Cooperative Body Channel Communications for Energy-Efficient Internet of Bodies

The Internet of Bodies (IoB) is a network formed by wearable, implantable, ingestible, and injectable smart devices to collect physiological, behavioral, and structural information from the human body. Thus, the IoB technology can revolutionize the quality of human life by using these context-rich data in myriad smart-health applications. Radio frequency (RF) transceivers have been typically preferred due to their availability and maturity. However, for most RF standards (e.g., Bluetooth low energy), the highly radiative omnidirectional RF propagation (even at the lowest settings) reaches tens of meters of coverage, thereby reducing energy efficiency, causing interference and co-existence issues, and raising privacy and security concerns. On the other hand, body channel communication (BCC) confines low-power and low-frequency (10 kHz–100 MHz) signals to the human body, leading to more secure and efficient communications. Since energy efficiency is one of the critical design parameters of IoB networks, this article focuses on energy-efficient orthogonal body channel access (OBA) and non-OBA (NOBA) schemes with and without cooperation. To this aim, three main BCC topologies are presented: 1) point-to-point channel; 2) medium access channel; and 3) broadcast channel. These topologies are then used as building blocks to create IoB networks relying on OBA and NOBA schemes for downlink (DL) and uplink (UL) traffic. For all schemes and traffic directions, optimal transmit power and phase time allocations are derived in closed-form, which is essential to reduce energy consumption by eliminating computational power. The closed-form expressions are further leveraged to obtain maximum network size as a function of data rate requirement, bandwidth, and hardware parameters.


Cooperative Body Channel Communications
for Energy-Efficient Internet of Bodies comprising wearable, ingestible, injectable, and implantable smart devices located in, on, and around the human body [1]. The IoB moves the focal point to the human body and places intrabody and interbody communication at the center stage of connectivity. IoB enables a myriad of applications, including but not limited to personalized medicine to offer proactive and preventative care; remote patient monitoring and rehabilitating patients; smart home assisted independent living for seniors and people with disabilities; self-care and welfare for a healthy and productive lifestyle; occupational health and safety to protect critical personnel from workplace injuries and work-related diseases; and sports and entertainment [2].
Since the IoB has its root in wireless body area networks, the IEEE 802.15.6 standard provides foundational physical (PHY) layer and medium access layer specifications for short-range, ultra-low power, and highly reliable wireless communication within the body area [3]. To this aim, it specifies three PHY layer options: narrowband (NB) and ultrawideband (UWB) radio frequency (RF) communications and body channel communications (BCCs), also known as the human body communications. Although the RF technology has gained a widespread use thanks to its maturity and availability, they are not always the best option to facilitate robust, secure, and scalable IoB networks due to the following major drawbacks [1], [4]. 1) Highly radiative and omnidirectional propagation of RF devices exposes sensitive data to the danger of eavesdroppers, bio-hackers, and interceptors, thereby imposing security and privacy threats. Even though the required coverage is 5-10 cm around the human body, RF systems, such as BLE can reach tens of meters of wireless coverage at the lowest transmit power setting [5], leading to energy loss, and potential security hazards. 2) Since most IoB devices operate on industrial, scientific, and medical (ISM) bands to avoid licensing issues, interference, and coexistence become major issues due to the overpopulated IoT devices in the ISM bands. 3) Complex and power-hungry radio front ends limit the node lifetime and necessitate larger area and battery sizes, which contradicts with the objective of small form-factor and energy self-sustainable IoB nodes. Alternative to RF communications, the BCC exploits the conductive properties of the human body by confining the transmitted signal to skin tissue at frequencies between 100 kHz and 100 MHz. Electrostatic fields are coupled to 2327-4662 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information. the body using galvanic coupling (GC) or capacitive coupling (CC). In the GC-BCC, both signal and ground electrodes of the transmitter and receiver are in contact with the skin. Transmission is initiated by passing small currents through the body and detecting signals at the receiver end. Since both the signal (forward) path and the return (backward) are formed through the body, the GC-BCC is mainly characterized by the dielectric properties of the body, granting GC-BCC immunity to environmental effects. However, since the operational frequency of GC-BCC is limited to below 1 MHz, it is incapable of supporting high-throughput and long-range communications [6]. As shown in Fig. 1, the CC-BCC requires only signal electrodes to be in contact with the body tissue to form the forward communication path. On the other hand, ground electrodes are left floating in the air to form the return path due to the parasitic capacitance between them and the Earth's ground [7]. Since the signal is subject to low attenuation in the forward path as a result of high tissue conductivity, the overall channel loss is influenced by over-the-air capacitive return (backward) paths [1]. Even though the CC-BCC is affected by the variations in the surrounding environment, it delivers a better channel gain than the GC-BCC scheme and can meet the QoS demand of IoB applications by exploiting higher frequencies. The advantages offered by BCC communications over RF systems can be summarized as follows [2], [4]: Coupling ultra-low power signals to the human body yields a negligible leakage, thus providing improved physicallayer security. Moreover, the BCC channels experience a better channel gain than over-the-air RF channels since the human body is more conductive than air. Furthermore, the human body does not behave as an antenna in the BCC frequency range (1 kHz-100 MHz), which mitigates body shadowing effects and yields a more stable wireless channel. The BCC frequency range also decouples the transceiver size from the carrier wavelength and eliminates the need for radio frontends. As shown in Fig. 2, putting all these virtues together paves the way for energy-efficient BCC transceivers in the order of pJ/bit levels, compared to nJ/bit energy efficiency levels of commercial-off-the-shelf RF transceivers.

