Efficient Downlink Channel Reconstruction for FDD Multi-Antenna Systems

Transcription

1 1 Efficient Downink Channe Reconstruction for FDD Muti-Antenna Systems Yu Han, Tien-Hao Hsu, Chao-Kai Wen, Kai-Kit Wong, and Shi Jin Nationa Mobie Communications Research Laboratory, Southeast University, Nanjing, China Institute of Communications Engineering, Nationa Sun Yat-sen University, Kaohsiung 804, Taiwan Department of Eectronic and Eectrica Engineering, University Coege London, United Kingdom arxiv: v1 [cs.it] 18 May 2018 Abstract In this paper, we propose an efficient downink channe reconstruction scheme for a frequency-division-dupex muti-antenna system by utiizing upink channe state information combined with imited feedback. Based on the spatia reciprocity in a wireess channe, the downink channe is reconstructed by using frequency-independent parameters. We first estimate the gains, deays, and anges during upink sounding. The gains are then refined through downink training and sent back to the base station (BS). With imited overhead, the refinement can substantiay improve the accuracy of the downink channe reconstruction. The BS can then reconstruct the downink channe with the upink-estimated deays and anges and the downink-refined gains. We aso introduce and extend the Newtonized orthogona matching pursuit (NOMP) agorithm to detect the deays and gains in a muti-antenna muti-subcarrier condition. The resuts of our anaysis show that the extended NOMP agorithm achieves high estimation accuracy. Simuations and over-the-air tests are performed to assess the performance of the efficient downink channe reconstruction scheme. The resuts show that the reconstructed channe is cose to the practica channe and that the accuracy is enhanced when the number of BS antennas increases, thereby highighting that the promising appication of the proposed scheme in arge-scae antenna array systems. Index Terms Mutipe antenna system, FDD, downink CSI acquisition, over-the-air test. I. INTRODUCTION Frequency division dupex (FDD) is one of the most widey used dupexing modes for mobie communication systems where both directions of communication can take pace simutaneousy without interference. The FDD mode in mutipeinput mutipe-output (MIMO) antenna systems has achieved great success in 3G and 4G commercia mobie communication networks. Recenty, arge-scae or massive MIMO, which is capabe of using spatia dimensions to guarantee an extraordinary spectra efficiency, has been identified as a key enaber for 5G networks. However, the use of arge-scae antenna arrays in 5G [1] [5] and future networks imposes huge chaenges to the acquisition of downink channe state information (CSI) at the base station (BS) for FDD-MIMO systems, and such information is crucia to an exceent system performance especiay in the downink. The ack of reciprocity between the upink and downink channes on different frequency bands makes the downink CSI acquisition process difficut to achieve. Therefore, downink CSI is usuay acquired through downink training and feedback. In previous generations of networks, ony a few antennas are used at the BS. An abundant amount of time and frequency resources are avaiabe to form orthogona piots and the amount of feedback is reativey sma. Whie in 5G and future networks, the use of hundreds or even thousands of antenna ports prevents the design of competey orthogona piot patterns.

2 2 In this case, reusing piots becomes inevitabe [5] [7], thereby affecting the accuracy of the CSI estimation. Moreover, using a huge amount of feedback for a high-dimensiona compex channe matrix is impractica. Thus, downink CSI acquisition remains a key probem in FDD massive MIMO systems. Many studies have been conducted to address the aforementioned probem [8] [19]. These studies generay appy two types of approaches. In the first approach, downink CSI is soey obtained from downink training and feedback but does not require orthogonaity among the piots that are transmitted from different antennas. For exampe, codebooks are used to quantize the space, and ony the codebook indices are sent back to the BS. In [8] and [9], treis-based and ange-of-departure-adaptive subspace codebooks were proposed, respectivey, to quantize the channe of FDD massive MIMO systems. These methods require training and produce feedback overhead. Other methods have attempted to expoit the sow-varying nature of the space. In [10], the authors assumed that the channes were correated in both time and space; they aso proposed an open- and cosedoop training with CSI memory that coud be derived from previous time instances. If the channe is sparse, then compressed sensing can effectivey reduce the training and feedback overhead [11], [12]. In the second approach, the spatia reciprocity between channes on two cose frequency bands is appied. In [13], the authors vaidated the spatia congruence by conducting measurements and demonstrated a sma deviation in the dominant directions of arriva at the frequencies 1935 MHz and 2125 MHz. Based on these findings, [14] reconstructed the downink channe with the aid of the downink channe covariance matrix that was inferred from the upink channe covariance matrix. Using the upink CSI aso heps reduce the downink training and feedback overhead. For instance, [15] and [16] proposed to expoit the channe sparsity for estimating the propagation directions via upink training and used the direction estimates in the downink training process to reduce the feedback overhead. In [17], the authors proposed a compressed downink CSI acquisition method that uses the partia support information obtained from the upink and demonstrated that this method coud significanty reduce the training overhead. Nevertheess, the aforementioned downink CSI acquisition methods have not been examined via over-the-air (OTA) tests. To verify the effectiveness of CSI on other bands in practica systems, [18] and [19] conducted OTA tests and found that the aforementioned method have promising appication in inferring the RF channes on one band by using the CSI on another band. In [19], the authors proposed to competey eiminate the downink training and feedback in ong-term evoution (LTE) systems because the gain of each path in a wireess channe was thought to be frequency-independent simiar to deay and ange. However, no sufficient evidence can confirm the frequency-independent feature of the gains, which may greaty degrade the performance of the scheme proposed in [19] when the upink and downink frequency bands are distincty separated. Motivated by spatia reciprocity, this paper proposes an efficient downink channe reconstruction scheme that utiizes the frequency-independent parameters of the deays and anges of the mutipath channe for a FDD muti-antenna orthogona frequency-division mutipexing (OFDM) system. 1 Given the uncertainty of their frequency-independence, we further refine the gains by using a imited amount of downink training and feedback. Our major contributions are summarized as foows. Downink channe reconstruction: An efficient downink channe reconstruction scheme is deveoped. The frequencyindependent deays and anges are initiay estimated during the upink training process by using the Newtonized orthogona matching pursuit (NOMP) agorithm that is extended in this paper. Afterward, the gains are refined by using the east squares (LS) agorithm in the downink with a sma amount of piots and feedback. With the upink-estimated deays and anges as we as the downink-refined gains, the downink channe can be reconstructed at the BS. The necessity of downink refinement is proven through theoretica anayses and simuations. The numerica resuts demonstrate that the 1 The proposed scheme can be straightforwardy appied to FDD massive MIMO systems.

