Interference-Nulling Time-Reversal Beamforming for mm-Wave Massive MIMO in Multi-User Frequency-Selective Indoor Channels
aa r X i v : . [ c s . I T ] J un Interference-Nulling Time-ReversalBeamforming for mm-Wave Massive MIMO inMulti-User Frequency-Selective IndoorChannels
Carlos A. Viteri-Mera and Fernando L. Teixeira
Abstract
Millimeter wave (mm-wave) and massive MIMO have been proposed for next generation wirelesssystems. However, there are many open problems for the implementation of those technologies. Inparticular, beamforming is necessary in mm-wave systems in order to counter high propagation losses.However, conventional beamsteering is not always appropriate in rich scattering multipath channels withfrequency selective fading, such as those found in indoor environments. In this context, time-reversal(TR) is considered a promising beamforming technique for such mm-wave massive MIMO systems. Inthis paper, we analyze a baseband TR beamforming system for mm-wave multi-user massive MIMO.We verify that, as the number of antennas increases, TR yields good equalization and interferencemitigation properties, but inter-user interference (IUI) remains a main impairment. Thus, we propose anovel technique called interference-nulling TR (INTR) to minimize IUI. We evaluate numerically theperformance of INTR and compare it with conventional TR and equalized TR beamforming. We usea 60 GHz MIMO channel model with spatial correlation based on the IEEE 802.11ad SISO NLoSmodel. We demonstrate that INTR outperforms conventional TR with respect to average BER peruser and achievable sum rate under diverse conditions, providing both diversity and multiplexing gainssimultaneously.
Index Terms
The authors are with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio StateUniversity, 1330 Kinnear Rd., Columbus, OH 43212 USA, (614) 292-6993 (e-mail: { viteri.5,teixeira.5 } @osu.edu).C. Viteri-Mera is also with the Department of Electronics Engineering, Univeridad de Nari˜no, Pasto, Colombia. Time-reversal, beamforming, mm-wave, massive MIMO, interference mitigation.
I. I
NTRODUCTION
Massive MIMO systems have been recently recognized as one of the technologies that canbring unprecedented performance gains for next generation wireless communications [1]. Amongits potential benefits, noise, fading and inter-user interference (IUI) effects have been shownto progressively reduce as the number of antennas in the system increases [2]. Thus, a largenumber of antennas simplifies the multiple access layer and increases the system’s capacity [3].However, many challenges remain for the implementation of massive MIMO systems such as:channel estimation and reciprocity issues, large pilot overheads, hardware cost, size and powerlimitations, network architecture adaptations, antennas and propagation aspects [4].Recently, millimeter wave (mm-wave) and massive MIMO have been proposed in tandem fornext generation systems [5], [6]. This can be easily justified because a large number of antennasoperating at mm-wave frequencies (e.g. 28, 38, 60 and 73 GHz) can be used in compact devicesdue to the small wavelength (4 to 10 mm approx.) and (hence) small antenna sizes. In addition,it has been shown that mm-wave networks are suitable for dense small cells (especially in indoorenvironments), as inter-cell interference is naturally mitigated due to high propagation losses atthose frequencies [7], [8]. Another benefit of using mm-wave is the huge bandwidth availability,with some standards planning to operate with more than 2 GHz bandwidth, e.g. [9].Nevertheless, a number of problems arise for mm-wave massive MIMO systems. In particular,their performance is highly dependent on the antenna array configuration and the propagationenvironment. Hence, factors such as the coupling between antennas and channel spatial corre-lation play a significant role on the actual capacity and diversity gain that this kind of systemscan achieve [4].Beamforming is needed in mm-wave systems because of large propagation losses. Tradi-tionally, antenna arrays use beamsteering techniques in order to increase the received powerin specific directions [10] and, consequently, achieve diversity gain. This is usually performedwith either analog (RF) or digital (baseband) phase shifters in each antenna. Recent approaches to beamforming in mm-wave massive MIMO use hybrid analog beamsteering combined withdigital precoding techniques assuming narrowband fading channels [5], [14]–[17]. This hybridanalog/digital solutions are necessary given that fully digital solutions requires one digital toanalog converter per antenna, which is extremely constly in terms of power. However, con-ventional beamsteering is not always appropriate in multipath channels with frequency selectivefading, such as those found in indoor environments. In those cases, more sophisticated techniquesare required to take full advantage of the number of elements in the array and also multipathpropagation.The specific propagation aspects of mm-wave systems have been recently studied. Statisticalmodels for mm-wave channels have been developed in [18], [19], where extremely narrowantenna radiation patterns are considered using massive MIMO. These models provide charac-terization of scattering clusters in the angular and delay domains, power-delay profiles (PDPs),and propagation losses in outdoor scenarios. Similar models can be found for indoor scenarios[20]. Another popular SISO channel model for indoor mm-wave systems is the IEEE 802.11ad[21], which considers extremely narrow radiation patterns and analog beamsteering. However,there are only few studies on the spatial correlation in mm-wave MIMO