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Impact of Half-duplex Radios and Decoding Latencies on MIMO Relay Channel Ashutosh Sabharwal Department of Electrical & Computer Engineering Rice University, Houston, TX 77005 ashu@rice.edu Abstract In this paper, we study achievable rates for the fading relay channel with multiple antennas, without channel state information at the transmitters. Our main motivation is understanding how much of theoretically predicted gains are achievable in real implementations of relay-based networks. Two “imperfections” are considered in this paper. First, we study how commonly used half-duplex radios aﬀect the performance of the relay channel. As shown in this paper, half-duplex radios can signiﬁcantly reduce the peak gains compared to the relay network using full-duplex radios, but still retain signiﬁcant superiority over the direct link without the relay. More importantly, we ﬁnd that simple data forwarding (sending to relay and relay forwarding to destination) performs reasonably close to upper bound on half-duplex relay channels, thus making multi-hop communication in distributed networks not too far from optimal. We also ﬁnd that decode-and-forward is near-optimal for the half-duplex relay channel in many geometries, and thus, is the coding scheme of choice to achieve high data rates. But decoding at the relay introduces decoding latencies, which is the topic of our second “imperfection.” Our analysis of full-duplex and half-duplex relay channels indicate that the full-duplex relay channel is more seriously aﬀected by ﬁnite decoding latencies compared to half-duplex. In fact, with appropriate choice of codelengths, the half-duplex relay channel need not suﬀer any rate losses. 1 Introduction Network coding oﬀers the possibility of using the distributed power and antenna resources of the system to increase data throughput and improve communication reliability. A number of results have addressed diﬀerent aspects of the distributed network coding, ranging from information theoretic limits [1, 4, 3, 15, 6, 2] to actual system design [12, 16, 9]. The main attraction of network coding, that is using the distributed resources of power, computation and antennas, is also its biggest challenge. Often, distributed systems have to contend with numerous issues not present in centralized systems, and which can limit their performance signiﬁcantly.1 In this paper, we look at the impact of two most immediate issues – half-duplex radios and ﬁnite processor capabilities of the relay node in a three-node network. Our driving question is two-fold. First, how much gain can we realistically expect from a relay- based system, if the system has to be built using realizable parts. It is not surprising that the system with “less-than-perfect” parts will not deliver performance comparable to a system with inﬁnitely capable parts. Thus, the more pressing question is what is the extent of these losses, which is 1 Examples are abound in parallel computing, where computational throughput does not always linearly scale with number of processors and can be limited by communication throughput between the processors or the amount of management overhead to maintain full processor utilization. 1 1802 what we explore in this paper. The second related question is that if the gains of using a relay are appreciable to warrant building a real system, then what is the form of best coding methods to optimally use the relay. We give some preliminary insights into this question. Our performance metric will be achievable rates, instead of diversity order. Most systems operate in low to medium SNR regimes, where the gain from large diversity orders is not truly relevant. Thus our practical focus is on achieving higher throughputs for reasonable packet losses. First, we analyze the impact of half-duplex radios on relay channel’s capacity. Half-duplex radios can either receive or transmit but not do both simultaneously. Most radios used in the practical systems are half-duplex radios since making them full-duplex requires very expensive, precise parts with high-dynamic range. We observe that half-duplex radios can lose signiﬁcantly compared to full-duplex radios in certain geometries, but can still deliver many-fold more throughput compared to a single-antenna non-relay based system. Furthermore, most of these large gains are achieved by a decode-and-forward architecture based on the work in [1]. The surprising