3 3 proposed efficient downink reconstruction scheme can be used to reconstruct a highy accurate downink channe. Extension of the NOMP agorithm: NOMP was originay designed to extract two parameters, namey, gains and frequencies, for a noisy mixture of sinusoids [20]. In this paper, we extend and adapt this agorithm to a trivariate case, where the gain, deay, and ange of each path are estimated. During each iteration of this agorithm, a 2D dictionary is utiized and the Newton step refines the deay and ange simutaneousy. After updating the stopping criteria, we evauate the accuracy of the estimations by deriving the ower bounds of the estimation errors and observe that the extracted deays and anges are very cose to the rea vaues. OTA test resuts: An OTA testbed is set up to assess the system performance of the proposed downink reconstruction scheme in practica wireess communication scenarios. We observe that the channe reconstructed by the proposed scheme is near the inear minimum mean square error (LMMSE)-estimated channe, thereby demonstrating the necessity of gain refinement and the effectiveness of the reconstruction. The OTA resuts aso show that with more antennas, the efficient channe reconstruction scheme demonstrates higher accuracy and can behave we in a massive MIMO scenario. The rest of this paper is organized as foows. Section II introduces the mutipath channe between the BS and a user and studies the frequency-independent spatia parameters over different frequency channes. Section III proposes an efficient downink channe reconstruction scheme based on the upink-estimated frequency-independent parameters and anayticay justifies the importance of refining the gains in the downink. Section IV presents the extended NOMP agorithm for estimating the gains, deays, and anges through the upink training process and anayzes its estimation accuracy. Section V discusses the simuation and OTA test resuts for the proposed efficient downink channe reconstruction scheme. Section VI concudes the paper. Notations In this paper, the matrices and vectors are denoted by uppercase and owercase bodface etters, respectivey, whie the superscripts ( ), ( ) H, and ( ) T denote the pseudo-inverse, conjugate-transpose, and transpose, respectivey. In addition, R{ } takes the rea component of a compex number, whie E{ } represents the expectation with respect to a random variabes inside the brackets. We aso use and to denote taking the absoute vaue and moduus operations, respectivey, and use the notations and to denote rounding a decima number to its nearest ower and higher integers, respectivey. II. CHANNEL MODEL In this section, we describe the wireess channe between the BS and its serving user by tracing the propagation paths of the signa. A singe ce of a mobie communication system operates in the FDD mode by empoying OFDM. We denote the difference between the upink and downink carrier frequencies by F and assume that each upink and downink frequency band has N sub-carriers with spacing f. We focus on the baseband and denote the upink centra sub-carrier by DC. The BS is equipped with a uniform inear array (ULA) with M antenna eements, whie the user has one antenna. The upink mutipath channe between the user and the BS antenna eement m on subcarrier n can be modeed as L u 1 h u m(n) = g u e j2πφu,n +j2πθu,m, (1) =0 in which m = M/2,..., 1, 0, 1,..., M/2 1, n = N/2,..., 1, 0, 1,..., N/2 1, L u is the number of propagation paths in the upink, g u is the gain of the th propagation path in the upink and is compex, Φ u,n is introduced by the deay of the th path in the upink, and Θ u,m is the phase difference between antenna eement m and 0 resuting from the time difference in the arriva of the th path in the upink.

4 4 d sin M 2 1 d d 0 1 M 1 2 Fig. 1. Difference of the propagation distances when the wireess signa arrives at two adjacent ULA eements. The back circe represents the reference antenna eement at the BS. τ u 0 < τ u denotes the deay of the th propagation path in the upink upon its arriva at antenna eement 0, which satisfies satisfies 0 < θ u < 1/ f. We know that Φ u,n = n fτu and we further denote the ange of the th path in upink by θ u, which < 2π. The wireess signa traves different distances when arriving at different BS antenna eements as iustrated in Fig. 1. The signa from direction θ u traves at a onger distance of d u,m = md sin θu upon arriving at eement m when compared with eement 0, where d denotes the distance between two adjacent antenna eements that equas to λ/2 and λ denotes the carrier waveength. The phase difference between eement m and 0 for the th path is cacuated as Θ u,m = fdu,m c = m d λ sin θu, (2) where f denotes the carrier frequency and c denotes the speed of ight. Therefore, (1) can be rewritten as L u 1 h u m(n) = g e j2πn fτu +j2πm d λ sin θu. (3) =0 By stacking the channes on a subcarriers and antennas into a vector, we obtain the muti-subcarrier muti-antenna channe between the user and the BS, which is expressed as where represents the Kronecker product, h u = L u 1 =0 g u p(τ u ) a(θ u ), (4) p(τ) = [e ] j2π N 2 fτ,..., e j2π( N 2 1) fτ H (5) denotes the deay-reated phase vector of the OFDM modue, and a(θ) = [e j2π M 2 d λ sin θ,..., e j2π( M 2 1) d λ sin θ] H (6) denotes the steering vector of the ULA. As a specia case, the BS ony has one antenna. Under this singe-input singe-output (SISO) condition, the anges of propagation paths are not modeed in this channe, but the upink channe vector on a subcarriers can be written as L u 1 h u SISO = g u p(τ u ). (7) =0 For the downink, by using the upink carrier frequency as the reference (i.e., 0 Hz), we denote the downink carrier frequency