channels. An interestingwork is [22], where it is demonstrated that correlation at 60 GHz can be very high due to thesmall number of multipath components (MPCs). It has also been recognized that the specificstructure of spatial correlation is highly dependent on the scattering environment. Given thisconclusions, it is not clear yet whether diversity and/or multiplexing schemes should be used inorder to maximize the system’s gain [23].In this paper we propose a novel time-reversal (TR) based [24] solution to the multi-userbeamforming problem for indoor scenarios in mm-wave massive MIMO, which provides bothdiversity and multiplexing gains. TR is a transmission technique that enables spatial focusing ofthe signal at the receiver by using the time-reversed channel impulse response (CIR) as a linearfilter applied to the transmitted signal [12], [13]. TR is considered a promising technique forfuture massive MIMO systems [4]. By using TR, all multipath components are added in phase at The term beamforming is traditionally used to denote phased array techniques for beam steering, i.e. operating in the 2Dmanifold spanned by the azimuth and elevation angles. In this paper, we shall use the term beamforming in a broader senseto denote signal processing techniques that allow spatial focusing of RF power in co-range as well (3D) or even in time (4Dspace-time beamforming) [11]–[13]. the receiver at a specific instant providing i ) an increase in the signal power in the surroundingsof the receiver (commonly referred to as spatial focusing), and ii ) a partial equalization effect(commonly known as time focusing) that reduces inter-symbol interference (ISI) caused by thechannel’s frequency selectivity [25], [26]. This features enable low computational complexityreceivers, which is a key advantage of TR with respect to multicarrier (OFDM-like) systems [27].Moreover, multipath components add incoherently at regions in space away from the receiver,mitigating interference to other users [28], [29].A number of works have addressed different aspects of TR beamforming, with particular focuson single user systems [12], [26], [28]–[30]. In these references, the spatial and temporal focusingproperties of TR have been considered, and both theoretical and empirical characterizations ofbit error rate (BER) have been made under specific scenarios and channel models. A commonfinding in the literature is that ISI is the main limiting factor of TR. This is because ISI imposesa lower bound in the achievable BER at high signal to noise ratios (SNR) in single user systems[29].The challenge of mitigating ISI in TR has also received increasing attention. Different equaliz-ing solutions have been proposed in [29], [31], [32] for single-user systems. An important resultof [29] is that the ratio between the desired signal power and the ISI power in TR increaseslinearly with the number of antennas. Thus, BER performance can potentially have a significantimprovement when TR is applied in massive MIMO, without additional equalization.For multiuser systems, TR for multiple access in the downlink was proposed in [33] and [34],where IUI is recognized as the main limiting factor of BER performance. Also, [24] proposesseveral multi-user TR techniques. A multiple access TR technique that uses rate-backoff isproposed in [35], where an approximation for the signal-to-interference-plus-noise-ratio (SINR)is given, showing that it increases with the number of antennas.However, previous works have not addressed the following aspects: ‚ Proposed beamsteering techniques in mm-wave massive MIMO are narrowband (for flatfading channels), and do not take take full advantage of multipath propagation to increasediversity gain. Thus, these techniques may not be appropriate for frequency selective chan-nels found in indoor scenarios. ‚ TR beamforming techniques, which have been thoughtfully analyzed in other scenariosand take advantage of rich scattering, have not been studied int he context of mm-wave massive MIMO. More specifically, [35] and [29] suggest that SINR in conventional TRgrows linearly with the number of antennas, enabling low complexity receivers.In this context, the contributions of this paper are the following: ‚ We introduce a simple channel model for 60 GHz massive-MIMO, which is based on theIEEE 802.11ad model [21]. We define the probability distribution of the channel taps, theirPDP, and spatial correlation. ‚ We study the performance of conventional TR in multi-user systems when the number ofantennas at the transmitter is very large. Moreover, we generalize the ETR [29] to multi-usersystems, and compare its performance with conventional TR. We demonstrate that, provideda sufficiently large number of transmit antennas, TR does not need further equalization,becoming an attractive beamforming alternative. ‚ Using the previous analysis, where we find that conventional TR performance is IUI-limited,we propose a novel TR multi-user beamforming technique that minimizes IUI and exploitsrich multipath commonly found in indoor environments. We call this technique interference-nulling time-reversal (INTR). ‚ We analyze and compare numerically the performance of these TR techniques using theproposed statistical MIMO channel model for 60 GHz.
Commonly Used Acronyms in this Paper
AP - Access Point; CB - cubicle scenario; CIR - channel impulse response; CR - conferenceroom scenario; ETR equalized timer-reversal; INTR - interference-nulling time-reversal; ISI -inter-symbol interference; IUI - inter-user interference; LR - living room scenario; MPC - mul-tipath component; PDP - power-delay profile; TR - time-reversal; US - uncorrelated scattering.