A. Related Work
The development of BCC is owed to the research efforts dedicated to modeling the channel characteristics and behavior of human tissue in the presence of electromagnetic (EM) fields. Hence, accurate channel models will facilitate efficient BCC transceivers [16] which are the key components of the physical layer. To address the impact of environmental effects, electrode placement on the body, intricate anatomy of the human tissues, channel length, and varying electrode specifications [6], [17], different models have been proposed. In the context of CC-BCC, the channel modeling techniques reported in the literature are analytical, circuit-based, numerical, and empirical models. We refer interested readers to [1] for a complete discussion on the different propagation characterizations and models of body channels.
Numerous design approaches were investigated in the literature to optimize BCC transceivers, including different coupling methods, operational frequency ranges, and modulation techniques to achieve enhanced performance with respect to achievable throughput and energy efficiency [17]. In addition, the small form-factor and long-battery lifetime requirements of IoB nodes suggest that IoB transceivers are designed to transmit at low power levels. Therefore, addressing the sensitivity of the BCC receivers, which is the minimum received power that guarantee the correct data retrieval, is crucial for reliable communication networks. Cho et al. [18] proposed a CC-BCC transceiver implemented in a 65-nm CMOS process with dual operational modes, namely, entertainment and healthcare. In the former, a dualband (40/160 MHz) full-duplex transceiver that is binary phase-shift keying (BPSK) based is considered to deliver −58 dBm sensitivity at 80 Mb/s with 79 pJ/bit efficiency. While in the latter mode, the super-regenerative transceiver adopting on-off-keying (OOK) achieves −72 dBm sensitivity at 100 kb/s for 42.5-μW power consumption. Unlike most of the narrowband transceivers operating at specific frequencies to alleviate interference at the expense of energy efficiency, Maity et al. [8] developed an energy-efficient broadband interference-tolerant transceiver. The transceiver utilizes timedomain interference rejection to attain a 30-Mb/s data rate and 6.3-pJ/b energy efficiency with −63.3 dBm. Moreover, [19] demonstrates the possibility of achieving −98.9 dBm sensitivity and energy efficiency of 3.8 nJ/bit for the operational range 164 kb/s to 1.13 Mb/s with the proposed standard mode operating transceiver. This performance was obtained using a frequency-selective digital transmission (FSDT) modulation scheme. In Fig. 2 performance comparison of state-of-the-art BCC transceivers and RF-based systems are illustrated. The aforementioned efforts validated the enhanced performance of CC-BCC over conventional RF systems, and the feasibility of efficient, highly sensitive BCC transceivers. This work tackles the networking aspect of multiple communicating nodes in order to realize an efficient, reliable, and connected BCC network.
The point-to-point (P2P) topology is widely investigated in WBANs where sensor nodes measure the physiological data of users and wirelessly convey the information to an access point (AP) [20], [21]. Ling et al. [20] studied duty-cycled-based P2P as they solve the throughput maximization problem by optimizing duty-cycle time-ratio and power allocation. To establish multipoint communication in WBANs, Huang et al. [22] proposed a transmission protocol based on time division multiple access for half-duplex mode. They divide the time slot into two phases active and inactive phase. In the former, nodes acquire energy wirelessly from AP, and in the latter, nodes communicate with the AP through TDMA. Thus, to solve the average sum-throughput, they jointly optimize the time allocation ratio and transmit power levels of the sensor nodes. Li et al. [23] presented a cooperative communication system in WBAN based on decode-and-forward (DF) relaying. The communication is facilitated by TDMA. Similar to the previous work, energy harvesting and data transmission phases exist. The difference is that the transmission phase is split between direct transmission from nodes to the hub, direct transmission from nodes toward the relay, and relay transmission toward the hub node. The purpose of their approach is to compare the direct and relayed links. Joint-objective linear programming methods are utilized to solve the joint optimization problem of power allocation at sensor nodes and relay, which maximizes the throughput of individual nodes. However, all these studies are based on WBAN communications and only consider uplink transmissions. In the realm of BCC, the work in [4] was the first to present a fullscale study of capacitive body channel access schemes for a generic IoB network. The purpose of the study was to evaluate regular and cooperative orthogonal and nonorthogonal access schemes with regards to max-min rate, max-sum rate, and QoS sufficient regimes. To account for the wide range of IoB applications, analytical expressions, numerical power control, and phase time allocations for the three operational regimes were provided. Albeit its valuable contributions, the work presented in [4] does not shed light into the energy efficiency and maximum achievable network size, which is one of the key design goals of IoB networks. Thus, the work in [24] focused on the energy efficiency of orthogonal and nonorthogonal capacitive body access schemes and derived optimal power allocations in closed-form for both uplink (UL) and downlink (DL) traffic. Moreover, the maximum network size is derived for both directions under orthogonal and nonorthogonal scenarios. This article extends [24] by introducing cooperative body channel access schemes for higher energy efficiency; deriving closedform power allocations under successive interference cancellation (SIC) imperfections; optimizing the phase time allocations between source nodes, relay, and hub; and analyzing the network size under both perfect and imperfect SIC conditions.

