, the per-antenna TO τ = R N + M can be related to the N M sum-offsets by γ = ¯ I NM τ , (11)where γ = ( τ , , ..., τ ,M , ..., τ N, , ..., τ N,M ) T ∈ R NM and ¯ I NM ∈ R ( NM ) × ( N + M ) is given by ¯ I NM = (cid:2) I N ⊗ M N ⊗ I M (cid:3) . (12)The column rank of ¯ I NM is only ( N + M − , thus thetransformation (11) is underdetermined and one additionalconstraint is required. The reference TO, i.e., τ rx = 0 withoutloss of generality, provides this additional constraint withwhich the per-antenna TO τ ∈ R N + M can be obtained bysolving the following LS problem: ˆ τ = arg min τ (cid:13)(cid:13) ¯ I NM τ − ˆ γ (cid:13)(cid:13) s . t . τ (1) = 0 . (13) C. Per-Antenna Timing Offset Compensation
In our system, the TOs are compensated with the corre-sponding transmit and receive pulse shaping filters, specifi-cally, with g ( t ) = g tx ( t ) ∗ g rx ( t ) , where g tx ( t ) and g rx ( t ) represents the transmit and receive pulse shaping filters, re-spectively. The TOs are compensated using g tx (cid:0) t + ˆ τ txj (cid:1) and g rx ( t + ˆ τ rxi ) at the j -th transmit and i -th receive antenna,respectively. Thus, the resulting equivalent channel after TOcompensation will be h ∆ τ i,j ( t ) = (cid:104) ˜ H ( t ) (cid:105) i,j ∗ g (cid:0) t − (cid:0) ∆ τ txj + ∆ τ rxi (cid:1) T (cid:1) , where ∆ τ txj = τ txj − ˆ τ txj and ∆ τ rxi = τ rxi − ˆ τ rxi representsthe residual TO at the j -th transmit and i -th receive antenna,respectively.Since the TOs of BS A (BS B) are identical in the downlinkand uplink transmission, the TO estimated during downlink(uplink) for BS A (BS B) as a receiver can also be used tocompensate the TO during uplink (downlink) for BS A (BSB) as a transmitter. Thus, the per-antenna TO compensationcan be implemented by only using the local information ineach BS without feedback. For example, consider an FDDsystem between BS A (with M antennas) and BS B (with N antennas), as illustrated in Fig. 4. The per-antenna TOcompensation scheme is described in Fig. 6. The received Sample shift can be regarded as the sample-level counterpart of lag inSection III-A.
BS A BS B
Correlator timing offset estimator !" $ Obtain by solving problem (11) %& !" ’( )!" %& )*" = %& )!" Compensate timing offset in BS BDuring ULCompensate timing offset in BS B during DL BS A BS B
Downlink transmission
Correlator timing offset estimator ! Obtain by solving problem (11) Compensate timing offset in BS ADuring DLCompensate timing offset in BS A during UL " %& ’ ( ) () ’ *$ = () ’ Implemented Locally
At BS A Implemented Locally
At BS B
Uplink transmission ! " = ! ! & ’$ = ! ’$
1: (
Fig. 6. Illustration of the decentralized per-antenna TO compensation. samples in BS B during uplink transmission are used tocorrelate with the preamble sequences U T transmitted by BSA to obtain the uplink sum-offsets ˆ γ ULB , which is then usedto obtain per antenna TO ˆ τ UL by solving the LS problemwith constraint τ B (1) = 0 (13). The TOs of the antennasof BS B are then compensated by the first N elements in ˆ τ UL , i.e., ˆ τ ULB in the corresponding transmit (receive) pulseshaping filters for the downlink (uplink) transmission. Thesame compensation procedure is implemented for BS A usingdownlink measurements.IV. R
OBUST
MIMO P
RECODER AND D ECORRELATOR D ESIGN
In this section, we propose the precoder/decorrelator de-sign for MAI and ISI suppression. After compensating theTO, there are two physical impairments that will hinder theperformance of the precoder/decorrelator. One is the residualof the TO estimation, which is within one symbol time,i.e., ∆ τ txj (cid:28) T and ∆ τ rxi (cid:28) T , and these residuals willintroduce ISI. Another is the PHN, as illustrated in (8), whichwill introduce MAI. For the convenience of the followingdiscussion, we define ∆ τ = (cid:104) (∆ τ rx ) T , (∆ τ tx ) T (cid:105) T =[∆ τ rx , ..., ∆ τ rxN , ∆ τ tx , ..., ∆ τ txM ] T as the aggregate residual ofthe TO estimation and the PHN matrix (cid:2) Λ [ k ] (cid:3) i,j = θ txj [ k ] + θ rxi [ k ] with θ txj [ k ] = θ txj ( kQ ) and θ rxi [ k ] = θ rxi ( kQ ) , asthe PHN of the k -th symbol at the j -th transmit antenna andthe i -th receive antenna. Since the operations of downlink anduplink are similar in this section, we omit the indicator forspecifying the downlink or the uplink for the channel and theprecoder/decorrelator. The channel state information (CSI) isassumed to be static, and the CSI drift caused by the PHNwill be discussed in Section V. A. Preliminary for Precoder/decorrelator Design