Notation
Lower and upper case symbols represent signals in the time and frequency domains, re-spectively. Boldface symbols represent vectors or matrices, whose dimensions are specifiedexplicitly. b is the convolution operator between two signals. E r¨s represents expectation over arandom variable. The operators p¨q T , p¨q ˚ , p¨q H and p¨q ´ represent transpose, complex conjugate,Hermitian transpose, and matrix inverse, respectively. The norm of the vector a is denoted as } a } “ a x a , a y , where x a , b y “ b H a represents the complex inner product of vectors a and b . The superscripts p¨q tr , p¨q eq , and p¨q in denote variables calculated using time-reversal, equalizedtime-reversal, and interference-nulling time-reversal pre-filters, respectively.II. T IME -R EVERSAL B EAMFORMING S YSTEM M ODEL
In this section, we present the general discrete signal model for TR beamforming. We first gen-eralize to the multi-user case two TR techniques for single-user scenarios [29]. These techniquesserve as a baseline comparison for the novel INTR introduced in Section III.
A. General TR Signal Model
Consider a digital baseband downlink wireless communication system, consisting of oneAccess Point (AP) with M transmit antennas and N single-antenna user terminals as depictedin Fig. 1. The transmitter has a very large number of antennas, so M " N . We denote thetransmit antenna set as M “ t , , . . . , M u and the user set as N “ t , , . . . , N u . Also, let m, m P M and n, n P N be arbitrary elements in those sets. The AP transmits simultaneouslyan independent data stream to each user. Let s n p t q be the complex random signal transmitted tothe n -th user, where t P Z ` is the discrete time index. These transmitted signals are assumedto have unit average power, i.e. E “ | s n p t q| ‰ “ , @ n, t , regardless of the modulation. In a TRmulti-user system, the transmitter sends independent signals simultaneously to the users usingdifferent pre-filters for each one of them. Thus, the baseband transmitted signal from the m -thantenna is x m p t q “ ? ρ N ÿ n “ s n p t q b p ˚ m,n p´ t q , (1)where ρ is the total average transmitted power in the AP, p m,n p t q is the power-normalizedpre-filter from the m -th transmit antenna to the n -th user (with a duration of L p samples, i.e. t “ , . . . , L p ´ ), and h m,n p t q is the random channel impulse response (CIR) from the m -thtransmit antenna to the n -th user (with a length of L samples). The random CIR vector to the n -th user is defined as h n p t q “ r h ,n p t q , . . . , h M,n p t qs T P C M . (2)In Section IV, we introduce the statistical characterization of h m,n p t q for mm-wave channels.Let H m,n p f q be the discrete Fourier transform (DFT) of h m,n p t q . In an analogous way to the Fig. 1. System model. An AP with M transmit antennas sends simultaneously an independent data stream to N single antennausers using time-reversed pre-filters p ˚ m,n p´ t q . time domain representation, the steering vector to the n -th user is H n p f q “ r H ,n p f q , . . . , H M,n p f qs T P C M . (3)The selection of p m,n p t q depends on the particular TR technique, as discussed later in thissection. We define the pre-filter vector to the n -th user as p n p t q “ r p ,n p t q , . . . , p M,n p t qs T P C M . (4)Let P m,n p f q be the DFT of p m,n p t q . Then, we define the frequency domain pre-filter vector tothe n -th user as P n p f q “ r P ,n p f q , . . . , P M,n p f qs T P C M . (5) The received baseband signal at user n is y n p t q “ ? ρ s n p t q b M ÿ m “ p ˚ m,n p´ t q b h m,n p t q looooooooooooooooooooomooooooooooooooooooooon signal directed to the n -th user ` ? ρ N ÿ n “ n ‰ n M ÿ m “ s n p t q b p ˚ m,n p´ t q b h m,n p t q looooooooooooooooooooooooomooooooooooooooooooooooooon IUI ` z n p t q loomoon noise , (6)where z n p t q represents additive white Gaussian noise (AWGN) with variance σ z . Next, we extendthe conventional TR and equalized TR single-user formulation in [29] to multi-user scenariosby explicitly defining the pre-filter p m,n p t q in terms of the CIR. Note that p m,n p t q is properlynormalized so the transmitted power ρ regardless of the number of antennas or users. B. Multiuser Conventional TR Beamforming
The general idea behind TR is to use the time-reversed CIR from every antenna to the receiveras a pre-filter for the transmitted signal. Such pre-filter acts as a beamformer in the spatial domain,focusing the RF signal around the receiver. In conventional TR, assuming perfect channel stateinformation (CSI) at the transmitter, the pre-filter vector is p trn p t q “ h n p L ´ ` t q a P trh , (7)where P trh is a normalization factor introduced to ensure that the total transmitted power remainsconstant in every realization, this is P trh “ N ÿ n “ L ´ ÿ l “ } h n p l q} . (8)Note that, in this case, the pre-filter’s length is equal to the CIR length, i.e. L trp “ L . Replacingthe conventional TR pre-filter into (6) and using the definitions in Section II-A, the time domain received signal in conventional TR is y trn p t q “ c ρP h L ´ ÿ l “ } h n p l q} s n p t ´ L ` q looooooooooooooooooomooooooooooooooooooon desired symbol directed to the n -th user ` c ρP h L ´ ÿ l “ l ‰ L ´ M ÿ m “ L ´ ÿ l “ h m,n p l q h ˚ m,n p L ´ ´ l ` l q s n p t ´ l q looooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooon ISI directed to the n -th user ` c ρP h N ÿ n “ n ‰ n M ÿ m “ h ˚ m,n p L ´ ´ t q b h m,n p t q b s n p t q looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon IUI ` z n p t q loomoon noise . (9)This received signal is composed of four terms: i ) the desired symbol multiplied by a realfactor resulting from coherent combination of multipath components in the CIR, ii ) ISI causedby incoherent addition of CIR components, iii ) IUI caused by the signals directed to otherusers (whose TR pre-filters do not match the CIR to the n -th user), and iv ) AWGN. Thus, in aconventional multiuser TR beamforming system, ISI and IUI are important problems that hamperdetection. In the single-user scenario, ETR was proposed before as a solution to mitigate the ISIcomponent in the received signal [29]. We extend ETR to the multiuser case next. C. Multi-user Equalized TR Beamforming
ETR uses the TR pre-filter in cascade with a ZF pre-equalizer in order to mitigate the ISI ofconventional TR. In [29], it is demonstrated that ETR outperforms conventional TR with respectto BER in a single-user scenario, with a marginal loss in the spatial focusing capability. We nowextend this technique to the multi-user scenario by defining the pre-filter vector components forthe n -th user as p eqm,n p t q “ h m,n p L ´ ` t q b g ˚ n p´ t q a P eqh , (10)where g n p t q represents a ZF linear equalizer with length L E . Thus, we have L eqp “ L ` L E ´ .The normalization factor is P eqh “ M ÿ m “ N ÿ n “ L ` L E ´ ÿ l “ ˇˇ h ˚ m,n p L ´ ´ l q b g n p l q ˇˇ . (11) One equalizer is required for each user, with the n -th equalizer designed to satisfy g n p t q b M ÿ m “ h ˚ m,n p L ´ ´ t q b h m,n p t q “ δ p t ´ t q , (12)where t is an arbitrary delay. Equation (12) can be written as an over-determined system oflinear equations on g n p t q , t “ , . . . , L E ´ . Thus, perfect ZF equalization is not possible witha finite equalizer’s length [36], but a good approximation can be achieved with a sufficientlylarge L E , eliminating the second term in (9). A detailed discussion on this subject is provided in[29]. Using the ETR pilot, and assuming perfect equalization, the time domain received signalat user n is y eqn p t q “ c ρP eqh s n p t ´ t q looooooooomooooooooon signal directed to the n -th user ` c ρP eqh N ÿ n “ n ‰ n M ÿ m “ s n p t q b g n p t q b h ˚ m,n p L ´ ´ t q b h m,n p t q looooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooon IUI ` z n p t q loomoon noise , (13)which has no ISI term, but still contains IUI. Thus, both conventional TR and ETR performanceis limited by ISI and/or IUI, as detailed next. D. Performance Analysis of TR and ETR
We now turn our attention to the power components in (9) and (13), following the same pro-cedure as in [29]. The fundamental assumptions are that the system operates under uncorrelatedscattering with uncorrelated channels between users, and normalized channel power, as stated inSection IV. In the derivations below, we also employ the approximation E r a { b s « E r a s{ E r b s fortwo random variables a and b , as analyzed in [29], [35] for TR systems. Complete derivations arenot shown due to space constraints. Here, we verify that TR is a suitable technique for massiveMIMO systems, where M " . Let P trs , P trisi , and P triui represent the power in the first, second,and third terms in (9), respectively. Then, the average desired signal power is E “ P trs ‰ “ E »– ρP trh ˇˇˇˇˇ L ´ ÿ l “ } h n p l q} s n p t ´ L ` q ˇˇˇˇˇ fifl « M ρ Γ N . (14) The average ISI power in (9) can be approximated as E “ P trisi ‰ “ E »—– ρP trh ˇˇˇˇˇˇˇ L ´ ÿ l “ l ‰ L ´ M ÿ m “ L ´ ÿ l “ h m,n p l q h ˚ m,n p L ´ ´ l ` l q s n p t ´ l q ˇˇˇˇˇˇˇ fiffifl « ρM N Γ L ´ ÿ l “ l ‰ L ´ M ÿ m “ M ÿ m “ L ´ ÿ l “ E “ h m,n p l q h ˚ m ,n p l q| ‰ ˆ E “ h ˚ m,n p L ´ ´ l ` l q h m ,n p L ´ ´ l ` l q ‰ . (15)An approximation to the average IUI power is E “ P triui ‰ “ E »—– ρP trh ˇˇˇˇˇˇˇ N ÿ n “ n ‰ n M ÿ m “ h ˚ m,n p L ´ ´ t q b h m,n p t q b s n p t q ˇˇˇˇˇˇˇ fiffifl « ρM N Γ N ÿ n “ n ‰ n L ´ ÿ l “ M ÿ m “ M ÿ m “ L ´ ÿ l “ E “ h m,n p l q h ˚ m ,n p l q ‰ ˆ E “ h ˚ m,n p L ´ ´ l ` l q h m ,n p L ´ ´ l ` l q ‰ . (16)For ETR, the average desired signal power is bounded by E r P eqs s “ E „ ρP eqh ď M ρ Γ N . (17)The average IUI power in ETR has a similar form to (16), but it is not shown here since it is notthe focus of this work. However, we analyze it numerically in Section V. From (14)-(16), wecan make the following remarks with respect to TR beamforming in massive MIMO systems: ‚ Both ISI and IUI powers are highly dependent on the propagation conditions. More specifi-cally, power delay profiles and correlation between antennas are present in the terms of theform E r h m,n p l q h ˚ m ,n p l qs . Thus, increasing spatial correlation would increase both ISI andIUI, degrading performance. ‚ In the case of uncorrelated antennas, E r h m,n p l q h ˚ m ,n p l qs “ if m ‰ m . Hence, the sumswould only depend on the power delay profile (which is the same for all antennas), andboth ISI and IUI powers would be independent of M . ‚ Desired signal power increases linearly with M . Thus, in uncorrelated channels E r P trs { P trisi s Ñ8 and E r P trs { P triui s Ñ 8 as M Ñ 8 . This implies that, with a sufficiently large number of antennas, a conventional TR beamforming system is noise limited instead of interferencelimited. ‚ However, if channels are spatially correlated (as in realistic scenarios), equalization andinterference mitigation provided by TR reduce. ‚ Note that, given that CIR statistics for users n and n are the same, i.e. E “ h ˚ m,n p l q h m ,n p l q ‰ “ E “ h ˚ m,n p l q h m ,n p l q ‰ @ l , then P triui is larger than P trisi by a factor on the order of the numberof users. Thus, IUI mitigation should be given priority over equalization when proposingimprovements over conventional TR.Given this characteristics of TR beamforming in massive MIMO, we now propose a novelTR extension to overcome the problems of IUI, even under highly correlated channels.III. I NTERFERENCE -N ULLING T IME -R EVERSAL B EAMFORMING
