Fueled by the ever increasing demand for bandwidth-hungry applications, global wireless traffic is expected to continue its rapid growth . However, due to scarce microwave bandwidth (i.e., sub- GHz spectrum) current 4G technologies are unlikely to be able to support the anticipated massive growth in traffic . To tackle this challenge, several new technologies are being studied to be potentially incorporated into 5G standards. Among these, a key technology is to integrate the vast bandwidth in the GHz frequency range, referred to as the millimeter wave (mm-wave) band, with sub- GHz spectrum [3, 4, 5], and provide cellular access jointly over these two bands.
Transmission in the mm-wave band differs from that in the conventional microwave band in that omnidirectional mm-wave transmission suffers from much higher power loss and absorption. Thus, a transmitter must use beamforming via highly directional antenna arrays to reach a receiver . Due to the small wavelength at mm-wave frequencies and large path loss, beamforming typically creates links that have a strong line-of-sight (LoS) component and only a few, if any, weak multi-path components. Such mm-wave links are inherently point-to-point, and are well modeled as AWGN links [7, 8, 9]. Although mm-wave links support high data rates due to their large bandwidths, they provide limited coverage, whereas microwave links typically provide reliable coverage and support only moderate data rates. Thus, in a dual-band setting, these two bands mutually complement each other: conventional traffic and control information can be reliably communicated in the microwave band, and high data-rate traffic can be communicated via the mm-wave links [10, 11, 3, 12, 13, 14, 4, 5, 15].
In future 5G networks, access via dual microwave and mm-wave bands will likely be a key technology, and hence they have been subject to much investigation recently. For example, studies as in [10, 12, 14, 11] focus on improving network layer metrics such as the number of served users, throughput, and link reliability, etc., while studies as in [16, 17, 18]
focus on improving physical layer metrics such as the achievable rates and outage probability. Moreover, the emergence of dual-band modems from Intel and Qualcomm , and practical demonstrations such as that in the GHz- GHz dual-bands in  clearly illustrate the immense potential of such networks. However, few studies have been reported on the information-theoretic limits of multi-user dual-band networks , which are crucial in identifying the limits of achievable rates, simplified encoding schemes, etc., in practical dual-band networks. For example, the study on the two-user interference channel over such integrated dual-bands  has shown that forwarding interference to the non-designated receivers through the mm-wave links can improve achievable rates considerably. Moreover, relay cooperation, which already plays a key role in microwave networks, will likely play a vital role in such dual-band networks as well, especially to offset impairments such as blockage in the mm-wave band [22, 12, 8, 23].
Thus motivated, we study the two-user Gaussian multiple-access relay channel (MARC) over dual microwave and mm-wave bands, which models uplink scenarios, e.g., fixed wireless access  which is expected to eliminate last mile wired connections to end users. In this case, the base station will communicate with a fixed access-point that is equipped with the hardware necessary for dual-band communication including mm-wave beamforming, which will likely be located outside a building and will provide high data rate access to users inside the building (end users). As such, the dual-band MARC can model relay-assisted uplink from two such fixed access-points located in nearby buildings. In the future, when mobile handsets are equipped with dual-band communication capable hardware, the dual-band MARC can also model relay-aided cellular uplink from mobile users.
In this MARC, two sources communicate to a destination with the help of a relay over dual microwave and mm-wave bands. In the microwave band, transmissions from both sources are superimposed at the relay and at the destination as in a conventional MARC (c-MARC) . In contrast, since mm-wave links are highly directional , when a transmitter in the mm-wave band transmits specifically to the relay or the destination, the resulting mm-wave link causes minimal to no interference to the unintended receiver [25, 9]. In fact, a mm-wave transmitter can create two parallel non-interfering links via beamforming, and then communicate with both relay and destination simultaneously [25, 6, 9]. Therefore, in this work a mm-wave transmitter is modeled as being able to create two such parallel non-interfering AWGN links to simultaneously transmit to the relay and the destination, while a mm-wave receiver is modeled as being able to simultaneously receive transmissions from multiple mm-wave transmitters via separate mm-wave links  with negligible inter-link interference.
