Energy consumption reduction in OFDMA wireless sensor networks in existence of interference hazards

Abstract

Energy consumption reduces Wireless Sensor Network’s (WSN’s) lifetime. Hence, this paper addresses energy saving problematic in cooperative Orthogonal Frequency Division Multiplexing Access (OFDMA) systems for WSNs (COFDMA-WSN). Analytical handlings are implemented. The performance improvement of COFDMA-WSNs is executed by considering two different COFDMA-WSNs schemes. These schemes represent classical COFDMA-WSNs and Relay Supported (R-S) COFDMA-WSNs. Additionally, there are different pondered configurations due to these schemes. These configurations denote classical OFDMA network with classical WSN, R-S OFDMA network with classical WSN, classical OFDMA network with R-S WSN, and R-S OFDMA network with R-S WSN. Moreover, each configuration is applied with four different fractional frequency reuse (FrFR) techniques. These techniques represent strict (St) FrFR technique with Frequency Reuse Factor (FReF) = 3, St FrFR technique with FReF = 4, sectored (Sc) FrFR technique and soft frequency reuse (SoFR) technique. Consequently, there are sixteen different patterns of COFDMA-WSNs are considered. Moreover, closed-form terms (CFTs) for cluster-head’s (C-H’s) signal to interference ratio (SIR) and sensor node’s (SN’s) SIR are presented. Additionally, different metrics are evaluated to contrast the performance of altered patterns using the obtained CFTs. The outcomes demonstrate, that St FrRF4 system outperforms other systems in prime and ensuing links. The cause of this outcome can be credited, to the frequency reuse process decrement due to FReF increment. As a result, the interference sources decrease. Hence, the interfering signals drop. Consequently, St FrRF4 system achieves the highest values of SIR. Moreover, St FrRF3 technique has the second usage priority in the prime link. But, it losses this preference in the ensuing link and SoFR technique that applied in the fourth configuration takes this significance. The work outcomes attain much higher C-H's and SN’s SIR improvements. Accordingly, the packet transmission and protocol behaviour are enhanced. So, the energy consumption is reduced. Consequently, WSN’s lifetime is maximized.

1 Introduction

Wireless Sensor Networks (WSNs) consist of large number of tiny devices called sensor nodes (SNs) [1]. These SNs can be employed in WSN surroundings through three ways. These ways represent regular, mobile SNs, and random deployment [2]. These connections variety helped WSNs to speedily gaining attractiveness in many critical areas [3]. Unfortunately, energy consumption issue represents a vast challenge in WSN. This is due to the impossibility of replacing or recharging the SNs' batteries in most hostile environments [4]. So, a method to reduce the energy consumption and increase the overall WSN life [4] is investigated. Furthermore, an approach to harvest and store energy in a machine to machine network through radio frequency signals adopting the wireless powered communication paradigm is proposed [5]. Additionally, a proposed way to reduce power consumption in wireless sensors is studied [6]. Although, these approaches reduce energy consumption problem to some extent it can’t completely eliminate this problem under interference condition. The hazard of interference is that it causes packet loss and so additional energy are consumed for the retransmission process [8]. Thus, some studies tended to use the cooperative Orthogonal Frequency Division Multiplexing Access (OFDMA) networks for WSNs (COFDMA-WSN) to reduce the interference impact [7]. In these networks, OFDMA network represents the backhaul of the WSN. Consequently, the secure resource distribution problematic is addressed for an OFDMA two-way relay WSN [10]. Additionally, joint neighbour discovery and coarse time-of-arrival estimation in WSNs via OFDMA [8] is introduced. Regrettably, it is noticed, that co-channel interference (CoCI) is originated in COFDMA-WSN due to frequency reuse technique deployment [9]. So, reducing CoCI is a necessity in order to improve the system performance. Thus, transmission performance enhancement is addressed by adjusting the transmission rate and the payload size in response to the change of CoCI [10]. Additionally, fractional frequency reuse (FrFR) scheme is assumed [11] to reduce CoCI and achieve high energy conservation. However, these papers have not considered the energy consumption reduction through incorporating FrFR technique and relaying technique together with the hierarchical protocol deployment. Therefore, based on this complex combination, there are sixteen patterns are studied in this paper.

This paper aims to offer an effective method for improving WSN performance through lessening the energy consumption. So, theoretical performance analysis of COFDMA-WSN is conducted via considering two different COFDMA-WSN schemes. These schemes represent classical COFDMA-WSN and relay supported (R-S) COFDMA-WSN. Furthermore, different configurations are considered due to these schemes. These configurations denote classical OFDMA network with classical WSN, R-S OFDMA network with classical WSN, classical OFDMA network with R-S WSN, and R-S OFDMA network with R-S WSN. Moreover, each configuration applies four different FrFR techniques. These techniques represent strict (St) FrFR technique with Frequency Reuse Factor (FReF) = 3 (it will be referred to by the acronym St FrFR3), St FrFR technique with FReF = 4 (it will be referred to by the acronym St FrFR4), sectored (Sc) FrFR technique and soft frequency reuse (SoFR) technique. Consequently, there are sixteen different patterns of COFDMA-WSN are addressed. Additionally, closed-form terms (CFTs) for cluster-head’s (C-H’s) signal to interference ratio (SIR) and SN’s SIR are presented. Furthermore, several performance metrics are valued to contrast all patterns using the gotten CFTs. The work outcomes contribute to enhancing the packet transmission, link stability and protocol behaviour. So, the WSNs’ performance is greatly improved. Accordingly, the WSN life time is maximized.

This paper is systematized as follows: Sect. 2 describes the basic assumptions and model description. Section 3 attentions on the metrics for performance estimation of different patterns. Section 4 offerings the outcomes and discussion. The work conclusion is presented in Sect. 5.

2 Basic assumptions and model description

This section presents the analysis of dissimilar configurations of COFDMA-WSN system. In this system, hierarchical protocol is considered due to its efficiency in reducing the required transmitted energy of data packets. Therefore, the communication is accomplished in two phases as revealed in Fig. 1. In first phase, BS transmits the signal to C-H through prime link.

Fig. 1

Concept illustration of COFDMA-WSN system

In second phase, C-H communicates with SNs through ensuing link. Moreover, the main parameters of experimentation are shown in Table 1. Also, the reference values of these parameters are taken from [11, 22] and [23]. Furthermore, the considered configurations denote classical OFDMA network with classical WSN, R-S OFDMA network with classical WSN, classical OFDMA network with R-S WSN, and R-S OFDMA network with R-S WSN. Each configuration is deployed with four different FrFR techniques to advance the performance of overall system through dropping the interference impact. These techniques represent strict St FrFR3 technique, St FrFR4 technique, Sc FrFR technique and SoFR technique as shown in Fig. 2.

Fig. 2

Idea diagram of COFDMA-WSN patterns for energy consumption reduction

Table 1 Simulation parameters of the proposed COFDMA-WSN configurations

Additionally, the analytical handling of the pondered configurations yielded CFTs for C-H’s SIR and SN’s SIR as presented in the following subsections. These CFTs are then evaluated to contrast the behaviour of dissimilar configurations by assess several metrics. For simplicity, definition of all notations are shown in Table 2.

Table 2 Notation definition

2.1 Analysis of Classical COFDMA-WSN

The first configuration considers a classical WSN implemented with a classical OFDMA network. Thus, OFDMA network based WSN consisting of two-tier FrFR system is constructed. In this system, there is a centralized BS and a set of N pre-assigned SNs denoted by C-H, SN₁, SN₂, and SN₃ in each cell of radius R_a (m). Each cell is distributed into inner and outer sections due to FrFR deployment. So, the deployed SNs are arranged in a regular arrangement in the outer area. Additionally, all SNs aim to exchange information over an OFDMA channel using clustering concept. Therefore, a C-H is implemented in each cluster to distribute synchronization signals within its cluster, keep the radio resources, and act as a gateway between the cluster and neighbouring clusters. Consequently, the C-H node takes the responsibility of managing the cluster as the cellular BS. In addition, it is assumed, that C-H is placed in front of the cellular BS at a distance equal to R_C-H (m) (R_C-H = 1/2 of R_a). This configuration contains different four patterns due to FrFR technique deployment as revealed in Fig. 3. So, the next subsections provide the analytical treatment for the considered patterns of this configuration.

Fig. 3

Concept illustration of classical COFDMA-WSN patterns for energy consumption reduction

2.1.1 Analysis of St FrFR system

Two- tier St FrFR based WSN structure is pondered as revealed in Fig. 4. This system consists of 19 cells. Every cell is separated in to outer and inner areas. Inner area deploys frequency band (BF) (BF₀). Furthermore, the outer region uses BFs (BF_1, BF₂ and BF₃) when FReF = 3 is applied as shown in Fig. 4(a). Moreover, these BFs is reformed to BF_1, BF_2, BF₃ and BF₄ when FReF = 4 is used as shown in Fig. 4(b). Additionally, in order to reduce the interference between the SNs themselves and between the SNs and BS, each outer BF in the outer area is distributed into five sub-band frequencies (S_Fs) as revealed in Fig. 4(c, d). Moreover, it is presumed, that SN of interest is positioned at the outermost location with coordinates (0, 1) as demonstrated in Fig. 4(e). Furthermore, signal to interference plus noise ratio (SINR) is used to governor CoCI between co-channel BSs and co-channel C-Hs that use the same frequency in the COFDMA-WSN. Consequently, in order to compute the SINR computation of C-H (X,Y) situated at a distance ${\text{s = }}\sqrt {{\text{X}}^{{2}} {\text{ + Y}}^{{2}} }$ from the aiding BS (BS₀), the following equality is used [12]

$$SINR(X,Y) = \frac{{J_{0a} \, E_{0\hbar \, } H_{0a\hbar } }}{{\zeta_{\hbar }^{2} + \sum\limits_{{\text{i = 1}}}^{{\text{g}}} {\left( {{\text{J}}_{{{\text{ia}}}} {\text{ E}}_{{{\text{i}}\hbar }} {\text{ H}}_{{{\text{ia}}\hbar }} } \right)} }}$$

(1)

where J_0a signifies the channel Propagation Path Gain (PPG) (dB) associated between SN a and BS₀. Furthermore, this parameter terms the inverse of Path-Loss (PLo) (dB) between BS₀ and SN a (J_0a = 1/PLo_0a), ζ_ħ ² indicates the Additive White Gaussian Noise (AWGN) channel noise power (watt), J_ia represents channel PPG between SN a and interfering BSs (dB), E_0ħ indicates BS₀’s transmit power on subcarrier ħ (watt), E_iħ designates interfering BSs’ transmit power (watt), H_0aħ represents channel Fast Fading (FaF) power (watt) between SN a and BS₀, and H_iaħ denotes the channel FaF power (watt) between SN a and interfering BSs. In addition, the factor J_0a is directly related to s^−η i.e. (J_0u α s^−η) where η characterizes the PLo Exponent (PLoE) and s denotes the space (m) between the BS and SN a. Consequently, this relative can be expressed as J_0a = J₀ s^−η where J₀ denotes the relational constant that is expressed as J₀ = (c/4πf_o)² where f_o and c indicate the employed BS’s centre frequency (Hz) and light speed (m/s), respectively. Additionally, the parameter J_ia is defined as J_ia = J_i s_i^−η where s_i represents the separation space (m) between the interfering BSs and SN a and J_i denotes the second relational constant. This relational constant is expressed by J_i = (c/4πf_i)² where f_i is the interfering BSs’ centre frequency (Hz). Moreover, symbols i and g represent the group of all interfering BSs (i.e. BSs that employing a similar BF as BS₀). Therefore, g signifies the number of co-channel cells and i indicates the co-channel cell index. Besides, the sources of interference are wholly surrounding BSs that deploy identical BF as BS₀. Thus, the sources of interference that affecting C-H are wholly surrounding BSs that employ an identical BF as BS₀. Accordingly, C-H suffers from 6 BSs with numbers (8, 10, 12, 14, 16 and 18) which usage same S_F (S_F1-1) as BS₀ in the outer section when St FrFR3 technique is employed. Furthermore, when St FrFR4 technique is deployed, the C-H suffers from 6 BSs with numbers (7, 9, 11, 13, 15 and 17). Furthermore, it is presumed, that fading powers of the channel are distinct with mean equal one, i.e. (H_0aħ = 1), and the key influence arises from interference rather than noise in COFDMA-WSNs. So, noise impact is ignored in this analysis. Accordingly, Eq. (1) is rewritten to calculate C-H's SIR [13] as follows

$$SIR\left( {X,Y} \right) = \frac{{E_{0} J_{0} s^{ - \eta } }}{{\sum\limits_{j = 1}^{j = 6} {E_{j} J_{j} s_{j}^{ - \eta } } }}$$

(2)

where E₀ denotes BS₀’s transmitted power (watt), j refers to interfering BSs set due to redeploying same S_F (S_F1-1) in the outer area, s_j is the distance (m) between 6 interfering BSs which employ S_F1-1 and C-H, E_j denotes interfering BSs’ transmitted power (watt) that disturb the C-H, J_j represents outer relational constant. This constant can be stated as J_j = (c/4πf_j)² where f_j is the interfering BSs’ centre frequency (Hz). In addition, it is postulate that the SNs’ interference is negligible and all BSs transmit with identical power. Accordingly, the common SIR calculation for C-H can be formulated as follows

$$SIR_{C - H} \left( {X,Y} \right) = \frac{{J_{o} s^{ - \eta } }}{{\sum\limits_{j = 1}^{j = 6} {J_{j} s_{j}^{ - \eta } } }}$$

(3)

where SIR_C-H is the C-H's SIR. Furthermore, values of J_o and J_j are equal because of using same S_F1-1 in the outer area by all BSs. Hence, Eq. (3) can be modified as follows

$$SIR_{C - H} \left( {X,Y} \right) = \frac{{s^{ - \eta } }}{{\sum\limits_{j = 1}^{j = 6} {s_{j}^{ - \eta } } }}$$

(4)

Fig. 4

Two-tier St FrFR network a St FrFR3 pattern b St FrFR4 pattern c Frequency allocation for St FrFR3 d frequency allocation for St FrFR4 (e) Cell layout

Moreover, it is supposed that (x, y) signifies the normalized coordinates to R_a i.e. (x, y) = (X/ R_a, Y/ R_a). Therefore, normalized separation space is equivalent to $R_{a} s = \sqrt {X^{2} + Y^{2} }$. Consequently, Eq. (4) can be modified to denote the C-H’s SIR term in normalized coordinates as follows

$$SIR_{C - H} \left( {x,y} \right) = \frac{{\left( {x^{2} + y^{2} } \right)^{\sigma } }}{{T_{FRF} \left( {x,y} \right)}}$$

(5)

where the factor T_FRF represents the interference parameter because of using FrFR technique, and σ designates the negative half of the PLoE (-η/2). Accordingly, this factor is calculated from the following relative when St FrFR3 technique is employed

$$T_{3} \left( {x,y} \right) = \sum\limits_{{g_{3} = 1}}^{{g_{3} = 6}} {\left[ {\left( {x + x_{{g_{3} }} } \right)^{2} + \left( {y + y_{{g_{3} }} } \right)^{2} } \right]^{\sigma } }$$