5 5 by F and mode the downink channe between the BS antenna array and the user on a subcarriers and antennas as h d = L d 1 =0 g d e d j2π F τ where L d represents the number of propagation paths in the downink, g d path, τ d p(τ d ) a(θ d ), (8) is the compex gain of the th downink propagation is the deay of the th downink path with respect to antenna eement 0 that satisfies 0 < τ d ange of the th path in the downink that satisfies 0 < θ d < 2π. < 1/ f, and θ d Reciprocity does not normay appy in FDD systems because of the different operating frequencies in the upink and downink. Nonetheess, the upink and downink channes share a common propagation space between the BS and the user, and some partia reciprocity is expected if the frequency bands are within a certain coherent bandwidth. The upink and downink signas propagate aong common paths and are refected by the same scatterers. Given that the wireess signas trave the same transmission distance and at the same speed, the deay is equa in both the upink and downink. According to the measurement resuts in [13] and [23], the spatia directions or anges in the upink channe are amost the same as those in the downink channe. Therefore, L u = L d = L, τ u = τ d = τ, and θ u = θ d is the = θ are obtained. The deays and anges {τ, θ } =0,...,L 1 are frequency-independent, thereby reveaing a spatia reciprocity between the upink and downink. III. EFFICIENT DOWNLINK CHANNEL RECONSTRUCTION The spatia reciprocity inspires us to reconstruct the FDD downink channes by using the frequency-independent parameters estimated in the upink instead of estimating the downink CSI via massive downink training and feedback. In this section, we propose an efficient downink channe reconstruction scheme for FDD muti-antenna systems based on spatia reciprocity by using the upink CSI with a sma amount of downink training and feedback overhead. A. Estimating Frequency-Independent Parameters During the upink sounding process, the BS receives piots sent from the user and is given the opportunity to estimate the frequency-independent parameters. The piot received by the BS antenna eement m on subcarrier n can be expressed as L 1 ym(n) u = h u m(n)s(n) + zm(n) u = g u e j2πn fτ +j2πm d λ sin θ s(n) + zm(n), u (9) =0 where s(n) is the transmitted piot on subcarrier n, and z u m(n) is the compex Gaussian noise vector on BS antenna eement m and subcarrier n with zero mean and unit variance. In this muti-subcarrier muti-antenna system, the BS can receive piots on each occupied subcarrier and antenna eement. N continuous subcarriers is assumed to be occupied by the piots, whie the centra subcarrier is assumed to be DC. The transmitted piots on a subcarriers are equa to 1, thereby satisfying s( N/2 ) = = s( N/2 1) = 1. To detect the two-tupes {τ, θ } =0,...,L 1 from the received piots, we stack the received piots on a subcarriers and antennas together into a vector and obtain where z u is the stacked upink noise vector with i.i.d. eements. Here, we denote L 1 y u = g u p(τ ) a(θ ) + z u, (10) =0 u(τ, θ) = p(τ) a(θ). (11)

6 6 Therefore, (10) can be rewritten as L 1 y u = g u u(τ, θ ) + z u. (12) Based on (12), the parameter detection probem can be transated to a frequency detection probem. =0 From [20], we find that the NOMP agorithm behaves we in detecting frequencies from a mixture of sinusoids. NOMP can extract (a, w ) =0,...,L 1 from where x(w) = [ 1, e jw,..., e j(n 1)w] H L 1 y = a x(w ) + z, (13) =0 and z is a compex Gaussian vector with i.i.d. eements. The NOMP agorithm introduced in [20] ony estimates one frequency parameter, namey, w. However, two frequency parameters need to be detected in our case, namey, (τ, θ). In other words, three-tupes must be extracted from (12), incuding g u. The origina NOMP agorithm cannot satisfy our requirement and we must extend it to fit the trivariate condition. A detaied description of the trivariate NOMP agorithm is provided in Section IV. In the foowing part of this section, we suppose that the fina resuts are obtained after the trivariate NOMP agorithm is appied to (12). The detected three-tupes are recorded as (ĝ u, ˆτ, ˆθ ) =0,..., ˆL 1. B. Necessity to Refine the Gains After obtaining the gain, deay, and ange of each path via upink training and trivariate NOMP estimation, we reconstruct the downink channe for the FDD transmission system. As mentioned before, the deays and anges are frequency-independent parameters, and their upink estimates can be appied in the reconstruction of the downink frequency band channe. Based on these facts, [19] proposed an R2-F2 system that extracts the information of the propagation paths from the channes on band 1 in order to reconstruct the corresponding channes on band 2. This system aows the LTE BSs to infer the downink channes by using the upink-derived CSI and underscores the need to eiminate CSI feedback, which wi significanty improve the time-frequency resource utiization. However, we are sti unsure whether the reconstructed downink channe is accurate enough if the upink estimates are directy appied to the downink channe mode without using any downink CSI. In [19], the precondition for eiminating downink training and feedback is that a spatia parameters, incuding the gains, deays, and anges, are frequency-independent. The gain of each path is viewed to be identica in both the upink and downink. However, the existing measurements for the correation of gains in different frequency bands do not provide sufficient evidence to confirm the frequency-independent feature of the gains. On the contrary, [23] demonstrated that due to phase difference, the power of a custer differs in upink and downink. A more commony accepted view is that the azimuth power spectrum, which can be regarded as the secondary moment of the compex gain, is highy correated in both the upink and downink. This view has been confirmed by the measurements in [13]. In [14], the authors proposed to mode the azimuth power spectrum based on a same shape but mutipied by a frequency-dependent factor, thereby suggesting that the ampitudes are not equa in different frequency bands. Previous studies generay hod that the instantaneous spatia compex gains are different in the upink and downink [21], [22]. Therefore, we cannot suppose that the gains are frequency-independent. Meanwhie, even if the gains are assumed to be frequency-independent, the estimation errors wi negativey affect the reconstructions on another frequency band. These errors are inevitabe for any detection method, incuding NOMP and the optimization method used in [19]. The simpest singe-antenna singe-path case is used as an exampe to determine the impact