ﬁnding is that a simple data forwarding method ( in which source sends to relay, relay completely decodes the packet and then forwards it to the destination) performs very close to decode-and-forward. Data forwarding is the fundamental communication modality in all distributed networks, like ad hoc and sensor networks. Our ﬁnding suggests that systems which perform appropriate route selection may have little to gain by moving to more complex relay channel based coding methods. The second aspect studied in this paper relates to an immediate implementation challenge in decode-and-forward methods, that is the decoding latency at the relay. In decode-and-forward, the relay has to completely decode the data, which implies that any realistic baseband processor operating at ﬁnite clock speeds is bound to introduce latencies into the system. Conventional relay formulation [1, and others] assume that the relay can instantly decode this data and can transmit in the very next symbol, maintaining completely synchronism with the source’s transmissions. However, actual processors performing decoding use ﬁnite clocks and have ﬁnite space-time complexity. In fact, they are fast enough to meet only the real time targets. Thus, if a packet arrives every S seconds, then the processors are fast enough to only decode a packet in no more than S seconds, thereby introducing a decoding latency of S seconds. We label such ”just-in-time” processors as maximal latency processors. We derive a simple achievable rate for full- and half-duplex relay channels where the relay uses a maximal latency processors. Two key observations are made. First, the full-duplex relay channel can have a large performance degradation for small number of coding blocks, since the full-duplex decode-and-forward is a Markovian code which requires inﬁnite number of blocks to attain the actual relay channel rate. Second, two time-slot decode-and-forward [7] is not possible in practice. However, if at least four time-slots are used, there is no loss in performance of half-duplex relay channels. We quickly note that we have not considered methods which are zero-latency in our model, like amplify-and-forward [7, 8], primarily none of them have been shown to outperform methods like decode-and-forward and estimate-and-forward [1, 5]. Though not explored in this paper, there appears to be an interesting tradeoﬀ between complexity of relay processing and resulting achievable rates. The rest of the paper is organized as follows. In Section 2, we introduce the system model and some notation related to the multiple antenna transmission. Section 3 derives the bounds on cheap relay channels and Section ?? introduces the problem of relay with vacations to account for decoding latencies. We conclude in Section 5. 2 1803 2 Preliminaries 2.1 System Model The signal transmitted by the transmitter in block t is represented by X1 (t), while the signal transmit- ted by relay is denoted by X2 (t). The received signals at destination and relay are given, respectively, by Y1 (t) = H1 X1 (t) + H2 X2 (t) + W1 (t), (1) Y2 (t) = H0 X1 (t) + W2 (t), (2) where H0 is the source-relay channel, H1 is the source-destination channel and H2 is the relay- destination channel. All channels are possibly matrix channels, i.e., they can have multiple antennas at both ends of each link. Speciﬁcally, the transmitter, relay and the receiver are assumed to have N, K, M antennas each, and the MIMO relay channel is denoted as N × K × M -relay channel. For a relay-channel with half-duplex radios, the network of three nodes, source, relay and desti- nation, can be one of two possible states, m1 and m2 . In state m1 , the relay listens and does not transmit anything. In state m2 , the relay transmits X2 (·) and cannot receive any signal from the source. 2.2 Multiple Antenna Basics The ergodic capacity of a t transmit and r receive antenna system is given by [11] P C(t, r) = E log det Ir + HH † , (3) t where Ir is r × r identity matrix and H is an r × t receive matrix. It is also well known that the capacity C(t, r) has an asymptotic growth of min(t, r) log P , where the multiplicative factor min(t, r) is commonly known as the multiplexing gain (see for example [17]). The sum-capacity of a two-user MIMO multiple access system [10] where the two-users have t1 and t2 antennas and the receiver has r antennas is equal to P1 † P2 † Cmac (t1 , t2 , r) = E log det Ir + H1 H1 + H2 H2 , (4) t1 t2 where Pi represents the average power available to user i and Hi is the channel between user i and the receiver. Since the optimal input distribution is i.i.d. for multiuser channel, which is also the optimal input distribution for single-user channel, the capacity of multiple access channel increases asymptotically as min(t1 + t2 , r) log P (if P1 = P2 = P ). Finally, we will label by Cbc (t, r1 , r2 ) as a MIMO broadcast channel whose sum rate is given by P1 † P2 † Cbc (t, r1 , r2 ) = E log det It + H1 H1 + H H2 , (5) t t 2 where Hi is the MIMO channel between the transmitter and user i and P1 + P2 ≤ P . For this channel, the sum-rate increases as min(t, r1 + r2 ) log P . 