(6)

where g₃ represents the interfering BSs group in the outer region that use S_F1-1. Additionally, Table 3 clarifies the outer area allotted BSs’ BFs. Furthermore, the BSs’ normalized coordinates are demonstrated in the same table. Hence, the parameter T₃ can be expressed as follows

$$\begin{aligned}{{\rm{T}}_3}\left( {{\rm{x,y}}} \right) = & {\left[ {{{\left( {{\rm{x}} \pm \frac{3}{2}\sqrt 3 } \right)}^2} + {{\left( {{\rm{y }} - \frac{3}{2}} \right)}^2}} \right]^\sigma } + {\left[ {{{\rm{x}}^2} + {{\left( {{\rm{y}} \pm 3} \right)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{{\left( {{\rm{x}} \pm \frac{3}{2}\sqrt 3 } \right)}^2} + {{\left( {{\rm{y + }}\frac{3}{2}} \right)}^2}} \right]^\sigma }\end{aligned}$$

(7)

Table 3 Coordinates outer region assigned frequencies of BSs in different FrFR systems

Therefore, by replacing the worth of Eq. (7) into Eq. (5), C-H's SIR when St FrFR3 technique is used can be computed as follows

$$SIR_{C - H3} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {\left( \frac{25}{4} \right)^{\sigma } + 2\left( \frac{31}{4} \right)^{\sigma } + 2\left( \frac{43}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(8)

where SIR_C-H3 denotes the attained C-H's SIR using St FrFR3 technique in the first configuration. Furthermore, the parameter T_FRF is calculated for St FrFR4 as follows

$$T_{4} \left( {x,y} \right) = \sum\limits_{{g_{4} = 1}}^{{g_{4} = 6}} {\left[ {\left( {x + x_{{g_{4} }} } \right)^{2} + \left( {y + y_{{g_{4} }} } \right)^{2} } \right]^{\sigma } }$$

(9)

where g₄ represents the interfering BSs set that use S_F1-1. Hence, this parameter can be expressed as follows

$$\begin{aligned}{T_4}\left( {x,y} \right) = & {\left[ {{{\left( {x + \sqrt 3 } \right)}^2} + {{\left( {y \pm 3} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x - \sqrt 3 } \right)}^2} + {{\left( {y \pm 3} \right)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{{\left( {x \pm 2\sqrt 3 } \right)}^2} + {y^2}} \right]^\sigma }\end{aligned}$$

(10)

Consequently, by replacing the worth of Eq. (10) into Eq. (5), C-H's SIR when St FrFR4 is used can be calculated as follows

$$SIR_{C - H4} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } 2\left( {\left( \frac{37}{4} \right)^{\sigma } + \left( \frac{61}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(11)

where SIR_C-H4 indicates the achieved C-H's SIR when St FrFR4 is applied in the first configuration. Moreover, it is supposed, that C-H transmits to SN that placed at the second corner with x = 0, y = 1. Accordingly, Eq. (5) is modified to definite the SN's SIR in normalized coordinates as follows

$$SIR_{SN} \left( {x,y} \right) = \frac{{\left( {x^{2} + (y - 0.5)^{2} } \right)^{\sigma } }}{{T_{SN} \left( {x,y} \right)}}$$

(12)

where T_SN represents the interference parameter because of using FrFR technique. Accordingly, this parameter is calculated, when St FrFR3 technique is used, as follows

$$T_{SN3} \left( {x,y} \right) = \sum\limits_{{t_{3} = 1}}^{{t_{3} = 6}} {\left[ {\left( {x + x_{{t_{3} }} } \right)^{2} + \left( {y + y_{{t_{3} }} } \right)^{2} } \right]^{\sigma } }$$

(13)

where t₃ symbolizes the interfering C-Hs that redeploy S_F1-2.

Furthermore, Table 4 shows the six interfering C-Hs coordinates. So, T_SN3 parameter is assessed as follows

$$\begin{aligned}{{\rm{T}}_{{\rm{SN3}}}}\left( {{\rm{x,y}}} \right) = & {\left[ {{{\left( {{\rm{x}} \pm \frac{3}{2}\sqrt 3 } \right)}^2} + {{\left( {{\rm{y + 1}}} \right)}^2}} \right]^\sigma } + {\left[ {{{\rm{x}}^2} + {{\left( {{\rm{y}} - 3.5} \right)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{{\left( {{\rm{x}} \pm \frac{3}{2}\sqrt 3 } \right)}^2} + {{\left( {{\rm{y}} - 2} \right)}^2}} \right]^\sigma } + {\left[ {{{\rm{x}}^2} + {{\left( {{\rm{y + 2}}.5} \right)}^2}} \right]^\sigma }\end{aligned}$$

(14)

Table 4 Interfering C-Hs' Coordinates for St FrFR technique

Therefore, by exchanging the worth of Eq. (14) into Eq. (5), SN's SIR for St FrFR3 technique can be evaluated as follows

$$SIR_{SN3} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {\left( \frac{25}{4} \right)^{\sigma } + 2\left( \frac{31}{4} \right)^{\sigma } + 2\left( \frac{43}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(15)

where SIR_SN3 denotes the attained SN's SIR when St FrFR3 technique is used in the first configuration. Moreover, the factor T_SN is valued when St FrFR4 technique is used as follows

$$T_{SN4} \left( {x,y} \right) = \sum\limits_{{t_{4} = 1}}^{{t_{4} = 6}} {\left[ {\left( {x + x_{{t_{4} }} } \right)^{2} + \left( {y + y_{{t_{4} }} } \right)^{2} } \right]^{\sigma } }$$

(16)

where t₄ denotes the interfering C-Hs set in the outer area that employ S_F1-2. Hence, this factor is represented as follows

$$\begin{aligned}{T_4}\left( {x,y} \right) = & {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{(y - 3.5)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm 2\sqrt 3 } \right)}^2} + {{(y - 0.5)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y + 2.5} \right)}^2}} \right]^\sigma }\end{aligned}$$

(17)

Consequently, by replacing the worth of Eq. (17) into Eq. (5), SN's SIR when St FrFR4 technique is used, can be estimated as follows

$$SIR_{SN4} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {2\left( \frac{37}{4} \right)^{\sigma } + \left( \frac{13}{4} \right)^{\sigma } + \left( {\frac{ - 47}{4}} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } + \left( \frac{61}{4} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(18)

where SIR_SN4 indicates SN's SIR when St FrFR4 technique is utilized in the first configuration.

2.1.2 Analysis of Sc FrFR system

Two tier Sc FrFR based WSN arrangement is considered as revealed in Fig. 5(a). Here, the outer area is separated into three parts. Consequently, the whole Bandwidth (BW) is distributed equally between each part in the outer and the inner areas. Thus, inner region employs BF₀ to cover cell centre users (CCUs). Additionally, outer region uses BFs (BF_1, BF₂ and BF₃) to serve cell edge users (CEUs) as shown in Fig. 5(b). Each outer region BF is more distributed into two dissimilar sets of S_Fs. The first set denotes S_F1-1, S_F2-1, and S_F3-1, which are assigned to BS. Additionally, the second group represents S_F1-2, S_F2-2, and S_F3-2, which are allocated to C-Hs as depicted in Fig. 5(c). In this system, the sector that deploys S_F1-1 covers the C-H. Consequently, the C-H hurts from 7 interfering cells (coloured cells) which employ similar BF as demonstrated in Fig. 5(a). So, Eq. (5) can be modernized to definite the C-H’s SIR formula as follows

$$SIR_{Sec} \left( {x,y} \right) = \frac{{\left( {x^{2} + y^{2} } \right)^{\sigma } }}{{T_{Sec} \left( {x,y} \right)}}$$

(19)

where T_Sec is the seven interfering BSs’ PLo distances summation. Accordingly, the factor T_Sec can be computed as follows

$$T_{Sec} \left( {x,y} \right) = \sum\limits_{{g_{S} = 1}}^{{g_{S} = 7}} {\left[ {\left( {x + x_{{g_{S} }} } \right)^{2} + \left( {y + y_{{g_{S} }} } \right)^{2} } \right]^{\sigma } }$$

(20)

where g_S indicates set of outer region interfering BSs that use S_F1-1. Hence, the interference parameter in Sc FrFR system is calculated using Table 3 as follows

$$\begin{aligned} & {T_{Sec}}\left( {x,y} \right) = {\left[ {{{\left( {x \pm \frac{{\sqrt 3 }}{2}} \right)}^2} + {{\left( {y + \frac{3}{2}} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \frac{{3\sqrt 3 }}{2}} \right)}^2} + {{\left( {y + \frac{3}{2}} \right)}^2}} \right]^\sigma }\\ & \quad {\rm{ + }}{\left[ {{x^2} + {{\left( {y + 3} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y + 3} \right)}^2}} \right]^\sigma }\end{aligned}$$

(21)

Fig. 5

Sectored FrFR network a System design, b Cell arrangement, c Frequency allocation partition

Consequently, by replacing the worth of Eq. (21) into Eq. (19), C-H's SIR can be formulated as follows

$$SIR_{Sec} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {2\left( \frac{19}{4} \right)^{\sigma } + 2\left( \frac{43}{4} \right)^{\sigma } + 2\left( \frac{61}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(22)

where SIR_Sec denotes the attained C-H's SIR when Sc FrFR technique is employed in the first configuration. On the other hand, when C-H transmits to SN through ensuing link, Eq. (19) can be adjusted to represent SN’s SIR as follows

$$SIR_{SCSN} \left( {x,y} \right) = \frac{{\left( {x^{2} + (y - 0.5)^{2} } \right)^{\sigma } }}{{T_{SCSN} \left( {x,y} \right)}}$$

(23)

where T_SCSN represents the interference factor. Accordingly, this parameter is assessed as follows

$$T_{SNSC} \left( {x,y} \right) = \sum\limits_{{t_{SC} = 1}}^{{t_{SC} = 7}} {\left[ {\left( {x + x_{{t_{SC} }} } \right)^{2} + \left( {y + y_{{t_{SC} }} } \right)^{2} } \right]^{\sigma } }$$

(24)

where t_SC signifies the interfering set of C-Hs. Moreover, Table 5 shows the seven interfering C-Hs coordinates. Hence, this interference factor is valued as follows

$$\begin{aligned} & {\text{T}}_{{{\text{SNSC}}}} \left( {\text{x,y}} \right) = \left[ {\left( {{\text{x}} \pm \frac{3}{{2}}\sqrt {3} } \right)^{{2}} { + }\left( {\text{y + 1}} \right)^{{2}} } \right]^{{\upsigma }} { + }\left[ {\left( {{\text{x}} \pm \frac{{\sqrt {3} }}{{2}}} \right)^{{2}} { + }\left( {\text{y + 1}} \right)^{{2}} } \right]^{{\upsigma }} \\ & \quad { + }\left[ {{\text{x}}^{{2}} { + }\left( {{\text{y + 2}}{.5}} \right)^{{2}} } \right]^{{\upsigma }} { + }\left[ {\left( {{\text{x}} \pm \sqrt {3} } \right)^{{2}} { + }\left( {{\text{y + 2}}{.5}} \right)^{{2}} } \right]^{{\upsigma }} \\ \end{aligned}$$

(25)

Table 5 Interfering C-Hs' Coordinates for Sc FrFR and SoFR techniques

Accordingly, by exchanging the worth of Eq. (25) into Eq. (23), SN's SIR equation can be evaluated as follows

$$SIR_{SNSC} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {2\left( \frac{19}{4} \right)^{\sigma } + 2\left( \frac{43}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } + 2\left( \frac{61}{4} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(26)

where SIR_SNSC denotes the attained SN's SIR when Sc FrFR technique is deployed in the first configuration.

2.1.3 Analysis of SoFR technique

Although Sc FrFR improves the C-H's SIR, the network BW usage is still not used with maximum efficiency. Consequently, Two Level Power Control (T_PC) strategy with reuse 1 structure is proposed to be the foundation stone of the SoFR technique. This technique divides the entire BW in into three equivalent portions [14]. One portion allotted to the outer area and two parts allocated to inner region as displayed in Fig. 6(a). Furthermore, it is presumed, that each subcarrier has output power P_o in the outer area. Furthermore, each subcarrier has output power P_in in the inner region. These powers can be expressed by P_o = ε P_in where ε signifies power control factor (ε ≥ 1). Each cell allocates the three BW parts (BF₁, BF₂ and BF₃) in different way considering a pseudo-reuse 3 arrangement between outer areas. Consequently, every cell uses the total network BW as depicted in Fig. 6(b). Furthermore, each outer region BF is more portioned into two dissimilar parts of S_Fs. One part denotes S_F1-1 which is assigned to BS. Besides, the other portion symbolizes S_F1-2 which is allotted to C-H as portrayed in Fig. 6(c). In this system, there are 18 interfering cells affecting C-H (6 BSs in outer area and 12 BSs in inner area) which deploys BF₁ as depicted in Fig. 6(a). The C-H's SIR expression of the SoFR network is formulated due to T_LPC utilization [15] as follows

$$SIR_{SFR} = \frac{{E_{0} J_{0u} }}{{\left[ {E_{in} \sum\limits_{k} {J_{ku} } } \right] + \left[ {E_{0} \sum\limits_{g} {J_{ju} } } \right]}}$$

(27)

where J_ku denotes channel PPG between C-H and its interfering BS_k in the inner area and J_ju represents channel PPG between C-H and its interfering BS_j in the outer area. Also, the parameter J_ku is defined as J_ku = J_k s_k^−η. Additionally, the factor J_ju can be represented by J_ju = J_g s_g^−η. Consequently, Eq. (27) is updated as follows

$$SIR_{SFR} = \frac{{\varepsilon J_{0} s^{ - \eta } }}{{\sum\limits_{k} {J_{k} s_{k}^{ - \eta } } + \varepsilon \sum\limits_{j} {J_{g} s_{g}^{ - \eta } } }}$$

(28)

Fig. 6

SoFR system a System design, b Cell arrangement, c Frequency allocation partition

In addition, the parameters J_k, J_g, and J_o use the same centre frequency value during their calculations. Accordingly, they have equivalent rates. Thus, Eq. (28) can be rewritten as follows

$$SIR_{SFR} = \frac{{\varepsilon s^{ - \eta } }}{{\sum\limits_{k} {s_{k}^{ - \eta } } + \varepsilon \sum\limits_{g} {s_{g}^{ - \eta } } }}$$

(29)

Accordingly, Eq. (29) is updated to definite the C-H's SIR in normalized coordinates as follows

$$SIR_{SFR} \left( {x,y} \right) = \frac{{\varepsilon \left( {x^{2} + y^{2} } \right)^{\sigma } }}{{T_{iF} \left( {x,y} \right) + \varepsilon T_{oF} \left( {x,y} \right)}}$$