7 7 2 f 2 2 f 1 (a) (b) Fig. 2. Impact of phase error: (a) Singe-path scenario. The soid ine with arrow represents the true phase of the path. The dotted ines with arrows are the phase errors on frequencies f 1 and f 2. (b) Muti-path scenario. The soid ines with arrows are the rea muti-path components and the rea superposition. The dotted ines with arrows are the estimated components and their superposition. of the estimation error. We denote the rea gain and deay by g and τ, respectivey, and denote the rea channe on frequency f 1 by h 1 = ge j2πf1τ. We assume that the estimated gain and deay on frequency f 1 are ĝ and ˆτ, respectivey, and that the estimation error of deay is τ = ˆτ τ. For frequency f 1, ĝ wi compensate for the phase error caused by τ because the gain is updated by using LS estimation at the end of the NOMP agorithm. The reconstructed channe ĥ1 = ĝe j2πf1 ˆτ wi be very much the same as the origina channe h 1. In this case, goba accuracy is obtained instead of oca accuracy. However, when ĝ and ˆτ are used directy to reconstruct the channe on frequency f 2, the phase error 2πf 2 τ is considerabe if either f 2 or τ is arge enough as shown in Fig. 2(a). Meanwhie, the ĝ derived on frequency f 1 is no onger abe to compensate for this phase error on frequency f 2. The reconstructed channe on frequency f 2 is expressed as ĥ 2 = ĝe j2πf2 ˆτ = h 2 ĝ g ej2πf2 τ, (14) where ĝ g e j2πf2 τ is the mutipicative estimation error from the goba perspective. We can find that the derived channe on frequency f 2 is far from the rea channe. Athough ĝ has the same absoute vaue as g, that is, ĝ = g, the phase difference between ĥ2 and h 2 becomes unacceptaby arge because the phase information of the wireess channe is of great importance to the transceiver design. The phase error wi severey affect the muti-path channe reconstruction. As shown in Fig. 2(b), the origina channe comprises two paths with different ampitudes and deays, which are denoted by soid ines with arrows. As a resut of the phase error, these muti-path components rotate and form an incorrect superposition. An anguar error aso takes pace in highy compicated muti-antenna muti-path scenarios, thereby further harming the reconstruction on another band. Therefore, we do not suggest to foow the approach in [19], which ony uses the upink CSI to reconstruct the downink channe for FDD transmission systems. C. Re-Estimation and Reconstruction Scheme Given the inevitabe estimation errors of the deays and anges, the gains are LS-estimated at the ast step of the NOMP agorithm to compensate for these errors in the reconstruction of the upink channe. Simiary, the gains can aso be reestimated via LS to compensate for the errors in reconstructing the downink channe. This approach requires additiona downink overhead. Fortunatey, ony the gains need to be refined. Both the deays and anges estimated in the upink are sti appicabe to the downink channe reconstruction. Therefore, a sma amount of overhead is required to refine the gains.

8 8 The gains are refined with the aid of piots that are transmitted in the downink. To retrieve the feature of the whoe downink frequency band, these piots are sparsey distributed in the downink band. We use comb-type a-one piots and insert one piot in every K subcarriers. Afterward, N p = N/K subcarriers are occupied by the piots, and the indices of the subcarriers are n 0, n 2,..., n Np 1. Unike the upink, mutipe antennas exist at the transmitter and a singe antenna exists at the receiver. The piots that are transmitted by the antenna array wi be additivey received at a singe antenna. To enhance the received power, the piots are beamformed before the transmission. Given the anges of the propagation paths {ˆθ } =0,..., ˆL 1 that are estimated in the upink, we target the piots to these directions and concentrate the transmit power onto the propagation path of the channe. The foowing beamforming types are considered here: Type 1: The piots in one OFDM symbo are beamformed to target one specific direction. We need ˆL OFDM symbos to send the piots, and different OFDM symbos correspond to different directions. For subcarrier n i on the jth OFDM symbo, the received piot can be expressed as L 1 yj d (n i ) = g d e j2π( F +ni f)τ a H (θ )a(ˆθ j ) + zj d (n i ), (15) =0 where z d j (n i) is the downink noise on subcarrier n i and OFDM symbo j, i = 0,..., N p 1, and j = 0,..., ˆL 1. Type 2: The piots are frequency-division mutipexed onto different directions, and ony one OFDM symbo is needed. Subcarriers n 0,..., n ˆL 1 correspond to directions ˆθ 0,..., ˆθ ˆL 1, respectivey. The received piot on subcarrier n i can be expressed as L 1 y d (n i ) = g d e j2π( F +ni f)τ a H (θ )a(ˆθ i ) + z d (n i ), (16) =0 where i = i mod ˆL, z d (n i ) is the downink noise on subcarrier n i, and i = 0,..., N p 1. To estimate the downink gains at the user side, the BS needs to inform the user with the upink estimated parameters (ˆτ, ˆθ ), = 0,..., ˆL 1 and the beamforming type. The user appies these estimates into (15) or (16) according to the beamforming type and rewrites the signa modes as and ˆL 1 yj d (n i ) = g d e j2π( F +ni f)ˆτ a H (ˆθ )a(ˆθ j ) + zj d (n i ) (17) =0 ˆL 1 y d (n i ) = =0 g d e j2π( F +ni f)ˆτ a H (ˆθ )a(ˆθ i ) + z d (n i ). (18) After stacking the received piots on a subcarriers and OFDM symbos, the foowing unified signa mode for both types is constructed as y d = Ag d + z d, (19) where y d and z d are the stacked M p 1 dimensiona received piots, gains, and noise vectors, respectivey, whie A denotes the M p ˆL dimensiona coefficient matrix.

9 9 For Type 1, M p = N p ˆL, and the matrix A comprises ˆL submatrices A = where the (i, )th entry of the submatrix A (j) is equa to A (0). A ( ˆL 1) where j = 0,..., ˆL 1, i = 0,..., N p 1 and = 0,..., ˆL 1. For Type 2, M p = N p, and the (i, )th entry of A is equa to T, (20) A (j) i, = e j2π( F +ni f)ˆτ a H (ˆθ )a(ˆθ j ), (21) A i, = e j2π( F +ni f)ˆτ a H (ˆθ )a(ˆθ i ), (22) where i = 0,..., N p 1, and = 0,..., ˆL 1. Given that the coefficient matrix A is aso known at the user side, the user can refine the gains via LS estimation as ĝ d = A y d, (23) where A represents the pseudo-inverse of A, and the dimension of the refined gain vector ĝ d is sti ˆL. The refined gains are then sent back to the BS. The feedback amount is independent of the number of antenna eements and subcarriers but is dependent on the number of detected propagation paths. The BS obtains a the information required for the reconstruction of the downink channe, namey, (ĝ d, ˆτ, ˆθ ), = 0,..., ˆL 1. Specificay, the downink mutipath channe on a subcarriers and antennas is reconstructed as ˆL 1 h d = =0 ĝ d e j2π F ˆτ p(ˆτ ) a(ˆθ ). (24) For carity, we briefy summarize the procedures used in the proposed efficient downink channe reconstruction scheme as foows: Step 1: Frequency-independent parameters estimation during the upink sounding. The user sends upink piots to the BS and then the BS uses the extended triviriate NOMP agorithm to estimate the gain, deay, and ange of each propagation path of the channe. Step 2: Downink gain refinement and feedback. The BS transmits the downink piots to the user and informs the user about the beamforming type and the upink-estimated deays and anges. The user re-estimates the gains and then feeds them back to the BS. Step 3: Downink channe reconstruction. The BS reconstructs the downink channe by using the upink-estimated deays and anges as we as the downink-refined gains. IV. UPLINK PARAMETERS EXTRACTION In this section, we describe the extension of the NOMP agorithm in detai in order to fit the trivariate condition and obtain the three-tupe (g u, τ, θ) from the upink as mentioned in Section III.A. We first introduce the rationae and stopping criterion of the trivariate NOMP agorithm and then evauate its accuracy by using the ower bounds of the estimation errors.