B. Main Contributions
The main contributions of this article can be summarized as follows.
1) Three primary BCC topologies are introduced: a) P2P topology; b) multiple access channel (MAC) topology; and c) broadcast channel (BC) topology. These topologies are then used as building blocks to facilitate four main body channel access schemes: a) orthogonal body channel access (OBA); b) non-OBA (NOBA); c) cooperative OBA (C-OBA); and d) cooperative NOBA (C-NOBA). 2) After formulating the optimization problem for energyefficient networking problems for regular and cooperative schemes, optimal power control levels are derived in closed-form under perfect and imperfect SIC conditions. Since phase times of source↔relay and relay↔hub links play a vital role in overall energy consumption, the derived power allocations are leveraged to find optimal phase time allocation numerically. 3) Finally, the maximum achievable network size of each topology is analyzed as a function of data rate requirement, SIC imperfections, and channel conditions. These analyses are further extended by numerically finding the optimal phase time allocation that gives the maximum network size under cooperation. Numerical results illustrate the circumstances under which regular and cooperative schemes improve the performance of the IoB network. Subject to different node deployment scenarios, C-NOBA was shown to sustain the least sum of transmit power among all other schemes, improving the overall energy efficiency in UL and DL traffic. Specifically, when moving the grouped source nodes away from the hub, C-NOBA exhibits up to 9% reduction in transmit power compared to NOBA schemes. Moreover, the results also indicate the importance of relay location regarding the hub and source nodes as an optimal distance was noted, which provided 14% less power consumption than regular schemes. C-NOBA schemes can exhibit a 13% better effectiveness in total transmission power than their regular counterparts at low QoS requirements as the network size increases. Conversely, C-NOBA performance is restrained to fewer nodes at higher QoS demands. It was noted that adopting relayed links constitutes a bottleneck on the maximum number of nodes K max compared to regular schemes, since the total power of the relay node must be shared by all. Compared to NOBA, the C-NOBA reduces the network size by 31.7% and 69% in UL and DL traffic, respectively. Numerical results also show that optimizing phase time allocation substantially improves the energy efficiency and network size by mitigating the negative impacts of node deployment.

C. Paper Organization
The remainder of this article is organized as follows. Section II presents BCC topologies followed by regular and cooperative body channel access schemes orthogonal and nonorthogonal for the capacitive channel. This will provide insights on channel resource allocation, maximum achievable throughput, and decoding processes. Next, in Section III, problems are formulated and algorithms for power control and phase time allocations are proposed, respectively. Section IV derives the maximum feasible network size for regular and cooperative orthogonal and nonorthogonal schemes. For nonorthogonal, the closed-form expressions were derived for both perfect and imperfect cancelations scenarios. Simulation results are illustrated in Section V. Finally, we conclude this article in Section VI.

II. SYSTEM MODEL
The system model considered in this article consists of a wearable hub device (e.g., smartwatch) communicating with K on-body IoB nodes in a time-slotted fashion in both uplink (UL) and downlink (DL) directions. The hub acts as an AP that coordinates transmissions in the network and with off-body entities (e.g., smartphones, base stations, routers, etc.) utilizing RF communication methods, e.g., cellular, Bluetooth, Wi-Fi, etc. The node deployment in this work is not subject to a specific arrangement as it is mainly determined by the underlying application, which is out of this article's scope. Throughout this article, we symbolize the total available bandwidth, timeslot duration, and thermal noise power spectral density with B, T, and N 0 , respectively. Moreover, P represents the maximum power of the nodes and hub and is selected to be within health and safety bounds. As illustrated in Fig. 3, we will discuss three main BCC topologies: 1) P2P channel; 2) MACs; and 3) BC. These topologies constitute the building blocks for regular and cooperative orthogonal and nonorthogonal body access schemes OBA, NOBA, C-OBA, and C-NOBA.

A. Body Channel Communication Topologies
In the remainder of this section, we present BCC topologies for two generic nodes, n i and n h . In the remainder of this article, we assume channel reciprocity and denote UL and DL channel gains by g h i and g i h , g h i = g i h , respectively. 1) Point-to-Point Channel: In the P2P topology, the information of a pair of transmitter and receiver IoB nodes is sent over a preassigned dedicated link. That is, the entire bandwidth is split equally between K available transmitters to form exclusive connections with receivers in the network design phase. As shown in Fig. 3(a), the interference is inherently avoided by such orthogonal network resource allocation. Accordingly, the signal transmitted by n i is received by n h as follows: where K is the set of IoB nodes sorted in descending order of the channel gains,ω h i ∈ [0, 1] is the power weight assigned to transmit the message of n i , x i , 1 and z h ∼ N (0, N 0 B/K) models the additive white Gaussian noise (AWGN) at receiver n h . Consequently, the signal-to-noise ratio (SNR) of UL-P2P is given byγ On the DL direction, n h communicates with the rest of the nodes utilizing its power budget equally for transmission. When the total bandwidth and power are equally distributed among K nodes, the SNR of DL-P2P can be obtained as in (2) and given bẏ