(30)

where T_oF and T_iF indicate the outer and inner interference factors in OFDMA network, respectively. The CoCI sources in the outer area resulted from BSs (8, 10, 12, 14, 16 and 18). Additionally, CoCI sources in the inner area originated from BSs (1, 4, 5, 2, 6, 13, 3, 7, 15, 11, 17 and 9). Consequently, the parameter T_iF using Table 3 is expressed as follows

$${T_{iF}}\left( {x,y} \right) = \left[ \begin{array}{l} {\left[ {{{\left( {x \pm 2\sqrt 3 } \right)}^2} + {y^2}} \right]^\sigma } + {\left[ {{{\left( {x + \frac{{\sqrt 3 }}{2}} \right)}^2} + {{\left( {y \pm \frac{3}{2}} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y \mp 3} \right)}^2}} \right]^\sigma }\\ + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {y^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y \pm 3} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x - \frac{{\sqrt 3 }}{2}} \right)}^2} + {{\left( {y \pm \frac{3}{2}} \right)}^2}} \right]^\sigma } \end{array} \right]$$

(31)

Accordingly, final expression of T_iF for a C-H sited at x = 0 and y = 0.5 is expressed as follows

$$T_{iF} = 2\left( \frac{7}{4} \right)^{\sigma } + 2\left( \frac{13}{4} \right)^{\sigma } + 2\left( \frac{19}{4} \right)^{\sigma } + 2\left( \frac{37}{4} \right)^{\sigma } + 2\left( \frac{49}{4} \right)^{\sigma } + 2\left( \frac{61}{4} \right)^{\sigma }$$

(32)

Moreover, the parameter T_oF can be computed as follows

$$\begin{aligned} T_{oF} \left( {x,y} \right) = & \left[ {\left( {x \pm \frac{3}{2}\sqrt 3 } \right)^{2} + \left( {y - 1.5} \right)^{2} } \right]^{\sigma } + \left[ {x^{2} + \left( {y \pm 3} \right)^{2} } \right]^{\sigma } \\ & \quad + \left[ {\left( {x \pm \frac{3\sqrt 3 }{2}} \right)^{2} + \left( {y + 1.5} \right)^{2} } \right]^{\sigma } \\ \end{aligned}$$

(33)

Accordingly, T_oF final form for the same C-H can be evaluated as follows

$$T_{oF} = \left( \frac{25}{4} \right)^{\sigma } + 2\left( \frac{31}{4} \right)^{\sigma } + 2\left( \frac{43}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma }$$

(34)

Subsequently, by replacing the values of Eqs. (32) and (34) into Eq. (30), C-H's SIR can be characterized as follows

$$SI{R_{SFR}} = {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( {2\left( {{{\left( {\frac{7}{4}} \right)}^\sigma } + {{\left( {\frac{{13}}{4}} \right)}^\sigma } + {{\left( {\frac{{19}}{4}} \right)}^\sigma } + {{\left( {\frac{{31}}{4}} \right)}^\sigma } + {{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{43}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma }} \right) + 3{{\left( {\frac{{49}}{4}} \right)}^\sigma } + {{\left( {\frac{{25}}{4}} \right)}^\sigma }} \right)} \right]^{ - 1}}$$

(35)

where SIR_SFR characterizes C-H's SIR of SoFR technique when first configuration is used. The above equation represents C-H's SIR CFT for ε = 1. The other CFTs for dissimilar ε values are executed, but not involved in this section. In the ensuing link, when C-H communicates with the SN located at coordinates (0, 1), Eq. (30) can be modified to characterize the cell edge SN’s SIR term as follows

$$SIR_{SNSFR} \left( {x,y} \right) = \frac{{\varepsilon \left( {x^{2} + (y - 0.5)^{2} } \right)^{\sigma } }}{{T_{iSN} \left( {x,y} \right) + \varepsilon T_{oSN} \left( {x,y} \right)}}$$

(36)

where T_oSN and T_iSN indicate the outer and inner interference factors in WSN, respectively. The CoCI sources in the outer area resulted from BSs (8, 10, 12, 14, 16 and 18). Additionally, CoCI sources in the inner area originated from BSs (1, 4, 5, 2, 6, 13, 3, 7, 15, 11, 17 and 9). Thus, the final expression of T_iSN using Table 5 for the SN is expressed as follows

$$I_{iF} = \frac{2}{5}\left( 4 \right)^{\sigma } + \frac{4}{5}\left( 7 \right)^{\sigma } + \frac{2}{5}\left( {13} \right)^{\sigma } + \frac{2}{5}\left( {19} \right)^{\sigma } + \frac{2}{5}$$

(37)

Moreover, the final expression of T_oSN can be formulated as follows

$$I_{oF} = \left( \frac{25}{4} \right)^{\sigma } + 2\left( \frac{31}{4} \right)^{\sigma } + 2\left( \frac{43}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma }$$

(38)

Hence, by replacing the values of Eqs. (37) and (38) into Eq. (36), SN's SIR can be characterized as follows

$$SI{R_{SNSFR}} = {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( {\frac{2}{5}{4^\sigma } + \frac{4}{5}{7^\sigma } + \frac{2}{5}{{13}^\sigma } + \frac{2}{5}{{19}^\sigma } + {{\left( {\frac{{25}}{4}} \right)}^\sigma } + 2{{\left( {\frac{{31}}{4}} \right)}^\sigma } + 2{{\left( {\frac{{43}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma } + \frac{2}{5}} \right)} \right]^{ - 1}}$$

(39)

where SIR_SNSFR denotes the attained SN's SIR using SoFR technique when first configuration is deployed.

2.2 Analysis of R-S OFDMA network with classical WSN

The second configuration considers a classical WSN realized with R-S OFDMA network. This configuration like classical COFDMA-WSN but with deploying relaying approach in the prim link to advance the reliability and performance of the linkage. Hence, the analysis is done into two main phases. The first phase represents R-S OFDMA network analysis to get the C-H’s SIR in the prim link. Furthermore, the second step signifies the analysis of classical WSN to compute the SN’s SIR in the ensuing link as shown in Fig. 7.

Fig. 7

Conception diagram of second configuration in COFDMA-WSN for energy consumption lessening

2.2.1 Analysis of R-S OFDMA network

In this network, every cell has a centralized BS, a definite number of SNs set in a regular arrangement, and relay station (RS) located at distance equal to R_d (m) in OFDMA network (R_d = 0.25 of R_a) from the BS. Furthermore, full duplex (FD) amplify and forward (AF) RSs are considered in this system because of their efficiency in employing spectrum resources [16]. Hence, the communication is carried out through two time slots (TSs) as shown in Fig. 8. In first TS, RS is inactive, while BS transmits the signal to both C-H and RS through prime and relay link, respectively. In second TS, RS transmits the formerly received signal to the C-H through access link after amplifying it whereas the BS is lazy. Consequently, Maximum Ratio Combined (MRC) procedure is used at C-H to combine the achieved two SIRs. Thus, the C-H's MRC formula can be stated as follows [17]

$$MRC_{SIR} = 1^{st} T.S._{SIR} + 2^{sc} T.S._{SIR}$$

(40)

where MRC_SIR denotes the MRC of the two TSs' SIRs, 1^st T.S._SIR represents obtained C-H's SIR from the first TS and 2^sc T.S._SIR indicates achieved C-H’s SIR from the second TS. Hence, two TSs analysis is conducted in the next subsections for MRC CFT computation using different FrFR techniques.

Fig. 8

Conception declaration of MRC technique in R-S OFDMA network

2.2.1.1 Analysis of R-S St FrFR system

Two-tier St FrFR system based R-S OFDMA network is considered as presented in Fig. 9(a, c). Accordingly, every outer area BF is additional separated in three unlike combinations of S_Fs to drop the interference impact between the RSs, SNs and BSs as portrayed in Fig. 9(b). The first part represents S_F1-1 that is allocated to BS. In addition, the second set signifies S_F1-2, S_F1-3, S_F1-4, and S_F1-5 that are allocated to the SNs. Furthermore, the third portion denotes S_F1-6 that is apportioned to RS as revealed in Fig. 9(d).

Fig. 9

Two-tier St FrFR based R-S OFDMA system a St FrFR3 system design, b Cell arrangement, c St FrFR4 system layout, d Allocated frequency partition

Analysis of first time slot: In this TS, the signal is transmitted by BS to both C-H and RS whereas RS keeps inactive. Thus, all neighbouring BSs that usage a BF similar to that of serving BS, represent an interfering sources. Consequently, 6 BSs in the outer section interfering with C-H when St FrFR3 technique is employed. These BSs represent cells number (8, 10, 12, 14, 16 and 18) that use S_F1-1 as BS₀. Furthermore, when St FrFR4 technique is deployed, there are 6 BSs in cells number (7, 9, 11, 13, 15 and 17) interfering with C-H. Therefore, the analysis of this TS is similar to classical St FrFR analysis because the BS in this TS does the communication completely. Thus, achieved C-H's SIR CFTs due to employing St FrFR3 technique and St FrFR4 technique can be evaluated from Eq. (8) and Eq. (11), respectively.

Analysis of second time slot: In this TS, C-H receives the amplified signal of RS whereas BSs are inactive. Consequently, C-H's SIR can be framed as follows

$$SIR_{R} \left( {x,y} \right) = \frac{{\left( {x^{2} + \left( {y - 0.25} \right)^{2} } \right)^{\sigma } }}{{T_{RF} \left( {x,y} \right)}}$$

(41)

where T_RF indicates the interference factor originated form co-channel RSs in surrounding cells. It is assumed, that RS covers C-H through S_F1-6 in the centre cell. Therefore, 6 RSs in cells with numbers (8, 10, 12, 14, 16 and 18) affecting the C-H when St FrFR3 technique is deployed as revealed in Fig. 9(a). Moreover, there are 6 RSs in cell number (7, 9, 11, 13, 15 and 17) that affecting C-H when St FrFR4 system is used as presented in Fig. 9(c). Furthermore, these RSs’ coordinates are declared in Table 6. Consequently, the factor T_RF for St FrFR3 system is formulated as follows

$$T_{RF3} \left( {x,y} \right) = \sum\limits_{{j_{RS} = 1}}^{{j_{RS} = 6}} {\left[ {\left( {x + x_{{j_{RS} }} } \right)^{2} + \left( {y + y_{{j_{RS} }} } \right)^{2} } \right]^{\sigma } }$$

(42)

where j_RS signifies interfering RSs group in the outer region that use S_F1-2. Thus, the parameter T_RF3 is valued as follows

$$\begin{aligned} & {T_{RF3}}\left( {x,y} \right) = {\left[ {{x^2} + {{\left( {y + 2.75} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \frac{{3\sqrt 3 }}{2}} \right)}^2} + {{\left( {y - 1.75} \right)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{x^2} + {{\left( {y - 3.25} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \frac{{3\sqrt 3 }}{2}} \right)}^2} + {{\left( {y + 1.25} \right)}^2}} \right]^\sigma }\end{aligned}$$

(43)

Table 6 Interfering RSs' Coordinates for St FrFR based R-S OFDMA network

Hence, via replacing the worth of Eq. (43) into Eq. (41), C-H's SIR can be obtained as follows

$$SI{R_{C - HR3}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( {{{\left( {\frac{{121}}{{16}}} \right)}^\sigma } + 2{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + 2{{\left( {\frac{{157}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{169}}{{16}}} \right)}^\sigma }} \right)} \right]^{ - 1}}$$

(44)

where SIR_C-HR3 denotes C-H's SIR attained through second TS when St FrFR3 technique is deployed. Consequently, by replacing the worth of Eq. (8) and Eq. (44) into Eq. (40), the CFT of C-H's MRC using St FrFR3 system is formulated as follows

$$\begin{aligned} & MR{C_{C - HR3}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( {2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {{\left( {\frac{{121}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{169}}{{16}}} \right)}^\sigma }} \right)} \right]^{ - 1}}\\ & \quad + {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( {2\left( {{{\left( {\frac{{31}}{4}} \right)}^\sigma } + {{\left( {\frac{{43}}{4}} \right)}^\sigma }} \right) + {{\left( {\frac{{25}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma }} \right)} \right]^{ - 1}}\end{aligned}$$

(45)

where MRC_C-HR3 represents the obtained C-H's MRC when St FrFR3 system is deployed in the second configuration. On the other hand, the parameter T_RF for St FrFR4 system is modelled as follows

$$T_{RF4} \left( {x,y} \right) = \sum\limits_{{j_{RS} = 1}}^{{j_{RS} = 6}} {\left[ {\left( {x + x_{{j_{RS} }} } \right)^{2} + \left( {y + y_{{j_{RS} }} } \right)^{2} } \right]^{\sigma } }$$

(46)

Consequently, the factor T_RF4 due to CoCI of surrounding RSs is expressed as follows

$$\begin{aligned}{T_{FR4}}\left( {x,y} \right) = & {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y + 2.75} \right)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y - 3.25} \right)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{{\left( {x \pm 2\sqrt 3 } \right)}^2} + {{\left( {y - \frac{1}{4}} \right)}^2}} \right]^\sigma }\end{aligned}$$

(47)

Hence, by replacing the worth of Eq. (47) into Eq. (41), C-H's SIR is symbolized by the following relative

$${SIR_{C - HR4}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( {{{\left( {\frac{{73}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{121}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{169}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{ - 191}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{193}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{217}}{{16}}} \right)}^\sigma }} \right)} \right]^{ - 1}}$$

(48)

where SIR_C-HR4 denotes achieved C-H's SIR from the second TS when St FrFR4 technique is employed.

Therefore, by replacing the values of Eq. (11) and Eq. (48) in Eq. (40), C-H's MRC using St FrFR4 system can be expressed as follows

$$\begin{aligned}MR{C_{C - HR4}} = & {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( {{{\left( {\frac{{73}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{121}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{169}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{ - 191}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{193}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{217}}{{16}}} \right)}^\sigma }} \right)} \right]^{ - 1}}\\ & \quad + {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( {2\left( {{{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma }} \right)} \right)} \right]^{ - 1}}\end{aligned}$$

(49)

where MRC_C-HR4 represents the achieved C-H's MRC when St FrFR4 system is used in the second configuration.

2.2.1.2 Analysis of R-S Sc FrFR system

Two tier Sc FFR system based R-S OFDMA network is assumed as presented in Fig. 10(a). Furthermore, each outer BF of each sector is further divided in three unlike S_Fs groups for interference influence reduction between the SNs, RSs and BSs as displayed in Fig. 10(b). The first portion represents S_F1-1 which is assigned to BSs. In addition, the second group indicates S_F1-2, S_F1-3, S_F1-4 and S_F1-5 which are allotted to SNs. Furthermore, the third part signifies S_F1-6 which is allocated to RS as illustrated in Fig. 10(c).

