Throughput of Cellular Systems with Conferencing Mobiles and Cooperative Base Stations
© O. Simeone et al. 2008
Received: 29 July 2007
Accepted: 15 February 2008
Published: 27 February 2008
This paper considers an enhancement to multicell processing for the uplink of a cellular system, whereby the mobile stations are allowed to exchange messages on orthogonal channels of fixed capacity (conferencing). Both conferencing among mobile stations in different cells and in the same cell (inter- and intracell conferencing, resp.) are studied. For both cases, it is shown that a rate-splitting transmission strategy, where part of the message is exchanged on the conferencing channels and then transmitted cooperatively to the base stations, is capacity achieving for sufficiently large conferencing capacity. In case of intercell conferencing, this strategy performs convolutional pre-equalization of the signal encoding the common messages in the spatial domain, where the number of taps of the finite-impulse response equalizer depends on the number of conferencing rounds. Analysis in the low signal-to-noise ratio regime and numerical results validate the advantages of conferencing as a complementary technology to multicell processing.
Recent information-theoretic results have shown that high-rate transmission over networks without any infrastructure (ad hoc networks) is bound to be infeasible over a large scale . Notice that this is envisaged to be true even if recent results show that, under demanding assumptions on channel state information availability and by resorting to complex transmission schemes, high-scale transmission on ad hoc networks can in principle be achieved . Therefore, the solution of choice for providing broadband communications necessarily implies the support of an infrastructure made of base stations (BSs or access points) connected by a high-capacity backbone. This class of solutions includes conventional cellular systems, where BSs are regularly placed in the area of interest ; distributed antenna systems, which are characterized by a less regular (e.g., random) deployment ; and hybrid networks, where infrastructure nodes coexist with multihopping . In all these networks, a solution that promises to greatly improve the overall throughput and that is gaining increasing interest in the community is multicell processing. This refers to the class of transmission/reception technologies that exploit the high-capacity backbone among the BSs to perform joint encoding/decoding at different cell sites (see [6, 7] for a list of references).
In this paper, we focus on the uplink of a cellular system and investigate a potential improvement to multicell processing. In particular, we consider a network where additional spectral resources allow nearby mobile stations (MSs) to exchange signals over finite-capacity channels that are orthogonal to the main uplink channel. This condition models the out-of-band relaying scenario for cooperative cellular networks discussed in, for example, , which can be realized when MSs are equipped with an orthogonal wireless interface (say Bluetooth or Wi-Fi) that is not available at the BSs. While with ordinary multicell processing only the BSs are enabled to cooperate (for joint decoding), in our setting MSs are allowed to collaborate as well, but only through finite-capacity and localized links. The limitation and localization of the inter-MS channels contrast with the typical assumption of unlimited and global connectivity among the BSs via the high-capacity backbone [3, 6, 7], which is reasonable due to topological and infrastructure constraints. However, see  for a recent work that considers multicell processing with limited backhaul capacity. Our goal is to bring insight into effective transmission strategies that exploit these additional system resources and into the performance gains that might be harnessed by deploying such technology.
1.1. Main Contributions
An achievable rate for the linear Wyner model with conferencing MSs is presented for both cases of intercell conferencing with intracell TDMA and intracell conferencing (Propositions 3 and 5). The considered transmission scheme prescribes rate splitting at the MSs, where part of the message (the "common" message) is exchanged during the conference phase among neighboring (out-of-cell or in-cell) MSs and transmitted cooperatively to the BSs.
In the case of intercell conferencing, the considered transmission scheme performs convolutional pre-equalization of the signal encoding the common messages in the spatial domain, where the equalizer is a finite-impulse response (FIR) filter whose number of taps depends on the number of conferencing rounds.
For both inter-and intracell conferencing, the considered transmission schemes are proved to be optimal as long as the conferencing capacity is large enough (Propositions 5 and 6).
An approximate analysis in the low signal-to-noise ratio regime is presented that gives further insight into the advantages of conferencing (Sections 2.5 and 3.5).
It is shown that intracell TDMA is not optimal in the presence of intracell conference channels as opposed to the basic scenario without conferencing studied in  (Section 3).
Finally, numerical results validate the relevant advantages of intercell and intracell conferencing (Sections 2.6 and 3.6).