We are now concerned with the design of pre-filter vectors that combine the spatial focusingproperties of conventional TR, while also providing additional IUI mitigation. We start from thefrequency representation of the received signal, and formulate an optimization problem for thedesign of the pre-filters. The frequency domain equivalent of (6) is Y n p f q “ ? ρ x H n p f q , P n p f qy S n p f q looooooooooooooomooooooooooooooon signal directed to the n -th user ` ? ρ N ÿ n “ n ‰ n x H n p f q , P n p f qy S n p f q loooooooooooooooooomoooooooooooooooooon IUI ` Z n p f q loomoon noise , (18)where S n p f q is the DFT of s n p t q , and Z n p f q is the DFT of z n p t q . Appropriate zero padding is usedin the time domain in order to represent linear convolution as a product in the frequency domain.The complex inner product defined above allows a convenient simplification in (18) with respectto (6), which is useful for the problem formulation. Let H p f q “ r H p f q . . . H N p f qs P C M ˆ N bethe matrix with columns given by the steering vectors to all users. Also, let H ´ n p f q P C M ˆ N ´ be the matrix formed by removing the n -th column from H p f q , i.e. removing the steering vectorto user n . For notational simplicity, we drop the frequency dependence in the remainder ofthis section. Note that the IUI power in (18) is proportional to ř n ‰ n | x H n , P n y | . Thus, our Fig. 2. Geometrical interpretation of the optimization procedure. The optimum pilot in the frequency domain is the conventionalTR prefilter projection onto the nullspace of H H ´ n . This ensures that IUI is set to zero for every user and every frequency. objective is to find the pre-filter P ‹ ,n which is closest to the conventional TR solution in thefrequency domain (providing partial equalization of the received signal), and such that the IUIis set to zero. Formally, this optimization problem can be formulated as P ‹ ,n “ arg min P n ›› P trn ´ P n ›› subject to H H ´ n P n “ , @ n P N , @ f P r , . . . , L ` L p ´ s , (19)whose solution is P ‹ ,n “ ´ I M ´ H ´ n ` H H ´ n H ´ n ˘ ´ H H ´ n ¯ P trn , @ n, f. (20)where I M is the M ˆ M identity matrix. Thus, we call P ‹ ,n the interference-nulling time-reversal(INTR) pre-filter in the frequency domain. Geometrically, the constraint in the problem ensuresthat the vector P ‹ ,n P null H H ´ n ( , and the solution is the projection of P trn into that null space.This is illustrated in Fig. 2.IV. C HANNEL M ODEL FOR
60 GH Z M ASSIVE
MIMOAs mentioned above, TR actually benefits from rich scattering, so it can be convenientlyapplied for indoor wireless communications. In this section, we briefly describe the IEEE802.11ad model for 60 GHz SISO systems in such scenarios [21], and extend it to the correlated multi-user massive MIMO case. In the following, we use a statistical description of h m,n p t q ,given by its probability distribution, power delay profile (PDP), and spatial correlation in thecontext of massive MIMO systems. A. Channel Tap Distribution
The most popular channel model for mm-wave propagation is the IEEE 802.11ad. This isa SISO double directional statistical channel model based on a limited set of measurements,complemented with ray-tracing simulations. This model is defined for three indoor scenarios:conference room (CR), living room (LR), and cubicle environment (CB). Some important modelfeatures include: support of two types of antennas (isotropic and basic steerable antenna array),support of polarization, wideband and pathloss modeling under LoS and NLoS situations.The IEEE 802.11ad channel model follows a scattering cluster structure, both in time andangular domains. Thus, several multipath components (MPCs) observed in the CIR have similarpropagation delays and angles of departure/arrival. More specifically, each central ray arriving atthe receiver has pre-cursor rays (which arrive earlier) and post-cursor rays (which arrive later).This is due to irregular scattering objects and geometrical features which are large compared tothe wavelength. Both pre-cursor and post-cursor rays have less amplitude than the central ray.The resolvability of those MPCs depend exclusively on the system’s sampling time (bandwidth).When those MPCs are not resolvable, they contribute to the same tap in the CIR. Given thispropagation characteristics, we assume that h m,n p t q has zero mean and that | h m,n p t q| is Nakagamidistributed, with parameters m and Ω , for all m , n and t [37]. Recall that the m parameter inthe Nakagami distribution is analogous to the K factor in the Rician distribution, and that alarger m implies a large power ratio between the central ray (specular component) and the otherrays (diffuse components). The parameter Ω depends also on the amplitudes of the specular anddiffuse components, and on the channel PDP (tap average power) [37]. Table I shows the valuesof m and RMS delay spread in the IEEE 802.11ad scenarios. The larger value of m in the CBscenario is due to the reduced scattering within the cubicles, which reduces the number andpower of diffuse components contributing to each channel tap. TABLE IN