Fig. 10

Two-tier Sc FrFR based R-S OFDMA structure a System design, b Cell arrangement, c Allotted frequency partition

Analysis of first time slot: In this TS, there are 7 BSs which deploy S_F1-1 and affecting the C-H. These BSs located in cells number (5, 6, 14, 15, 16, 17, and 18). Therefore, the analysis of this TS is similar to classical Sc FrFR analysis because BS in this TS does the communication totally. Therefore, C-H's SIR CFT can be calculated from Eq. (22).

Analysis of second time slot: In this TS, every RS transmits the amplified signal to C-H whilst BSs are inactive. Hence, C-H's SIR is valued as follows

$$SIR_{R\sec } \left( {x,y} \right) = \frac{{\left( {x^{2} + \left( {y - 0.25} \right)^{2} } \right)^{\sigma } }}{{T_{{\text{Rsec}}} \left( {x,y} \right)}}$$

(50)

where T_Rsec is the interference parameter because of co-channel RSs that deploy S_F1-6 in surrounding cells as revealed in Fig. 10(a). The coordinates of these RSs that located in cells number (5, 6, 14, 15, 16, 17 and 18) are presented in Table 7. Hence, the factor T_Rsec is framed as follows

$$T_{{\text{Rsec}}} \left( {x,y} \right) = \sum\limits_{{j_{R} = 1}}^{{j_{R} = 7}} {\left[ {\left( {x + x_{{j_{R} }} } \right)^{2} + \left( {y + y_{{j_{R} }} } \right)^{2} } \right]^{\sigma } }$$

(51)

where j_R denotes the interfering RSs group. Hence, the interference parameter can be evaluated using Table 7 as follows

$$\begin{aligned} T_{{\text{Rsec}}} \left( {x,y} \right) = & \left[ {\left( {x \pm \frac{\sqrt 3 }{2}} \right)^{2} + \left( {y + 1.25} \right)^{2} } \right]^{\sigma } + \left[ {\left( {x \pm \sqrt 3 } \right)^{2} + \left( {y + 2.75} \right)^{2} } \right]^{\sigma } \\ & \quad + \left[ {\left( x \right)^{2} + \left( {y + 2.75} \right)^{2} } \right]^{\sigma } + \left[ {\left( {x \pm \frac{3\sqrt 3 }{2}} \right)^{2} + \left( {y + 1.25} \right)^{2} } \right]^{\sigma } \\ \end{aligned}$$

(52)

Table 7 Interfering RSs' Coordinates for Sc FrFR and SoFR based R-S OFDMA network

Consequently, by replacing the values of Eq. (52) into Eq. (50), SIR of C-H located at coordinates (0, 0.5) is formulated as follows

$$SIR_{R\sec } = \left[ {\left( \frac{1}{16} \right)^{ - \sigma } \left( {2\left( {\left( {\frac{61}{{16}}} \right)^{\sigma } + \left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{217}{{16}}} \right)^{\sigma } } \right) + \left( {\frac{169}{{16}}} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(53)

where SIR_Rsec signifies C-H's SIR of the second TS using Sc FrFR system in second configuration. So, by replacing the values of Eq. (22) and Eq. (53) into Eq. (40), MRC CFT for Sc FrFR can be computed as follows

$$\begin{aligned} MRC_{Sec} = & \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {2\left( {\left( \frac{19}{4} \right)^{\sigma } + \left( \frac{43}{4} \right)^{\sigma } + \left( \frac{61}{4} \right)^{\sigma } } \right) + \left( \frac{49}{4} \right)^{\sigma } } \right)} \right]^{ - 1} \\ & \quad + \left[ {\left( \frac{1}{16} \right)^{ - \sigma } \left( {2\left( {\left( {\frac{61}{{16}}} \right)^{\sigma } + \left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{217}{{16}}} \right)^{\sigma } } \right) + \left( {\frac{169}{{16}}} \right)^{\sigma } } \right)} \right]^{ - 1} \\ \end{aligned}$$

(54)

where MRC_Sec represents the obtained C-H's MRC when Sc FrFR system is deployed in the second configuration.

2.2.1.3 Analysis of SoFR networks

Two tier SoFR system based R-S OFDMA network is assumed as revealed in Fig. 11(a). Consequently, each outer BF is additional portioned in three different parts of S_Fs for interference reduction between the SNs, RSs and BSs as shown in Fig. 11(b). One part represents S_F1-1 which is allocated to BSs. In addition, the second part indicates S_F1-2, S_F1-3, S_F1-4 and S_F1-5 which are assigned to SNs. Furthermore, the third part denotes S_F1-6 which is allotted to RS as portrayed in Fig. 11(c).

Fig. 11

Two-tier SoFR based R-S OFDMA structure a System design, b Cell arrangement, c Allotted frequency partition

Analysis of first time slot: In this TS, there are18 interfering cells that use S_F1-1 and affecting C-H. Therefore, the analysis of this TS is similar to the classical SoFR analysis because BS does the communication totally in this TS. Accordingly, C-H's SIR CFT can be evaluated from Eq. (35).

Analysis of second time slot: In this TS, the communication is done between the RS and C-H through the access link. Thus, Eq. (30) can be rewritten to evaluate C-H's SIR in this TS as follows

$$SIR_{SFR - R} \left( {x,y} \right) = \frac{{\varepsilon \left( {x^{2} + \left( {y - 0.25} \right)^{2} } \right)^{\sigma } }}{{T_{iR} \left( {x,y} \right) + \varepsilon T_{oR} \left( {x,y} \right)}}$$

(55)

where T_oR and T_iR signify the outer and inner interference parameters, respectively. The interfering BSs’ normalized coordinates that deploy BF₁ in the inner region are illustrated by Table 3. Accordingly, the factor T_iR is stated as follows

$${T_{iR}}\left( {x,y} \right) = \frac{1}{6}\left[ \begin{aligned}{\left[ {{{\left( {x \pm 2\sqrt 3 } \right)}^2} + {y^2}} \right]^\sigma } + {\left[ {{{\left( {x + \frac{{\sqrt 3 }}{2}} \right)}^2} + {{\left( {y \pm \frac{3}{2}} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y \mp 3} \right)}^2}} \right]^\sigma }\\ + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {y^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y \pm 3} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x - \frac{{\sqrt 3 }}{2}} \right)}^2} + {{\left( {y \pm \frac{3}{2}} \right)}^2}} \right]^\sigma }\end{aligned} \right]$$

(56)

Therefore, inner interference parameter expression for C-H located at x = 0 and y = 0.5 is finalized as follows

$$T_{iR} = 3\left( \frac{7}{4} \right)^{\sigma } + 3\left( \frac{13}{4} \right)^{\sigma } + 3\left( \frac{19}{4} \right)^{\sigma } + 3\left( \frac{37}{4} \right)^{\sigma } + 3\left( \frac{49}{4} \right)^{\sigma } + 3\left( \frac{61}{4} \right)^{\sigma }$$

(57)

Furthermore, there are 6 RSs that use S_F1-6 and interfering with C-H. These RSs’ coordinates are presented in Table 7. Subsequently, the factor T_oR is valued as follows

$$\begin{aligned} &T_{oR} \left( {x,y} \right) = \left[ {x^{2} + \left( {y - 3.25} \right)^{2} } \right]^{\sigma } + \left[ {\left( {x \pm \frac{3\sqrt 3 }{2}} \right)^{2} + \left( {y - 1.75} \right)^{2} } \right]^{\sigma } \\ & \quad + \left[ {\left( {x \pm \frac{3\sqrt 3 }{2}} \right)^{2} + \left( {y + 1.25} \right)^{2} } \right]^{\sigma } + \left[ {x^{2} + \left( {y + 2.75} \right)^{2} } \right]^{\sigma } \\ \end{aligned}$$

(58)

Thus, the outer interference factor formula for the C-H is finalized as follows

$$T_{oR} = \left( {\frac{121}{{16}}} \right)^{\sigma } + 2\left( {\frac{133}{{16}}} \right)^{\sigma } + 2\left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{169}{{16}}} \right)^{\sigma }$$

(59)

Accordingly, by replacing the values of Eqs. (57) and (59) into Eq. (55), C-H's SIR of the SoFR technique can be characterized as follows

$$SI{R_{SRC - H}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( \begin{aligned}3\left( {{{\left( {\frac{7}{4}} \right)}^\sigma } + {{\left( {\frac{{13}}{4}} \right)}^\sigma } + {{\left( {\frac{{19}}{4}} \right)}^\sigma } + {{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma }} \right)\\ + {\left( {\frac{{121}}{{16}}} \right)^\sigma } + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {\left( {\frac{{169}}{{16}}} \right)^\sigma }\end{aligned} \right)} \right]^{ - 1}}$$

(60)

where SIR_SRC-H characterizes C-H's SIR of SoFR technique in the second configuration. Consequently, by substituting the worth of Eqs. (35) and (60) into Eq. (40), the CFT of MRC for the C-H with ε = 1 is calculated as follows

$$\begin{aligned}MR{C_{SRC - H}} = & {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( {2\left( {{{\left( {\frac{7}{4}} \right)}^\sigma } + {{\left( {\frac{{13}}{4}} \right)}^\sigma } + {{\left( {\frac{{19}}{4}} \right)}^\sigma } + {{\left( {\frac{{31}}{4}} \right)}^\sigma } + {{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{43}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma }} \right) + 3{{\left( {\frac{{49}}{4}} \right)}^\sigma } + {{\left( {\frac{{25}}{4}} \right)}^\sigma }} \right)} \right]^{ - 1}}\\ & \quad + {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( {3\left( {{{\left( {\frac{7}{4}} \right)}^\sigma } + {{\left( {\frac{{13}}{4}} \right)}^\sigma } + {{\left( {\frac{{19}}{4}} \right)}^\sigma } + {{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma }} \right) + {{\left( {\frac{{121}}{{16}}} \right)}^\sigma } + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {{\left( {\frac{{169}}{{16}}} \right)}^\sigma }} \right)} \right]^{ - 1}}\end{aligned}$$

(61)

where MRC_SRC-H symbolizes C-H's MRC when SoFR technique is used in the second configuration.

2.2.2 Analysis of classical WSN

The analysis of this network is identical with that in the first configuration. Consequently, Eqs. (15) and (18) can be used to represent SN's SIR CFTs when St FrFR3 and St FrFR4 techniques are applied, respectively. Furthermore, Eqs. (26) and (39) can be utilized to indicate SN's SIR CFTs when Sc FrFR and SoFR techniques are deployed, respectively. Additionally, the analysis of third configuration (classical OFDMA network with R-S WSN) and fourth configuration (R-S COFDMA-WSN) are proposed and investigated in Appendix A and B, respectively. The comparison of WSN behaviour in different COFDMA-WSN configurations is revealed in Table 8. Furthermore, three different metrics are used to compare different configurations. These metrics represent complexity, achieved SIR and cell cost. Additionally, it is observed, that achieving high SIR requires high complexity and cost. Therefore, the outcomes of this work, in some cases, provide high network performance with low complexity and cost.

Table 8 Practical evaluation of WSN behaviour in different COFDMA-WSN configurations

3 Metrics for performance estimation of different patterns

In this section, different metrics for performance estimation are valued to contrast all configurations and patterns. These metrics involve the data rate, Energy Efficiency (EE), successful decoding probability, link throughput, packet transmission rate, maximum number of bits per symbols and outage probability deeming the dissimilar channel characteristics in wireless communication.

3.1 Outage probability

The probability of outage is a vital measure for cellular networks’ performance. It is definite as the probability which the reachable rate can’t reach a specified rate of transmission δ [18]. The significance of this probability acts when an outage happens, the decoding process is more likely to fail. Thus, it represents a typical error. Therefore, different channel propagation influences are considered when calculating the outage probability using the attained SIR expression for each pattern. These influences represent (1) collective influence of shadowing and PLo, (2) joint impact of shadowing, FaF and PLo and (3) combined effect of FaF and PLo.

3.1.1 The collective influence of the shadowing and PLo

For C-H positioned at a distance d (m) from its working BS, the probability of outage by deeming collective shadowing and PLo effect can be valued from the succeeding formula [19]

$${\rm P}\left( { \, SIR_{C} \prec \delta } \right) = Q\left[ {\frac{{10\log_{10} \left( {\frac{1}{\delta }} \right) - X_{m} }}{{X_{s} }}} \right]$$

(62)

where Q denotes the error function which is expressed as $Q\left( u \right) = 0.5erfc\left( {1.414u} \right)$ and SIR_C characterizes the calculated SIR_C in each pattern. Additionally, X_m and X_s factors denote the mean and standard deviation of G_F factor which is expressed as G_F = (SIR_C)⁻¹, respectively. Furthermore, mean and standard deviation can be formulated as follows

$$X_{m} = \frac{1}{\tau }\ln \left( {G_{f} \left( {d,\eta } \right)H\left( {d,Q_{sd} } \right)} \right)$$

(63)

$$X_{s}^{2} = 2\left( {Q_{sd}^{2} - \frac{1}{{\tau^{2} }}\ln H\left( {d,Q_{sd} } \right)} \right)$$

(64)

where $H\left( {d,Q_{sd} } \right) = e^{{{\raise0.7ex\hbox{${^{{\tau^{2} Q_{sd}^{2} }} }$} \!\mathord{\left/ {\vphantom {{^{{\tau^{2} Q_{sd}^{2} }} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \left( {A\left( {d,\eta } \right)\left( {e^{{^{{\tau^{2} Q_{sd}^{2} }} }} - 1} \right) + 1} \right)^{{\frac{ - 1}{2}}}$, $A\left( {d,\eta } \right) = \frac{{\sum\limits_{j} {d_{j}^{ - 2\eta } } }}{{\left( {\sum\limits_{j} {d_{j}^{ - \eta } } } \right)^{2} }}$, $G_{f} \left( {d,\eta } \right) = \frac{{\sum\limits_{j} {d_{j}^{ - \eta } } }}{{d^{ - \eta } }}$, $\tau = \frac{\ln (10)}{{10}}$ and Q_sd represents mean received signal logarithmic standard deviation under shadowing effect (0 ≤ Q_sd ≤ 8 dB).

3.1.2 Joint shadowing, FaF and PLo impact

Outage probability tacking in to account the joint effect of PLo, shadowing and FaF is considered. Consequently, all configurations and patterns can be contrasted under worst case propagation situation.

