Throughput of Cellular Systems with Conferencing Mobiles and Cooperative Base Stations

In modeling the interaction among the terminals, we follow the framework of conferencing encoders first studied in [10] in the context of a two-user multiple access channel and then extended in a number of recent works to other scenarios (see, e.g., [11, 12] and references therein). Moreover, the topology of a cellular system is abstracted according to the linear version of the model introduced in [3] (see [6, 7] for a review of related papers). We will refer to this model in the following as the linear Wyner model. Under such assumptions, we consider two scenarios: in the first, only one MS is active in each cell at any given time (intracell time-division multiple access (TDMA)) and conferencing channels exist between MSs belonging to adjacent cells (intercell conferencing); in the second, simultaneous uplink transmission by multiple MSs per cell is allowed and conferencing channels are present only among MSs sharing the same cell (intracell conferencing). These two scenarios conceivably correspond to limiting situations with either small cells, so as to enable intercell conferencings or large cells, where only connections among same-cell MSs are feasible. Our main contributions are as follows.

Finally, numerical results validate the relevant advantages of intercell and intracell conferencing (Sections 2.6 and 3.6).

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 [14], and the respective merits of multicell processing and MS cooperative transmission technologies, and combinations thereof, were discussed. Another related work is [16], 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).

We consider the uplink of a cellular system abstracted according to the linear Wyner model as sketched in the upper part of Figure 1. cells are arranged into a linear array, where each cell contains MSs ( in the figure). Following [3], the signal transmitted by each MS is received only by the same-cell BS, with unitary gain, and by the two adjacent BSs with intercell gain As anticipated, we consider at first the case, where only one MS transmits in each cell at any give time in a TDMA fashion (intracell TDMA). It should be remarked that this choice does not entail any loss of optimality in a basic Wyner model with no conferencing, as shown in [3]. Overall, defining as the input symbol of the MS active in the th cell, the signal received by the th BS reads ( for and )

where is an independent and identically distributed (i.i.d.) sequence of complex noise samples. The noise samples are Gaussian with independent real and imaginary parts that each have zero mean and variance 1/2, and we write this as ~ CN(0,1). Notice that we assume full (symbol and codeword) synchronization among the cells. We focus on multi-cell processing, that is we assume that the signals received by the BSs, , are jointly processed by a central unit that detects the transmitted signals. Finally, each MS has an average power constraint of so that the available power per cell is . With intracell TDMA, each MS is active for a fraction of the time, wherein it can transmit with power , still satisfying the average power constraint. The power constraint then is given by which can be interpreted as the signal-to-noise-ratio (SNR) for the system at hand. We remark at this point that in the following we will be interested in limiting results for a very large number of cells ( ); edge effects can be handled as in [3] and we will neglect them in the presentation below, unless explicitly stated otherwise. We refer the reader to [7] for a discussion of the relevance of this asymptotic regime in practical scenarios with a limited number of cells.

We now extend the basic linear Wyner model discussed above to include conferencing among the active MSs in adjacent cells (intercell conferencing). A different variation of the Wyner model where intracell conferencing is enabled is discussed in Section 3. As shown in the lower part of Figure 1, with intercell conferencing, orthogonal channels with capacity (bits/symbol) are assumed to exist; each links the MS currently active in any th cell to the active MS in an adjacent cell (i.e., the or th cell, unless or We assume block transmission, as shown in Figure 2. Within any th block and in any th cell, the MS currently active generates a message meant to be decoded by the central processor connecting the BSs, where is the number of channel uses per block and is the per-cell rate (bits/channel use). According to a standard information-theoretic assumption, we will consider a large block length Transmission of a given set of messages takes place in two successive phases (or slots). In the first phase (conferencing phase), during the th block, the MSs exchange information on the conferencing channels during rounds (see further details below). This information collected during the conferencing phase by each active MS is then leveraged to encode the local message for transmission to the BSs in the ( th block (transmission phase). Notice that, as shown in Figure 2, the conferencing phase corresponding to can be carried out at the same time as the transmission phase for messages given the orthogonality between the conferencing channels and uplink channel.

To formalize the model discussed in the previous paragraph, we need to specify the coding/decoding operations allowed at different terminals. Given our intracell TDMA assumption, each set of active MSs uses both the conferencing channels and the uplink channels for a fraction of the time. In the following, we focus on a specific set of active MSs and, furthermore, we drop the dependence on the block index for simplicity of notation. For the conferencing phase, following [10], we consider rounds of conference. In each th round ( ), any active th MS transmits a message to the adjacent MSs with This depends on the messages received by the th MS during the previous rounds and similarly defined) as

where is a given deterministic function and a given alphabet For convenience of notation, the messages transmitted on each link are collected in vectors The finite capacity of the conferencing links imposes the following constraint on the alphabets [10]:

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

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 [3] and does not assume a priori intracell TDMA.

Proposition 1 (lower bound, no conferencing [3]).

The per-cell capacity (i.e., maximum achievable per-cell rate) in a basic linear Wyner model with no conferencing ( ) and is achieved with intracell TDMA and is given by

with

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).

An upper bound on the rate achievable with intercell conferencing and intracell TDMA in the linear Wyner model (with ) is given by

with

Proposition 2 follows by considering this results followed by considering the cut-set bound [17] 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) [18]. 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.

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).

The following per-cell rate is achievable for the linear Wyner model with intercell conferencing and intracell TDMA for and any :

with constraints

definitions: and

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.

As shown by Proposition 3, the impact of intercell conferencing, according to the scheme discussed above, is equivalent to that of allowing precoding (pre-equalization) of the common information by a FIR filter with frequency response (10). The equivalent channel seen by the input symbols encoding the common information (say is shown for illustration in Figure 3 for conference rounds. We emphasize that, while the number of taps increases with the number of conference rounds, the overall achievable rate may suffer according to (8). We further explore this trade-off in Section 2.6 with a numerical example.

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).

The transmission scheme achieving the rate (8) is optimal for and

Proof.

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 [19] (which corresponds to perfect cooperation among the MSs).

Remark 1.

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

Remark 2.

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 [20], 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 [20]). The channel considered in this paper belongs to this class when .

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 [21], 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., [22]).

According to [21], for low SNRs the rate of a given transmission scheme can be described by the minimum transmit energy per bit required for reliable communication (normalized to the background noise level) (which is obtained for ) and by the slope at (measured in In the following, we focus for simplicity on the minimum energy per bit , and use this criterion to compare the performance of intercell conferencing with the lower and upper bounds (4) and (6) in the low-SNR regime. Starting with the bounds, the minimum energy per bit is given by

for the lower bound (4) (see [7]) and

for the upper bound (6). The latter can be proved by noticing, similarly to [21], 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 [21], equality (12) is easily shown.

Let us now consider the rate (8) achievable by intercell conferencing. We start with the observation that for and any finite we have so that the first term in (8) is dominant and rate (8) is given by

The optimization problem (13) (with constraints (9a) and (9b)) is generally not convex so that finding a global optimal solution is not an easy task [23]. For this reason, we focus on a suboptimal feasible solution that is asymptotically (in the sense of Proposition 4) optimal and allows to gain insight into the performance of intercell conferencing. This solution is based on the observation that, from Remark 1 and from the discussion above, the asymptotically optimal power allocation is (and and the optimal filter satisfies . Accordingly, with the stated power allocation, here we design for any finite (but large) the filter so as to approximate the (asymptotically) optimal by an ideal low-pass filter with frequency response,

where the bandwidth satisfies . Clearly, frequency response (14) can only be approximated by a FIR filter, but the approximation is acceptable for large . Hence, under the low-SNR condition and assuming large , the rate (13) is given by

so that the minimum energy can be calculated following [21] and after some algebra (We use the second-order approximation: ), as

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

In this section, we present some numerical examples in order to assess the performance of the discussed intercell conferencing scheme. Similarly to the previous section, since the optimization problem (8) that yields the considered achievable rate is generally nonconvex, here we focus on a simple feasible solution that is asymptotically (in the sense of Proposition 4 optimal and allows to gain interesting insight into the system performance. As discussed in Remark 1, for and the (global) optimal power allocation is (and ) and the optimal frequency response satisfies . Based on this result, for any choice of the parameters, first the taps of filter are generated according to the frequency sampling method with target amplitude of the frequency response given by the waterfilling solution [19] (the filter is scaled to satisfy the constraint (9b)). Then, for fixed filter the optimization problem (8) is convex in the powers ( ) and can be solved efficiently by using standard numerical methods [23]. Illustration of the performance of the frequency sampling filter design for different values of is shown in Figure 4 for and It can be seen that with large enough, the FIR filter in (10) is able to approximate closely the (asymptotically) optimal waterfilling solution

As discussed above, increasing is always beneficial to obtain a better approximation of the waterfilling strategy (7). However, due to the finite conferencing capacity , it is not necessarily advantageous in terms of the achievable rate (8). To show this, Figures 5 and 6 present the achievable rate (8) versus the intercell gain along with the lower bound (4) and upper bound (6) for and respectively. Figure 5 shows that, with while increasing the conferencing rounds from to increases the achievable rate, further increments of the number of conferencing rounds are disadvantageous, according to the trade-off mentioned above. With a larger capacity Figure 6 shows that substantial performance gains can be harnessed by increasing the number of conference rounds, especially from to . Moreover, as expected from Proposition 4, having sufficiently large conference capacity and sufficiently many conference rounds (with enables the upper bound (6) to be approached.

The basic linear Wyner model with multiple active users per cell, say is defined as follows. Denoting as the input symbol of the th MS ( ) in the th cell, the signal received by the th BS is given by ( for and ,

As in Section 2, the per-user power constraint is so that a total power constraint per cell of is enforced.

The basic Wyner model is now extended to allow intracell conferencing. We consider intracell multicast channels with capacity one per cell; each such channel connects an MS to all the other same-cell MSs, and is accessed by only one MS at each time in a TDMA fashion (see Figure 7). Such channels are orthogonal for different cells and with respect to the main uplink channel. As in Section 2.1, transmission of a given set of messages for the ( )th MS with and occur in two phases that are arranged in a frame structure as shown in Figure 2.

Notice that in Section 2, we considered intracell TDMA so that the total number of conferencing rounds was . We again assume rounds of conferencing. Each th MS at any th round transmits a message to all the other MSs in the th cell (see Figure 7), which is a deterministic function of the previously received messages (recall (2)),

for a given deterministic mapping and alphabet Notice that, in order to deal with multiple access to the conferencing channels of each cell by the local MSs (only one MS in each cell can access the conferencing channels at any given round), we have extended the alphabet of symbols used for conferencing with a symbol , which represent no transmission. Moreover, similarly to (3), the finite capacity of the conferencing links imposes the condition,

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 .

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).

An upper bound on the rate achievable with intracell conferencing on the Wyner model (with ) is given by

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.

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).

The following rate is achievable on the linear Wyner model with intracell processing and :

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

Remark 3.

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 [3] for the case of no conferencing (see Proposition 1).

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).

The transmission scheme achieving the rate (8) is optimal if

Proof.

We need to prove that the rate (21) equals the upper bound (20) under the conditions in the proposition above. This follows easily by setting (and .

Remark 4.

The argument in the proof above shows that for it is optimal to allocate all the available power to the common messages ( and ).

For the sake of completeness, similarly to Section 2.5, here we assess the performance of intracell conferencing in the low-SNR regime by calculating the minimum energy per bit required for reliable communications. This task is pretty straightforward since the advantages of intracell conferencing are related to the power gain achievable through coherent power combining, which differently from the waterfilling advantage of intercell conferencing is immediate to account for in the low-SNR regime. In particular, the energy obtained by the upper bound (20) is given by

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

Figure 8 shows the achievable rate (21) versus the transmitted power per cell along with the lower bound (4) and the upper bounds (6) and (20) for and MSs per cell. Notice that (21) is a convex problem so that global optimality can be attained by using standard numerical methods [23]. From the figure, it is seen that increasing the intracell conferencing capacity allows the upper bound to be approached and eventually reached (as stated in Proposition 7). Moreover, it is interesting to observe that the best performance achievable with intracell conferencing ( ) is preferable to the best rate attainable with intercell conferencing ( ) lending evidence to the effectiveness of coherent power combining.

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.

Proposition 8.

Assume that and

then the upper bound (6)-(7) becomes

Proof.

Under the assumption that the power is sufficiently large so that

(i.e., the high-power regime), the constraint (7) can be written as

where the last equality follows from [25, formula 3.661.4] and some algebra. Hence, from (A.3) and (A.4) the high-power regime is defined by condition (A.1), and the waterfilling constant is given by

Finally, the rate expression is given by

where the last equality is achieved by applying [25, formula 4.224.12].

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.

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.