10 10 d For simpification, fτ is treated as a whoe that is simpified by µ [0, 1). Simiary, λ sin θ is simpified by ν [0, 1). Then, vector u is represented by u(µ, ν) = p(µ) a(ν) (25) in the subsequent part of this section, where p(µ) = [e j2π N 2 µ,..., e j2π( N 1)µ] H 2 (26) and a(ν) = [ e j2π M 2 ν,..., e j2π( M 2 1)ν] H. (27) The three-tupes to be detected are transformed to (g u, µ, ν) in the extended trivariate NOMP agorithm. A. Trivariate NOMP Agorithm NOMP is an iteration-based agorithm. In our extended version of this agorithm, a three-tupe of (g u, µ, ν) is estimated in each iteration. The component made by this three-tupe is then removed from the observed piot. At the end of the ith iteration, the residua is cacuated as y u r i 1 = y u =0 g u u( µ, ν ), (28) where ( g u, µ, ν ), = 0,..., i 1 are the estimated three-tupes in the previous i iterations. Afterward, in the (i + 1)th iteration, we estimate a new three-tupe by minimizing the new residua power yr u g u u(µ, ν) 2, which is further transated to maximize the foowing function S(g u, µ, ν) = 2R { y uh r g u u(µ, ν) } g u 2 u(µ, ν) 2. (29) The working steps in the (i + 1)th iteration of the extended trivariate NOMP agorithm are simiar to those of the origina agorithm in [20]. We first briefy introduce the steps in the (i + 1)th iteration of the extended agorithm, which are isted beow. Step 1: New Detection. Seect the coarse estimates µ i and ν i from a 2D over-samped ange-and-deay grid and then cacuate g u i from µ i and ν i. Step 2: Singe Refinement. Soey refine the coarsey estimated three-tupe ( g u i, µ i, ν i ) through the Newton refinement steps and add the obtained ( g u i, µ i, ν i ) into the set of the estimated three-tupes. Step 3: Cycic Refinement. Cycicay refine the set of estimated three-tupes through the Newton refinement steps and obtain ( g u, µ, ν ), = 0,..., i. Step 4: Gains Update. Retain the estimated deays and anges and update a the ampitudes through LS estimation [ g u 0, g u 1,..., g u i ] T = U y u, where U = [u( µ 0, ν 0 ), u( µ 1, ν 1 ),..., u( µ i, ν i )]. Detais about the extensions of this work are then provided. Given that the deay and the ange jointy determine the channe phase and can be represented using a common vector u(µ, ν), these two parameters are estimated and refined together in our design. This combination resuts in the 2D grid and the extended Newton step.

11 11 1) 2D Grid: The coarse estimates ( µ i, ν i ) in Step 1 are chosen from a 2D ange-and-deay grid Ω, which consists of γ 1 N γ 2 M over-samped grid points {( ) k1 Ω = γ 1 N, k 2 γ 2 M } : k 1 = 0, 1,..., γ 1 N 1; k 2 = 0, 1,..., γ 2 M 1, (30) where γ 1 and γ 2 are the over-samping rates for the deay grid and the ange grid, respectivey. Each point in the grid forms a vector u(k 1 /(γ 1 N), k 2 /(γ 2 M)). The coarsey estimated deay and ange are obtained by exhaustivey searching the grid points as foows: ( µ i, ν i ) = arg max (µ,ν) Ω u H (µ, ν)y u r 2 u(µ, ν) 2. (31) Next, the gain is cacuated as g u i = uh ( µ i, ν i )yr u u( µ i, ν i ) 2. (32) 2) Extended Newton Step: With one more parameter than the origina Newton step, the extended Newton step in Steps 2 and 3 can refine the deay and the ange simutaneousy. In this bivariate probem, the coarsey estimated ( µ i, ν i ) are refined through µ i = µ i ν i ν i ) u 1Ṡ ) u S ( g i, µ i, ν i ( g i, µ i, ν i, (33) where is the first-order partia derivative vector, and Ṡ ( g u, µ, ν ) = S ( g u, µ, ν ) = S µ S ν 2 S 2 S µ 2 µ ν 2 S ν µ (34) 2 S ν 2 (35) is the second-order partia derivative matrix. According to (29), we can write the first-order partia derivatives of S(g u, µ, ν) as { S x = 2R g u ( yr u g u u ) } H u, (36) x where x can be µ or ν. The second-order partia derivative of S(g u, τ, θ) is cacuated as 2 { S = 2R g u ( yr u g u u ) H 2 } u g u 2 uh u, (37) x 1 x 2 x 1 x 2 x 2 x 1 where x 1 and x 2 can be µ or ν. One requirement is that S(g u, µ, ν) is ocay convex in the neighborhood of ( µ, ν) because we are pursuing its maximum vaue. Therefore, the Newton refinement (33) wi be carried out if, and ony if, det ( S ( g u, µ, ν )) > 0 and the first eement of S ( g u, µ, ν ) is ower than 0. At the end of each Newton step, the gain is aso updated using (32). Note that the 2D grid and the extended Newton step are the required major extensions to the origina NOMP agorithm. Other minor modifications to fit the trivariate condition are trivia and omitted here. B. Stopping Criterion One major chaenge is that BS does not know the number of propagation paths in the rea channe, a detai which directy determines when the iteration process is terminated. If the estimated three-tupes are precise enough, a the paths wi be