2) Multiple Access Channel:
The MAC topology allows multiple source nodes sharing the communication medium to exploit the entire bandwidth B and transmit their data to the same destination at the same time. This implies that the received signal at the receiver node n h is a superposed version of all transmit signals as follows: whereω h i ∈ [0, 1] is the power factor allocated to the message of n i , x i , and z h ∼ N (0, N 0 B) is the AWGN at n h . Since y h is a composition of all transmit signals, interference alleviation techniques is necessary to decode each message. To this aim, n h is equipped with an SIC receiver that decodes messages in the descending order of their reception power strength. To improve the spectral efficiency of the MAC channel, a higher power weight must be allocated to nodes with better channel conditions. In this case, the node with ith strongest reception , the first term in the denominator represent residual interference coming from lower rank nodes due to the SIC imperfections, which is modeled by ∈ [0, 1] to capture channel estimation errors and hardware limitations. On the other hand, the second term in the denominator represents the uncancellable interference originated from the lower rank nodes.
3) Broadcast Channel: Similar to the MAC topology, the entire available bandwidth is exploited in the BC topology, where n h broadcasts superposition of messages intended for the rest of the nodes. In this case, the signal received by n i is given byÿ where power weights,ω k h ∈ [0, 1], is subject to the total energy constraint of n h , i.e., k∈Kω k h ≤ 1. Similar to the MAC topology, the receiver node n i ∀i ∈ K performs the SIC procedure to decode the message intended for itself. However, the power allocation weights and decoding order must be in reverse order. In this case, the weakest channel node is allocated with the highest power weight and subject to interference from the rest of nodes, which yields the following relation: Accordingly, the SINR of n i is given bÿ , the first and second terms in the denominator represent the uncancellable and cancellable interference originated from the lower and higher rank nodes, respectively.

B. Regular Capacitive Body Channel Access
In this section, we explain how P2P, MAC, and BC topologies can be leveraged to facilitate orthogonal and nonorthogonal capacitive body channel access schemes.

1) Orthogonal Body Channel Access:
In the OBA, multiple access interference (MAI) is avoided by dedicating K P2P links in both UL and DL over the entire time slot duration T, as shown in Fig. 3(a). By substituting the SNR of UL-P2P given in (2) into the Shannon-Hartley channel capacity formula, the maximum achievable UL-OBA rate is expressed as follows: Similarly, the maximum achievable DL rate,Ṙ i h (ω h ) ∀i ∈ K, can be obtained by substituting DL-P2P given in (3) into (10).
2) Nonorthogonal Body Channel Access: As shown in Fig. 3(b), the UL and DL traffic is facilitated by MAC and BC topologies, respectively. In this case, the maximum achievable UL-NOBA rate can be obtained by using the SINR expression of MAC topology given in (6) as follows: Similarly, the maximum achievable DL-NOBA rate can be obtained by using the SINR expression of BC topology given in (9) as follows:

C. Cooperative Body Channel Access
In this section, we explain how P2P, MAC, and BC topologies can be leveraged to facilitate cooperative orthogonal and nonorthogonal capacitive body channel access schemes. In the UL/DL direction, the cooperation is performed in two phases as shown in Fig. 3. In the former, the cooperating IoB node (i.e., relay) remains idle to receive the transmitted signals from hub/source nodes over λT duration, where λ ∈ [0, 1] is the phase time allocation factor. In the latter, the relay node n r forward decoded messages sent by hub/source nodes along with its own message to source/hub nodes over the remaining time slot duration, (1 − λ)T.
1) Cooperative Orthogonal Body Channel Access: As shown in Fig. 3(a), the first and second phases of C-OBA schemes consist of K − 1 and K P2P links. Hence, received signals from K − 1 source nodes, whose set is denoted by K −r , follows the definition in (1) by replacing (·) h , (·) h i , and K with (·) r , (·) r i , and K −r , respectively. Applying the same notational changes to (2) and (10) yields the maximum achievable UL-OBA rates during the first phase as follows: is the power allocation vector of nodes in the first phase. In the second phase, received signals by the hub node follows the definition in (1) by replacing (·) h i with (·) h r . Applying the same notational changes to (2) and (10) yields the maximum achievable UL-OBA rates during the second phase as follows: is the power allocation vector of nodes in the second phase. Accordingly, end-to-end UL-OBA rate for n i is given bẏ Following similar steps above, end-to-end rate of n i in the DL-OBA scheme is given by: Cooperative Nonorthogonal Body Channel Access: As shown in Fig. 3(b), the both phases of UL and DL C-NOBA schemes consist of MAC and BC topologies, respectively. In the UL direction, the signals received by n r from K − 1 source nodes follows the definition in (4) by replacing (·) h , (·) h i , and K with (·) r , (·) r i , and K −r , respectively. Applying the same notational changes to (6) and (11) yields the achievable rates during the first phase as follows: In the second phase, received signals by the hub node follows the definition in (4) by replacing (·) h i with (·) h r . Applying the same notational changes to (6) and (11) yields the maximum achievable UL-OBA rates during the second phase as follows: . Accordingly, end-to-end UL-NOBA rate for n i is given bÿ Following similar steps above, end-to-end rate of n i in the DL-NOBA scheme is given by:

III. PROBLEM FORMULATION AND SOLUTION METHODOLOGY
In this section, we first formulate the total energy consumption minimization problem then provide a solution methodology to obtain optimal power weights and phase time allocation. Throughout this section we allow nodes to have distinct (QoS) demands and channel gains.