1.2. Related Work
In addition to the quickly growing body of work on multicell processing for cellular systems [6, 7], there has recently been some activity around the basic idea of complementing and comparing the advantages of cooperation between BSs with some form of collaboration at the MS level as well. In [13–15], the basic linear Wyner model was extended by including a layer of dedicated relay terminals, one for each cell, that forward traffic from MSs to BSs (uplink). Focusing on intracell TDMA, different transmission schemes were considered, namely half-duplex and full-duplex amplify-and-forward in [13, 15], respectively, and decode-and-forward in , and the respective merits of multicell processing and MS cooperative transmission technologies, and combinations thereof, were discussed. Another related work is , where the linear Wyner model with intracell TDMA and single-cell processing was modified by assuming that the active MS in a given cell knows (noncausally) the messages to be sent by a number of its neighbors (as might be the case in some implementations of the principle of cognitive radio).
Notation: throughout the paper, bold letters denote either vectors or matrices; upper-case letters are used for random variables, while lower-case letters indicate specific realizations of the corresponding random variable.
2. Intercell Conferencing with Intracell TDMA
In this section, we consider the first scenario of interest, which consists of a modification of the linear Wyner model with intracell TDMA where intercell conferencing channels are present.
2.1. System Model
All the logarithms are to be assumed base-2 in keeping with our measure of information in bits/symbol. For the transmission phase, encoding at each th MS takes place according to a deterministic mapping from the message set and the received conferencing messages to sequences of (complex) channel symbols (codewords) as for Decoding at the central processor is based on the signal received by the BSs over the channel uses according to the deterministic mapping : as . Following standard definitions, a per-cell rate is said to be achievable if there exists a sequence of encoders and decoders such that the probability of error tends to zero for block length
2.2. Reference Results
In this section, we discuss lower and upper bounds on the per-cell achievable rate in the presence of intercell conferencing. The first result is due to  and does not assume a priori intracell TDMA.
Proposition 1 (lower bound, no conferencing ).
It should be noted that the rate (4) can be understood by regarding the Wyner model of Figure 1 as an intersymbol interference (ISI) channel in the spatial domain, characterized by the channel impulse response ( denotes the Kronecker delta function) and corresponding transfer function in (5) Moreover, we emphasize that the rate (4) clearly sets a lower bound on the performance achievable with intercell conferencing since it assumes .
The following proposition defines a useful upper bound on the performance attainable with intercell conferencing and intracell TDMA.
Proposition 2 (upper bound, perfect conferencing).
Proposition 2 follows by considering this results followed by considering the cut-set bound  applied to the cut that divides MSs and BSs or equivalently by assuming a perfect conferencing phase ( ), where each th active MS is able to exchange the local message with all the other active MSs in other cells. In fact, in such an asymptotic regime, joint encoding of the set of messages by all the MSs is feasible, and recalling the equivalence of (1) with an ISI channel, we can conclude that the optimal transmission strategy is defined by the waterfilling solution (7) . Notice that the waterfilling solution is obtained for a sum-power constraint over the MSs, but given the symmetry of our setting, it also applies to the considered per-MS power constraint. It should also be remarked that this result shows that, in the limit a stationary input in the spatial domain with power spectral density is capacity achieving. This conclusion will be used in the next section to bring insight into the performance of intercell conferencing with finite capacity. While the upper bound (6)-(7) is reported here in integral form, in Appendix A we present a closed-form expression for (6) that holds in a specific regime of interest.
2.3. An Achievable Rate
In this section, we derive an achievable rate for the Wyner model with intercell conferencing and intracell TDMA and discuss some of the implications of this result.
Proposition 3 (achievable rate).
We briefly discuss the transmission scheme that attains the rate (8) and point out some implications of this result, leaving the details of the proof of achievability to Appendix B. Again, to fix the ideas, consider the set of active MSs at a given time, one per cell, which employ a fraction of time of both the uplink and the conferencing channels. The proposed scheme works as follows. In the conference phase, each th MS first splits its message into two parts, say private ( ) and common ( ). Then it shares the common part with the neighboring MSs in cells with during conferencing rounds. More precisely, in the first round, the th MS transmits its local common information to the two adjacent MSs and , which then propagate the information towards the two edges of the network, and so on. Notice that, after the conference phase, each th MS is aware of the common messages During the transmission phase, each common message can be then transmitted cooperatively by all the MSs that have acquired the information on in the conferencing phase. On top of the cooperative signal encoding common information, each MS jointly encodes the private message Gaussian codebooks are employed and the total power is divided as (9a) between the common ( ) and private ( ) parts.
2.4. Asymptotic Optimality of the Considered Scheme
From Proposition 3, it is easy to see that the proposed scheme is optimal under a specific asymptotic regime, as stated in the following Proposition.
Proposition 4 (asymptotic optimality).