It is natural to ask whether a user (or source) in the mm-wave band should transmit to the relay, the destination, or both. Depending on whether each of the two sources transmits to only the relay, only the destination, both, or none, different models are possible. The general model that includes all microwave and mm-wave links is referred to as the destination-and-relay-linked MARC (DR-MARC), where the two sources ( and ) simultaneously communicate to the destination () via the mm-wave - and - direct links as well as to the relay () via the mm-wave - and - relay links. Since all other models with varying mm-wave link connectivity can be obtained from the DR-MARC by setting the relevant transmit powers to zero, they are not defined explicitly. However, the model where transmit powers in the mm-wave direct links are set to zero is an important one and referred to as the relay-linked MARC (R-MARC).
In addition to mm-wave links, the dual-band MARC also consists of an underlying conventional microwave band c-MARC. The capacity of such an individual c-MARC was partially characterized under phase and Rayleigh fading [24, 27], and therefore, we assume that the dual-band MARC is subject to a general ergodic fading where the phase of the fading coefficients are i.i.d. uniform in , similar to phase and Rayleigh fading. The general fading contains phase and Rayleigh fading as special cases, and can model a range of channel impairments. For example, phase fading models the effect of oscillator phase noise in high-speed time-invariant communications , the effect of phase-change due to slight transmitter-receiver misalignment in LoS dominant links , etc., while Rayleigh fading models the effect of rich scattering .
, the conventional c-MARC was classified into thenear c-MARC and the far c-MARC cases. In the near c-MARC, the sources are near the relay in that the source-relay channels are stronger than the source-destination channels in the sense of [24, Theorem 9], and thus the capacity of the near c-MARC was characterized. Naturally, the far c-MARC case is complementary to the near case. Here, we similarly classify the dual-band MARCs (DR-MARC and R-MARC) based on whether the underlying c-MARC in the microwave band is a near or a far c-MARC in the sense of .
First, we consider the DR-MARC where the sources simultaneously transmit in both the mm-wave relay and mm-wave direct links. We show that irrespective of whether the underlying c-MARC is a near or a far c-MARC, its capacity can be decomposed into the capacity of the underlying R-MARC (that consists of the c-MARC and the two mm-wave relay links) and the two mm-wave direct links. Hence, it is sufficient to focus on the R-MARC. The capacity of the R-MARC with near underlying c-MARC are characterized under the same conditions as in  and thus does not need additional conditions on the mm-wave links. Therefore, we focus primarily on R-MARCs with far underlying c-MARC where the mm-wave links play a key role, and for such R-MARCs, we find sufficient channel conditions under which its capacity is characterized by an achievable scheme.
The DR-MARC is a building block for future dual-band multiuser networks. Since, its performance will be significantly affected by the mm-wave links due to their large bandwidths [21, 18, 11], it is useful to understand how allocating the mm-wave band resources optimizes the performance, similar to other multiuser networks [21, 8, 31]. Hence, to quantify the impact of the mm-wave spectrum on the performance of the DR-MARC, we study the power allocation strategy for the mm-wave direct and relay links (subject to a power budget) that maximizes the achievable sum-rate.
The contributions of this paper is summarized as follows.
We decompose the capacity of the DR-MARC into the capacity of the underlying R-MARC and two direct links. This shows that irrespective of whether the underlying c-MARC is a near or a far c-MARC, operating the R-MARC independently of the direct links is optimal.
We derive an achievable region for the R-MARC. Then, for R-MARCs with far underlying c-MARC, we obtain sufficient conditions under which this achievable scheme is capacity achieving.
We characterize the optimal power allocation scheme (OA) for the mm-wave direct and relay links that maximizes the sum-rate achievable on the DR-MARC with the aforementioned achievable scheme. For intuition, we partition the range of the total power budget () into several link gain regimes (LGR) based on whether satisfies certain channel conditions, and show that the OA allocates link powers in different modes in each LGR. We obtain all such LGRs and modes of power allocation which reveal useful insights.
We observe that for DR-MARCs with near underlying c-MARC, the OA allocates entirely to the direct links for all . However, for DR-MARCs with far underlying c-MARC, we observe the following:
when is smaller than a certain saturation threshold (), for the direct and relay links of each source, the OA allocates powers following a Waterfilling (WF) approach. Specifically, for sufficiently small , the OA allocates entirely to the strongest of the direct and relay links of a source, and as increases, power is eventually allocated to the remaining links. Thus, for , each link-power either increases piecewise linearly with , or remains zero.
when , saturation occurs where the relay link powers are constrained to satisfy a certain saturation condition. As increases beyond , the direct link powers grow unbounded with , while the relay link powers vary with as follows. There exists a threshold , such that if one relay link is significantly stronger than the other (in a sense to be defined later), then for all , power in the stronger relay link remains fixed at a constant level and that in the weaker relay link at zero, and if the relay link is only stronger but not significantly stronger, for all , power in the stronger and the weaker relay links monotonically increase and decrease respectively, and approach constant levels.
if the mm-wave bandwidth is large and the power received at the destination from the relay via the mm-wave link is also large, allocating power as in the WF-like solution is optimal for all practical , and saturation only occurs for large values of .