Before presenting the proposed precoder/decorrelator de-sign, the channel estimation procedure is introduced in thissubsection since the CSI is a fundamental preliminary forthe precoder/decorrelator design. Due to the existence of the residual of the TO estimation, there is ISI at each symboltransmission. Hence, we need to estimate the CSI for severaltaps to support the precoder/decorrelator to suppress the ISI.We consider the channels within a finite length window witha W + 1 symbol length. Using the discrete expression of thechannel (6), the aggregate channel to be estimated is given by ¯H = [ H [ − W ] , ..., H [0] , ..., H [ W ]] ∈ C N × (2 W +1) M , (14)where H [0] ∈ C N × M represents the principal channel cor-responding to the current transmitted symbol, while H [ w ] isdefined as the ISI channel with w ∈ {− W, ..., − , ..., W } corresponding to the adjacent interference symbols. Eachsubchannel in ¯H is given by [ H [ w ]] i,j = h ∆ τi,j [ w ] e j [ Λ [ k ∗ + w ]] i,j , ∀ w ∈ {− W, ..., ..., W } , (15)where h ∆ τi,j [ w ] = h ∆ τi,j ( wT ) . k ∗ is the reference symbol; inother words, we assume the principal channel and ISI channelsfor each symbol are identical to the principal channel andISI channels for the k ∗ -th symbol. Since the residual of theTO estimation is within one symbol time and the channelis assumed to be static, the input-output relationship at the k -th symbol in the preamble for channel estimation can beexpressed as y ( k ) = ¯HU T ( k ) + ν [ k ] (16)where U T ( k ) = (cid:104) a ( k + W ) T , ..., a ( k ) T , ..., a ( k − W ) T (cid:105) T ∈ C (2 W +1) M × is the reshaped preamble for channel estimationand ν [ k ] = ν ( kQ ) is the AWGN vector at the k -th symbolwith variance σ . Using L t preamble symbols at the beginningof each frame (except the initial frame for timing) to estimatethe CSI, the received signal Y ∈ R N × L t is given by Y = ¯H [ U T (0) , . . . U T ( L t − ν [0] , . . . , ν [ L t − ¯H ¯ U T + ¯ ν . Then, the CSI estimation is obtained by the LS solution ˆ¯ H = argmin ¯H ∈ C N × (2 W +1) M (cid:13)(cid:13) Y − ¯HU T (cid:13)(cid:13) F = YU T H (cid:0) U T U HT (cid:1) − , (17)where ˆ¯ H = (cid:104) ˆH [ − W ] , ..., ˆH [0] , ..., ˆH [ W ] (cid:105) ∈ C N × (2 W +1) M . B. Optimization Problem Formulation for Pre-coder/decorrelator Design
We use the estimated CSI to optimize the pre-coder/decorrelator. Consider a MIMO system, which cansupport N s = min { N, M } parallel transmission streams.In order to suppress MAI and ISI, we apply a memorylessprecoder, F ∈ C M × N s , at the transmitter, and a memory decor-relator, ˜W =[ W ( − D ) H , ... W ( D ) H ] H ∈ C N (2 D +1) × N s , at thereceiver, where W ( d ) ∈ C N × N s is the decorrelator at the d -thtap. The received signal in the k -th symbol time at the m -thstream can be represented as r m ( k ) = ˜w Hm ( D + W ) (cid:88) w (cid:48) = − ( D + W ) ˜H [ w (cid:48) ] Fs ( k + w (cid:48) ) + ˜w Hm ˜ ν [ k ] , (18)where ˜w m is the m -th column of ˜W , and ˜H [ w (cid:48) ] = ˆH ( w (cid:48) + D ) ... ˆH ( w (cid:48) − D ) , ˜ ν [ w (cid:48) ] = ν ( w (cid:48) + D ) ... ν ( w (cid:48) − D ) is the aggregatechannel and noise, respectively, after considering the memoryeffect at the decorrelator. s ( k ) = [ s ( k ) , . . . s N s ( k )] is thetransmitted symbol vector at the k -th symbol time in the datasection, and we assume s j ( k ) ∀ j and ∀ k are i.i.d with zeromean unit and variance.The existence of both ISI and MAI in Eq.(18) cannot behandled by traditional solutions, such as singular value de-composition (SVD) or naive water filling due to the existenceof both ISI and MAI. Furthermore, considering the practi-cal maximum modulation level constraint (e.g., we assumethat the maximum modulation level that can be supportedin implementation is 4096-QAM), the design of an optimalprecoder/decorrelator is challenging. To achieve a high spec-trum efficiency and incorporate ISI, MAI, and the modulationconstraint, we formulate the following max-sum-rate problem: maximize F , ˜W N s (cid:88) m =1 min { log (1 + SINR m ) , (cid:36) } s.t T r ( F H F ) ≤ P, (19)where (cid:36) is the capacity under a given QAM modulation level, Q M , with a small error (e.g., symbol error rate (SER) < − ). P is the total transmit power, and SINR m is the signal tointerference plus noise ratio (SINR) of the m -th stream, whichcan be expressed as C. Precoder/decorrelator Design via Alternative Optimization