12 12 accuratey identified and the residua wi be reduced to the noise at the end of the NOMP agorithm, i.e., y u r study, this assumption is utiized to design the stopping criterion. z u. In this 1) Power-based Criterion: One choice is to terminate the NOMP iterations when the residua power is ess than the tota noise power. Since the noise power is normaized to 1, if y u r 2 < κ, (38) where κ = E{ z u 2 } = MN, (39) then the trivariate NOMP agorithm wi be stopped. 2) Fase-Aarm-Rate-based Criterion: Aternativey, we can design the stopping criterion based on the fase aarm rate. If we detect a fake path that does not exist, then we say that a faut appears. This situation happens when a the paths have been detected but the agorithm is sti not working. The foowing theorem introduces the fase-aarm-rate-based stopping criterion. Theorem 1: If the trivariate NOMP agorithm terminates when u(µ, ν) H y u r 2 < κ (40) hods for a grid points where (µ, ν) {( k1 N, k ) 2 M } : k 1 = 0, 1,..., N 1; k 2 = 0, 1,..., M 1, (41) κ = n(mn) n( n(1 P fa )), (42) then the fase aarm rate can be approximated by P fa. Proof: As the grid points isted in (41) are non-over-samped points, the corresponding vaues of u(µ, ν) H y u r viewed as the Fourier transformed vaues of y u r paths are precisey detected, the condition (40) can be transated to can be and remain the same statistic property of yr u. Since yr u z u when a the z u 2 < κ. (43) From [24], we know that E { z u 2 } n(mn) (44) when MN grows arge. Denoting z = z u 2 n(mn), we can derive that P { z Z} = P { z u 2 < Z + n(mn) }. (45) Given that each eement of z is i.i.d., it hods that P { z u 2 < Z + n(mn) } = P { z 1 2 Z + n(mn) } ( MN = 1 1 ) MN MN e Z, (46)

13 13 where z 1 is the first eement of z u and is a Gaussian variabe with zero mean and unit variance. As ( exp{ξ} = im 1 + ξ n, (47) n n) we have ( 1 1 ) MN MN e Z exp{ exp{ Z}} (48) when MN grows without imit. By appying (46) (48) into (45) and denoting κ = Z + n(mn), we can obtain P { z u 2 κ } exp{ exp{n(mn) κ }}. (49) If we further appy (42), then it hods that P { z u 2 > κ } = 1 P { z u 2 κ } P fa, (50) which means that the fase aarm rate approximates P fa. After the iterations stop, the fina estimation resuts of the trivariate NOMP agorithm are denoted as (ĝ u, ˆµ, ˆν ), = 0,..., ˆL 1. These estimation resuts are further transated to (ĝ u, ˆτ, ˆθ ), = 0,..., ˆL 1, as mentioned in Section III. C. Estimation Accuracy To evauate the estimation accuracy of the deay and the ange, we cacuate the respective normaized mean square errors (MSEs) of µ and ν by { } ˆµ µ 2 ε µ = E 1/N 2 (51) and { } ˆν ν 2 ε ν = E 1/M 2. (52) The foowing theorem is used to study the estimation accuracy of the extended trivariate NOMP agorithm. Theorem 2: The normaized MSEs of the deay and ange are ower bounded, respectivey, by ε µ 3N SNR 2π 2 M(N 2 1) (53) and 3M ε ν SNR 2π 2 N(M 2 1). (54) Proof: Cramer-Rao bound (CRB) can be interpreted as a ower bound of the variance of the estimator. The CRBs of the singe path case, y u = g u u(µ, ν) + z u, are introduced, where each eement of z is i.i.d Gaussian with zero mean and unit variance. According to [25], the Fisher information matrix is cacuated by F(µ, ν) = 2 g u 2 R uh u µ µ Appying (11) into (55), we can get an anaytica expression of the Fisher information matrix as F(µ, ν) = 2 g u 2 u H ν u µ u H µ u H ν u ν u ν π 2 MN(M 2 1) π 2 MN(N 2 1) 3. (55). (56)

14 Normaized MSE of u (db) Normaized MSE of v (db) case 1 rea case 1 bound case 2 rea case 2 bound SNR (db) case 1 rea case 1 bound case 2 rea case 2 bound SNR (db) Fig. 3. Comparison of the practica normaized MSEs and the ower bounds for µ and ν. Case 1: M = 32, N = 128; Case 2: M = 32, N = 64. Then, the CRB of the deay is expressed as Simiary, the CRB of the ange is CRB µ = F 1 1,1 (µ, ν) = 3 2 g u 2 π 2 MN(N 2 1). (57) CRB ν = F 1 2,2 (µ, ν) = 3 2 g u 2 π 2 MN(M 2 1). (58) With a containing the piot and the noise power equaing 1, the signa-to-noise ratio (SNR) here is measured through g u 2, that is, g u 2 = SNR. Moreover, CRB µ E { ˆµ µ 2} and CRB ν E { ˆν ν 2}, from where we obtain (53) and (54), respectivey. When we set M = 1, the probem is reduced to the bivariate case that ony the gain and the deay are to be estimated. If p(µ) = [ e j N/2 µ,..., e j( N/2 1)µ] H / N, then the CRB of deay is written as CRB (M=1) 6 µ = SNR(N 2 1), (59) which is exacty in accordance with the CRB bound given in [20]. It proves the correctness of Theorem 2. Coroary 1: When N or M grows arge, the ower bounds of the normaized MSEs of the deay and ange coincide, i.e., 3 ε µ, ε ν SNR 2π 2 MN. (60) Proof: It hods that N 2 /(N 2 1) 1 when N grows arge. Then (53) approaches (60). Simiary, (54) approaches (60) when M grows arge. Remark: From Theorem 2 and Coroary 1, we can find that the bounds can be further owered if the number of subcarriers occupied by the piots or the number of BS antenna eements increases. This is because with more observed sampes, we can see more detais about the spatia channe. What shoud be emphasized are the preconditions of high estimation accuracy, i.e., the anges and deays of different paths are we separated and that the number of paths is far ess than M or N. In addition, ony if the channe satisfies these preconditions can the agorithm achieve ower-bound performances. The resuts in Fig. 3 provide an intuitive comparison of the normaized MSEs with the derived ower bounds. The circes and stars represent the practica MSEs of the trivariate NOMP agorithm, and the dotted ines are the ower bounds. We set γ 1 = γ 2 = 2. A tota of 15 equa-power paths are present in the channe. The minimum separations among the deays and