This paper is organized as follows. The system model is defined in Section II. The results on the DR-MARC and the R-MARC are presented in Section III and Section IV respectively. The optimum sum-rate problem is presented in Section V, while in Section VI insights are derived from the link gain regimes. Finally, conclusions are drawn in Section VII.
Notation: The sets of real, non-negative real and complex numbers are denoted by and
. Vectors are generally denoted in bold (e.g., ) with denoting that each
. Random variables (RVs) and their realizations are denoted by upper and lower cases (e.g., and ). Specifically, denotes a circularly symmetric complex Gaussian (CSCG) RV with mean
and variance, and
denotes a uniformly distributed RV in. Also, denotes expectation, while denotes the greatest integer no larger than , and .
Ii System Model
We consider a relay-assisted two-user uplink scenario as in Fig. (a)a which is modeled as the DR-MARC as in Fig. (c)c. Note that a bandwidth mismatch factor (BMF) may exist between the two bands such that for accesses of the microwave band, the mm-wave band is accessed times. To communicate a message from source , it is encoded into three codewords, and , of lengths and respectively. Then, is transmitted towards by using the microwave (first) channel times, and due to the nature of this band, and superimpose at and at as in the c-MARC . Meanwhile, in the mm-wave (second) band, is transmitted to through the - relay link and to through the - direct link simultaneously by using the links times. The relay aids by creating codewords and from its received signals and transmitting them to in both bands.
We now define the channel model of the Gaussian DR-MARC. As in , in the first band, the channel outputs at and at the -th use of the band are given by
where are channel fading coefficients from the transmitter at node to the receiver at , , and input are block power constrained, . Also, the noise RVs are , i.i.d., and , i.i.d.
In the second band, the outputs of the - relay links at the relay are modeled as
and the outputs of the - direct links and the - link at are modeled respectively as
where are the fading coefficients of the - relay links, while and are the same for the - direct links and the - mm-wave link respectively. The input symbols, and , are block power constrained as follows: , . Also, the noise RVs are , i.i.d., and , i.i.d.
We assume that the DR-MARC is subject to an ergodic fading process where, across channel uses, the phase of the fading coefficients are i.i.d. Specifically, the fading coefficients from node to node , , in the first band are denoted by , while those in the second band by , with . Here, i.i.d., and are i.i.d. RVs that depend on the inter-node distance , as well as the pathloss exponent (for the first band) and (for the second band). For example, when specializing to phase fading, we take and to be constant, and for Rayleigh fading, we take and i.i.d., where
is an exponential distribution with mean.
We also assume that the long term parameters, i.e., the distances and the pathloss exponents, are known at all nodes; the instantaneous channel state information (CSI), i.e., the phase and magnitude of the fading coefficients, are not available to any transmitter; and
each receiver knows the CSI on all its incoming channels, but has no CSI of other channels. This models practical scenarios where CSI feedback to a transmitter is unavailable, while a receiver can reliably estimate the CSI. Also, this is less restrictive than where full or partial CSI is also available at a transmitter.
Note that given a BMF , for uses of the microwave band, the mm-wave band is used times, while for uses in the mm-wave band, the microwave band is used times. We define a code for the DR-MARC that consists of () two independent, uniformly distributed message sets , one for each source; () two encoders and such that ; () a set of relay encoding functions, and , such that and , ; and () a decoder at such that .
The relay helps by computing and causally by applying functions and on its past received signals and CSI as above and transmitting them to . A rate tuple is said to be achievable if there exists a sequence of codes such that the average probability of error as [32, Chap. 15.3]. Finally, the system model of the R-MARC is defined from that of the DR-MARC by setting .
Iii Decomposition Result on the DR-MARC
TWe show that the capacity of the DR-MARC with BMF , denoted , can be decomposed into the capacity of the underlying R-MARC, denoted , and the two - direct links.
is given by the set of all non-negative rate tuples that satisfy
where , and the expectations are taken over the corresponding RVs.
The proof is relegated to Appendix A. For the special case of phase fading where are constant, expectations in Theorem 1 are not needed, while for Rayleigh fading expectations are over . Any in the DR-MARC can be achieved by achieving in the underlying R-MARC and supplementing it with the capacity of the direct links. Hence, operating the direct links independently of the R-MARC is optimal, which simplifies the transmission. Since can be determined from , it is sufficient to focus on , considered next.
Iv Capacity of a Class of R-MARC
Unlike the DR-MARC where separating the operation of the underlying R-MARC from the mm-wave direct links is optimal, in the R-MARC separating the underlying c-MARC and the mm-wave relay links is suboptimal in general. In fact, capacity of the R-MARC is derived by operating the c-MARC jointly with the relay links. First, we characterize an achievable rate region for the R-MARC.
An achievable region of the R-MARC with BMF , denoted , is given by the set of all non-negative rate tuples that satisfy
where expectations are over the channel gains and , .
The achievable region is obtained by performing block Markov encoding and backward decoding for the relay, as outlined in Appendix B. Moreover, the same message is jointly encoded into codewords that are transmitted simultaneously in both bands. Interestingly, the bounds in (8)-(10) can be interpreted as that of the MAC from the sources to the destination aided by the relay.
In , the capacity of the near c-MARC, where the source-relay links can support higher rates than source-destination links, was characterized. In contrast, for R-MARCs with far underlying c-MARC, if the following conditions hold, then the scheme of Theorem 2 is also capacity achieving.
While the proof is relegated to Appendix C, we discuss the key steps here. First, in the proof of the outer bounds in steps (e)-(f) of (30), the cross-correlation coefficients between the source and relay signals are set to zero. Since instantaneous CSI are not available to the transmitters and the phase of the fading coefficients , i.i.d., setting the cross-correlation to zero proves optimal, resulting in outer bounds (8)-(10). Next, in Theorem 2, if conditions (11)-(13) hold, the achievable rates (8)-(10) for the destination are smaller than those in (5)-(7) for the relay. Hence, the relay can decode both messages without becoming a bottleneck to the rates. Thus, under (11)-(13), rates (8)-(10) are achievable and they match the outer bounds.
Note that the rates in Theorem 2 are achieved by encoding jointly over both bands. Hence, while capacity of the c-MARC is known only when the source-relay links are stronger in the microwave band (near case), in the R-MARC, they only need to be stronger jointly over both bands. Thus, even if sources are not near the relay in the microwave band, for sufficiently strong mm-wave relay links, they can become “jointly near” over both bands, where the scheme of Theorem 2 achieves capacity.
The above result applies directly to phase and Rayleigh fading: for phase fading, and are geometry determined constants, and thus the expectations in Theorem 2 are not needed, while for Rayleigh fading, the expectations are over and .
Numerical Examples: To illustrate the impact of mm-wave links on the capacity of the R-MARC, we consider a two-dimensional topology as in Fig. (b)b where and are located on the x-axis at and , and and are located symmetrically at , with being the angle between a source and and the resulting source-destination distance. We take both bands in the R-MARC to be under phase fading as in . Hence, expectations in conditions (11)-(13) and Theorem 2 are not needed, and observations can be interpreted in terms of distances. Also, power constraints in the R-MARC are set to and .
First, note that under condition (13), the sum-rate outer bound (OB), given by the r.h.s. of (10), matches the achievable sum-rate (ASR) in Theorem 2, given by the minimum of r.h.s. of (7) and (10). For ease of exposition, we fix and BMF . Hence, condition (13) is equivalent to for some threshold source-destination distance . We verify this for fixed and and two cases of by plotting the ASR and the OB as functions of in Fig. (a)a. We observe that the ASR matches the OB if with for , and for , otherwise the ASR is strictly smaller. As reduces from to , for condition (13) to hold, also reduces from to .
Next, to illustrate the impact of the mm-wave links, in Fig. (b)b we depict the source locations relative to the relay and the destination for which all of conditions (11)-(13) are satisfied and therefore the scheme of Theorem 2 achieves capacity. As such, we fix , vary and to vary source locations, and plot the resulting regions: we overlay the region for the case without mm-wave links () on those with mm-wave links with BMF as well as .
