Wideband Distributed Transmit Beamforming using Channel Reciprocity and Relative Calibration

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1 Wideband Distributed Transmit Beamorming using Channel Reciprocity and Relative Calibration T. Patrick Bidigare BBN Technologies Arlington, VA Upamanyu Madhow Dept. o Electrical and Computer Engineering University o Caliornia, Santa Barbara, CA madhow@ece.ucsb.edu D. Richard Brown III Dept. o Electrical and Computer Engineering Worcester Polytechnic Institute, Worcester, MA drb@wpi.edu R. Mudumbai, A. Kumar, B. Peier, and S. Dasgupta Dept. o Electrical and Computer Engineering University o Iowa, Iowa City, IA {rmudumbai,amy-kumar,benjamin-peier,dasgupta}@uiowa.edu Abstract This paper describes a technique or transmit beamorming rom the nodes in a distributed radio network to a distant target node across a requency-selective channel. The approach exploits signals transmitted by the target node and channel reciprocity to avoid requiring explicit channel state eedback rom the target. Channel estimates between network radios are used or relative calibration to address non-reciprocal eects due to independent clocks and electronic component variability. Variants o the technique allow wideband coherent beamorming to the target when the target signal is known or when it is unknown. Index Terms distributed beamorming, channel reciprocity, relative calibration, synchronization I. INTRODUCTION It is well-known that channel state inormation at the transmitters (CSIT) can improve the eiciency o wireless communication and can acilitate coherent communication techniques including distributed beamorming (see [1] and the reerences therein) and distributed nullorming, e.g., [2] [6]. The diversity and power growth associated with distributed beamorming allows a network o low-power radios to communicate more reliably and over much longer distances than single radios. In the context o the system model shown in Fig. 1, each node j {1,..., N} in the transmit cluster needs an estimate o the orward link (uplink) channel H j 0 () to acilitate coherent transmission to the target node. One common approach is eedback-based techniques, e.g., [7] [13], where the target node estimates the channels and eeds back quantized versions o these estimates to the transmit cluster. Feedback techniques suer rom overhead and latency which can be prohibitive, especially in large-scale massive MIMO [14] systems. Direct estimation o the uplink channel by the target quickly becomes SNR-limited as network sizes and communication ranges increase since isotropic channel sounding doesn t beneit rom beamorming gain. This work was supported by the National Science Foundation awards CCF , EPS , ECCS , CNS and CCF , CCF , CCF and ONR grant N An alternative approach are reciprocity-based techniques, e.g., [15], [16], where the transmit cluster estimates the uplink channels rom signals emitted by the target node on the reverse link (downlink). These techniques assume the uplink and downlink are time division duplexed (TDD) and accessed on the same requency. Reciprocity-based techniques are especially attractive in asymmetric links such as cellular systems where the transmit power available at the target node is considerably higher than the individual radios in the transmit cluster and downlink channels can be accurately estimated at much longer ranges. 1 cross links transmit cluster orward link (uplink) reverse link (downlink) target node Fig. 1. System model with and N node transmit cluster and a single target node. Each node is assumed to have a single antenna. Node 0 corresponding to the target and node 1 corresponding to the master transmit node assumed to have a direct link to transmit nodes j {2,..., N}. Basic electromagnetic principles have long established that channel reciprocity holds at the antennas when the channel is accessed at the same requency in both directions [17]. While propagation is inherently reciprocal, the radio requency chains in each transceiver are generally not reciprocal and may vary with temperature, aging, and other eects. These eects can sometimes be mitigated with specialized transceiver architectures such as a reciprocal transceiver architecture, e.g., [18]. To compensate or time-varying eects, however, it is usually preerred to perorm some type o transceiver calibration, 0