Problem (19) is non-convex and non-smooth, hence isvery challenging to solve. Inspired by [22], we introducean auxiliary variable Γ ∈ C N s × N s and use an alternativeoptimization scheme to find a simple solution to problem (19).Using the variable Γ , problem (19) can be well approximatedby minimize F , ˜W , Γ T r (cid:0) Γ N s (cid:88) m =1 I Tm E ( W , F ) I m (cid:1) − log det( Γ ) s.t T r ( F H F ) ≤ P, (21) [ Γ ] m,m ≤ (cid:36) , ∀ m, Note that signal model (18) can also be applied to pilot symbols bysubstituting s ( k ) with b ( k ) . SINR m = ˜w Hm ˜H [0] f m f Hm ˜H [0] H ˜w m (cid:80) N s m (cid:48) (cid:54) = m ˜w Hm ˜H [0] f m (cid:48) f Hm (cid:48) ˜H [0] ˜w m + (cid:80) ( D + W ) w (cid:48) = − ( D + W ); w (cid:48) (cid:54) =0 ˜w Hm ˜H [ w (cid:48) ] FF H ˜H [ w (cid:48) ] H ˜w m + σ ˜w Hm ˜w m . (20) where I m is the diagonal matrix with diagonal elements as theelements in the m -th column of the identity matrix, and E ( ˜W , F ) = ( ˜W H ˜H [0] F − I )( F H ˜H [0] H ˜W − I )+ ˜W H (cid:0) ( D + W ) (cid:88) w (cid:48) = − ( D + W ); w (cid:48) (cid:54) =0 ˜H [ w (cid:48) ] FF H ˜H [ w (cid:48) ] H + σ I (cid:1) ˜W . (22)The details of this approximation can be found in AppendixA.We then present the detailed update rules using the alter-native optimization method for problem (21). At the t iter -thiteration of the optimization, we alternatively update ˜W , Γ and F by the following 3 steps: • Step 1: Update ˜W ( t iter ) given Γ ( t iter − and F ( t iter − by ˜W ( t iter ) = B − ˜H [0] F ( t iter − , (23)where B = ˜H [0] F ( t iter − ( F ( t iter − ) H ˜H H [0]+ ( D + W ) (cid:88) w (cid:48) = − ( D + W ); w (cid:48) (cid:54) =0 ˜H [ w (cid:48) ] F ( t iter − ( F ( t iter − ) H ˜H [ w (cid:48) ] H (24) + σ I . • Step 2: Update Γ ( t iter ) given ˜W ( t iter ) and F ( t iter − by Γ ( t iter ) = min { [ E ( ˜W ( t iter ) , F ( t iter − )] − m,m , (cid:36) } . (25) • Step 3: Update F ( t iter ) given ˜W ( t iter ) and Γ ( t iter ) bysolving minimize F T r (cid:0) Γ ( t iter ) N s (cid:88) m =1 I Tm E ( ˜W ( t iter ) , F ) I m (cid:1) s.t T r ( F H F ) ≤ P. (26)Problem (26) is convex, which can be efficiently solvedby any solver for convex problems, e.g., CVX Matlabpackage [37].These 3 update rules are obtained by the property that problem(21) becomes convex in terms of any individual variable in F , ˜W , Γ when fixing the other two. This property guaranteesthe sequence generated by the above alternative optimizationconverges to a stationary point of problem (21). To summarize,the precoder/decorrelator can be obtained via Algorithm 1.V. P HASE N OISE E STIMATION AND C OMPENSATION
After the precoder and decorrelator, the MAI and ISI issuppressed to enable high spectral efficiency transmission.However, the phase of the effective channel changes after the
Algorithm 1
Alternative optimization algorithm for pre-coder/decorrelator design Input:
Estimated CSI ˆ¯ H , total power P and noisevariance σ . Output: ˆ F , ˆ W . Initialize:
Construct ˜H [ w (cid:48) ] , w (cid:48) ∈ {− ( D + W ) , . . . , ( D + W ) } by ˆH [ w ] . t iter = 0 , any F (0) satisfy the powerconstraint, Γ (0) = diag (2 (cid:36) , . . . (cid:36) ) . while not converge do t iter = t iter + 1 Update ˜W ( t iter ) according to Eq.(23) and Eq.(24) with F ( t iter − , Γ ( t iter − . Update Γ ( t iter ) according to Eq.(25) and Eq.(22) with F ( t iter − , W ( t iter ) . Update F ( t iter ) by solving problem (26) with ˜W ( t iter ) , Γ ( t iter ) by any solver for convex problems. end while ˆ F = F ( t iter ) , ˆ W = ˜W ( t iter ) .channel estimation stage due to the drifting of the PHN, andthus there will be accumulating MAI caused by the coherenceloss in the precoder and decorrelator. As a result, the PHNneeds to be tracked and compensated in the data transmissionstage, which is the main target of this section. A. Per-Antenna PHN Estimation based on Pilots
From equation (15) and (16), the channel estimation stageabsorbs the initial PHN into the channel phase response. Butas the PHN continues to drift, the effect of coherency losswill become dominant and introduce more MAI. To reset thephase error over one frame, we utilize the N sf − pilotsinserted in a frame after the preamble to estimate the PHNincrement from the preamble to each subframe. However, thepresence of TO error and multipath creates non-negligibleISI, which deteriorates the PHN estimation quality. Thoughwe can accurately estimate the sum-PHN in each ISI channelmatrix as in the channel estimation stage, it requires a longpilot sequence to achieve high accuracy, which introducessignificant overhead. In this section, we exploit the interferencesuppression capability of the precoder/decorrelator to improvethe estimation accuracy.Since the PHN varies very slowly within each pilot se-quence, we assume that the PHN is constant during pi-lot sequence transmission in each subframe. We define the accumulated PHN increment from the preamble to the q -th subframe as φ txj [ q ] = θ txj [ q ( L d + L p )] − θ txj [ k ∗ ] and φ rxi [ q ] = θ rxi [ q ( L d + L p )] − θ rxi [ k ∗ ] for the j -th transmitantenna and the i -th receive antenna, where k ∗ is the referencesymbol we picked in the preamble for channel estimation (aselaborated in Eq.