AKAGAMI m PARAMETER AND
RMS
DELAY SPREAD OF
IEEE 802.11
AD SCENARIOS
Scenario Nakagami m parameter RMS delay spread [ns]CB 4.34 3.47CR 2.56 4.82LR 1.74 7.81 B. Power Delay Profile
We are particularly interested in the PDP, a second order statistic defined as A h p t q “ E “ | h m,n p t q| ‰ , @ m, n, (21)where the expectation is calculated over CIRs that are subject to the same large-scale fading[38]. Signal power components depend on the PDP and spatial correlation, as seen in SectionII-D. We assume that all CIR in the system have the same PDP. This is valid for mm-waveindoor environments, where APs are usually positioned on or close to the ceiling and similarshadowing affects all elements in the transmit array. We also define the following constraint onthe CIR total power: L ´ ÿ t “ E “ | h m,n p t q| ‰ “ L ´ ÿ t “ A h p t q “ Γ , (22)where Γ ! is a constant accounting for channel induced propagation losses. This constraintimplies that all channels between the transmit antennas and each receiver have same averagepower. Fig. 3 shows PDPs obtained over realizations of the IEEE 802.11ad model for thethree scenarios simulated under NLoS and isotropic antennas. We do not consider LoS situationssince they correspond to flat-fading channels, which are of no interest here. Isotropic antennas areassumed so the system can take advantage of all MPCs in the channel. In practice, planar omni-directional antennas (e.g. [39]) would be a good alternative for implementation. We observe thatRMS delay spread is minimum for the CB scenario, where an AP is located in the ceiling of anoffice populated with cubicles. In that case, scattering is confined within the cubicle’s structureand other delayed paths (e.g. reflections from outer walls) are obstructed. On the other hand,CR and LR scenarios correspond to more open spaces, where first and second order reflections from walls are considered. Those reflections cause long tails in their PDPs, increasing their delayspread. C. Spatial Correlation Model
Consider m, m P M , n, n P N , and t, t P t , . . . , L ´ u . We make the following assumptionswith respect to CIRs in the systems: ‚ CIR are correlated across transmit antennas, i.e. the spatial channel autocorrelation functionis R h p ∆ d q ‰ , where ∆ d is the distance between two measured CIRs. The specificcorrelation structure depends on the array configuration, but it is assumed that the processin wide sense stationary with respect to the space. This implies that E r h m,n p t q h ˚ m ,n p t qs ‰ . ‚ Different users have uncorrelated CIRs to the AP, i.e. h m,n p t q and h m,n p t q are uncorrelatedif n ‰ n , @ m, t . This is due to the fact that MPCs are independent for different users. Thiscan be clearly seen in the CB environment, where each user is assumed to be in its owncubicle. ‚ CIR taps are uncorrelated, i.e. h m,n p t q and h m,n p t q are uncorrelated if t ‰ t , @ m, n . Thisis the conventional uncorrelated scattering (US) assumption widely used in the literature[40], and implies that contributions to different taps come from different scatterers.Nakagami correlated variables (across antennas) are generated according to the method de-scribed in [37], as follows. Consider the setting in Fig. 4. A planar randomly-oriented arraywith M isotropic elements is located in the environment according to the standard [21]. Anisotropic receiving antenna is randomly located in the environment as well. Each tap is assumedto have specular and diffuse contributions from an irregular scatterer (located according tothe corresponding delay), whose amplitudes depend on the PDP and the desired m parameter.All contributions to a fixed tap in a given CIR come from the same scatterer, with differenttaps corresponding to different scatterers. Using this procedure, the resulting normalized spatialcorrelation function R h p ∆ d q is shown in Fig. 5. These results are consistent with measured andsimulated spatial correlations in 60 GHz channels, e.g. [22]. High correlation values are causedby the reduced number of dominant MPC contributing to each tap. The specific correlationbetween transmit antenna elements depends only on the geometry of the array. For the numericalvalidation shown in Section V, we use rectangular arrays with 32 ( ˆ ), 64 ( ˆ ), or 128( ˆ ) elements with a uniform separation of 20 mm. t [ns] A h ( t ) CubicleConference RoomLiving Room
Fig. 3. Power delay profile of IEEE 802.11ad channel model scenarios with isotropic antennas. RMS delay spreads are 3.47ns for the CB scenario, 4.82 ns for CR and 7.81 ns for LR.Fig. 4. Method to generate correlated Nakagami CIR. Different taps are assumed to have contributions from specular anddiffuse reflections from different objects. The transmit array is planar (rectangular) with uniformly distributed elements.