2 e.g., [19]. A particularly interesting approach to compensating or non-reciprocal transceivers is relative calibration [20]. To summarize this approach, let H j 0 () and H 0 j () denote the uplink and downlink channels, respectively, including the eects o the transceivers and clock oset. By exchanging messages between node j and node 0 and receiving eedback 1 rom node 0 regarding the j 0 channel, node j can estimate the relative calibration unction H j 0 () (1) H 0 j () using, or example, a structured total least squares solver [21]. Then, given a downlink signal rom node 0 to node j, node j can simply estimate the downlink channel H 0 j () multiply this estimate by the relative calibration unction in (1) to generate a corresponding estimate o the uplink channel or subsequent use during coherent transmission. The main contribution o this paper is an explicit description o a techniques or implementing distributed beamorming using channel reciprocity in a scenario where the nodes in the transmit cluster can not perorm relative calibration with the target node. The techniques described in this paper can be used in scenarios where the target node is unable to exchange calibration messages with the transmit cluster. For example, the target node may not have the capability to estimate the uplink channels and provide eedback or relative calibration. Our techniques are based on indirect relative calibration where the nodes in the transmit cluster exchange messages only among themselves and pre-calibrate prior to the target node emitting a signal. The two variants o this approach, distinguished by whether the signal emitted by the target node is known or unknown, are described. In both cases, we show that the transmit cluster can compute appropriate precoder ilters to pre-compensate or clock osets, electronics and propagation delays and achieve wideband coherent combining at the target node. Numerical results characterize the perormance o the variants in terms o the uplink channel capacity when compared to the capacity when perect CSIT is available. II. SYSTEM MODEL We consider the system shown in Figure 1 with nodes numbered {0,..., N}. Each node is assumed to have a single antenna. Node 0 corresponds to the intended target. Nodes j {1,..., N} comprise the transmit cluster and node 1 corresponds to the master transmit node. It is assumed that node 1 can directly communicate with each o the other transmit nodes j {2,..., N} in the transmit cluster. The eective channel between any pair o nodes i j can be expressed as CLK i TX i PROP i j RX j CLK 1 j. (2) For simplicity, we assume all o these eects are linear and slowly time varying so that the requency response over time intervals o interest is approximately constant. We can also 1 Note that the relative calibration eedback rom node 0 is inrequent since it is only used to compensate or transceiver non-reciprocity. lump the eect o the clock and transmit electronics at node i as T i () and, similarly, the eect o the clock and receive electronics at node j as R j (). Then, in the requency domain, we have H i j () = T i () }{{} transmitter G i j () R j (). (3) }{{}}{{} propagation receiver In general, while propagation is reciprocal such that G i j () = G j i (), the transmitter and receiver unctions are non-reciprocal and H j i () H i j (). From (3) it ollows immediately that or any sequence o nodes, the clockwise and counter-clockwise concatenation o their channels are equal [15], [16]. In particular, or every node j {2,..., N} H j 0 ()H 0 1 ()H 1 j () = H 1 0 ()H j 1 ()H 0 j () (4) which can be rewritten in two ways as H j 0 () = H 1 0() H 0 1 () H j 1()H 0 j () H 1 j () = H 1 0 () H j 1()H 0 j () H 1 j ()H 0 1 (), (5a) (5b) showing that the uplink channels are products o a term that is common across all the uplink channels (let term) times a term involving only the downlink and crosslink channels (right term). The two key insights behind our beamorming approach are 1) Beamorming coherence only requires precoding to correct or the relative uplink channels; distortions common to all uplink channels (let terms) can be equalized by the target node. 2) In addition to the downlink channels, we are also able to measure the crosslink channels via node-to-node channel sounding (right terms). III. RECIPROCAL BEAMFORMING PROTOCOLS In this section, we develop reciprocal beamorming protocols or two cases: (i) known downlink signals and (ii) unknown downlink signals. In the ormer case, the transmit cluster can use the known downlink signal to directly estimate the downlink eective channels H 0 j () or j {1,..., N}. In the latter case, we assume the transmit cluster can not directly estimate the downlink eective channel due to the act that the downlink signal is unknown. Both protocols can be described in our phases: Phase I: Transmit cluster pre-calibration Phase II: Transmit cluster reception o waveorm rom target Phase III: Precoder estimation Phase IV: Transmit cluster beamorming Both cases have have essentially the same transmit cluster precalibration phase (Phase I) but dier in the remaining phases. The ollowing sections provide details on the our phases or the cases with known and unknown downlink signals.