(15)). The collection of the accumulated PHNincrements at the q -th subframe from all M transmit antennas and from all N receive antennas are denoted by φ tx [ q ] (cid:44) [ φ tx [ q ] , ..., φ txM [ q ]] T and φ rx [ q ] (cid:44) [ φ rx [ q ] , ..., φ rxN [ q ]] T . Weassume that after the processing by the precoder/decorrelator,the ISI in the pilot transmissions is canceled, hence thereceived symbols at the ( q + 1) -th pilot is given by R [ q + 1]= D (cid:88) d = − D W [ d ] H D rx ∆ φ [ q ] H [ − d ] D tx ∆ φ [ q ] (cid:124) (cid:123)(cid:122) (cid:125) H q +1 [ d ] FU P + V [ q + 1]= D (cid:88) d = − D W [ d ] H H q +1 [ d ] X + V [ q + 1] , where • D rx ∆ φ [ q ] (cid:44) diag (cid:0) e j ∆ φ rx [ q ] (cid:1) is an M × M diagonalmatrix, where ∆ φ rx [ q ] (cid:44) φ rx [ q + 1] − ˆ φ rx [ q ] collectsthe PHN increments at the receiver; • D tx ∆ φ [ q ] (cid:44) diag (cid:16) e j ∆ φ tx [ q ] (cid:17) is an N × N diagonalmatrix, where ∆ φ tx [ q ] (cid:44) φ tx [ q + 1] − ˆ φ tx [ q ] collectsthe PHN increments at the transmitter; • H q +1 [ d ] (cid:44) D rx ∆ φ [ q ] H [ − d ] D tx ∆ φ [ q ] is the channel cor-responding to the desired pilot signal for tap d at thebeginning of the ( q + 1) -th subframe; • X = FU P denotes the transmitted signal after applyingthe precoder F to the pilots U P and • V [ q + 1] ∈ C N s × L p contains the AWGN noise.To estimate of the PHN increments at R [ q + 1] , we usefirst order Taylor expansion e j ∆ φ ≈ j ∆ φ to approxi-mate D rx ∆ φ [ q ] and D tx ∆ φ [ q ] by I + diag( j ∆ φ tx [ q ]) and I +diag( j ∆ φ rx [ q ]) . We also define Ω m ( k ) ∈ R NM × NM , with (cid:2) Ω dm ( k ) (cid:3) i,j = [ W ( d ) H ] i,m H [ − d ] i,j [ X ] j,k . The collectionof the PHN increments ∆ φ [ q ] (cid:44) (cid:104) ∆ φ rx [ q ] T , ∆ φ tx [ q ] T (cid:105) T ∈ R M + N can be obtained by solving the following LS problem: ∆ ˆ φ [ q ] = arg min ∆ φ [ q ] (cid:107) Ξ ∆ φ [ q ] − (vec( R [ q + 1]) − ζ ) (cid:107) s . t . ∆ φ [ q ] = 0 , (27)where Ξ = (cid:2) ξ T (1) , ..., ξ TN s (1) , . . . , ξ T ( L p ) , . . . , ξ TN s ( L p ) (cid:3) with ξ Tm ( k ) = TMN (2 D +1) diag (cid:110) vec (cid:104) j (cid:0) Ω − Dm ( k ) (cid:1) T (cid:105)(cid:111) ... diag (cid:110) vec (cid:104) j (cid:0) Ω Dm ( k ) (cid:1) T (cid:105)(cid:111) ¯ I NM (28) ∀ m = 1 , . . . N s , k = 1 , . . . L p , and we defined ζ = (cid:34) TMN (2 D +1) η (1) , . . . , TMN (2 D +1) η N s ( L p ) (cid:35) T with η m ( k ) = (cid:104) vec (cid:2) Ω − Dm ( k ) (cid:3) T , ..., vec (cid:2) Ω Dm ( k ) (cid:3) T (cid:105) T (29) ∀ m = 1 , . . . N s , k = 1 , . . . L p . Problem (27) is derived byexpanding R [ q + 1] elementwisely, and the detailed derivationcan be found in Appendix B. The PHN at ( q + 1) -th pilot is then updated by ˆ φ [ q + 1] = ˆ φ [ q ] + ∆ ˆ φ [ q ] for q = 0 , ..., N sf − , where ˆ φ [0] = 0 as we assume PHN at the preamble isabsorbed to the channel and is perfectly estimated during thechannel estimation stage. B. Per-Antenna PHN Compensation
We focus on the per-antenna PHN compensation in thissubsection. Traditionally, the PHN information obtained at thereceiver is feedback to the transmitter for PHN compensation.However, per-symbol feedback of PHN estimates will incurtoo much overhead. Fortunately, the transmitter and receivershare the same oscillator in an FDD system so that the PHNis identical in the uplink and downlink at the same site, whichenables local compensation of the PHN and avoids the over-head for PHN feedback. Consider the transmission datapath inFig. 4, the PHN estimated during downlink (uplink) for BS A(BS B) as a receiver can also be used to compensate the PHNduring uplink (downlink) for BS A (BS B) as a transmitter.Thus, PHN compensation can be implemented by using localinformation only without feedback.However, unlike the TO compensation, where the reference τ is usually known a priori, ∆ φ [ q ] in (27) is not availablein advance to both BS A and BS B. Thus, there will be a common PHN error across all the antennas at the receiver. Ifwe directly use the result ∆ ˆ φ [ q ] to compensate the PHN andthe common PHN error is identical to the PHN of the firstlink, i.e., the sum-PHN at the first receive antenna and firsttransmit antenna. For example, let ∆ ˆ φ UL [ q ] = (cid:20)(cid:16) ∆ ˆ φ ULB [ q ] (cid:17) T , (cid:16) ∆ ˆ φ ULA [ q ] (cid:17) T (cid:21) T and ∆ ˆ φ DL [ q ] = (cid:20)(cid:16) ∆ ˆ φ DLA [ q ] (cid:17) T , (cid:16) ∆ ˆ φ DLB [ q ] (cid:17) T (cid:21) T be the estimated ( M + N ) PHN vector obtained by solvingproblem (27) in BS B during uplink and BS A duringdownlink, respectively. When BS A and BS B compensatethe PHN using local information ∆ ˆ φ DLA [ q ] and ∆ ˆ φ ULB [ q ] respectively, the common PHN error at BS A and BS B can becalculated locally by ˆ φ DLA,CM [ q + 1] = ∆ ˆ φ DLA, [ q ] + ∆ ˆ φ DLB, [ q ] and ˆ φ ULB,CM [ q + 1] = ∆ ˆ φ ULB, [ q ] + ∆ ˆ φ ULA, [ q ] , where ∆ ˆ φ DLA,i [ q ] denotes the i -th element of ∆ ˆ φ DLA [ q ] .The per-antenna PHN compensation scheme is described inFig. 