V. N
UMERICAL R ESULTS AND D ISCUSSION
In this section, we present numerical results for the performance analysis of the multiuser TR,ETR and INTR techniques, as described in Sections II and III.
A. Pilot Length and Channel Correlation
First, we analyze the impact of pre-filter’s length L p and spatial correlation on the signalpower components. We calculate the values of P s , P isi , and P iui for the three techniques over Distance ∆ d [mm] | R h ( ∆ d ) | Fig. 5. CIR spatial correlation as a function of distance ∆ d , calculated over 1000 CIR realizations. L p is a typical consequence of zero-forcing equalization [29]. However, ETR is not designed tomitigate IUI. Thus, BER performance of TR and ETR are expected to be very similar since thescenarios we consider are clearly IUI limited.For INTR, IUI mitigation improves by increasing L p . This is due to the discarding of L ´ time samples when performing the transformation between the frequency domain prefilter (oflength L ` L p ´ ) and the time domain prefilter (of length L p ). Such discarding is necessary dueto the circular convolution theorem. Thus, the time domain prefilter is a least squares projectionof the optimum frequency domain solution. The error in the projection reduces as L p increases.We observe the impact of signal power components over the BER performance in Fig. 6, L p
60 80 100 120 P e q i s i / ( ρ Γ ) M = 32 M = 64 M = 128 CorrelatedUncorrelated (a) L p
60 80 100 120 P i n i u i / ( ρ Γ ) M = 32 M = 64 M = 128 CorrelatedUncorrelated (b)Fig. 6. Prefilter length ( L p ) vs (a) ISI power in ETR, and (b) IUI power in INTR. These results were obtained with L “ and N “ in the CB scenario. Other signal components in each technique remained approximately constant vs. L p . It isnoted that increasing L p reduces ISI power in ETR and IUI in INTR. This is due to the discarding of L ´ time samples whenperforming the transformation between the frequency domain prefilter (of length L ` L p ´ ) and the time domain prefilter(of length L p ). Such discarding is necessary due to the circular convolution theorem. Thus, the time domain prefilter is a leastsquares projection of the optimum frequency domain solution. The error in the projection reduces as L p increases. where the influence of channel spatial correlation is also shown. Signal to noise ratio is definedas SNR “ ρ Γ { σ z , where σ z is the variance of z n p t q @ n, t . These results were obtained for 5 usersand 32 antennas in the CB scenario, with a transmission of BPSK symbols over 1000 channelrealizations. Performance of both TR and ETR is limited by IUI, which causes a lower boundin the BER. We notice that ETR does not provide a significant improvement over conventionalTR in the case of multi-user massive MIMO systems. Thus, ETR does not offer any advantagefor such scenarios, given its greater computational complexity with respect to TR. In the caseof INTR, IUI is successfully mitigated and hence INTR outperforms the other techniques. Wealso observe that channel correlation degrades system performance in all cases.