15 15 the anges are no ess than 1/N and 1/M, respectivey. We first evauate case 1, where M = 32 and N = 128. The vaues of M and N satisfy the condition in Coroary 1 and we find that the ower bounds of the deay and the ange are neary the same. Besides, the estimation accuracy is enhanced proportionay with the increase of SNR. The practica MSEs cosey coincide with their theoretica ower bounds for both µ and ν, which demonstrates the high accuracy of the trivariate NOMP agorithm. Moreover, even though M N, the practica estimation accuracy of ν is not inferior to that of µ because of the we-separated spatia anges of each path and the significanty ower number of paths compared to M or N. The resuts in Fig. 3 aso compare the performances of the agorithm in case 2, where M = 32 and N = 64, that is, the number of subcarriers is haf of that in case 1. The resuts demonstrate that the practica estimation accuracy degrades and the MSE ines deviate with the theoretica ower bounds. The ower bounds are accessed when the observations are far more than the propagation paths. Despite this, the MSEs of µ and ν are beow 30 db, demonstrating that the practica estimation accuracy of the deay and ange remain high. V. PERFORMANCE EVALUATION In this section, we evauate the performance of the proposed efficient downink channe reconstruction scheme. We first discuss our computer simuation resuts and then move on to our hardware OTA tests for vaidation. A. Simuation Resuts Computer simuations are reaized through MATLAB. For the NOMP agorithm, the over-samping rates of the deay and ange are set to 2 and 4, respectivey. One round of singe refinement and three rounds of cycic refinement are impemented during each NOMP iteration. The number of FFT points is set as 2048 and the subcarrier spacing is set as 75 khz. Note that we infer the out-of-band or downink channe soey using the in-band or upink derived gains, deays and anges as suggested in [19], whie we reconstruct the channe by utiizing the out-of-band or downink refined gains and in-band or upink derived deays and anges as suggested by the proposed efficient reconstruction scheme. Channe inference is equivaent to channe reconstruction for the in-band or the upink, and both are reaized by appying the NOMP estimated gains, deays, and anges in the channe mode. The necessity of the gain refinement is first vaidated through a comparison of the reconstructed channe s ampitude with that of the rea channe. The tota bandwidth is MHz. The center frequency of 45 MHz is regarded as the in-band to estimate the frequency-independent parameters. The out-of-band channes on the other bands are then inferred or reconstructed using these parameters. We consider a simpe exampe where the BS is equipped with one antenna. Fig. 4(a) reveas the difference between the actua fu-band channe and the inferred channe. Within the 45 MHz in-band, we note that the inferred channe matches we with the actua channe, corroborating the precision of the NOMP agorithm. On the other hand, for the out-of-band, an obvious deviation can be seen between the inferred and actua channes. The arge performance degradation indicates that the gains derived from the in-band estimation are insufficienty accurate in inferring the out-of-band channe. Therefore, the gains are refine using LS estimation with the aid of the out-of-band piots which are inserted in every four subcarriers. The resuts of the refinement are given in Fig. 4(b). Resuts demonstrate that the reconstructed out-of-band channe matches cosey with the actua channe, thereby vaidating the necessity and effectiveness of the gain refinement. Now, we examine the MSE performance of the proposed efficient downink channe reconstruction scheme in FDD-OFDM systems. In both upink and downink OFDM modues, the centra 1200 subcarriers around DC compose the transmission band whose bandwidth equas 90 MHz. The separation between the upink and downink centra subcarriers is 300 MHz. In

16 NOMP inferred channe Practica channe 1 Out of band In band Out of band 0.8 Ampitude Frequency bands (a) NOMP-LS reconstructed channe Practica channe Ampitude Frequency bands (b) Fig. 4. Fu-band channe inference and reconstruction resuts, where (a) presents the inferred channe in the fu band, and (b) shows the reconstructed channe in the out-of-band. the upink, a 1200 subcarriers are fied with piots for the NOMP agorithm. Moreover, in the downink, piots are sparsey and uniformy inserted in every K subcarriers. We focus on two propagation scenarios. Scenario (a) is a sparsey scattering scenario, where two distinct paths exist in the channe. The anges of the two paths are i.i.d. and randomy generated in [0, 360 ). Scenario (b) is the custering channe, where there is one custer with six cose paths. The anguar spread of the custer is 30. In addition, the SNR measures the ratio of the piot power versus the noise power on one antenna and for each subcarrier. The LS and LMMSE channe estimation resuts are introduced as the ower and upper benchmarks, respectivey. LS is a commony used estimation method with ow compexity, but has a drawback of increasing the noise. LMMSE is an improved estimation agorithm that fixes this drawback and achieves consideraby higher accuracy. When conducting LS and LMMSE estimation agorithms, we use the piots on every four subcarriers. We ikewise compare the reconstructed downink and reconstructed upink channes with the actua channe by evauating their MSE performance, which is cacuated as } E { ĥ h 2 MSE =, (61) where ĥ denotes the reconstructed channe on one subcarrier, h is the rea channe, and σ2 z is the addition of the noise power on mutipe antennas which equas M here. The case when the BS is equipped with four antennas is tested first. Fig. 5(a) demonstrates the MSE performances of the reconstructed upink and downink channes in Scenario (a). Resuts show that the LS estimated channe has the worst MSE, whereas the upink reconstructed channe has the best MSE. Furthermore, the precision of the upink-reconstructed σ 2 z