3 A. Known Downlink Signals 1) Phase I: Transmit cluster pre-calibration: In this phase, each node j {2,..., N} exchanges known messages with the master node (node 1) to synchronize their clock requencies and estimate the relative calibration unction H1 j() H. j 1() Speciically, each node j {2,..., N} sends a known message to the master node and the master node estimates the eective channels H j 1 (). Node 1 then broadcasts one or more known messages and each node j {2,..., N} estimates the eective channel H 1 j (). By also eeding back estimates o H j 1 () rom the master node, node j can then estimate the relative calibration unction H j 1 () H 1 j () = T j()r 1 () T 1 ()R j (), (6) In the presence o noise, these estimates can be generated with a structured total least squares (STLS) solver as discussed in [20]. Accurate requency synchronization can also be achieved during this phase since each node j {2,..., N} can adjust its clock rate according to the observed requency oset in messages received rom node 1. 2) Phase II: Transmit cluster reception o waveorm rom target: The target (node 0) transmits the known signal X 0 (). Nodes j {1,..., N} then receive Y j () = H 0 j ()X 0 (). (7) We assume X j () is non-zero on all. In this case, node j can directly estimate the quantity Y j () X 0 () = H 0 j(). (8) 3) Phase III: Precoder selection: From (5a) we see that the uplink channel H j 0 () is, up to a common term, the product o (6) and (8). We a priori assume the unknown common term is the identity H1 0() H 0 1() = 1 which represents the an inormation-less prior. The nodes can now compute precoders or the resulting uplink channels: { W KD j 0 = H 0 1 () i j = 1 H j 1() H H 1 j() 0 j() i j = 2... N In a distributed application, the transmit power at each node will be ixed and thus we will have a unity-gain constraint on each o the precoders. The capacity-maximizing precoder ormulation in this scenario is derived in [22]. A sub-optimal but eective alternative, with a closed orm, is to select the precoder to be the scaled conjugate o the relative uplink channel, i.e. P KD j () = W KD j 0() KD j 0 () 2 d (9) (10) A practical precoding scheme must introduce a bulk beamorming delay across all the nodes to insure the resulting precoders are causal. 4) Phase IV: Transmit cluster beamorming: Upon some local clock trigger, each node j {1,..., N} transmits the common signal X() to the target with precoder ilter Pj KD () to orm a beam at the target. Note that the precoder ilters align the signals so that they combine coherently at node 0, including compensating or clock osets among the transmit nodes. For the precoders deined in (10), the aggregate SISO channel seen at node 0 is HBF 0() KD = Pj KD ()H j 0 () = H 1 0() H 0 1 () KD j 0 () 2 KD j 0 () 2 d H 1 0() H 0 1() (11) For reasonable transceiver designs, is approximately lat and should not typically introduce deep nulls or other undesirable artiacts into the received signal at node 0. Hence, in the case with known downlink signals, the transmit cluster eectively equalizes the uplink channels to node 0. Residual amplitude variation can be accommodated via receive-side equalization. B. Unknown Downlink Signals = R0()T1() T 0()R 1() 1) Phase I: Transmit cluster pre-calibration: This phase is identical to the case with known downlink received signals. One minor dierence is that nodes j {2,..., N} do not need to estimate the relative calibration unction as in (6) but, rather, as will be seen in Phase III, only need to orm an estimate o H j 1 () which can be obtained via eedback rom node 1. 2) Phase II: Transmit cluster reception o waveorm rom target: The target (node 0) transmits the unknown signal X 0 (). Nodes j {1,..., N} then receive Y j () = H 0 j ()X 0 (). Since X 0 () is unknown, node j can not directly separate H 0 j () rom X 0 (). Node 1 now rebroadcasts its received signal in Phase II to nodes j {2,..., N}. The delay between target signal reception and rebroadcast is known to all nodes and ignored here. Node j then receives Z j () = H 1 j Y 1 () = H 1 j ()H 0 1 ()X 0 () (12) Node j {2,..., N} now computes the quotient channel Q j () = Y j() Z j () H 0 j () = H 1 j ()H 0 1 (). (13) 3) Phase III: Precoder selection: From (5b) we see that the uplink channel H j 0 () is, up to the common term H 1 0 (), the product o H j 1 () and (13). As beore we assume the inormation-less prior H j 0 () = 1 and compute precoders or the resulting uplink channels: W UD j 0 = { 1 i j = 1 Q j ()H j 1 () i j = 2... N (14)