B. Number of Antennas and Number of Users
Fig. 8 shows the average BER performance results per user for varying number of antennasand users. These results where obtained with the transmission of BPSK symbols over 1000spatially correlated channel realizations. We used a fixed pre-filter length L p “ and theCB scenario PDP. Conventional TR performance results are consistent with the analysis made TABLE IIR
ECEIVED SIGNAL POWER COMPONENTS FOR M “ . V ALUES ARE NORMALIZED {p ρ Γ q .Number of users N Technique L p “ L “ L p “ L p “ P s P isi P iui P s P isi P iui P s P isi P iui Uncorrelated channel2 TR 32 0.15 0.51 - - - - - -ETR - - - 31.9 0.01 0.52 31.9 0.001 0.52INTR 31.7 0.15 0.09 31.6 0.16 0.01 31.6 0.15 0.00210 TR 6.4 0.03 0.9 - - - - - -ETR - - - 6.37 0.002 0.9 6.37 0.0003 0.9INTR 5.77 0.04 0.17 5.58 0.04 0.02 5.55 0.04 0.004Correlated channel2 TR 31.4 0.6 0.88 - - - - - -ETR - - - 30.3 0.02 0.88 30.3 0.003 0.88INTR 30.8 0.61 0.15 30.6 0.6 0.02 30.6 0.6 0.00310 TR 6.39 0.15 1.48 - - - - - -ETR - - - 6.07 0.004 1.47 6.07 0.0005 1.47INTR 5.32 0.14 0.25 5.06 0.14 0.04 5.02 0.14 0.006 in Section II-D. The desired signal power increases linearly with the number of antennas whileinterference components remain constant. Thus, the minimum achievable BER per user improvesby increasing the number of antennas, providing diversity gain. On the other hand, increasingthe number of users with a fixed number of antennas decreases the desired signal power andincreases IUI. This is reflected in a higher BER for larger N . INTR outperforms conventionalTR in every simulated scenario. Nevertheless, the performance improvement provided by INTRis more evident with a large number of users or a limited number of antennas. C. Average Achievable Sum Rate
The achievable sum rate measures the downlink spectral efficiency in a multiple-access system.Assuming that each user treats ISI and IUI as Gaussian interferences, and according to the modeldefined in Sections II and III, the average achievable sum rate for a multi-user TR system is R “ E « N ÿ n “ log ˆ ` P s,n P isi,n ` P iui,n ` σ z ˙ff , (23) SNR [dB] -20 -10 0 10 20 B E R -6 -4 -2 TR L p = 60ETR L p = 90ETR L p = 120INTR L p = 60INTR L p = 90INTR L p = 120 CorrelatedUncorrelatedCorrelatedUncorrelated
Fig. 7. Average BER per user comparison of TR, ETR, and INTR, under correlated and uncorrelated channels (across antennaelements) with M “ , and N “ . Results are shown for different pilot lengths ( L p ). It is observed that spatial correlationincreases ISI and IUI, degrading performance. Also, increasing prefilter’s length improves IUI mitigation in INTR. where P s,n , P isi,n , and P iui,n are the desired signal power, ISI power, and IUI power, respectively,calculated at user n for a given realization. For simplicity, it is also assumed that the channel isused for downlink transmission all the time. A proper reduction factor can be used to accountfor uplink time in a TDD system or channel estimation overheads. Numerical results for theaverage achievable sum rate are shown in Fig. 9. These results were obtained in the LR scenariowith correlated channels and L p “ . As seen, INTR offers a significant improvement overconventional TR, doubling its rate in some cases and providing a remarkable multiplexing gain.In addition, we simulated a more extreme case with N “ , and N “ and 128 antennas,with the purpose of further demonstrate the capabilities of TR to handle IUI. Results are shownFig. 9c. Even though the assumption of uncorrelated CIR between users is hardly met when N is that large, results show that an outstanding efficiency of more than 170 bps/Hz can be SNR [dB] -20 -10 0 10 20 B E R -6 -4 -2 TRINTR M = 32 M = 64 M = 128 (a) SNR [dB] -20 -10 0 10 20 B E R -6 -4 -2 TRINTR N = 2 N = 5 N = 10 (b)Fig. 8. Avera BER per user for TR and INTR. (a) Different number of antennas M with L p “ and N “ . (b) Differentnumber of users N with L p “ and M “ . An important diversity gain is achieved even in spatially correlated channels.The effect of IUI is mitigated by increasing the number of antennas. SNR [dB] -30 -20 -10 0 10 20 30 R [ bp s / H z ] TRINTR N = 10 N = 5 (a) SNR [dB] -30 -20 -10 0 10 20 30 R [ bp s / H z ] TRINTR N = 5 N = 10 (b) SNR [dB] -30 -20 -10 0 10 20 30 R [ bp s / H z ] TRINTR N = 25 N = 50 (c)Fig. 9. Achievable rate of TR and INTR in the LR scenario. (a) M “ antennas, (b) M “ antennas, (c) M “ antennas with an extreme number of users. The multiplexing gain increases with the number of antennas. achieved with INTR. In all the simulated scenarios our proposed INTR technique outperformsconventional TR, as it can better withstand an increase in user load.VI. C ONCLUSION
We have analyzed a baseband TR beamforming system for mm-wave multi-user massiveMIMO. We studied conventional TR and equalized TR and found that their performance is IUI limited. We also noticed that, when the number of antennas is large, the ratio between thedesired signal power and ISI or IUI power increases. Thus, we confirm the potential of TR asa beamforming technology for massive MIMO. We also note that equalizing solutions such asETR are not necessary when the number of transmit antennas is large. After identifying IUI asthe main detection impairment for TR systems, we propose a modified technique called INTR.This technique calculates the transmit pre-filters in the frequency domain that set the IUI to zeroand are closest to the original TR solution. We proposed a 60 GHz MIMO channel model, whereCIR taps are modeled with Nakagami distributed amplitudes. In addition, we use PDPs given bythe IEEE 802.11ad SISO NLoS model, and generate spatial correlation in the CIRs according toa geometrical model. By means of numerical simulations, we verified that the proposed INTRoutperforms conventional TR with respect to average BER and achievable sum rate. In particular,we note that INTR performance is extremely tolerant to increases in the number of users, andprovides both diversity and multiplexing gains simultaneously.R
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