3 Half-duplex Radios In this section, we will derive an upper and lower bound on the capacity of the MIMO relay channel with half-duplex radios, and compare them with those derived for full-duplex radios in [15]. Our 3 1804 main comparisons will be with (a) relay channel with full-duplex radios, (b) data forwarding, which relies on relay decoding and simply forwarding the data (much like in most ad hoc networks), and (c) direct link communication which does not use the relay. 3.1 Upper and Lower Bounds The upper bound is derived using the multi-state cut-set theorem derived in [3], and bounds the performance of any relay protocol which uses half-duplex cheap relays. The lower bound is derived using the decode and forward protocol proposed for half-duplex relay channels in [4]. Much like in the case of full-duplex MIMO relay channel without CSIT and full CSIR [15], decode and forward is capacity achieving in certain cases. Theorem 1 The cut-set upper bound for N × K × M cheap relay channel is given by Chd,upper = max min {tCbc (N, K, M ) + (1 − t)C(N, M ), tC(N, M ) + (1 − t)Cmac (N, K, M )} . (6) t∈[0,1] Further, Chd,upper has the multiplexing gain of min(N, M ). Proof : The proof is similar to that for full-duplex radios [15] and is omitted here. The key step is to show that the optimal coding uses independent codebooks at the source and relay, and each of the codebooks is i.i.d. Gaussian. Finally, note that the minimum multiplexing gain of all the constituent terms in Cupper is min(N, M ), achieved by the C(N, M ). Thus, the multiplexing gain of Cupper is min(N, M ). For comparison, the cut-set bound for full-duplex radios is given as [15] Cf d,upper = min {Cbc (N, K, M ), Cmac (N, K, M )} . (7) To ﬁnd a lower bound on the capacity of the cheap relay channel, we use a ﬁxed encoding scheme based on decode-and-forward in [1] and extended to half-duplex radios in [4]. Theorem 2 The decode-and-forward lower bound for N × K × M cheap relay channel is given by Chd,df = max C(N, M ), max min {tC(N, K) + (1 − t)C(N, M ), tC(N, M ) + (1 − t)Cmac (N, K, M )} . t∈[0,1] Direct Decode and Forward (8) Also, Chd,df has the multiplexing gain of min(N, M ). Proof : Similar to the proof of Theorem 1. Again, as a comparison the full-duplex lower bound is given as [15] Cf d,df = max C(N, M ), min{C(N, M ), Cmac (N, K, M )} . (9) Direct Decode and Forward Note that both lower bounds choose between direct communication (without the relay) and relay- based communication, hence their minimum multiplexing gain is min(N, M ). Further, since the cut- set upper bound has a multiplexing gain of min(N, M ), the multiplexing gain of both lower bounds is 4 1805 min(N, M ). Further observe that both the decode-and-forward part of the lower bounds, (8) and (9) diﬀer only in the ﬁrst terms in the minimums when compared to their respective upper bounds (6) and (7). The ﬁrst term in the upper bounds and decode-and-forward (the ﬁrst term in the minima) represents the rate at which the source can send the data out, while the second represents how fast the data can be sunk into the destination. Thus, while decode-and-forward can sink the data in the fastest possible way, there is a possibility that getting data out of the source may not be at the possibly highest rate . The biggest diﬀerence between relay channel-based utilization of resources and conventional meth- ods is that both source-relay-destination and source-destination channels are used simultaneously throughout the whole communication. Thus, our two points of comparisons are using either of two “routes” to the destination while ignoring the second. The two schemes of particular interest will be as follows: 1. Direct Link : In this case, the system ignores source-relay-destination route and uses only the direct route between source and destination. The capacity of this system is simply C(N, M ) with power P1 at the source, resulting in a multiplexing gain of min(N, M ). 