A. Problem Formulation
Our goal in this article is to maximize the network longevity while satisfying various QoS demands to meet the requirements of different applications. To this end, we formulate our optimization problems to minimize total energy consumption by optimizing power and phase time allocations. Throughout this section, we omit (˙ ) and (¨ ) notations to keep formulations and solutions generic to both OBA and NOBA schemes.
1) OBA and NOBA: The optimization problem that optimizes the power allocation weights to minimize total UL power consumption can be formulated as follows: where C 1 is the QoS constraints that ensure that n i is provided with a data rate not less than its demandR i and denotes the pairwise inequality. Similarly, the DL problem can be formulated as follows: where C 2 is an additional constraint to ensure total DL transmission power is less than the maximum transmission power of n h .
2) C-OBA and C-NOBA: Apart from regular OBA and NOBA schemes, we formulate the optimization problem to jointly obtain power and phase time allocations which minimize the total energy and maximize the network lifetime while satisfying QoS demands as follows: where C 1 and C 2 are the QoS constraints of the first and second phases satisfying the end-to-end data rate demands, respectively. In (23), the phase time allocation plays a vital role in overall energy consumption since λ requires QoS constraint to be scaled by the phase time duration. Hence, λ determines the overall energy consumed at both phases as shown in the objective function. Likewise, the DL problem is formulated as where C 3 and C 4 are additional constraints to guarantee that the total DL transmission power in both stages is within the maximum transmission power limit.

B. Solution Methodology
Problems formulated above can be transformed into geometric programming (GP) problems [25], [26], [27], [28], [29], which can be readily solved by convex optimization solvers such as GP toolbox of CVX [30]. However, the low-cost and ultra-low-power design goals of IoB nodes necessitate the derivation of closed-form optimal power allocations to reduce hardware cost and power consumption related to the computational complexity.

1) OBA and NOBA:
The optimal solution of both P UL REG and P DL REG is attained by satisfying the QoS constraints at equality since operating at data rate above the threshold would increase the overall power consumption. Therefore, the optimal power weight allocations can be obtained by solving the following system of equations: where the vectors are of size K ×1; matrices are of size K ×K; p is the column vector of received powers; σ is the column vector of receiver noise with identical elements of N 0 B;¯ = diag(¯ 1 , . . . ,¯ i , . . . ,¯ K ) is the diagonal matrix of the SINR demands with respect to QoS demands; I is the identity matrix; and J is the interference channel matrix whose entries are given by where the cases 1, 0, and refer to no interference, clusterinterference, and residual interference, respectively [31]. We also draw attention that power levels in (25) (25) is generic to provides closed-form power allocation to reach the minimum energy consumption objective. In what follows, we provide closed-form power allocations for regular OBA and NOBA schemes.
2) C-OBA and C-NOBA: As explained in the problem formulation, the joint optimization of phase time allocation and power weights is crucial as they determined the overall energy consumption in a time slot duration. For a given λ, each phase behaves as an individual time slot and optimal power weights that minimizes the overall consumption can be obtained by Lemmas 1-3 for OBA, UL-NOBA, and DL-NOBA schemes, respectively. Therefore, optimal λ can be expeditiously obtained by Golden section search as explained in Algorithm 1, which is explained as follows.
In line 2, we initialize golden ratio τ , iteration index t, lower bound parameter lb, and upper bound parameter ub. Then, two initial points, λ 1 and λ 2 , are calculated based on the golden ratio and evaluated by EVALUATE OBJECTIVE FUNCTION(λ, ) procedure. This procedure sets parameters for each phase (i.e., number of nodes, QoS constraint, etc.) and obtain the minimum energy consumption for the given λ by using the corresponding lemma as mentioned above. Based on the evaluation of these initial probe points, the while loop between lines 6 and 25 iteratively founds optimal λ by evaluating the objective for two intervals, discarding the one with higher energy consumption, resetting bounds as per the new interval, and calculating new probe points for the next iterations. The loop terminates if the step tolerance (i.e., the absolute value of difference between two selected λ values) is less than a accuracy of interest μ or the maximum of the number of iterations is reached.

IV. MAXIMUM IOB NETWORK SIZE ANALYSIS
The IoB applications may differ in required QoS and number of nodes to provide a sufficient service. Therefore, this Algorithm 1 Optimal Phase Time Allocation 1: Input: section analyzes the maximum feasible number of nodes and derive closed-form network size, K max , as a function of key parameters, such as QoS demand, SIC error factor, channel gain, and available bandwidth. Throughout this section, we assume all nodes have a common data rate requirement,R, for the sake of analytical tractability.