7. The received symbols R UL [ q + 1] in BS B during up-link transmission are used to compute the estimation for PHNincrement ∆ ˆ φ UL [ q ] by solving problem (27), which is thenused to obtain per antenna PHN accumulation ˆ φ ULB [ q + 1] and common phase error accumulation ˆ φ ULB,CM [ q + 1] at BSB. The PHN of the antennas of BS B are then compensatedby ˆ φ ULB [ q + 1] in the downlink, while they are compensatedby ˆ φ ULB [ q + 1] + ˆ φ ULB,CM [ q + 1] during the uplink. The samecompensation procedure is implemented for BS A using thedownlink measurements. BS A BS B
Uplink transmission
PHN Estimator by solving problem (24)
Update PHN Compensate PHN in BS B During UL
Update common phase error
BS A BS B
Compensate PHN in BS B During DL
PHN Estimatorby solving problem (24)Update PHN Compensate PHN in BS A During DLUpdate common phase error
Compensate PHN in BS A During UL ! " % Implemented LocallyAt BS A
Implemented LocallyAt BS B !" (cid:28607)(cid:28595) $ % (cid:28607) & (cid:28607) ’ (" (cid:28607)(cid:28595) $ % (cid:28607) & (cid:28607) ’ ) !" *+1 ) (" *+1 ,- . !" *+1 =,- . !" * +/,- . !" * ,- .023 !" *+1 =,- .023 !" *+/,- .04 !" * (cid:28606) /,- !" * ,- .!" *+1 , - .023!" *+1 ,- . (" *+1 (cid:28624) , - .!" *+1 /, - (" * ,- (" *+1 =,- (" * +/,- (" * ,- (" *+1 =,- (" *+/,- (" * (cid:28606) /,- .04 (" * ,- *+1 , - *+1 ,- !" *+1 (cid:28624) , - *+1 Downlink transmission
Fig. 7. Per-antenna PHN compensation scheme.
C. Per-Antenna PHN Tracking based on Decision Feedback
To track the PHN parameter of each antenna in a subframe,we proposed a decision-feedback (DFB) estimator to supportPHN estimation during data transmission between consecutivepilots.The L d symbols in a subframe is split to N b data blockswhere each data block contains L b symbols such that L d = N b L b . In the practical scenario of interest, the PHN is varyingslowly compared to symbol time, the PHNs during each datablock are assumed to be constant and we denote the PHNof the p -th data block in the q -th subframe by φ tx [ p ; q ] and φ rx [ p ; q ] for the transmitter and receiver, respectively, anddefine φ [ p ; q ] = (cid:104) φ rx [ p ; q ] T , φ tx [ p ; q ] T (cid:105) T . Therefore, in theDFB case, the L b previously detected data symbols ¯ S [ p ; q ] inthe p -th data block of the q -th subframe are used to estimatethe PHN parameters φ [ p ; q ] . For simplicity, we assume perfectdecision-feedback during data transmission.Based on the above assumptions, the estimate of PHNincrement ∆ φ [ p ; q ] = (cid:104) ∆ φ rx [ p ; q ] T , ∆ φ tx [ p ; q ] T (cid:105) T fromthe p -th to the ( p + 1) -th data block is obtained by solving ∆ ˆ φ DFB [ p ; q ] = arg min ∆ φ (cid:107) R [ p + 1; q ] − D (cid:88) d = − D W ( d ) H D rx ∆ φ H [ − d ] D tx ∆ φ F ¯ X [ p + 1; q ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (30) where R [ p + 1; q ] is the received symbols of the ( p + 1) -th data block in the q -th subframe, and ¯ X [ p + 1; q ] = F ¯ S [ p + 1; q ] . The problem can also be approximated by theTaylor expansion on small PHN increments between datablocks and solved using the same technique introduced in pilotPHN estimation as specified in Section V-A. The estimationfor accumulated PHN via DFB at the ( p + 1) -th data block isthen updated by ˆ φ DF B [ p + 1; q ] = ˆ φ [ p ; q ] + ∆ ˆ φ DF B [ p ; q ] . However, the DFB estimator may be prone to errors inthe detection of symbols. Instead, we use moving windowaveraging to track the evolution of the PHN, i.e., the estimatedPHN of φ [ p + 1; q ] is given by ˆ φ [ p + 1; q ] = (1 − α ) ˆ φ [ p ; q ] + α ˆ φ DF B [ p + 1; q ] , (31) -30 -25 -20 -15 -10 -5 0 XPD (in dB) -3 -2 -1 T i m i ng o ff s e t e rr o r Proposed preamble with LSProposed preamble without LS Walsh sequenceZC sequence
Fig. 8. TO estimation error using the proposed preamble sequence,the ZCsequence, and theWalsh sequence. where ≤ α ≤ , p = 0 , , ..., N b , and ˆ φ [0; q ] = ˆ φ [ q ] takesthe pilot-aided PHN estimation result. In (31), ˆ φ [ p ; q ] is theestimated PHN from the history, while ˆ φ DF B [ p + 1; q ] is thenew PHN estimate from the DFB in the ( p + 1) th data block.By adjusting α , we strike a balance between the history andthe innovation from the DFB.VI. S IMULATIONS AND D ISCUSSIONS