2. Data Forwarding: In this case, the system only uses the source-relay-destination route while ignoring the source-destination route. For the system with half-duplex radios, the capacity of data forwarding is [3] Cf = max min {tC(N, K), (1 − t)C(K, M )} t∈[0,1] C(N, K)C(K, M ) = , (10) C(N, K) + C(K, M ) with a multiplexing gain of min(N, K, M ). 3.2 Numerical examples In this section, we look at numerical examples to study the performance of diﬀerent methods as a function of relay location, amount of power at source and destination, and number of antennas at diﬀerent nodes. For each case, full-duplex bounds, half-duplex bounds, data forwarding and direct link are used as comparison. Furthermore, actual achievable rates are not of particular interest. Instead, we are more interested in where we gain the most over a simple no-relay based system. Hence, we have normalized all rates by direct link rate C(N, M ). 1. Relay location: Figure 1 shows the performance of diﬀerent methods as the relay location is changed (d measures the distance between source and relay, and D is the distance between source and destination). It is clear that the half-duplex radios can result in large peak losses (∼ 40%) over full-duplex. But can still retain large gains, in general, over direct link (2 to 5 times, depending on the number of antennas at the relay and pathloss exponents). Though the gain over direct link is substantial, the gain over simple data forwarding is not substantial, especially if the relay is equipped with the same number of antennas as the source and destination. Often in the literature, the comparisons are performed only with the direct link (may be with more antennas or more power), but not with data forwarding. However, one of the major reasons for gains in relay channel is eﬀective reduction in pathloss due to presence of relay in the middle, and actively “regenerating” degraded source’s signal. That feature is not present if the comparisons are only performed with direct link (even with more power or antennas). 5 1806 8 5 Cut−set Bound (Full−duplex) Cut−set Bound (Full−duplex) Decode & Forward (Full−duplex) Decode & Forward (Full−duplex) Cut−set Bound (Half−duplex) Normalized Gain (Actual Rate/Direct Link) 4.5 Cut−set Bound (Half−duplex) 7 Decode & Forward (Half−duplex) Normalized Gain (Actual Rate/Direct Link) Decode & Forward (Half−duplex) Data Forwarding Data Forwarding Direct Link 4 Direct Link 6 3.5 5 3 4 2.5 2 3 1.5 2 1 1 0.5 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized distance (d/D) Normalized distance (d/D) (a) (b) Figure 1: Impact of half-duplex radios on relay channel with P1 = P2 = 10 dB, and pathloss exponent of 3 for (a) 3 × 1 × 3 relay channel and (b) 3 × 3 × 3 relay channel. The loss due to half-duplex nature of radios ranges from negligible to more than 40% depending on relay’s location. 2. SNR: Figure 2 shows the impact of increasing power at both source and destination. Though the rates achieved by each of the methods are increasing, their gain over direct link is quickly vanishing as the SNR increasing. Direct path is close to the best one can do for high SNRs. Alternately, the biggest advantage of relaying is at low to medium SNRs. Finally, data for- warding has reasonable performance when compared to half-duplex decode-and-forward across the whole SNR range, making it an attractive low-complexity alternative. 3. Number of Antennas: Figure 3 shows how increasing the number of antennas at one node while keeping others constant impacts the gain over direct link. The impact of increasing number of source antennas is minimal (the normalized gain is almost constant over the whole range) and hence that result is not shown here. Figure 3(a) shows that increasing number of relay antennas results in a (seemingly) monotonic growth over direct link. On the other hand, increasing destination antennas has the opposite eﬀect, cf Figure 3(b), and the gain over direct link steadily reduces. Thus, spatial diversity is most useful for the relay channel when present at relay node, since it is visible to both source and destination. From the above limited evaluation of diﬀerent methods, we can conclude the following. Relaying is most beneﬁcial when relay has the many antennas and the system is operating at low to medium SNRs. Further, though true relay codes perform very well, simple data forwarding performs very well compared to direct link in most cases. 4 Impact of Decoding Latencies The major gain from relay comes from the coherent combining of source’s and relay’s transmissions at the destination. Thus, it is important that the relay and source are time-aligned. The decode and forward architecture requires the relay to decode the data completely before re-encoding it for transmission. With a ﬁnite speed processor, decoding at the relay introduces latencies, which implies that the source cannot cooperate with relay instantly after ﬁnishing its transmission of a codeword. In this section, we study how the decoding latency aﬀects the full- and half-duplex relay channels. Often the baseband processors on receivers are built to meet the real-time decoding requirements 6 1807 9 4 Cut−set Bound (Full−duplex) Cut−set Bound (Full−duplex) Normalized Gain (Actual Rate/Direct Link Rate) Decode & Forward (Full−duplex) Normalized Gain (Actual Rate/Direct Link Rate) Decode & Forward (Full−duplex) 8 Cut−set Bound (Half−duplex) Cut−set Bound (Half−duplex) 3.5 Decode & Forward (Half−duplex) Decode & Forward (Half−duplex) Data Forwarding Data Forwarding Direct Link Direct Link 7 3 6 2.5 5 2 4 1.5 3 1 2 0.5 1 0 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 Signal to Noise Ratio (dB) Signal to Noise Ratio (dB) (a) (b) Figure 2: Achievable gains as a function of power at the source and relay. Assuming P1 = P2 , pathloss exponent of 3 and relay in the middle for (a) 3 × 1 × 3 relay channel and (b) 3 × 3 × 3 relay channel. The biggest gain from relaying is at low to medium SNRs, where relaying results in a reduction in eﬀective path loss. based on the data rates. Thus, if a packet is received every S seconds, then the previous packet should be completely decoded in no more than S seconds to ensure that there is no build-up in the input buﬀer of the receiver. We will label such receivers as maximal latency processors. Analogously, if a processor can ﬁnish decoding packets in less than S seconds, then it will be labeled as fractional latency processor. The maximal latency for decode-and-forward coding method for full- and half- duplex relay channels can be computed as follows. Full-duplex relay channel: The Markovian decode-and-forward proceeds with B blocks of length n codewords. Thus, the maximal latency which can be tolerated to achieve the asymptotic relay chan- nel rate is no more than n symbols. Half-duplex relay channel: In half-duplex channels, for the smallest end-to-end delay, the relay switches between the two modes of receive and transmit. It is in the receive mode for n symbols n followed by transmit mode for next m symbols, such that t = n+m , where t is the fraction of time to be spent in receive mode. To ensure same rate as inﬁnite processor relay, the encoding can be modiﬁed to have two receive blocks of n symbols followed by two transmit blocks m. In this case, again, the maximal latency allowable is n symbols. Proposition 1 (Rate with maximal latency processor) Let the decoding latency of the relay be nd < n symbols. Further assume that n symbol codewords are used in both relay channels (in receive mode for half-duplex relays), and a total of T symbols can be transmitted. Then an achievable rate for the two relay channels is given as follows: 1. Full-duplex channels: Assume T > 2n and let k be the largest integer such that kn < T , then k−2 T −T Rf d = Rf d + Rdirect (11) k T where T = (k − 2)n. The rates Rf d and Rdirect are the achievable rates for block lengths n with inﬁnitely growing number of blocks k and length (T − T ), respectively. 7 1808 12 9 Cut−set Bound (Full−duplex) Cut−set Bound (Full−duplex) Normalized Gain (Actual Rate/Direct Link Rate) Decode & Forward (Full−duplex) Normalized Gain (Actual Rate/Direct Link Rate) Decode & Forward (Full−duplex) Cut−set Bound (Half−duplex) 8 Cut−set Bound (Half−duplex) Decode & Forward (Half−duplex) Decode & Forward (Half−duplex) Data Forwarding Data Forwarding 10 Direct Link Direct Link 7 8 6 5 6 4 4 3 2 2 1 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Number of Relay Antennas Number of Destination Antennas (a) (b) Figure 3: Impact of number of antennas for P1 = P2 = 10dB and relay in the middle. (a) Increasing antennas at the relay leads to improved gain over direct link, while (b) increasing antennas at the destination has the opposite eﬀect. In both plots (a) and (b), the other nodes have 3 antennas. 