4 The same precoder approach as in the known-downlink signal case can be used here as well. The simple but sub-optimal scaled conjugate precoder is P UD j () = W UD j 0() UD j 0 () 2 d (15) 4) Phase IV: Transmit cluster beamorming: This phase is also identical to the case with known downlink signals. For the precoders deined in (15), the aggregate SISO channel seen at node 0 is H UD BF 0() = = H 1 0 () Pj UD ()H j 0 () UD j 0 () 2 UD j 0 () 2 d (16) Unlike the case with known downlink signal, we observe that this aggregate channel includes the eect o the propagation channel G 0 1 (). The uplink channels are not equalized (except to match the 1 0 eective channel). Hence, the perormance o this technique may be sensitive to the choice o the master node (node 1). C. Remarks Most practical channels have a maximum possible delay spread which imposes a smoothness constraint on their transer unctions. This smoothness constraint can be imposed to insure the stability o estimators (8) and (13) which involve the ratio o noisy samples. IV. NUMERICAL RESULTS The beamorming perormance o our techniques can be quantiied in terms o the resultant ergodic capacities o the beamorming channels (11) and (16). This ergodic capacity [23] is given by: ( ( E Hi j log HBF 0 () 2) ) d (17) Fig. 2 shows the ergodic capacity vs. per-transmitter SNR or known and unknown downlink signals or a N = 10 node network with L = 8 independent requency subcarriers. The transmitter and receiver transer unctions T j () and R j () were chosen to be unity gain and to have independent random phases between nodes and subcarriers. The propagation channels G i j () were modeled as independent, identically distributed complex Gaussian random variables with variance SN R/L. In this simple scenario, we model the propagation channels as IID across subcarriers and between nodes. For this scenario we note that there is less than a 0.25 bits/s/hz reduction in capacity associated with not knowing the downlink signal. The results highlight the potential beneit o distributed transmit beamorming or extending communications range: spectral eiciencies greater than 1 bit/s/hz can theoretically be achieved even or per-transmitter SNRs below -10dB. Fig. 2. Ergodic capacities o 10-node requency-selective beamorming channels or known downlink (KD=blue) and unknown downlink (UD=green) signals. Solid line shows capacity-maximizing precoder [22], dashed line shows simple scaled-conjugate precoder (10, 15). V. CONCLUSION The prolieration o networked wireless devices enables the possibility o distributed coherent communications which can provide much longer uplink ranges and better qualityo-service than possible with single devices. While much o the early work in this area has looked at eedback-based techniques, direct uplink CSI estimation at the target node becomes SNR-limited as communications ranges increase. Reciprocity-based techniques that estimate downlink CSI potentially beneit rom higher transmit powers at the target node, but independent clocks and electronic components introduce non-reciprocal components which must be addressed. The contribution o this paper is a practical method o using crosslink channel estimates to provide an indirect relative calibration or these eects. We show that this technique is applicable whether or not the downlink signal rom the target node is known. The technique in this paper relies on a star-topology within the radio network; uture work could address the generalization o this technique to more arbitrary network topologies. REFERENCES [1] R. Mudumbai, D. Brown, U. Madhow, and H. Poor, Distributed transmit beamorming: challenges and recent progress, Communications Magazine, IEEE, vol. 47, no. 2, pp , February [2] D.R. Brown III and U. Madhow, Receiver-coordinated distributed transmit nullorming with channel state uncertainty, in Con. In. Sciences and Systems (CISS2012), Mar [3] D.R. Brown III and R. David, Receiver-coordinated distributed transmit nullorming with local and uniied tracking, in Proc. o the 39th International Con. on Acoustics, Speech, and Signal Processing (ICASSP2014), Florence, Italy, May [4] D.R. Brown III, P. Bidigare, S. Dasgupta, and U. Madhow, Receivercoordinated zero-orcing distributed transmit nullorming, in Statistical Signal Processing Workshop (SSP), 2012 IEEE, Aug. 2012, pp

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