A. Maximum Network Size Analysis of OBA Schemes
Since the OBA scheme consists of P2P links, the maximum number of nodes can be directly obtained from the SINR constraint of the node with the weakest channel gain, g min . By limiting the optimal power weights provided in Lemma 1 to unity, K max can be obtained as in Lemma 4.
Lemma 4: K max for the UL-OBA scheme is given by where W −1 (·) is the −1th branch of the Lambert-W function.
On the other hand, K max for the DL-OBA scheme is given by Proof: Please see Appendix-B.

B. Maximum Network Size Analysis of NOBA Schemes
In the UL-NOBA scheme, the IoB node with the strongest channel is required to transmit with the highest power. Therefore, as we add more users to the network, the strongest node needs to increase its transmission power to satisfy QoS constraints, which may not be feasible after a certain network size. Accordingly, K max can be obtained as in Lemma 5 by limiting the optimal power weight of the strongest node given in (30) Lemma 2 to unity, i.e.,ω h, 1 ≤ 1. Although the IoB node with the weakest channel is required to transmit with the highest power in the DL-NOBA, the overall network feasibility is mainly determined by the total power consumption constraint, i.e., K max i=1ω h, i ≤ 1, rather than the weakest channel node's individual feasibility. Accordingly, K max can be obtained as in Lemma 5 by limiting the sum of optimal power weight of the strongest node given in (33) Lemma 3 to unity.
Lemma 5: K max for the UL-NOBA scheme under perfect SIC case ( → 0) is given by where g max is the maximum channel gain in the network. On the other hand, K max for the DL-NOBA scheme under perfect SIC case ( → 0) is given by where ρ = ([N 0 B]/Pḡ) and all nodes are assumed to have a channel gain ofḡ = g min /2, which represent a hypothetical node located in the middle between the hub node and the source node with the weakest channel gain. Proof: Please see Appendix-C. The maximum network size analysis can be further extended to NOBA schemes under imperfect SIC conditions as follows.
Lemma 6: K max for the UL-NOBA scheme under imperfect SIC conditions ( > 0) is given by On the other hand, K max for the DL-NOBA scheme under perfect SIC case ( > 0) is given by where ϕ = ([N 0 Bγ ]/Pḡ) and all nodes are assumed to have a channel gain ofḡ = g min /2, which represent a hypothetical node located in the middle between the hub node and the source node with the weakest channel gain. Proof: Please see Appendix-D.

C. Maximum Network Size Analysis of Cooperative Schemes
The analyses presented clearly show that QoS demand and channel gains play a vital role in the maximum network size of regular OBA and NOBA schemes. In the case of cooperation, the maximum network size is determined by the minimum of network size of both phases, each of which heavily depends on channel gains to/from the relay node and phase time allocation, which determines the QoS demand to be met at each phase, i.e.,R/λ andR/(1 − λ). In light of the above discussions, the maximum network size of cooperative schemes can be obtained as shown in the following corollary.
Corollary 1: Denote g 1 max and g 1 min as the maximum and minimum channel gain between source nodes and relay node, respectively. Likewise, denote g h r as the channel gain between the relay and the hub nodes. Following from Lemma 4, K max for OBA schemes are given by: where the inner terms of min(·, ·) function is obtained from (34) and (35) for UL-OBA and DL-OBA schemes, respectively. The K max for NOBA schemes similarly follows from Lemma 6 as: where the inner terms of min(·, ·) function is obtained from (38) and (39) for UL-NOBA and DL-NOBA schemes, respectively. The optimal cooperative network size K max can be numerically obtained by substituting optimal phase time allocation, λ , into (40) and (41) for C-OBA and C-NOBA schemes, respectively. Proof: The corollary directly follows from Lemmas 4-6 by considering the bottleneck of two phases. The λ can obtained as shown in Algorithm 2, following Algorithm 1.

V. SIMULATION RESULTS
This section evaluates the performance of the proposed energy-efficient body channel access topologies and schemes in terms of transmit power and network size for different node deployment scenarios, QoS requirements, and SIC imperfections. The simulation parameters summarized in Table I will be utilized throughout this section unless explicitly stated otherwise. Moreover, throughout simulations we exploit the proposed frequency-dependent parametric path loss model in [4] to calculate PL h i the path loss between n i and n h . The linear channel gain between n i and n h is then obtained by

