Capacity Scaling for Cloud Radio Access Networks with Limited Orthogonal Fronthaul

Abstract

Uplink and downlink cloud radio access networks are modeled as two-hop K-user L-relay networks, whereby small base-stations act as relays for end-to-end communications and are connected to a central processor via orthogonal fronthaul links of finite capacities. Simplified versions of network compressforward (or noisy network coding) and distributed decode-forward are presented to establish inner bounds on the capacity region for uplink and downlink communications, that match the respective cutset bounds to within a finite gap independent of the channel gains and signal to noise ratios. These approximate capacity regions are then compared with the capacity regions for networks with no capacity limit on the fronthaul. Although it takes infinite fronthaul link capacities to achieve these "fronthaul-unlimited" capacity regions exactly, these capacity regions can be approached approximately with finite-capacity fronthaul. The total fronthaul link capacities required to approach the fronthaul-unlimited sum-rates (for uplink and downlink) are characterized. Based on these results, the capacity scaling law in the large network size limit is established under certain uplink and downlink network models, both theoretically and via simulations. A. Uplink C-RANSeveral coding schemes have been proposed in the literature for the uplink C-RAN with K users (senders) and L relays. Zhou and Yu [5] applied the network compress-forward relaying scheme [7] to this model and showed, by optimizing over quantizers, that under some symmetry assumptions, this scheme achieves the optimal sum-rate within L/2 bits per real dimension uniformly over all K and all channel parameters. Sanderovich, Someskh, Poor, and Shamai [8] used the same scheme and analyzed the large-user asymptotics (i.e., the scaling law) of symmetric achievable rates when all fronthaul links have equal capacities. Zhou, Xu, Yu, and Chen [6] subsequently showed that under a sum-capacity constraint on the fronthaul links, the coding scheme in [5] and [8] can be simplified through successive cancellation decoding, generalizing an earlier result for the single-sender multiple relay network [9]. Aguerri and Zaidi [10] proposed a hybrid coding scheme of network compress-forward and compute-forward [11], and demonstrated that it outperforms the better of the two in general. Aguerri, Zaidi, Caire, and Shamai [12] specialized the noisy network coding scheme [13] to the uplink C-RAN, the achievable rate region of which coincides with that of network compress-forward [5], [8]. The most general outer bound on the capacity region of the uplink C-RAN can be obtained by specializing the cutset bound [14]; see, for example, [5] and the references therein, as well as Proposition 2 in this paper. The cutset bound has been further tightened under additional assumptions. Aguerri, Zaidi, Caire, and Shamai [12] studied the uplink C-RAN in which the relays are oblivious of the codebooks of the senders, and demonstrated that network compress-forward (or noisy network coding) achieves the capacit...

Magnetic Fields and Asymptotic Limits of Current Loops

“…Remark 1. The integrals in eqns (7) and (8) are referred to as elliptic integrals [26]. They have no finite closed form expressions in terms of elementary functions.…”

“…This result, commonly known as the Biot-Savart law, was used by Maxwell to formulate the so-called Ampère's circuital law [6,7], which states that the circulation of the magnetic field B produced by a steady (time-invariant) current over a closed loop is proportional to the total current flowing through the surface, or, equivalently in differential form, ∇ × B = µ 0 J, where J is the volume current density and ∇ × A for a vector field A(r) ≡ A x (r) x + A y (r) ŷ + A z (r) ẑ is defined as…”