It is enough to prove that the rate (8) equals the upper bound (6) under the conditions in the proposition above. This follows easily by setting (and and recalling that the optimal power spectral density (7) can be approximated arbitrarily well by the frequency response in (10) as the number of taps increases  (which corresponds to perfect cooperation among the MSs).
The argument in the proof above shows that under the asymptotic conditions stated in Proposition 4, it is optimal to allocate all the power to the common messages, (and and to select the filter so that
While in this paper we do not consider fading channels, it is apparent from the discussion above that the advantages of intercell conferencing are related to the possibility of optimizing the transmission strategy based on the knowledge of the channel structure at the MSs. Therefore, intercell conferencing is expected not to provide any performance gain over fading channels in the absence of channel state information at the MSs. This claim can be substantiated by using the results in , where it is shown that, in case of independent fading channels even in the presence of statistical channel state information at the transmitter (i.e., at the MSs), the optimal power allocation is asymptotically (in ) uniform so that cooperation at the MSs does not provide any advantage. This result holds for channels with column-regular gain matrices (see definition in ). The channel considered in this paper belongs to this class when .
2.5. Discussion: the Low-SNR Regime
In this section and Section 2.6, we elaborate on the performance of the considered scheme that exploits intercell conferencing. Here, this goal is pursued via an (approximate) analytical approach that focuses on the low-SNR regime according to the framework in , whereas in the next section we resort to numerical simulations to study the case of arbitrary SNR. The attention to the low-SNR regime is justified by the fact that, as discussed above, the advantages of intercell conferencing are (asymptotically) related to the opportunity of performing waterfilling power allocation, which is known to provide relevant gains only for low to moderate SNRs (see, e.g., ).
for the upper bound (6). The latter can be proved by noticing, similarly to , that when the SNR tends to zero ( , it is optimal to allocate all the available power around the maximum value of the channel transfer function, which occurs at In other words, the optimal waterfilling power allocation is , where is a Dirac delta function. Plugging into (6) and using tools from , equality (12) is easily shown.
From (16), it is clear that the minimum energy per bit of intercell conferencing (16) is a decreasing function of the number of conferencing rounds as expected from Proposition 4, tends to the optimal performance (12) for
2.6. Numerical Results
3. Intracell Conferencing
In this section, we study a different extension of the linear Wyner model, where there exist conferencing channels that link MSs within the same cell so as to enable intracell conferencing. Due to the proximity of same-cell MSs, as detailed below, here it is assumed that a signal transmitted on the conferencing channel within any cell is overheard by all other MSs within the cell. Moreover, unlike the previous section, in the following we do not assume intracell TDMA, that is, same-cell MSs are allowed to transmit to the BSs at the same time.
3.1. System Model
where with a slight abuse of notation, we have defined as the MS that uses the conferencing channel in the th cell at round . Finally, since only one th MS in cell can transmit in a given round , we have that if then for all
In the transmission phase, encoding at each th MS takes place according to a deterministic mapping from the message set and the received conferencing messages to the codebook as for Finally, decoding is based on the signal according to the deterministic mapping : as .
3.2. Reference Results
In this section, we present relevant upper and lower bounds on the achievable rate of the linear Wyner model with intracell conferencing presented above. We first notice that a lower bound on the achievable rate is still set by (4), which corresponds to the case of no conferencing ( ). We now discuss a useful upper bound.
Proposition 5 (upper bound, perfect conferencing).
Similarly to Proposition 2, Proposition 5 follows by assuming a perfect conferencing phase, where each th MS is able to deliver the entire message to all the other in-cell MSs. In fact, under such assumption, we observe that all the MSs in any th cell can be seen as a "super-MS" with input symbol (recall (17)) and power constraint due to coherent power combining.
3.3. An Achievable Rate
Here, we provide an achievable rate for the linear Wyner model with intracell conferencing and describe the transmission scheme that is able to attain it.
Proposition 6 (achievable rate).
with constraint (9a) and definition (5).
A brief sketch of the proof of achievability is in order. The details are worked out in Appendix C. Each th MS first splits its message into two parts: say private ( ) and common ( ). The common part is then communicated to all the MSs belonging to the same cell in one conference round (a total number of conference rounds is thus employed). In the transmission phase, all the MSs in a cell cooperate to achieve coherent power combining on the common part of the message, which is transmitted by each user with power and received with power . The private message is instead jointly encoded by each MS on top of the common message and carries power
It should be noticed that rate (21) is achieved with multiple MSs simultaneously active in each cell. By comparison with rate (8), which is achievable with intracell TDMA, it can be seen that, in case intracell conferencing is allowed, intracell TDMA is not optimal. In fact, as explained above, simultaneous transmission of multiple MSs after intracell conferencing allows coherent power combining to be achieved. This lack of optimality of intracell TDMA in the presence of intracell conferencing clearly contrasts with the results in  for the case of no conferencing (see Proposition 1).
3.4. Conditional Optimality of the Considered Scheme
Similarly to the case of intercell processing, the considered scheme based on rate splitting is optimal if the conferencing capacity is large enough. However, in contrast with the previously considered scenario (see Proposition 3), here optimality is obtained for finite conferencing capacity
Proposition 7 (conditional optimality).
3.5. Discussion: the Low-SNR Regime
which when compared to the lower bound (11), clearly shows the coherent power gain by due to cooperation. As proved in Proposition 7, under the assumption the achievable rate (21) attains the upper bound so that we clearly have for
3.6. Numerical Results
Most of the current proposals for the enhancement of cellular-based wireless networks, such as the IEEE 802.16j standard are based on cooperative technologies. Among such solutions, multicell processing, where cooperation is at the BS level, is receiving an increasing attention for its significant potential enabled by the high-capacity backbone connecting the BSs. In this paper, we have looked at an extension to this technology, where besides multicell processing, partial cooperation is allowed at the MS level as well. In particular, additional system resources are assumed to be available to provide conferencing channels of finite capacity between nearby MSs. Two limiting scenarios have been considered: one in which conferencing is allowed between MSs belonging to adjacent cells (as is reasonable for small cells) and another where conferencing is possible only among MSs belonging to the same cell. In both cases, a transmission scheme based on rate splitting and cooperative transmission has been proven to be optimal when the conferencing capacity is large enough.
A relevant extension of this work, that is currently under study, is to consider achievable rates for a two-dimensional cellular systems in the spirit of the hexagonal-cell models presented in . The main problem in such scenarios is the propagation of the conferencing messages, which, given the geometry at hand, could possibly benefit from network coding.
A second open problem is that of optimal resource allocation between the conferencing and uplink channels, similar to .
A final interesting issue left open by this work is the establishment of capacity-achieving schemes for any value of the conferencing capacity and finite number of cell sites. The main challenge in this regard appears to be the extension of the converse result in  to the scenario at hand. In particular, it remains to be determined whether unlike the simpler model in , interactive communications among the MSs during the conferencing phase is necessary to achieve capacity. The results of this paper have shown that this is not the case in the regime of high conferencing capacity.
In this section, we reconsider the upper bound given in Proposition 1 based on waterfilling power allocation and present a closed-form analytical expression of (6)-(7) that hold in a specific regime of low intercell gain and high power. We remark that in other regimes (large and/or small power), we were not able to obtain such compact expressions.
where the last equality is achieved by applying [25, formula 4.224.12].
B. Proof of Proposition 3
In this section, the proof of achievability of rate (8) stated in Proposition 3 is provided. For simplicity of notation, we consider since the extension to requires only straightforward modifications given the intracell TDMA assumption. We consider conference and transmission phases separately.
B.1. Conference Phase
B.2. Transmission Phase (After Conference)
The codebooksare generated as follows. For each we generate a codebook of independent codewords according to a distribution . We label these sequences as Now, for each and common messages sets we generate ( ) independent codewords according to a distribution and label them as
where , independent of all and " " denotes convolution. From (B.2) it is clear that the overall impact of conferencing here is to enable linear precoding via the FIR filter of the signal encoding common messages ( ) in the spatial domain (recall Figure 3).
Analysis of Probability of Error
(The reader is referred to  for a thorough discussion of the border effects in this argument.) Now consider the remaining bounds (B.4). In the following, we would like to show that, coupled with (B.9), these conditions identify the achievable ( ) region sketched in Figure 9, which is characterized by the conditions
where (a) is a consequence of the Markov condition Now, since the channel seen by is an ISI channel with channel response and additive noise, we can again apply the same approach as in  (see proof of Theorem 2.1 therein) to show that is nonincreasing with
C. Proof of Proposition 5
As for the proof of Proposition 3, we consider conference and transmission phases separately. The treatment follows closely Appendix B so that here we emphasize only the major differences.
C.1. Conference Phase
C.2. Transmission Phase (After Conference)
The codebooks are generated as follows. For each we generate independent codewords according to a distribution and label these sequences as Now, for each and for each common message we generate ( ) independent codewords according to a distribution and label them as
Analysis of Probability of Error
This work was supported by the National Science Foundation under Grants CNS-06-26611 and CNS-06-25637, and by a Marie Curie Outgoing International Fellowship within the 6th European Community Framework Program and by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM++.
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