A. Impact of Node Deployment on Energy Efficiency
The impact of the source node and relay deployment on the transmit power is investigated in Figs. 5 and 6. In both simulations, the network comprises three BCC-enabled IoB nodes that exploit regular OBA, NOBA, C-OBA, and C-NOBA schemes. To further elucidate the difference between the two cases, both node deployment scenarios are illustrated in Fig. 4 for OBA schemes which also apply to NOBA schemes. Fig. 5(a) plots the sum of transmit power for UL (left y-axis) and DL (right y-axis) traffic in OBA and NOBA schemes with and without cooperation against the change in group distance. The group distance denotes the cumulative change in distance for source nodes n 2 and n 3 due to sweeping n 1 . This is implemented by setting the channel lengths l 1 2 and l 1 3 at 20 and 40 cm, respectively. Then, the channel length between n 1 and n h is increased from 20 up to 160 cm. As a result the grouped  source nodes with n 1 are also swept and the end-to-end channel lengths for n 2 and n 3 in regular schemes are obtained by l h 2 = l h 1 +l 1 2 and l h 3 = l h 1 +l 1 3 , respectively. For cooperative links, n 1 is selected to act as a relay since it is the closest node to the hub. This means that source nodes n 2 and n 3 will maintain the same distance in reference to the relay throughout the simulations. As can be observed, cooperative schemes improved energy efficiency, i.e., provided a reduction in transmission power compared to regular schemes. The reduction in transmit power ranges between 1% and 9% in both directions. Indeed, the improvement is more notable with pushing the nodes further away from the hub. This is because by increasing the channel length between source nodes and the hub, more power will be allocated in regular schemes to mitigate the channel conditions and communicate directly with the hub. Further, the performance of C-NOBA in both UL and DL directions, which is matched with C-OBA in UL when operating at low QoS, has the least power consumption over the entire channel length range. Whereas in DL transmission, OBA schemes perform worse than their nonorthogonal counterparts at all times. Because the maximum transmission power of n h in the first phase and n r in the second phase is equally split between source nodes. Fig. 5(b) demonstrates the change in optimal phase time allocation λ that is adjusted to minimize the end-to-end transmit power in UL and DL against change in group distance.
In UL traffic, when l h 1 < l r i the duration of the first phase is longer compared to the second phase, and conversely, when the relay is placed very far-away from n h , the second phase occurs over a longer time slot. While in DL, the more significant the difference between n h and n r is, the longer the duration of the first phase is. Fig. 6(a) demonstrates the effect of changing relay distance, n 1 , on the sum of transmit power for a network consisting of three IoB nodes. In this case, the channel lengths l h 2 and l h 3 are set to 120 and 160 cm, respectively. To investigate the effect of the relay, we increase the channel length from the first node to the hub node l h 1 up to 100 cm. Accordingly, the channel lengths in the first phase of cooperation will change depending on the relay location. Thus, in cooperative schemes, the channel lengths of the first phase are obtained by l r i = l h i − l h r . As shown in Fig. 6(a), the cooperative schemes improve the sum of transmit power by an 11% average reduction compared to their regular counterparts. Yet the most important observation is that when the relay is somewhat between the hub and source nodes, transmit power reaches a minimum in cooperative schemes. Which corresponds to achieving a 14% power reduction in cooperative schemes. However, pushing the relay closer to source nodes, the transmit power will increase with a further increase in the hub and relay separation. On the other hand, regular schemes maintain their performance over the entire distance vector since l h 2 and l h 3 are fixed throughout the simulation.
Likewise, Fig. 6(b) demonstrates the change in optimal phase time allocation λ * that is adjusted to minimize the End-To-End transmission power in UL and DL with respect to group localization. In UL traffic, when l h 1 < l r i time allocated for the first phase is longer compared to the second phase and conversely when the relay is placed very faraway from n h the second phase occurs over a longer time slot. The opposite is true in DL as a larger distance between n h and n r is translated into a longer phase. Both Figs. 5(b) and 6(b) clearly show how λ is optimized by Algorithm 1 to minimize overall energy consumption by manipulating λ to mitigate the adverse effects of node deployment.

B. Impact of QoS, K, and on Energy Efficiency
The impact of network size and QoS requirements on the transmit power consumption is investigated in Fig. 7(a) and (b) for orthogonal and nonorthogonal schemes, respectively. A network supporting up to nine IoB nodes with all nodes demanding the same QoS levels is simulated. We fix the channel length l h 1 at 60 cm and place additional nodes to the network at 5 × i cm from n 1 . At low QoS demands, particularly below 0.5 Mb/s, the effectiveness of cooperation is realized at a larger network size. For instance, at 100 kb/s, C-NOBA improved the total transmit power performance from 9% when K = 1% to 13.7% when K = 9 compared to NOBA. However, at high QoS, as we add more nodes to the network, the performance of cooperative schemes deteriorate, depending on λ nodes are required to satisfy ≥R, which suggests that higher powers will be allocated to meet the rate requirements.    Note that the orthogonal schemes are independent of , unlike their nonorthogonal counterparts. Accordingly, they sustain the same sum of transmit power at all times. First, we note that in regular schemes, NOBA has a better performance compared to OBA as long as the SIC error is kept below 0.03. Which is anticipated since power weights in NOBA are dependent on . Second, at QoS 500 kb/s and 1 Mb/s, C-NOBA is found to outperform NOBA in terms of the total transmit power by 8.6% and 12.6%, respectively. However, at = 0.02 the situation is reversed and NOBA schemes will be more effective. It is worth noting that, beyond = 0.03, it becomes infeasible for C-NOBA to operate, which is mainly since, in C-NOBA, SIC mitigation is performed in two phases. Hence, low SIC efficiency will increase the power weights until it approaches infeasibility as inhibits achieving the SINR thresholds.  The results are plotted for analytically derived K max expressions and the maximum feasible number of nodes determined by simulations. The figures exhibit that as a result of NOBA's throughput efficiency, it can support more nodes when compared to other schemes. It was determined that OBA and C-NOBA, on average, can support 55.6% and 31.7% less nodes, respectively, compared to the NOBA scheme. The reason for such performance in cooperation is that during the second phase, all nodes share the relay's power to achieve their SINR requirements, which constitutes a bottleneck on the overall network size. Similarly, in DL, both OBA and C-NOBA, on average, support 69% less nodes than the NOBA scheme. Fig. 10 plot K max obtained by CF and simulation for a network adopting NOBA schemes with respect to cancelation error for three different QoS requirements 0.5, 1, and 10 Mb/s. In generating this results l h 1 was fixed at 50 cm and the additional source nodes where uniformly distributed as l h K = l h 1 + 4 (K = 1). It is obvious that in both directions NOBA scheme can tolerate up to = 1e −5 to deliver the maximum network size at both 0.5 and 1 Mb/s. Which again demonstrates NOBA's SINR efficiency. One can note the plusminus one deviation between CF and simulation for DL K max . This is because we assumed that all nodes have the same channel gain to approximate the sum of weights and come up with the CF K max expression.