2. Half-duplex channels: Let k be the largest integer such that T = k(n + m) ≤ T , then T T −T T Rhd + T Rdirect k>1 Rhd = (12) Rdirect k = 1 and (T − T ) < n is achievable. The rates Rhd and Rdirect are the rates achievable with block lengths (n, m) and (T − T ), respectively. The diﬀerence in encoding method for the two relay channels also impacts how latency aﬀects the achievable rates. Since the full-duplex scheme are Markovian, the decode-and-forward rates are achievable only with inﬁnite number of code-blocks (k → ∞). However, the half-duplex relay channels do not require a Markovian code, which implies that with proper choice of codelength T for a given (n, m), there is no rate loss compared to a inﬁnite speed processor. Note that the case of no rate loss occurs only if a minimum of two consecutive cooperation blocks are used in the relay channel, unlike the cases studied in the literature where cooperation occurs over only codeblock [7]. 5 Conclusions In this paper, we studied the impact of half-duplex radios on the capacity of relay-channel. The signiﬁcant ﬁnding were that the biggest gains from relaying occur when the relay has as many antennas as possible, the system is operating at low to medium SNRs and relay has a very eﬃcient decoding implementation. An immediate conclusion of the results in this paper seems to be that sophisticated relay based methods may not hold much promise, but in author’s opinion, it will be too soon to conclude that. We conjecture that a system which has multiple relays could oﬀer substantial gains, high enough to warrant actually building that system. References [1] T. M. Cover and A. El Gamal. Capacity theorems for the relay channel. IEEE Info. Theory, 25(5):572–584, September 1979. 8 1809 [2] N. Jindal, U. Mitra, and A. J. Goldsmith. Capacity of ad hoc networks with node cooperation. In Proc. of ISIT, June 2004. [3] M. A. Khojastepour, A. Sabharwal, and B. Aazhang. Bounds on achievable rates for general multi-terminal networks with practical constraints. In Proc. of 2nd International Workshop on Information Processing (IPSN), April 2002. [4] M. A. Khojastepour, A. Sabharwal, and B. Aazhang. On the capacity of ’cheap’ relay networks. In Proc. of 37th Annual Conf. Information Sciences and Systems (CISS), March 2002. [5] M. A. Khojastepour, A. Sabharwal, and B. Aazhang. On capacity of gaussian cheap relay channel. In Proc. of GLOBECOM, volume 3, pages 1776–1780, December 2003. [6] G. Kramer, M. Gastpar, and P. Gupta. Cooperative strategies capacity theorems for relay networks. In submitted to IEEE Transactions on Information Theory, Feb 2004. [7] J. N. Laneman, D. N. C. Tse, and G. W. Wornell. Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior. IEEE Info. Theory, 2004. to appear. [8] U. Mitra and A. Sabharwal. On achievable rates of complexity constrained relay channels. In Proceedings of the Allerton Conference, Monticello, IL, October 2003. [9] R. U. Nabar, H. Boelcskei, and F. W. Kneubhueler. Fading relay channels: Performance limits and space-time signal design. IEEE J. Sel. Areas Comm (special issue on fundamental perfor- mance limits of wireless sensor networks). [10] W. Rhee and J. M Cioﬃ. On the capacity of multiuser wireless channels with multiple antennas. IEEE Transactions on Information Theory, 49(10):2580–2595, October 2003. [11] I. E. Telatar. Capacity of multiple-antenna Gaussian channels. Eur. Trans. Tel., 10(6):585–595, 1999. [12] M. C. Valenti and Bin Zhao. Distributed turbo codes: Towards the capacity of the relay channel. In Proc. of VTC, volume 1, pages 322–326, October 2003. [13] S. Vishwanath, N. Jindal, and A. J. Goldsmith. Duality, achievable rates, and sum-rate ca- pacity of Gaussian MIMO broadcast channels. IEEE Transactions on Information Theory, 49(10):2658–2668, October 2003. [14] P. Viswanath and D. Tse. Sum capacity of the multiple antenna Gaussian broadcast channel and uplink-downlink duality. IEEE Transactions on Information Theory, 49(8):1912–1921, August 2003. [15] Bo Wang, Junshan Zhang, and Anders Host-Madsen. On capacity of MIMO relay channel. submitted for publication. [16] Bin Zhao and M. C. Valenti. Distributed turbo coded diversity for relay channel. Electronics Letters, 39(10):786–787, May 2003. [17] L. Zheng and D. Tse. Diverstiy and multiplexing: A fundamental tradeoﬀ in multiple antenna channels. IEEE Transactions on Information Theory, 49(5), 2003. 9 1810

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