Additionally, the succeeding formula can express outage probability in this case as follows [20]

$${\rm P}\left( {SIR_{C} \prec \delta } \right) = \int\limits_{0}^{\infty } {Q\left[ {\frac{{10\log_{10} \left( {\frac{x}{\delta }} \right) - X_{m} }}{{X_{s} }}} \right]} e^{ - x} dx$$

(65)

3.1.3 The combined PLo and FaF effect

The probability of outage under combined PLo and FaF effect due to positioning C-H at a parting point dc (m) from the serving BS, can be formulated as in the next expression [21]

$${\rm P}\left( {SIR_{C} \prec \delta } \right) = 1 - \frac{1}{{\left( {\frac{\lambda }{{dc^{ - \eta } }}\delta + 1} \right)^{v} }}$$

(66)

where $v = {\raise0.7ex\hbox{${\left( {\sum\limits_{j = 1}^{N} {dc_{j}^{ - \eta } } } \right)^{2} }$} \!\mathord{\left/ {\vphantom {{\left( {\sum\limits_{j = 1}^{N} {dc_{j}^{ - \eta } } } \right)^{2} } {\sum\limits_{j = 1}^{N} {dc_{j}^{ - 2\eta } } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sum\limits_{j = 1}^{N} {dc_{j}^{ - 2\eta } } }$}}$ and $\lambda = {\raise0.7ex\hbox{${\sum\limits_{j = 1}^N {dc_j^{ - 2\eta }} }$} \!\mathord{\left/ {\vphantom {{\sum\limits_{j = 1}^N {dc_j^{ - 2\eta }} } {\sum\limits_{j = 1}^N {dc_j^{ - \eta }} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sum\limits_{j = 1}^N {dc_j^{ - \eta }} }$}}.$

3.2 Successful transmission probability

Successful transmission probability (STP) refers to the probability of achieving a positively decoded message [22]. Accordingly, the STP expression can be formulated as follows

$$P_{SC} = 1 - P_{otg}^{1 + L} \,$$

(67)

where P_SC represents STP of the message, P_otg signifies the outage probability for each transmission attempt and L denotes number of retransmission attempts.

3.3 Link throughput valuation

The end-to-end network throughput is a crucial performance metric in cellular networks. It measures the number of packets per second received at the destination. Furthermore, this metric is evaluated from the following formula [22].

$$T_{L} = \frac{{\log \left( {1 + \delta } \right)}}{{1 + L_{avg} }}\left( {1 - P_{otg}^{1 + L} } \right)\quad ({\text{bit/sec}})$$

(68)

where T_L represents the link throughput and 1 + L_avg is the average number of transmissions for a successful transmission.

3.4 Maximum bits number per symbol

Maximum bits number per symbol that a subcarrier can transmit per unit time at a specific TS can be represented as a function of δ and maximum bit error rate (BER) [23] as below

$$B_{Sy} = \left[ {\log_{2} \left( {1 + \frac{ - 1.5}{{\ln \left( {5B_{ER} } \right)}}\delta } \right)} \right]\quad ({\text{bit/symbol}})$$

(69)

3.5 Data rate evaluation

Data Rate indicates the transmitted data amount through an identified time period over a network. It can be considered to represent the rapidity at which data is moved from transmitter to the receiver [22]. Accordingly, Eq. (68) can be modified to evaluate the data rate as follows

$$\log \left( {1 + \delta } \right) = \frac{{T_{L} \left( {1 + L_{avg} } \right)}}{{\left( {1 - P_{otg}^{1 + L} } \right)}}\quad ({\text{bit/sec)}}$$

(70)

3.6 Assessment of energy efficiency for cellular networks

Energy efficiency represents the compatibility process between the total energy consumption per bit (P_T) in a network and the throughput [22]. Accordingly, EE can be represented by the following formula

$$EE = \frac{{\left( {1 - P_{otg}^{1 + L} } \right)\log \left( {1 + \delta } \right)}}{{\left( {1 + L_{avg} } \right)\left( {\frac{\delta }{\beta } + P_{T} } \right)}}$$

(71)

where P_T represents the total energy consumption per bit. Hence, this parameter can be evaluated from the following formula

$$P_{T} = \sum\limits_{A = 1}^{L + 1} {\frac{{P_{p} + P_{TX} + P_{RX} }}{{\log \left( {1 + \delta } \right)}}} ({\text{mW}})$$

(72)

where P_TX is the transmission power consumption, P_RX is the consumed power during reception, and P_P is the power consumed by the power amplifier in an one-hop communication network. This consumed power is itself a function of the drain efficiency parameter of the amplifier (β). So, the power consumed by the power amplifier can be represented by the following formula

$$P_{p} = \frac{\delta }{\beta } (\text{mW})$$

(73)

3.7 Packet transmission rate

Packet transmission rate is an efficient way to handle transmissions on a connectionless network. Therefore, if channel has B_BW bandwidth and N_sc subcarriers, then each subcarrier has a bandwidth of Δf = B_BW / N_sc. If the length of a TS is L_TS, having B_Sy bit rate from Eq. (69) with a packet of length P_L bits, then packet transmission rate P_TR per TS for this Node and subcarrier [23] is,

$$P_{TR} = \frac{{L_{TS} .B_{Sy} .\Delta f}}{{P_{L} }}\quad ({\text{packet/sec}})$$

(74)

3.8 Average number of retransmissions

An outage event occurs if a communicated message is not positively decoded at the receiver side. So, the retransmission is a very effective method for reducing the link error rate and outage probability [22]. Hence, Eq. (68) can be reformulated to compute the average number of retransmissions as follows

$$1 + L_{avg} = \frac{{\log \left( {1 + \delta } \right)}}{{T_{L} }}P_{SC}$$

(75)

4 Outcomes and discussion

This study presents an analytical conduct to reduce energy consumption in OFDMA-WSNs through different interference mitigation systems. This treatment yielded CFTs of the C-H's SIR and worst-case SN's SIR. The obtained CFTs are deployed to value several performance metrics. Consequently, different patterns are compared in this section. This comparison is arranged into two main phases. The first phase compares the effect of using different interference techniques on the behaviour of the C-H through the prim link. So, the attained values of C-H’s SIR form first, second, and forth configurations are considered in this stage. Furthermore, it is noticed, that the achieved C-H’s SIR of St FrFR3, St FrFR4 and Sc FrFR techniques in second configuration and forth configuration are equal. So, second configuration is considered to represent these two configurations in the comparison from the mentioned techniques perspective. Moreover, the second phase studies the effect of interference reduction techniques on SNs through the ensuing link. Consequently, the attained values of SN’s SIR form first, third, and forth configurations are deliberated in this phase. Additionally, it is observed, that the realized SN’s SIR of St FrFR3, St FrFR4 and Sc FrFR techniques in third configuration and forth configuration are identical. Therefore, third configuration is deliberated to characterize these two configurations in the comparison from the declared techniques viewpoint. Accordingly, the consequences of this work introduce an obvious concept about the behaviour of cooperative OFDMA-WSN systems. Consequently, this work contribute to improving the link throughput, STP, and EE of these networks.

The outage probability variant with SIR threshold is shown in Figs. 12, 13. From these figures, it is noted, that probability of coverage surges as the obtained SIR value surpasses the SIR threshold worth. Consequently, rising SIR threshold worth cause, a decreasing in the probability at which network can cover C-H and SNs. Accordingly, probability of outage increases. Additionally, the outage probability valuation deeming different propagation conditions is done. The attained outcomes definite, that the collective influence of the PLo, shadowing and FaF has the poorest influence on achieved SIR worth. Therefore, the following consequences are attained considering this circumstance. In addition, the outcomes under joint influence of (PLo and shadowing) and combined effect of (PLo and FaF) situations are investigated but not involved in this section. Also, it is observed, that in general, the first configuration applies dissimilar FrFR methods attains the lowest performance in the prime link compared to the second and fourth configurations as revealed in Fig. 12. The cause of this consequence can be ascribed, to the presence of RSs that significantly improve the signal strength in the second and fourth configurations. Moreover, the performance significant drop of first configuration using SoFR can be accredited, to the high interference rates caused by deployment of T_PC scheme. Additionally, it is noted, that the performance of second configuration employs St FrFR4 is superior to other patterns. The cause of this outcome, thanks to the decrement of frequency reuse process due to FReF increment. Consequently, the interference sources reduces. Accordingly, the strength of interfering signals decreases. Therefore, this pattern achieves the highest value of the SIR. On the other hand, it is noticed, that the first configuration comes late in terms of performance in the ensuing link compared to the third and fourth configurations as revealed in Fig. 13.

Fig. 12

Outage probability of different configurations against SIR threshold in the prime link with Ra = 1000 m, BW = 10 MHz, σ = 4 and Qsd = 8 dB

Fig. 13

Outage probability of different configurations against SIR threshold in the ensuing link with Ra = 1000 m, BW = 10 MHz, σ = 4 and Qsd = 8 dB

The cause for this consequence can be ascribed, to the strong interference effect, which greatly disturbs the SIR in the first configuration. Consequently, the probability of outage increases. Furthermore, it is noticed, compared to Fig. 12, the performance of St FrFR3 system deployed in the second configuration was retracted, and the SoFR technique using fourth configuration took its place as the second best pattern gives coverage probability for the SNs as revealed in Fig. 13. The cause of this outcome, thanks to the MRC increment in the two TSs due to RSs deployment.The change of EE value with PLoE is represented in Figs. 14, 15. From these figures, it is noted that, PLo gain surges due to PLoE rise which leads to the compensation of the signal attenuation. Accordingly, the obtained SIR from the two TSs increases. Therefore, the gotten MRC increases. Consequently, coverage probability of C-H and SNs increases. Accordingly, the retransmission attempts for reducing the link error rate reduces. Therefore, value of EE increases. Furthermore, it is noticed, that fourth configuration applies SoFR system exhibits better performance than pattern of second configuration deploys St FrFR3 in a low PLoE range as revealed in Fig. 14. Moreover, SoFR technique lost its supremacy in high PLoE worth and second configuration uses St FrFR3 attains the highest values of EE. The cause of this outcome can be attributed, to the increment rate of SIR worth with PLoE in the second TS of second configuration uses St FrFR3 is much higher than that in fourth configuration applies SoFR. So, St FrFR3 in second configuration resulted in better achieved MRC. Consequently, it realizes much higher EE values than those of SoFR system in fourth configuration under high PLoE values condition. Additionally, it is observed, that Sc FrFR technique performs worse than other techniques when deployed in prim or ensuing link as revealed in Figs. 14, 15. The cause of this outcome can be ascribed, to Sc FrFR technique inability to reach low the outage probability value. Therefore, STP decreases. This leads to more energy consumption. Accordingly, EE value decreases. Furthermore, it is noticed, that the performance of SoFR technique in third configuration is close to that in fourth configuration as revealed in Fig. 15.

Fig. 14

Energy efficiency of different configurations against PLoE through prime link with R_a = 1000 m, BW = 10 MHz, P_TX = 97.9 mW, P_RX = 112.2 mW and β = 0.35

Fig. 15

Energy efficiency of different configurations against PLoE through ensuing link with R_a = 1000 m, BW = 10 MHz, P_TX = 97.9 mW, P_RX = 112.2 mW and β = 0.35

Accordingly, applying SoFR technique with third configuration has the periority of usage from the network cost point of view. The cause of this outcome, thanks to the ability of third configuration employs SoFR pattern to provide nearly the same EE values as using SoFR in fourth configuration with lower cost. These outcomes are suitable in defining the deployment significance level of dissimilar interference mitigation techniques after making a trade-off between desirable performance and network cost. The alteration of data rate with PLoE is depicted in Figs. 16, 17. From these figures, it is noticed, that transmitted data rate over the system BW increases with SIR increase. Consequently, data rate increase with PLoE increase. Furthermore, it is observed, that negative interference effect decreases as the FReF increases. Accordingly, St FrFR4 attains highest SIR values in the two TSs.

Fig. 16

Data rate of different configurations through prime link against PLoE with R_a = 1000 m, BW = 10 MHz, 1 + L_avg = 5, T_L = 5000 bit/sec and Q_sd = 8 dB

Fig. 17

Data rate of different configurations through ensuing link against PLoE with R_a = 1000 m, BW = 10 MHz, 1 + L_avg = 5, T_L = 5000 bit/sec and Q_sd = 8 dB

Hence, St FrFR4 technique outperforms the techniques in achieving high MRC values. As a result, patterns of using St FrFR4 with second and third configurations outdoes other patterns. Also, it is noticed, that the adjoining in distance between interfering sources and C-H in the second TS reasons the deploying of SoFR system with fourth configuration has preference in use than using St FrFR3 technique with second configuration under low PLoE values as revealed in Fig. 16. However, the pattern of deploying St FrFR3 technique in the second configuration has the superiority in a high PLoE range. This result can be accredited, to the high interference impact caused by T_PC employment which negatively affects the data transfer rate. Furthermore, it is observed, that applying SoFR technique with third configuration has the usage priority than using it with fourth configuration. The cause of this consequence can be accredited, to the aptitude of SoFR technique in third configuration to achieve closely similar data rate as in fourth configuration as shown in Fig. 17 with lower power consumption and network budget.

Furthermore, the variation of link throughput with PLoE is portrayed in Figs. 18, 19. From these figures, it is observed, that SIR increases with PLoE increase due to the signal lessening compensation.

Fig. 18

Link throughput of different configurations against PLoE through prim link with R_a = 1000 m, BW = 10 MHz, 1 + L_avg = 5, log (1 + δ) = 4000 bit/sec and Q_sd = 8 dB

Fig. 19

Link throughput of different configurations against PLoE through ensuing link with R_a = 1000 m, BW = 10 MHz, 1 + L_avg = 5, log (1 + δ) = 4000 bit/sec and Q_sd = 8 dB

Therefore, the number of packets per second received at the destination upsurge. Hence, the throughput rises with PLoE surge. It is noticed, that patterns of second configuration employs FrFR3 and fourth configuration deploys SoFR have approximately the same performance in low PLoE range as revealed in Fig. 18. However, the superiority of the second configuration employs St FrFR3 technique is evident in the high PLoE range. The reason of this outcome can be attributed, to the SIR worth rise rate with PLoE in the second TS of second configuration uses St FrFR3 is much higher than it in fourth configuration deploys SoFR. Therefore, it excels in terms of achieving a high C-H’s throughput. However, it was noticed, that the use of St FrFR3 technique with the third configuration in ensuing link led to a decline in the level of performance for the link throughput, to the fourth place in the low PLoE values and the third place in the high PLoE values as depicted in Fig. 19. Therefore, it is not preferable to use the St FrFR3 technique when high throughput values are required in the ensuing link.

The change of average number of retransmission attempts with SIR threshold is shown in Figs. 20, 21. From these figures, it is observed, that probability of outage surges, as the obtained SIR worth cannot exceed the SIR threshold worth. So, the transmitted message failing probability to reach the receiver increases. Hence, the number of retransmission attempts increase to overcome the outage probability impact. Consequently, the number of retransmission attempts increase as SIR threshold increase. Additionally, the performance difference of the patterns in terms of deliver the message capability is obvious in the high SIR threshold values. Moreover, it is noticed, applying St FrFR4 technique in second and third configurations outperforms the other patterns. The cause of this outcome can be ascribed, to high-attained MRC due to SIR increase. Thus, the coverage probability increases. Therefore, these two patterns attain highest data consistent transfer. Accordingly, retransmission attempts decreases. In addition, it is noted, that deploying SoFR in fourth configuration is superlative than using it in other configurations as portrayed in Figs. 20, 21. This outcome can be credited, to the improved desired signal power due to using relaying technique in both prime and ensuing links. Therefore, employing SoFR technique in fourth configuration is superior to its peers in other configurations. However, this pattern loss its preference in the low SIR threshold range and it is preferable to use the pattern of second and third configurations applying SoFR sysetm. The reason of this consequence can be ascribed, to the aptitude of these two patterns to do approximately equal performance with reduced cost.