In this section, the performance of the proposed massiveMIMO mmWave backhaul system is evaluated. In the fol-lowing results, we consider multiple practical cases where GHz carrier is used and × flat-panel dual-polarized MIMOantennas arrays are equipped at the transmitter and receiver totransmit data at a distance D = 3 km, as illustrated in Fig.1. Within a single the antenna array, antenna element spacingis d λ , and the XPD between H mode and V mode ineach dual polar element varies from dB to dB. All thetransmitted signals have been normalized to have unit averagepower, with symbol duration T = 40 ns. The propagationdelay associated with the interpath is τ d = 6 . ns and the notchdepth ρ in the reflected path varies from dB to dB. Thevariance of the AWGN, is set to σ = SNR with SNR= dB. The maximum TOs is τ max · T = 200ns ( symbolstime). The variance of the PHN increment process over onesymbol duration is σ = 10 − rad and we set α = 0 . forPHN tracking. The frame structure parameters are chosen as L t = 256 , L p = 64 , L d = 1280 and N sf = 100 . The numberof taps for the ISI channel and the memory decorrelator arerespectively, W = 3 and D = 3 . In the simulation results, wewill first demonstrate the efficiency of the proposed methodsin each component of this system, i.e., TO estimation andcompensation, precoder and decorrelator, PHN estimation andcompensation. Finally, we will show the end-to-end simulationresults, which incorporate all the components proposed in thiswork. A. Performance of Individual Design
Figure. 8 shows the performance comparison of the LS-based TO estimator with the proposed preamble sequence, Proposed method
SVD
30 40 50 60 70 bits/s/Hz
Fig. 9. Theoretical sum-rate performance (Eq.(19)) of the proposed methodand SVD under different physical impairments, ρ = 10 , SNR= dB, N = M = 8 . A × grid is constructed for each algorithm with the performanceaveraged by 100 independent trials at each grid point. traditional ZC sequence, and Walsh sequence of same length L t = 256 . The performance metric for TO estimation is thesquare root of the MSE of the effective sum-offset in eachMIMO link, which is defined as (cid:114) E (cid:104) || ˆ γ − γ || N (cid:105) , where γ isthe sum-offsets as defined in Section III, and the expecta-tion is taken over 500 realizations. ZC sequence and
Walshsequence act as the baselines to estimate the sum-offsetsvia the LS-based method.
Proposed preamble without LS isanother baseline that only adopts the proposed preamble todirectly estimate the sum-offsets without the LS procedure in(13). The proposed preamble with LS is the proposed TOsestimation method, in which ˆ γ ∈ R N × is reconstructedfrom ˆ τ ∈ R N × by equation (13). As shown in Fig. 8,the proposed method shows superior performance over theother baselines with XPD ranging from dB to dB,and the proposed preamble is more robust to higher XPD.The proposed preamble only shows a significant performancedecay when XPD ≥ dB. However, the performance of theZC sequence and Walsh sequence degrades significantly forXPD ≥ dB. The LS procedure gives us more than a 2 timeshigher accuracy in terms of square root MSE for the proposedpreamble.To verify the robustness of the precoder/decorrelator, multi-ple simulations are performed under different physical impair-ments. Specifically, the received signal is generated by (8) con-sidering one sample per symbol with τ max /T ∈ (0 . , . and σ ∆ ∈ (0 . , . . The estimated CSI ¯ H required for theprecoder/decorrelator is obtained by the channel estimation insubsection IV-A. In the optimization problem for the proposedmethod, (cid:36) is set as the capacity under the -QAM and(SER) < . . Fig. 9 presents the theoretical sum-rateperformance (Eq.(19)) comparisons between the proposed pre-coder/decorrelator and the SVD solution. The results show thatthe proposed method outperforms SVD in all the consideredscenarios.Fig. 10 shows the performance of PHN estimation in pilotsover a frame for 100 Monte-Carlo trials. The performancemetric is the rooted MSE of the effective sum-PHN in eachMIMO link. The accumulated sum-PHNs at q -th pilot, denoted XPD (dB) -3 -2 -1 s uqa r e r oo t o f M SE o f P HN e s t i m a t i on Extract sum-PHN, ignore ISIExtract sum-PHN, consider ISIProposed PHN estimation