17 17 (a) (b) Fig. 5. MSE performances of the reconstructions when M = 4 (a) in Scenario (a), and (b) in Scenario (b). channe is even higher than the LMMSE-estimated downink channe. This finding is attributed to the empoyment of in-band piots in the estimation of in-band CSI, and obtaining an accurate composition of the muti-path components through the trivariate NOMP agorithm. Hence, the upink-reconstructed channe is amost the same as the actua channe. To evauate the efficient downink reconstruction scheme, we adopt both beamforming types and compare their MSE performances. As expected, the MSE performance of the downink reconstruction is inferior to that of the upink reconstruction. Especiay when using beamforming type 2 and setting K = 4, a significant performance gap appears between the upink and downink reconstructions. If we increase the density of downink piots by setting K = 2 or switch to beamforming type 1, the MSE resuts are improved. As an overy high performance is not necessary and the cost is arge, a baance must be reached between performance and cost. The numerica resuts of the four-antenna case in Scenario (b) are presented in Fig. 5(b). Extracting each path from their spatia superposition is difficut because the paths are custered within a sma anguar-spread area. This condition is particuary true if the anges cannot be accuratey estimated, and the number of estimated paths may be more or ess than the paths that actua channe has. Hence, the performance of upink channe reconstruction degrades when compared with that in Fig. 5(a). Regarding the downink reconstructions, using beamforming type 1 sti achieves exceent MSE performance due to the arge amount of downink beamformed piots. By contrast, using beamforming type 2 resuts in about 9 db oss in MSE when compared with type 1 if we set K = 4 because a reativey arge number of estimated paths exist and each estimated direction cannot be aocated with enough piots. The accuracy is significanty enhanced when setting K = 2. Therefore, the amount of downink piots shoud be increased in proportion to the number of detected propagation paths.

18 18 Fig. 6. MSE performance of the reconstructions when M = 32 in Scenario (b). TABLE I OTA CONFIGURATIONS Parameter Vaue Antenna Bandwidth 90 MHz Carrier Frequency 3.5 GHz Samping Rate MHz Number of FFT Points 2048 Subcarrier Spacing 75 khz Transmit Power 20dBm Now, we scae up the computer simuations by considering the 32-antenna configuration in the more compicated and commony seen Scenario (b). This simuation aims to assess the performance of the proposed reconstruction scheme in massive MIMO environments. The over-samping rates are reduced by setting them to 1. The resuts are shown in Fig. 6. Ceary, when the number of antenna eements grows arge, the reconstructed channe has exceent performance as we. By comparing Fig. 6 with Fig. 5(b), we first find that the MSE performance of the upink reconstruction in the 32-antenna case is obviousy better than that in the four-antenna case owing to the high spatia resoution of a arge-scae antenna array. With the hep of the muti-antenna gain, the downink reconstructions are significanty improved as we. Therefore, the numerica resuts indicate that both the upink reconstruction and the efficient downink reconstruction perform we in reconstructing the actua channe. Tx Wardrobe Photocopier Tabe for equipment 4.5m (1) (2) Working area with desks and computers Transmitter Receiver Cabinet Equipment Rx (a) (b) Fig. 7. (a) OTA test environment in a aboratory and radio devices paced aong the tabe for equipment. (b) The eft and the right subfigures show the user and the BS, respectivey.

19 19 B. OTA Test Resuts We aso set up an OTA testbed [Fig. 7(a)] to vaidate the resuts in practica environments. The radio devices are paced aong the tabe for equipment. The yeow circe represents the position of the user, and the red squares are the potentia positions of the BS. The BS and the user are equipped as shown in Fig. 7(b). The user works as the transmitter and has a singe antenna controed by a RF vector signa generator. The BS is the receiver. The received signa at the BS antenna array is first transported to a digita oscioscope. After down-converting, synchronizing, and samping, the received signa is imported to the computer and processed through MATLAB. Fig. 7(b) iustrates the BS antenna array, which is a four-eement ULA where one coumn of the array is combined to form a ULA eement. When evauating the singe-antenna case, the ULA is repaced by one antenna eement ike the user antenna. The configurations of the OTA tests are isted in Tabe I. Owing to the imitations of the hardware equipments, in-band versus out-of-band tests are used to imitate the upink versus downink tests. Considering the antenna bandwidth, we seect the in-band and out-of-band regions within the centra 90 MHz band and separate them to the greatest extent. As shown in Fig. 8, we regard the red region with 45 MHz bandwidth as the in-band to imitate the upink. The 15 MHz-bandwidth region coored in bue is chosen as the out-of-band to imitate the downink. The centra frequency of the out-of-band region is 60 MHz away from the centra frequency of the in-band region. In the gain refinement stage, K = 2 or 4, which means that one-haf or onefourth of the subcarriers are aocated for the out-of-band piots. Given the high accuracy of LMMSE estimation agorithm, we regard the LMMSE-estimated channe as the rea channe when evauating the out-of-band reconstruction scheme. The out-of-band channe inference method is aso evauated, which represents the method introduced in [19]. Simiary, MSE is used as the metric, which is cacuated as { E ĥ ĥlmmse 2} MSE =, (62) where ĥlmmse is the LMMSE-estimated channe and is regarded as the rea channe. The test resuts are dispayed in the form of a cumuative distribution function (CDF). We start by reconstructing the simpest channe when the BS is equipped with a singe antenna. In the singe-antenna tests, K = 4. Fig. 9(a) provides the CDF of the MSE (in db) when the BS is ocated at Position (1). The figure shows that inband reconstruction achieves the highest accuracy, with a 90% probabiity that the MSE is beow 9.65 db. However, the performance of the out-of-band inference is poor, with a 90% probabiity that MSE is ower than db, demonstrating that the inferred channe can not accuratey depict the actua channe. Fortunatey, the accuracy is greaty improved when the gains are refined with the aid of the out-of-band piots. The out-of-band reconstruction scheme functions we, with a 90% probabiity that the MSE is beow 6.64 db. These OTA resuts aign with the previous numerica resuts. We can further investigate the power ratio of the propagation paths through the resuts in Fig. 9(b). The first detected path occupies 75.2% power of the channe because a strong ine-of-sight (LoS) propagation path can be detected at Position (1). The power ratio increases to 94.3% after the second detected paths is added. The resuts aso indicate that the possibiity of detecting more than four paths is beow The performances are then tested when the BS is ocated at Position (2), and the resuts are given in Fig. 10. We find that both in-band and out-of-band reconstructions sti have exceent performances in the non-los (NLoS) propagation scenario. The 90%-probabiity MSEs of the two reconstructions are 9.00 and 6.17 db, respectivey. Athough the MSE performance is inferior to that of the LoS case, this performance degradation is reativey sma. As for the power ratio of the propagation σ 2 z