Fig. 20

Average number of retransmission attempts of different configurations against PLoE through prim link with R_a = 1000 m, BW = 10 MHz, log (1 + δ) = 4000 bit/sec, T_L = 5000 bit/sec and Q_sd = 8 dB

Fig. 21

Average number of retransmission attempts of different configurations against PLoE through ensuing link with R_a = 1000 m, BW = 10 MHz, log (1 + δ) = 4000 bit/sec, T_L = 5000 bit/sec and Q_sd = 8 dB

Moreover, it is noted, that applying Sc FrFR in the second, third and fourth configurations leads to an increase in the number of retransmissions. Therefore, the average time taken between a packet initially sent by the source, and the time for successfully receiving the message at the destination increases. Consequently, the network delay increases. However, the SoFR technique achieves the highest network delay when the traditional configuration is used in the prim link. On the other hand, the Sc FrFR technique maintains the performance of the latecomer in terms of achieving the highest network delay when used with the traditional configuration in the ensuing link. Moreover, it is noted, that deploying St FrFR4 technique with all configuration achieves the lowest network delay. Moreover, applying St FrFR3 technique in the second configuration achieves the second place in terms of speed of successful message delivery in the prime link as shown in Fig. 20. However, the second priority in the ensuing link is attained by fourth configuration deploys SoFR technique as revealed in Fig. 21. Additionally, it is observed, that the variance rate between dissimilar configurations in low PLoE part is fewer than it in high PLoE part. The cause of this effect can be ascribed, to the decreasing of signal lessening influences in high PLoE worth because of the rise in PLo gain worth.

The successful transmission probability change with SIR threshold is depicted in Figs. 22, 23. From these figures, it is noted, that outage probability increases as the SIR threshold worth increase. Thus, the probability of achieving successful decoded message at the receiver decreases. Therefore, successful transmission probability decreases. Furthermore, it is noticed, that using SoFR in the prime link with the second configuration provides deprived behavior, but this performance improves with the use of the fourth configuration as revealed in Fig. 22. The origin of this effect can be accredited, to the increased SIR caused by employing relaying technique in both links. Accordingly, STP increases. Additionally, it is noted, that pattern of third configuration deploying Sc FrFR achieves the highest outage probability as shown in Fig. 23. Accordingly, it cannot cover the SNs in an efficient way. Consequently, it achieves the lowest probability of having a successfully decoded message at SN. Additionally, it is noticed, that all patterns that employ St FrFR4 technique are superior to other patterns. The cause of this consequence can be ascribed, to the reduced interference influence due to high FReF. Therefore, obtained SIR in the two TSs rises. Consequently, the achieved MRC exceeds greatly the attained MRC from other techniques. Accordingly, St FrFR4 technique has the priority to use when high STP is desirable.

Fig. 22

STP of different configurations against SIR threshold through prime link with R_a = 1000 m, BW = 10 MHz, σ = 4 and Q_sd = 8 dB

Fig. 23

STP of different configurations against SIR threshold through ensuing link with R_a = 1000 m, BW = 10 MHz, σ = 4 and Q_sd = 8 dB

The variation of packet transmission rate with PLoE is shown in Figs. 24, 25. From these figures, it is observed, that effect of signal attenuation decreases with PLoE increase. Hence, the data quantity moved positively from one point to another in a specified time period increases. Accordingly, the packet transmission rate increases. Furthermore, it is noted that pattern of second configuration uses St FrRF3 technique has preference in usage than other techniques in high PLoE values as shown in Fig. 24. This consequence may be ascribed, to the improved C-H's SIR in the second TS. Accordingly, this technique attains the highest value of MRC, which allow it from realizing the maximum P_TR rates.

Fig. 24

Packet transmission rate variation in prime link against PLoE for different network configurations with R_a = 1000 m, P_L = 500 bytes, Δf = 120 kHz and L_TS = 10 ms

Fig. 25

Packet transmission rate variation in ensuing link against PLoE for different network configurations with R_a = 1000 m, P_L = 500 bytes, Δf = 120 kHz and L_TS = 10 ms

Additionally, it is noticed, that the near in distance between interfering sources and C-H in the second TS reasons the pattern of second configuration uses St FrRF4 has preference in usage in low PLoE value only. Moreover, it is noticed, that pattern of third configuration uses St FrRF4 technique is superior to other techniques as shown in Fig. 25. This outcome may be accredited, to its aptitude to achieve the highest SN's MRC. Consequently, this pattern attains the maximum P_TR values. Moreover, it is noted, that the preference for using St FrRF3 technique has fallen to the third position in the high PLoE range and to the fourth position in low PLoE part. Furthermore, a comparison with existing techniques in terms of several key performance parameters is executed. Specifically, in terms of the SIR, the proposed methodology achieves an increment of 29% in prime link and 26% rise in ensuing link compared to the current state-of-the-art approaches [11]. This increment indicates a higher accuracy and precision in receiving the desired signal. Furthermore, the proposed methodology exhibits an increase in link throughput by 17% in prime link and by 13% in ensuing link compared to [11]. This increase signifies a better overall consistency and reliability in the data transfer. Additionally, the proposed methodology outperforms existing techniques in terms of energy efficiency, increasing the energy efficiency by 25% in prime link and by 15% in ensuing link compared to [11]. This improvement allows for lower energy consumption, which is crucial in WSN life time. Table 9 shows the performance analysis of the proposed COFDMA-WSN methodology compared to [11]. Overall, the analysis of the proposed methodology highlights its superiority in terms of increased SIR, link throughput and improved energy efficiency compared to existing techniques [11]. These advancements contribute to enhanced energy saving and so maximizing network life time making the proposed methodology a promising and valuable approach for the targeted area.

Table 9 Performance analysis of the Proposed COFDMA-WSN methodology compared to [11] applying St FrFR3 technique with σ = 3.2

5 Conclusion

In this paper, the WSN performance enhancements by decreasing energy consumption is deliberated. These enhancements are studied through considering two different COFDMA-WSN schemes. These schemes represent classical COFDMA-WSN and R-S COFDMA-WSN. Additionally, there are different pondered configurations due to these schemes. These configurations denote classical OFDMA network with classical WSN, R-S OFDMA network with classical WSN, classical OFDMA network with R-S WSN, and R-S OFDMA network with R-S WSN. Moreover, each configuration is applied with four different FrFR techniques. These techniques represent St FrFR4 technique, St FrFR3 technique, Sc FrFR technique and SoFR technique. Consequently, there are sixteen different patterns of COFDMA-WSN are studied. A comparison of all patterns is presented to improve the network QoS. Analytical dealings are realized. These treatments yielded CFTs for C-H’s SIR and SN’s SIR. Additionally, dissimilar metrics of performance valuation are applied to contrast the performance of altered patterns. The consequences show, that as unpredicted in prime link, SoFR using fourth configuration outperforms St FrFR3 deploying second configuration in low PLoE part only. Furthermore, this technique lost its superiority in high PLoE part and St FrFR3 using second configuration is preferable in use from EE perspectives. Furthermore, it is noticed, that SoFR using third configuration in ensuing link shows performance close to SoFR using fourth configuration from EE viewpoint. Therefore, applying SoFR in third configuration is preferable with respect to the energy preserve and network budget in ensuing link. Moreover, it is noted, that deploying St FrFR4 technique with all configuration attains the lowest average number of retransmission attempts. Consequently, St FrFR4 technique achieves the lowest network delay. Furthermore, it is noticed, that variance worth between dissimilar techniques in low PLoE part is less than it in high PLoE part from average number of retransmission attempts standpoint. The cause of this outcome can be credited, to the decreasing of signal attenuation impacts in high PLoE value due to the increase in PLo gain value. Furthermore, it is noted, that St FrFR4 technique outdoes the other techniques when high STP is needed. This outcome can be accredited, to its capability to attain lower outage probability. Furthermore, it is noted, that pattern of second configuration uses St FrRF3 technique has the first priority in use in high PLoE range from P_TR viewpoint in prim link. This outcome may be credited, to the improved C-H's SIR in the second TS. Accordingly, this technique attains the highest values of MRC, which permits it from realizing the maximum P_TR values. Additionally, it is noted, that the close in distance between interfering sources and C-H in the second TS reasons the pattern of second configuration uses St FrRF4 has preference in use in low PLoE range only. Additionally, it is noticed, that pattern of third configuration uses St FrRF4 technique is superior to other techniques in ensuing link. This consequence may be credited, to its aptitude to achieve the highest SN's MRC. Consequently, this pattern attains the maximum P_TR values. Moreover, it is observed, that the preference of using St FrRF3 technique has fallen to the third position in the low PLoE range and to the fourth position in low PLoE part in ensuing link. So, there is a necessity to do a trade-off between all different patterns to attain the best network performance. These consequences introduce much higher SIR improvements. Hence, the system will experience to lower outage probability. Therefore, successful transmission probability is enhanced. So, packet loss is minimized. Thus, average number of retransmissions is reduced. Therefore, packet transmission rate is maximized. Accordingly, link throughput and data rate are improved. Also, the value of EE is enhanced. Consequently, the networks lifetime is maximized. The future work will give a special care to more complex developments for further energy consumption reduction in WSNs. These developments represent COFDMA-WSNs with directional antennas apply different beamforming techniques.

Appendices

A: Analysis of classical OFDMA network with R-S WSN

The third configuration considers R-S WSN executed with a classical OFDMA network. This configuration is the same as COFDMA-WSN but with using FD AF RSs in the ensuing connection. Consequently, the analysis of this configuration is completed via two key steps. The first stage represents the analysis of a classical OFDMA network to compute the C-H’s SIR. Furthermore, the second step signifies the analysis of R-S WSN for the SN’s MRC calculation as revealed in Fig. 

Fig. 26

Conception diagram of third configuration in COFDMA-WSN for energy consumption dropping

26.

1.1 Analysis of classical OFDMA network

The analysis of this network is similar to that in the first configuration. Consequently, Eqs. (8), and (11) can be used to represent C-H's SIR CFTs for St FrFR3 and St FrFR4 techniques, respectively. Furthermore, Eqs. (22), and (35) can be utilized to designate C-H's SIR CFTs when Sc FrFR and SoFR techniques are deployed, respectively.

1.2 Analysis of R-S WSN

In this network, every cell has a centralized BS, a specified numeral of SNs planned in a regular pattern, and a FD AF RS located at distance equal to R_dR (m) (R_dR = 0.75 of R_Cell) from the BS. Hence, the communication is accomplished in two TSs as shown in Fig. 

Fig. 27

Idea clarification of MRC technique in R-S WSN

27. In first TS, RS is inactive, while C-H transmits the signal to both RS and SN through relay and ensuing link, respectively. In second TS, RS transmits the formerly received signal to the SN through access link after amplification process whereas the C-H keeps lazy. Accordingly, SN combines these two TSs' SIRs by MRC technique as displayed in Fig. 27. Consequently, Eq. (40) can be modified to symbolize the SN's MRC calculation as follows

$$MRC_{SN\_SIR} = SIR_{1} + SIR_{2}$$

(76)

where MRC_{SN_SIR} denotes the MRC of the two TSs' SIRs, SIR₁ characterizes reached SN's SIR from the first TS and SIR₂ designates obtained SN’s SIR from the second TS. Consequently, the analysis of two TSs is done in the succeeding subsections to compute MRC CFT using different FrFR techniques.

1.2.1 Analysis of St FrFR technique

Two-tier St FrFR system based R-S WSN is considered as displayed in Fig. 

Fig. 28

Two-tier St FrFR based R-S WSN system a St FrFR3 system design, b Cell arrangement, c St FrFR4 system layout, d Assigned frequency division

28(a, c). Consequently, every outer BF is more distributed in three different parts of S_Fs to reduce the CoCI influence between cell elements as represented in Fig. 28(b). The first section symbolizes S_F1-1 which is allotted to BS. Besides, the next group signifies S_F1-2, S_F1-3, S_F1-4 and S_F1-5 which are apportioned to SNs. Furthermore, the third sector indicates S_F1-6 which is allocated to RS as portrayed in Fig. 28(d).

1.2.1.1 Analysis of first time slot

In this TS, C-H transfers the signal to both RS and SN whilst RS preserves inactive. Accordingly, wholly neighboring C-Hs that employ S_F1-3 as the serving C-H represent an interfering sources. Hence, SN of interest hurts from 6 C-Hs in cells number (8, 10, 12, 14, 16 and 18). These C-Hs usage S_F1-3 as serving C-H in the outer section when St FrFR3 technique is employed. Furthermore, when St FrFR4 technique is deployed, the SN of interest suffers from 6 interfering C-Hs that characterize cells number (7, 9, 11, 13, 15 and 17). Therefore, the analysis of this TS is similar to the classical St FrFR analysis because in this TS the C-H does the communication completely. Therefore, SN's SIR CFTs for St FrFR3 technique and St FrFR4 technique can be valued from Eq. (15) and Eq. (18), respectively.