First, for the case without mm-wave links (), conditions (11)-(13) hold only when sources are within the innermost black region in Fig. (b)b. Noting that for each , the resulting threshold distance is at the boundary of this region, as increases from to , decreases monotonically from to . We thus observe that conditions (11)-(13) hold for much larger threshold distance when sources are located far away from destination (i.e., ), and threshold distance reduces considerably when sources are closer to the destination (i.e., ).
We note that the above trends continue to hold when mm-wave links are used (), however, the resulting region (union of the inner black and outer gray regions) now extends much closer to the destination. For example, for the region with , reduces to only near the destination, compared to with . Moreover, the resulting region grows with but the growth saturates for higher values of , with producing almost the same region as that for .
V The Optimal Sum-Rate Problem
Since mm-wave links can have significantly larger bandwidth than the microwave links, they can significantly affect the performance limits of the DR-MARC. To understand this impact, we study how the sum-rate achievable on the DR-MARC (with the scheme of Theorem 2) is maximized by optimally allocating power to the mm-wave direct and relay links. We observe that the resulting scheme allocates power to the mm-wave links in different modes depending on whether certain channel conditions hold. This characterization reveals insights into the nature of the scheme, and can serve as an effective resource allocation strategy for such dual-band networks in practice.
For ease of exposition, the mm-wave band is assumed to be under phase fading while the microwave band is assumed to be under the general fading of Section II. Here, phase fading is a good model for mm-wave links such as those in , as phase fading is a special case of the general fading model  when the diffuse component associated with the non-LoS propagation is not present. Furthermore, this simplification reveals useful insights into the optimal power allocation.
Under phase fading, the link gain in the - direct link (referred to as ) is , and that in the - relay link (referred to as ) is , which are constants. For convenience, we denote the link gains of and by and . We assume that the transmit power in () and () from source satisfy a total power budget
For a fixed power allocation , is an achievable sum-rate of the DR-MARC iff
Here, and denote the sum-rates achievable at the relay and destination, and are given by
where and , with the expectations taken over the RVs involved. Note that and are obtained as follows. For direct link powers , it follows from the decomposition result in Theorem 1 that the sum-rate of the DR-MARC is given by the sum of the sum-rate of the R-MARC and the total rate of the direct links, i.e., . Now, for given relay link powers , the sum-rate of the R-MARC is given by the minimum of r.h.s. of (7) and (10). Hence, is given by the sum of the r.h.s. of (7) and as expressed in (16), while is obtained by the sum of the r.h.s. of (10) and , as given in (17).
The problem of maximizing over the transmit powers () is then
Note that is a convex optimization problem as the objective is linear, constraints in (20) are affine, and those in (18)–(19) are convex. Hence, it can be solved by formulating the Lagrangian function of by associating a Lagrange multiplier to each constraint in (18)-(21), and then deriving and solving the KKT conditions . See Appendix D for details.
V-a Link Gain Regimes and Optimal Power Allocation
To gain insights, we derive the optimal power allocation in closed form, and describe it in terms of link-gain regimes (LGR) which are partitions of the set of all tuples of link gains and power budget , found while solving the KKT conditions for . Specifically, we derive the KKT conditions and solve for the optimal primal variables (i.e., transmit powers) and the optimal Lagrange multipliers (OLM). To simplify the procedure, we consider the set of tuples of OLMs associated with inequality constraints in (18), (19) and (21), and partition this set into a few subsets based on whether the OLMs in the set are positive or zero, i.e., whether the associated primal constraints are tight or not (detailed in Appendix D). For each resulting partition of the set of OLM tuples, we first derive the expression for the optimal powers in closed form. However, the conditions that define these partitions are still characterized in terms of the OLMs. Therefore, to express the optimal power allocation explicitly in terms of link gains and power budget , we express the conditions that partition the set of the OLM tuples in terms of link gains, , and parameter , defined as
As a result, the set of -tuples is partitioned into a few subsets (LGRs), each corresponding to one and only one subset of OLM tuples. The conditions for each LGR is then simplified and expressed as upper and lower bounds (threshold powers) on power budget where the threshold powers depend on . This results in partitioning the power budget into a few intervals, each describing an LGR. Specifically, we consider two cases and , which are equivalent to and respectively.
|Definition of LGR||Optimal power allocation|