Fig. 10. PHN estimation error in pilots over a frame. as vector ϑ [ q ] , can also be obtained from the per-antenna PHN φ [ q ] via the transformation ϑ [ q ] = ¯ I NM φ [ q ] . The baselinesextract the PHN by first getting an estimate of the principalchannel during the q -th pilot transmission ˆ H q [0] , then ϑ [ q ] isestimated by [6] ˆ ϑ [ q ] = ∠ vec (cid:16) ˆ H q [0] (cid:17) − ∠ vec (cid:16) ˆ H [0] (cid:17) . (32)In getting the principal channel, the first baseline ignoresthe ISI and uses an LS algorithm [16][6] ˆ H q [0] = Y [ q ] X H (cid:0) XX H (cid:1) − to directly estimate the principal chan-nel, while the second baseline uses the pilots to estimate theprincipal and ISI channels with the LS-based channel esti-mation method introduced in Section IV-A, and then extractsthe desired principal channel ˆ H q [0] . As observed from Fig.10, for 0 dB ¡ XPD ¡ 20 dB, the proposed PHN estimation method reaches an estimation error at about 0.002 rad, whilethe errors of the two baselines increase rapidly to 0.01 rad asXPD increases. This is because, with higher XPD, the phaseof cross-polar links is more sensitive to the MAI and noisesdue to small channel power in these links. It is noted that atlow XPD (around 0dB), the proposed PHN estimation methodstill outperforms the baselines because it leverages the ISIsuppression capability of the precoder and decorrelator. Theperformance of the proposed PHN estimation method starts todecay for high XPD ( > dB) due to loss of measurementsin the cross-polar links, but it still outperforms the baselines. B. End-to-End Performance
For the end-to-end performance evaluation, we perform 200Monte-Carlo trials (i.e., independent frames) with XPD = 20 dB and ρ = 10 dB in which our proposed holistic method iscompared to several baselines. These baselines partially adoptdifferent techniques in the system: • Baseline 1 [6]: This baseline first estimates the accu-mulated sum-PHN ˆ ϑ [ q ] by (32). Then, the sum-PHN isconverted back to the original transmitter and receiverPHN ˆ φ [ q ] using the pseudo-inverse of A NM . Based onthe information of ˆ φ [ q ] , PHN de-rotation operations atboth the receiver and the transmitter are applied beforeand after an equalizer to compensate per antenna PHN. The equalizer is calculated based on the principal channelestimated from the received preamble to suppress theMAI. Finally, an MMSE-FIR filter per stream is per-formed to suppress the ISI. • Baseline 2 [16]: This baseline extracts accumulated sum-PHN ϑ [ q ] in the estimated channel matrix during pilottransmission by (32). This sum-PHN is then combinedwith the principal channel matrix estimated during thepreamble transmission. An MMSE linear decorrelator isconstructed based on the combined channel to equalizethe effect of the PHN and channel gain. However, sincethis method pays no attention to the coherence loss effectand the ISI, thus it shows an inferior performance. • Baseline 3 (SVD): This baseline uses the unitary matricesof the SVD on the estimated principal channel as theprecoder and decorrelator. Because this method ignoresthe ISI and MAI, thus it suffers a poor performance inour considered scenario.In this experiment, channel coding is not considered andall the approaches adopt the same timing synchronization.To perform an end-to-end experiment and show reasonableresults, we take an adaptive modulation strategy by estimatingthe SINR via Eq. 20 before data transmission. After the pre-coder/decorrelator calculation, modulation levels according tothe estimated SINR are suggested according to the relationshipbetween the modulation level, SINR, and SER where we fixSER < . for each stream. Then, the data is transmittedwith the Gray mapped Q M -QAM symbol where Q M is thesuggested modulation level and is demodulated with the harddecision. Note that practical adaptive modulation may not beoperated in this way, but taking this strategy will not influencethe performance comparisons.To verify the superiority of the proposed scheme,we first show per stream SINR to verify the reliabilityand efficiency of the system in a detailed view.Specifically, for the m -th stream, SINR m ( dB ) =10 log E k (cid:20) || w Hm (cid:16) e j diag ( ∆ φ rx [ k ] ) H [0] e j diag ( ∆ φ tx [ k ] ) (cid:17) s ( k ) || || r m ( b ) − w Hm (cid:16) e j diag(∆ φ rx [ k ]) H [0] e j diag ( ∆ φ tx [ k ] ) (cid:17) s ( k ) || (cid:21) .In Fig. 11, per stream SINR performance is given amongdifferent methods. The highest QAM modulation level thatcan be achieved by the proposed method at each stream isalso given. It is obvious that the proposed method outperformsthe baselines of each stream with an over 10 dB gain, and themodulation level achieved by the proposed method can not beafforded by baselines. To show an overall performance, wethen compare the BER and
Spectrum Efficiency in Table. I.Note that both results are averaged over uplink and downlinkeffective transmissions. From Table. I, the results show thatthe proposed method can achieve an over times spectrumefficiency gain compare to all the baselines with a nearlyidentical BER. VII. C ONCLUSION