1.2.1.2 Analysis of second time slot

In this TS, every RS transmits its signal to SN whereas serving C-H is lazy. Hence, SN's SIR in this TS is assessed as follows

$$SIR_{R} \left( {x,y} \right) = \frac{{\left( {x^{2} + \left( {y - 0.75} \right)^{2} } \right)^{\sigma } }}{{T_{RW} \left( {x,y} \right)}}$$

(77)

where T_RW stands the interference parameter due to CoCI from co-channel RSs that use S_F1-6 as the serving RS. Therefore, when St FrFR3 system is used, there are 6 RSs in cell number (8, 10, 12, 14, 16 and 18) affecting SN of interest as revealed in Fig. 28(a). Furthermore, when St FrFR4 system is used, there are 6 RSs in cell number (7, 9, 11, 13, 15 and 17) affecting SN of interest as presented in Fig. 28(c). The interfering RSs’ coordinates are revealed in Table

Table 10 Interfering RSs' Coordinates for St FrFR3 technique and St FrFR4 technique based R-S WSN

10. Accordingly, the factor T_RW for St FrFR3 system is valued as follows

$$T_{RW3} \left( {x,y} \right) = \sum\limits_{{j_{RW} = 1}}^{{j_{RW} = 6}} {\left[ {\left( {x + x_{{j_{RW} }} } \right)^{2} + \left( {y + y_{{j_{RW} }} } \right)^{2} } \right]^{\sigma } }$$

(78)

where j_RW signifies interfering RSs group. Accordingly, this interference factor can be expressed as follows

$$\begin{aligned} T_{RW3} \left( {x,y} \right) = & \left[ {\left( {x \pm \frac{3\sqrt 3 }{2}} \right)^{2} + \left( {y - 2.25} \right)^{2} } \right]^{\sigma } \\ & \quad + \left[ {\left( {x \pm \frac{3\sqrt 3 }{2}} \right)^{2} + \left( {y + 0.75} \right)^{2} } \right]^{\sigma } \\ & \quad + \left[ {x^{2} + \left( {y - 3.75} \right)^{2} } \right]^{\sigma } + \left[ {x^{2} + \left( {y + 2.5} \right)^{2} } \right]^{\sigma } \\ \end{aligned}$$

(79)

Therefore, by replacing the account of Eq. (79) into Eq. (77) with x = 0 and y = 1, SN's SIR of St FrFR3 system can be obtained as follows

$$SIR_{RW3} = \left[ {\left( \frac{1}{16} \right)^{ - \sigma } \left( {\left( {\frac{121}{{16}}} \right)^{\sigma } + 2\left( {\frac{133}{{16}}} \right)^{\sigma } + 2\left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{169}{{16}}} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(80)

where SIR_RW3 denotes SN's SIR achieved by using St FrFR3 technique in the third configuration. Accordingly, by replacing the values of Eq. (15) and Eq. (80) into Eq. (76), the SN's MRC CFT for St FrFR3 system is expressed as follows

$$\begin{aligned} & {\text{MRC}}_{{{\text{W3}}}} = \left[ {\left( {\frac{{1}}{{4}}} \right)^{{ - {\upsigma }}} \left( {\left( {\frac{{{25}}}{{4}}} \right)^{{\upsigma }} { + 2}\left( {\frac{{{31}}}{{4}}} \right)^{{\upsigma }} { + 2}\left( {\frac{{{43}}}{{4}}} \right)^{{\upsigma }} { + }\left( {\frac{{{49}}}{{4}}} \right)^{{\upsigma }} } \right)} \right]^{{ - 1}} \\ & \quad { + }\left[ {\left( {\frac{{1}}{{{16}}}} \right)^{{ - {\upsigma }}} \left( {\left( {\frac{{{121}}}{{{16}}}} \right)^{{\upsigma }} { + 2}\left( {\frac{{{133}}}{{{16}}}} \right)^{{\upsigma }} { + 2}\left( {\frac{{{157}}}{{{16}}}} \right)^{{\upsigma }} { + }\left( {\frac{{{169}}}{{{16}}}} \right)^{{\upsigma }} } \right)} \right]^{{ - 1}} \\ \end{aligned}$$

(81)

where MRC_W3 signifies SN's MRC attained by using St FrFR3 technique in the third configuration. Additionally, the factor T_RW for St FrFR4 technique is definite as follows

$$T_{RW4} \left( {x,y} \right) = \sum\limits_{{j_{RS} = 1}}^{{j_{RS} = 6}} {\left[ {\left( {x + x_{{j_{RS} }} } \right)^{2} + \left( {y + y_{{j_{RS} }} } \right)^{2} } \right]^{\sigma } }$$

(82)

Consequently, by using interfering RSs’ coordinates that are revealed in Table 10, this interference parameter can be evaluated as follows

$$\begin{aligned} T_{RW4} \left( {x,y} \right) = & \left[ {\left( {x \pm 2\sqrt 3 } \right)^{2} + \left( {y - \frac{3}{4}} \right)^{2} } \right]^{\sigma } + \left[ {\left( {x \pm \sqrt 3 } \right)^{2} + \left( {y + 2.25} \right)^{2} } \right]^{\sigma } \\ & \quad + \left[ {\left( {x \pm \sqrt 3 } \right)^{2} + \left( {y - 3.75} \right)^{2} } \right]^{\sigma } \\ \end{aligned}$$

(83)

Thus, by replacing the rate of Eq. (83) into Eq. (77), SN's SIR of St FrFR4 technique is expressed as follows

$$SI{R_{RW4}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( {{{\left( {\frac{{73}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{121}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{169}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{ - 191}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{193}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{217}}{{16}}} \right)}^\sigma }} \right)} \right]^{ - 1}}$$

(84)

where SIR_RW4 denotes attained SN's SIR by using St FrFR4 technique in the third configuration. Therefore, by replacing the worth of Eq. (18) and Eq. (84) in Eq. (76), the CFT of the SN's MRC for St FrFR4 system is computed as follows

$$\begin{aligned} &MRC_{W4} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {2\left( \frac{37}{4} \right)^{\sigma } + \left( \frac{13}{4} \right)^{\sigma } + \left( {\frac{ - 47}{4}} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } + \left( \frac{61}{4} \right)^{\sigma } } \right)} \right]^{ - 1} \\ & \quad + \left[ {\left( \frac{1}{16} \right)^{ - \sigma } \left( {\left( {\frac{73}{{16}}} \right)^{\sigma } + \left( {\frac{121}{{16}}} \right)^{\sigma } + \left( {\frac{169}{{16}}} \right)^{\sigma } + \left( {\frac{ - 191}{{16}}} \right)^{\sigma } + \left( {\frac{193}{{16}}} \right)^{\sigma } + \left( {\frac{217}{{16}}} \right)^{\sigma } } \right)} \right]^{ - 1} \\ \end{aligned}$$

(85)

where MRC_W4 indicates SN's MRC reached by using St FrFR4 technique in the third configuration.

1.2.2 Analysis of Sc FrFR technique

Two tier Sc FrFR system based R-S WSN is assumed as shown in Fig. 

Fig. 29

Two-tier Sc FrFR based R-S WSN structure a System design, b Cell arrangement, c Allotted frequency partition

29(a). Additionally, each outer sector’s BF is scattered in three unlike S_Fs groups for the purpose of reducing CoCI as shown in Fig. 29(b). The first set characterizes S_F1-1, which is allocated to BSs. In addition, the second set symbolizes S_F1-2, S_F1-3, S_F1-4, and S_F1-5, which are allotted to RSs. Furthermore, the third set denotes S_F1-6 which is allocated to RS as showed in Fig. 29(c).

1.2.2.1 Analysis of first time slot

In this TS, SN of interest hurts from 7 interfering C-Hs that usage the S_F1-3 in cells number (5, 6, 14, 15, 16, 17, and 18). Therefore, the analysis of this TS is similar to that in classical Sc FrFR because in this TS the communication is completed totally by the serving C-H. So, SN's SIR CFT can be evaluated from Eq. (26).

1.2.2.2 Analysis of second time slot

In this TS, every RS transfers its signal to SN of interest while C-Hs are still. Hence, SN's SIR can be assessed as follows

$$SIR_{RW\sec } \left( {x,y} \right) = \frac{{\left( {x^{2} + \left( {y - 0.75} \right)^{2} } \right)^{\sigma } }}{{T_{{\text{RWsec}}} \left( {x,y} \right)}}$$

(86)

where T_RWsec is the interference parameter because of all co-channel RSs that use S_F1-2 similar to the aiding RS. Therefore, there are 7 RSs in cells number (5, 6, 14, 15, 16, 17 and 18) cause CoCI as demonstrated in Fig. 29(a). The coordinates of these 7 RSs that affecting SN of interest are presented in Table

Table 11 Interfering RSs' Coordinates for Sc FrFR and SoFR based R-S WSNs

11. Consequently, the factor T_RWsec is valued as follows

$$T_{{\text{RWsec}}} \left( {x,y} \right) = \sum\limits_{{j_{RWS} = 1}}^{{j_{RWS} = 7}} {\left[ {\left( {x + x_{{j_{RWS} }} } \right)^{2} + \left( {y + y_{{j_{RWS} }} } \right)^{2} } \right]^{\sigma } }$$

(87)

where j_RWS denotes the group of interfering RSs. Hence, this interference factor can be calculated as follows

$$\begin{aligned} &T_{{\text{RWsec}}} \left( {x,y} \right) = \left[ {\left( {x \pm \sqrt 3 } \right)^{2} + \left( {y + 2.5} \right)^{2} } \right]^{\sigma } + \left[ {\left( x \right)^{2} + \left( {y + 2.5} \right)^{2} } \right]^{\sigma } \\ & \quad + \left[ {\left( {x \pm \frac{\sqrt 3 }{2}} \right)^{2} + \left( {y + 0.75} \right)^{2} } \right]^{\sigma } + \left[ {\left( {x \pm \frac{3\sqrt 3 }{2}} \right)^{2} + \left( {y + 0.75} \right)^{2} } \right]^{\sigma } \\ \end{aligned}$$

(88)

Thus, by replacing the rate of Eq. (88) into Eq. (86), the SIR of SN located at x = 0, y = 1 is formulated as follows

$$SIR_{RW\sec } = \left[ {\left( \frac{1}{16} \right)^{ - \sigma } \left( {2\left( {\left( {\frac{61}{{16}}} \right)^{\sigma } + \left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{217}{{16}}} \right)^{\sigma } } \right) + \left( {\frac{169}{{16}}} \right)^{\sigma } } \right)} \right]^{ - 1}$$

(89)

where SIR_RWsec signifies SN's SIR of the second TS for Sc FrFR system applied with the third configuration. Accordingly, by placing the values of Eq. (26) and Eq. (89) into Eq. (76), the CFT of SN's MRC for Sc FrFR system can be computed as follows

$$\begin{aligned} & MRC_{WSec} = \left[ {\left( \frac{1}{4} \right)^{ - \sigma } \left( {2\left( {\left( \frac{19}{4} \right)^{\sigma } + \left( \frac{43}{4} \right)^{\sigma } + \left( \frac{61}{4} \right)^{\sigma } } \right) + \left( \frac{49}{4} \right)^{\sigma } } \right)} \right]^{ - 1} \\ & \quad + \left[ {\left( \frac{1}{16} \right)^{ - \sigma } \left( {2\left( {\left( {\frac{61}{{16}}} \right)^{\sigma } + \left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{217}{{16}}} \right)^{\sigma } } \right) + \left( {\frac{169}{{16}}} \right)^{\sigma } } \right)} \right]^{ - 1} \\ \end{aligned}$$

(90)

where MRC_WSec designates SN's MRC gotten by using Sc FrFR technique in the third configuration.

1.2.3 Analysis of SoFR network

Two tier SoFR system based R-S WSN is assumed as shown in Fig. 

Fig. 30

Two-tier SoFR based R-S WSN structure a System design, b Cell arrangement, c Allotted frequency partition

30(a). Consequently, the outer BF is additional portioned in three dissimilar groups of S_Fs for interference reduction purpose as shown in Fig. 30(b). The first portion represents S_F1-1 which is assigned to BSs. In addition, the second set represents S_F1-2, S_F1-3, S_F1-4 and S_F1-5 which are allocated to SNs. Furthermore, the third set signifies S_F1-6 which is allotted to RS as portrayed in Fig. 30(c).

1.2.3.1 Analysis of first time slot

In this TS, there are 18 interfering cells affecting origin BS. These cells can be divided to 12 BSs in inner area and 6 C-Hs in outer area. Therefore, the analysis of this TS is similar to the classical SoFR analysis because the C-H does the communication wholly in this TS. So, SN's SIR CFT can be valued from Eq. (39).

1.2.3.2 Analysis of second time slot

In this TS, the communication is completed via the access link, so the RS transmits its amplified signal to SN of interest. Thus, Eq. (36) can be modernized to determine SN's SIR as follows

$$SIR_{SFR - RW} \left( {x,y} \right) = \frac{{\varepsilon \left( {x^{2} + \left( {y - 0.75} \right)^{2} } \right)^{\sigma } }}{{T_{iRW} \left( {x,y} \right) + \varepsilon T_{oRW} \left( {x,y} \right)}}$$

(91)

where T_oRW and T_iRW signify the outer and inner interference parameters, respectively. The interfering BSs’ normalized coordinates that use BF₁ in the inner area are demonstrated by Table 3. Thus, the inner interference parameter expression for SN positioned at coordinates (0,1) is finalized as follows

$$T_{iRW} = \frac{1}{3}4^{\sigma } + \frac{2}{3}7^{\sigma } + \frac{1}{3}13^{\sigma } + \frac{1}{3}19^{\sigma } + \frac{1}{3}$$

(92)

Additionally, the outer interference parameter in final form can be expressed as follows

$$T_{oWR} = \left( {\frac{121}{{16}}} \right)^{\sigma } + 2\left( {\frac{133}{{16}}} \right)^{\sigma } + 2\left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{169}{{16}}} \right)^{\sigma }$$

(93)

Therefore, by replacing the worth of Eqs. (92) and (93) into Eq. (91), the SN's SIR due to applying SoFR technique in the third configuration can be characterized as follows

$$SI{R_{WSFR}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( \begin{aligned}\frac{1}{3}{4^\sigma } + \frac{2}{3}{7^\sigma } + \frac{1}{3}{13^\sigma } + \frac{1}{3}{19^\sigma } + {\left( {\frac{{121}}{{16}}} \right)^\sigma }\\ + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {\left( {\frac{{169}}{{16}}} \right)^\sigma } + \frac{1}{3}\end{aligned} \right)} \right]^{ - 1}}$$

(94)

where SIR_WSFR characterizes SN's SIR of SoFR technique in the third configuration. Consequently, by.

replacing the value of Eqs. (39) and (94) into Eq. (76), the MRC CFT for SN of interest with ε = 1 can be computed as follows

$$\begin{aligned} MR{C_{WS}} = & {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( {\frac{2}{5}{4^\sigma } + \frac{4}{5}{7^\sigma } + \frac{2}{5}{{13}^\sigma } + \frac{2}{5}{{19}^\sigma } + {{\left( {\frac{{25}}{4}} \right)}^\sigma } + 2{{\left( {\frac{{31}}{4}} \right)}^\sigma } + 2{{\left( {\frac{{43}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma } + \frac{2}{5}} \right)} \right]^{ - 1}} + \\ & \quad {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( {\frac{1}{3}{4^\sigma } + \frac{2}{3}{7^\sigma } + \frac{1}{3}{{13}^\sigma } + \frac{1}{3}{{19}^\sigma } + {{\left( {\frac{{121}}{{16}}} \right)}^\sigma } + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {{\left( {\frac{{169}}{{16}}} \right)}^\sigma } + \frac{1}{3}} \right)} \right]^{ - 1}}\end{aligned}$$

(95)

where MRC_WS denote the attained MRC of SN of interest due to deploying SoFR technique in the third configuration.

B: Analysis of R-S COFDMA-WSN

The fourth configuration considers R-S WSN employed with R-S OFDMA network. This network uses FD AF RSs in the prime and ensuing links. Thus, in this configuration, each cell has centralized BS, a specific figure of SNs organized in a regular form, and two RSs positioned at distance equal to R_d and R_dR from the BS, respectively. Consequently, R-S OFDMA network and R-S WSN analytical handlings are conducted in the following subsections. These treatments are done through two principal steps. The first stage represents the analysis of R-S OFDMA network to calculate the C-H’s MRC in the prim link. Moreover, the second step signifies the analysis of R-S WSN for the SN’s MRC calculation in the ensuing link as revealed in Fig. 

Fig. 31

Conception diagram of fourth configuration in COFDMA-WSN for energy consumption reducing

31.

