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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.


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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.




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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

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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

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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).


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                                                                                                                                                                                      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

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                                                                                                                                                                                                  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.

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                                                     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.


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