This paper proposes a holistic solution to achieve high spec-tral efficiency for an LoS MIMO system equipped with dual-polarized antenna arrays for wireless backhaul using FDD.Various physical impairments such as TOs and phase noises
Stream Index S I NR ( d B ) Proposed MethodBaseline1 Baseline2Baseline3
Fig. 11. Per-stream SINR performance in the End-to-End experiment.TABLE IE ND - TO -E ND BER
AND S PECTRUM E FFICIENCY PERFORMANCE . BER( − ) Spectrum Efficiency(bits/s/Hz) Proposed 0.2221 60Baseline 1 0.6740 15Baseline 2 0.2036 8Baseline 3 0.4401 8 as well as the effect of multipath are taken into account. AnLS-based TO estimator is proposed with a new set of preamblesequences to perform timing synchronization. After channelestimation, an optimization-driven precoder/decorrelator de-sign is proposed to suppress the MAI and ISI introduced by themultipath to maximize the theoretical sum-rate. Additionally,a PHN estimation, tracking, and compensation scheme isproposed to tackle the problem of the coherence loss ofprecoder/decorrelator due to the drifting of the PHN overa frame. With a simulation setup in a × LoS MIMOdemonstrator, the proposed method can achieve high spectralefficiency at 60 bits/s/Hz. Simulation results also show thatour proposed solution outperforms the existing baselines.A
PPENDIX AD ERIVATION OF PROBLEM (21)The derivation is similar to [22, Appendix A], exceptfor the procedure used to approximate the non-smoothnessin the original problem (19). First, when fixing the othervariables, the optimal decorrelator, ˜W is the MMSE decor-relator as shown in (23). Second, to update the auxiliaryvariable Γ ∈ C N s × N s , we first check the first-order optimalityof Γ and then do the projection by considering the boxconstraint for it. The optimal Γ is a diagonal matrix with Γ m,m = min { [ E ( ˜W , F )] − m,m , (cid:36) } . Substituting the optimal ˜W and Γ into (19) we have maximize F − N s (cid:88) m =1 min { [ E ( ˜W ( F ) , F )] − m,m , (cid:36) } [ E ( ˜W ( F ) , F )] m,m + N s (cid:88) m =1 min { log(1 + SINR m ) , (cid:36) } s.t T r ( F H F ) ≤ P, where − N s (cid:80) m =1 min { [ E ( ˜W ( F ) , F )] − m,m , (cid:36) } [ E ( ˜W ( F ) , F )] m,m is a constant when min { [ E ( ˜W ( F ) , F )] − m,m , (cid:36) } =[ E ( ˜W ( F ) , F )] − m,m , ∀ m . When a stream has a verysmall MMSE (or high SINR), there is truncation, i.e., min { [ E ( ˜W ( F ) , F )] − m,m , (cid:36) } = 2 (cid:36) . However, this truncationusually causes a small error when (cid:36) is very large, since inthis case, [ E ( ˜W ( F ) , F )] m,m is very small.A PPENDIX BD ERIVATION OF PROBLEM (27)Elementwisely, the { m, k } -th element of R [ q + 1] can beexpressed as r m ( k | q + 1)= D (cid:88) d = − D N (cid:88) i =1 M (cid:88) j =1 W ∗ im ( d ) h ij [ − d ] x j ( k ) e j ( ∆ φ rxi [ q ]+∆ φ txj [ q ] ) + v m ( k ) ≈ D (cid:88) d = − D N (cid:88) i =1 M (cid:88) j =1 α dm,i,j ( k ) (cid:2) j (cid:0) ∆ φ rxi [ q ] + ∆ φ txj [ q ] (cid:1)(cid:3) + v m ( k ) , (33) where ∆ φ rxi [ q ] and ∆ φ txj [ q ] represents the i -th and j -th ele-ment of ∆ φ rx [ q ] and ∆ φ tx [ q ] , respectively, x j ( k ) = [ X ] j,k is the k -th transmitted signal at the j -th transmit antenna, andwe define α dm,i,j ( k ) (cid:44) W ∗ im ( d ) h ij [ − d ] x j ( k ) . In (33), weused first-order Taylor expansion e j ∆ φ ≈ j ∆ φ for smallPHN variation ∆ φ for the approximation.As seen from (33), each element r m ( k | q + 1) is an observa-tion for the M + N PHN parameters, but each transmitted sig-nal is distorted by the sum-PHN process ∆ φ rxi [ q ] + ∆ φ txj [ q ] .The M × N sum-PHN can be calculated in principle if wehave adequate observations, but the transformation back totheir original form ∆ φ [ q ] = (cid:104) (∆ φ rx [ q ]) T , (∆ φ tx [ q ]) T (cid:105) isimpossible [15] as we lack a reference phase. To eliminatethe degree of freedom, we can just fix ∆ φ rx [ q ] = 0 whilepreserving the effective sum-PHN in each link.Using matrix representation, we define the matrix (cid:2) Ω dm ( k ) (cid:3) i,j = α dm,i,j ( k ) , the vector ξ m ( k ) and η m ( k ) in(28) and (29), respectively, and the approximation in (33) canbe rewritten as r m ( k | q + 1) − TMN (2 D +1) η m ( k ) ≈ ξ Tm ( k ) ∆ φ [ q ] + v m ( k ) , ∀ m = 1 , ..., N s , k = 1 , ..., L p , where ¯ I NM is defined in Eq.12. Stacking the observations intoa big column vector during the ( q + 1) -th pilot transmission,we shall arrive at the LS problem as (27). A BOUT THE A UTHORS R EFERENCES[1] Naga Bhushan, Junyi Li, Durga Malladi, Rob Gilmore, Dean Brenner,Aleksandar Damnjanovic, Ravi Teja Sukhavasi, Chirag Patel, and StefanGeirhofer. Network densification: the dominant theme for wirelessevolution into 5G.
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