2.1 Analysis of R-S OFDMA network

Analytical treatments of R-S OFDMA network is presented in this subsection. Accordingly, the analysis of the two TSs is implemented to compute C-H’s MRC CFT using different FrFR techniques.

2.1.1 Analysis of St FrFR technique

Two-tier St FrFR system based R-S OFDMA network is considered as revealed in Fig. 

Fig. 32

Two-tier St FrFR based R-S COFDMA-WSN system a St FrFR3 system design, b Cell arrangement, c St FrFR4 system layout, d Designated frequency partition

32(a, c). Accordingly, every outer BF is advance portioned in seven S_Fs for interference level controlling between the SNs, RSs and BSs as displayed in Fig. 32(b). These S_FS are grouped into three sets. The first set represents S_F1-1, which is allocated to BS. Moreover, the second set signifies S_F1-2, S_F1-3, S_F1-4, and S_F1-5 which are assigned to the SNs. Additionally, the third set denotes S_F1-6, and S_F1-7, which are assigned to first RS (RS₁) and second RS (RS₂), respectively as depicted in Fig. 32(d).

Furthermore, the CFTs for St FrFR3 and St FrFR4 techniques are derived. It is observed, that current configuration provides equal C-H's SIR in the two TSs as the second configuration. This outcome can be attributed, to presence of the same interference sources that negatively affecting the centralized C-H. Consequently, Eqs. (45), and (49) can be used to represent C-H's MRC CFTs when St FrFR3 and St FrFR4 techniques are applied, respectively.

2.1.2 Analysis of Sc FrFR system

R-S OFDMA network that applies Sc FrFR system is considered as revealed in Fig. 

Fig. 33

Two-tier Sc FrFR based R-S COFDMA-WSN structure a System design, b Cell arrangement, c Assigned frequency partition

33(a). Accordingly, every outer BF is further portioned by the same manner as in St FrFR technique as revealed in Fig. 33(b, c). Also, CFTs for Sc FrFR technique are implemented. It is noticed, that Sc FrFR technique attains equal C-H's SIR in the two TSs for the second and fourth configurations. Consequently, Eq. (54) can be used to signify C-H's MRC CFTs when Sc FrFR technique is employed in the fourth configuration.

2.1.3 Analysis of SoFR technique

R-S OFDMA network that employs SoFR system is considered as depicted in Fig. 

Fig. 34

Two-tier SoFR based R-S COFDMA-WSN structure a System design, b Cell arrangement, c Allotted frequency division

34(a). Consequently, each outer BF is more portioned by similar method as in St FrFR technique as revealed in Fig. 34(b, c). Also, CFTs for SoFR technique are implemented. It is noticed, that this pattern of SoFR achieves C-H's SIR CFTs different from those in the second configuration. This outcome can be ascribed, to the utilization of SoFR power control factor. Hence, the fourth configuration applies SoFR technique analysis is completed through the following subsections.

2.1.3.1 Analysis of first time slot

In this TS, there are 18 cells (6 BSs in outer area and 12 BSs in inner area) which deploy S_F1-1. So, these 18 cells affecting the centralized BS. Therefore, the analysis of this TS is identical with the classical SoFR analysis because in this TS the communication is done completely by the BS. So, C-H's SIR CFT can be evaluated from Eq. (35).

2.1.3.2 Analysis of second time slot

In this TS, the communication is done via the access link, so RS transmits its amplified signal to the C-H. Hence, Eq. (55) can be updated to evaluate C-H's SIR in this TS as follows

$$SIR_{SFR - R4} \left( {x,y} \right) = \frac{{\varepsilon \left( {x^{2} + \left( {y - 0.25} \right)^{2} } \right)^{\sigma } }}{{T_{iR4} \left( {x,y} \right) + \varepsilon T_{oR4} \left( {x,y} \right)}}$$

(96)

where T_oR4 and T_iR4 symbolize the outer and inner interference parameters, respectively. The BSs’ normalized coordinates that use BF₁ in the inner area are demonstrated by Table 3. Consequently, the factor T_iR4 is stated as follows

$${T_{iR}}\left( {x,y} \right) = \frac{1}{7}\left[ \begin{array}{l}{\left[ {{{\left( {x \pm 2\sqrt 3 } \right)}^2} + {y^2}} \right]^\sigma }\\ + {\left[ {{{\left( {x + \frac{{\sqrt 3 }}{2}} \right)}^2} + {{\left( {y \pm \frac{3}{2}} \right)}^2}} \right]^\sigma }\\ + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y \mp 3} \right)}^2}} \right]^\sigma }\\ + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {y^2}} \right]^\sigma }\\ + {\left[ {{{\left( {x \pm \sqrt 3 } \right)}^2} + {{\left( {y \pm 3} \right)}^2}} \right]^\sigma }\\ + {\left[ {{{\left( {x - \frac{{\sqrt 3 }}{2}} \right)}^2} + {{\left( {y \pm \frac{3}{2}} \right)}^2}} \right]^\sigma }\end{array} \right]$$

(97)

Thus, the inner interference parameter for C-H sited at (0, 1) point is expressed by the succeeding equation

$$T_{iR4} = \frac{2}{7}\left[ {\left( \frac{7}{4} \right)^{\sigma } + \left( \frac{13}{4} \right)^{\sigma } + \left( \frac{19}{4} \right)^{\sigma } + \left( \frac{37}{4} \right)^{\sigma } + \left( \frac{49}{4} \right)^{\sigma } + \left( \frac{61}{4} \right)^{\sigma } } \right]$$

(98)

Furthermore, there are 6 RSs deploy S_F1-6 and interfering with serving RS. These RSs’ coordinates of are presented in Table 7. Thus, the factor T_oR4 is calculated as follows

$$\begin{aligned} & {T_{oR4}}\left( {x,y} \right) = {\left[ {{x^2} + {{\left( {y - 3.25} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \frac{3}{2}\sqrt 3 } \right)}^2} + {{\left( {y - 1.75} \right)}^2}} \right]^\sigma }\\ & \quad + {\left[ {{x^2} + {{\left( {y + 2.75} \right)}^2}} \right]^\sigma } + {\left[ {{{\left( {x \pm \frac{3}{2}\sqrt 3 } \right)}^2} + {{\left( {y + 1.25} \right)}^2}} \right]^\sigma }\end{aligned}$$

(99)

Accordingly, the outer interference factor for the C-H is expressed as follows

$$T_{oR} = \left( {\frac{169}{{16}}} \right)^{\sigma } + 2\left( {\frac{133}{{16}}} \right)^{\sigma } + 2\left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{121}{{16}}} \right)^{\sigma }$$

(100)

Subsequently, by replacing the value of Eqs. (98) and (100) into Eq. (96), C-H's SIR of the SoFR technique can be characterized as follows

$$SI{R_{SRC - H4}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( \begin{aligned}\frac{2}{7}\left( {{{\left( {\frac{7}{4}} \right)}^\sigma } + {{\left( {\frac{{13}}{4}} \right)}^\sigma } + {{\left( {\frac{{19}}{4}} \right)}^\sigma } + {{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma }} \right) + {\left( {\frac{{121}}{{16}}} \right)^\sigma } + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {\left( {\frac{{169}}{{16}}} \right)^\sigma }\end{aligned} \right)} \right]^{ - 1}}$$

(101)

where SIR_SRC-H4 characterizes C-H's SIR of SoFR technique applied in the fourth configuration. Consequently, by replacing the rates of Eqs. (35) and (101) into Eq. (40), the MRC CFT of the C-H with.

ε = 1 is calculated as follows

$$\begin{aligned}&MR{C_{SRC - H4}} = {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( \begin{array}{l}2\left( {{{\left( {\frac{7}{4}} \right)}^\sigma } + {{\left( {\frac{{13}}{4}} \right)}^\sigma } + {{\left( {\frac{{19}}{4}} \right)}^\sigma } + {{\left( {\frac{{31}}{4}} \right)}^\sigma } + {{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{43}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma }} \right)\\ + 3{\left( {\frac{{49}}{4}} \right)^\sigma } + {\left( {\frac{{25}}{4}} \right)^\sigma }\end{array} \right)} \right]^{ - 1}}\\ & \quad + {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( \begin{array}{l}\frac{2}{7}\left( {{{\left( {\frac{7}{4}} \right)}^\sigma } + {{\left( {\frac{{13}}{4}} \right)}^\sigma } + {{\left( {\frac{{19}}{4}} \right)}^\sigma } + {{\left( {\frac{{37}}{4}} \right)}^\sigma } + {{\left( {\frac{{49}}{4}} \right)}^\sigma } + {{\left( {\frac{{61}}{4}} \right)}^\sigma }} \right) + {\left( {\frac{{121}}{{16}}} \right)^\sigma }\\ + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {\left( {\frac{{169}}{{16}}} \right)^\sigma }\end{array} \right)} \right]^{ - 1}}\end{aligned}$$

(102)

where MRC_SRC-H4 denotes C-H’s MRC of R-S OFDMA network uses SoFR technique.

2.2 Analysis of R-S WSN

Analytical treatments of R-S WSN is introduced in this subsection. Accordingly, the two TSs are analyzed to evaluate SN's MRC CFT using different FrFR techniques.

2.2.1 Analysis of St FrFR system

Two-tier St FrFR system based R-S WSN network is considered as shown in Fig. 32(a, c). Furthermore, as mentioned above, different sets of S_Fs in the outer BF are allocated to BS, RS and SNs as depicted in Fig. 32(d). Furthermore, the CFTs for St FrFR3 and St FrFR4 techniques are derived. It is observed, that current configuration provides equal SN's SIR in the two TSs as the third configuration. This consequence can be attributed, to presence of the same interference sources that negatively affecting the SN of interest. Consequently, Eqs. (81), and (85) can be used to represent SN's MRC CFTs when St FrFR3 and St FrFR4 techniques are applied, respectively.

2.2.2 Analysis of Sc FrFR technique

R-S WSN that applies Sc FrFR system is considered as shown in Fig. 33(a). Therefore, outer BF is divided as presented in Fig. 33(c) to reduce the interference negative influence between cell elements as revealed in Fig. 33(b). Also, CFTs for Sc FrFR technique are executed. It is noticed, that Sc FrFR technique attains equal SN's SIR in the two TSs for the third and fourth configurations. Consequently, Eq. (90) can be used to indicate SN's MRC CFTs when Sc FrFR technique is employed in the fourth configuration.

2.2.3 Analysis of SoFR system

R-S WSN that deploys SoFR system is deliberated as depicted in Fig. 34(a). Therefore, BF allocation of outer region is revealed in Fig. 34(c) for cell layout shown in Fig. 34(b). In addition, CFTs for present pattern are carried out. It is noted, that SoFR technique in this configuration achieves SN's SIR CFTs unlike from those in the third configuration. Thus, the analysis of present pattern is completed in the following subsections.

2.2.3.1 Analysis of first time slot

In this TS, C-H hurts from 18 interfering cells. These cells contain 12 BSs in inner area and 6 C-Hs in outer area. Therefore, because of communicating by the C-H in this TS, this TS analysis is same as that in the classical WSN deploys SoFR technique. Accordingly, SN's SIR CFT can be computed from Eq. (39).

2.2.3.2 Analysis of second time slot

In this TS, RS transfers its signal through the access linkage after amplification process to the SN of interest. Hence, Eq. (91) can be revised to represent SN's SIR in this TS as follows

$$SIR_{SFR - RW4} \left( {x,y} \right) = \frac{{\varepsilon \left( {x^{2} + \left( {y - 0.75} \right)^{2} } \right)^{\sigma } }}{{T_{iRW4} \left( {x,y} \right) + \varepsilon T_{oRW4} \left( {x,y} \right)}}$$

(103)

where T_oRW4 and T_iRW4 indicate the outer and inner interference parameters, respectively. Additionally, Table 3 illustrates the normalized coordinates of the 12 interfering cells that use BF1 in the inner region. Accordingly, the inner interference parameter for SN of interest is valued as follows

$$T_{iRW4} = \frac{2}{7}4^{\sigma } + \frac{4}{7}7^{\sigma } + \frac{2}{7}13^{\sigma } + \frac{2}{7}19^{\sigma } + \frac{2}{7}$$

(104)

Additionally, the outer interference parameter can be expressed as follows

$$T_{oWR4} = \left( {\frac{121}{{16}}} \right)^{\sigma } + 2\left( {\frac{133}{{16}}} \right)^{\sigma } + 2\left( {\frac{157}{{16}}} \right)^{\sigma } + \left( {\frac{169}{{16}}} \right)^{\sigma }$$

(105)

Hence, by replacing the rates of Eqs. (104) and (105) into Eq. (103), SN's SIR of the SoFR technique can be characterized as follows

$$SI{R_{WSFR4}} = {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( \begin{aligned}{l}\frac{2}{7}{4^\sigma } + \frac{4}{7}{7^\sigma } + \frac{2}{7}{13^\sigma } + \frac{2}{7}{19^\sigma } + {\left( {\frac{{121}}{{16}}} \right)^\sigma }\\ + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {\left( {\frac{{169}}{{16}}} \right)^\sigma } + \frac{2}{7}\end{aligned} \right)} \right]^{ - 1}}$$

(106)

where SIR_WSFR4 characterizes SN's SIR of SoFR technique in the fourth configuration. Consequently, by substituting the rates of Eqs. (39) and (106) into Eq. (76), the MRC CFT of the SN of interest with ε = 1 is computed as follows

$$\begin{aligned} MR{C_{WS4}} = & {\left[ {{{\left( {\frac{1}{4}} \right)}^{ - \sigma }}\left( \begin{array}{l}\frac{2}{5}{4^\sigma } + \frac{4}{5}{7^\sigma } + \frac{2}{5}{13^\sigma } + \frac{2}{5}{19^\sigma } + {\left( {\frac{{25}}{4}} \right)^\sigma }\\ + 2{\left( {\frac{{31}}{4}} \right)^\sigma } + 2{\left( {\frac{{43}}{4}} \right)^\sigma } + {\left( {\frac{{49}}{4}} \right)^\sigma } + \frac{2}{5}\end{array} \right)} \right]^{ - 1}}\\ & \quad + {\left[ {{{\left( {\frac{1}{{16}}} \right)}^{ - \sigma }}\left( \begin{array}{l}\frac{2}{7}{4^\sigma } + \frac{4}{7}{7^\sigma } + \frac{2}{7}{13^\sigma } + \frac{2}{7}{19^\sigma } + {\left( {\frac{{121}}{{16}}} \right)^\sigma }\\ + 2\left( {{{\left( {\frac{{133}}{{16}}} \right)}^\sigma } + {{\left( {\frac{{157}}{{16}}} \right)}^\sigma }} \right) + {\left( {\frac{{169}}{{16}}} \right)^\sigma } + \frac{2}{7}\end{array} \right)} \right]^{ - 1}}\end{aligned}$$

(107)

where MRC_WS4 represents SN's MRC achieved by R-S WSN deploys SoFR technique.