D. Impact of Network Traffic
We model the effect of traffic generated by sensor nodes on power consumption based on a duty-cycled transmission scheme. In our simulation in Fig. 11, a network containing five sensor nodes operating at 1 Mb/s is considered, where nodes have five operational scenarios. These operational modes include active nodes transmitting for 10%-30%, 30%-70%, 70%-100%, and 100% of the time. Indeed, the results agree with the intuition that nodes operating at higher duty cycles will contribute more to power consumption. Moreover, R-NOBA, as expected, has better performance than C-NOBA in almost all cases because, in our evaluation of the impact of  network size and QoS requirements, we found that higher QoS and larger network size limit the performance of cooperative schemes. However, when nodes are only active for 10%-30% of the time, C-NOBA performs better because, at each time slot, the probability of having all five nodes active simultaneously is low. Hence, it maintains a small network over the entire time duration. Fig. 12(a) validates the proposed CF solution by comparing calculated UL total transmit power with the CVX solution. On the other hand, Fig. 12(b) compares the running time of the CF and CVX solution for various networks sizes. It is obvious that the CF solution provides at least four orders of magnitude time reduction over the CVX solution. It is worth noting that the cooperative schemes require more calculation time to find optimal phase time allocations as explained in Algorithm 41. The proposed CF solution is practical and helpful especially considering the limited hardware, battery, and computational power available at IoB nodes.

VI. CONCLUSION
Toward accelerating the adoption of IoB in multiple sectors while embracing the technological advancements in BCC, we model and evaluate the performance of orthogonal and nonorthogonal schemes with and without cooperation to establish highly secure and efficient networks. Hence, P2P, MAC, and BCC topologies are presented along with optimal closedform power control techniques to permit multipoint communication and meet the different QoS requirements. Further, line search algorithms are presented for joint optimal power and phase time allocations to facilitate cooperative communications. Finally, to address the full capacity in each scheme, the maximum number of supportable nodes was analytically obtained. Namely, closed-form K max expressions were extended to consider SIC imperfections and cooperative schemes. Thoroughly performed simulations identified the network settings under which cooperation is more beneficial than regular schemes in terms of the sum of transmit power. Conversely, regular NOBA schemes illustrated better capabilities to improve the network size.

A. Derivation of Optimal Power Weights
It is worth noting that interference matrix J in (25) can be assumed to be irreducible since it has nonnegative elements and = 0 is unattainable in practice due to the SIC imperfections [31], [32]. Following from the Perron-Frobenius theorem [33], the maximum modulus eigenvalue of J is real and positive, while the corresponding eigenvector is positive componentwise. Thereby, a feasible solution to (25) exists if and only if the magnitude of the maximum eigenvalue of H ¯ J is less than unity, i.e., ρ H < 1 [31], [32].
Accordingly, optimal power weights ω can be obtained from p = (I − J) −1¯ σ = P ωg, where g is the vector of channel gains sorted in descending order. By using the eigenvalue equation Hν = λ H ν, the optimal power allocations can be obtained as described in [27].

B. K max for OBA
For UL-OBA scheme, K max can be obtained by constraining optimal power weight in (27) as shown in Lemma 1, i.e., ([γ i N 0 B]/ḡKP) ≤ 1 whereγ = 2 ([RK]/B) − 1. For the sake of analytical tractability, assumingγ 1 yields γ ≈ 2 ([RK]/B) and allows us to obtain K max by leveraging Lambert-W function as shown in (34). For the DL-OBA, above approximation is not necessary and K max can be directly derived from ([γ N 0 B]/ḡP) ≤ 1 as shown in (35).

C. K max for NOBA Under Perfect SIC Conditions
Following from (30) and discussions before Lemma 5, K max for UL perfect NOBA case can be obtained by solving ([N 0 B]/[Pg max ])γ (1 +γ ) K max −1 ≤ 1 for K max . To obtain K max in the DL perfect NOBA case, we first obtain the sum of power weights as follows: By setting m = 1 and n = K in (43), (42) can be rewritten as follows: By substituting (1/[(γ + 1)]) into (44), we can rewrite the total power constraint as follows: Solving (45) for K yields the K max as follows: