Ergodic Capacity Evaluation of Multi-Hop Decode-and-Forward MIMO-OFDM Relaying Network

: Spatial diversity plays a significant role in wireless communication systems, including the Fourth Generation (4G) and Fifth Generation (5G) systems, and it is expected to be a fundamental part of the future wireless communication systems as well. The Multiple-Input Multiple-Output (MIMO) technology, which is included in the IEEE 802.16jstandard, still holds the most crucial position in the 4G spectrum as it promises to improve the throughput, capacity, spectral, and energy efficiency of wireless communication systems in the 2020s. This makes MIMO a viable technology for delay constrained medical and health care facilities. This paper presents an approximate closed-form expression of the ergodic capacity for the Decode-and-Forward (DF) protocol MIMO-Orthogonal Frequency Division Multiplexing (OFDM) relaying network. Although MIMO-OFDM is highly valuable for modern high-speed wireless communication systems, especially in the medical sciences, its performance degrades in multi-hop relay networks. Therefore, in this paper, an approximate closed-form expression is derived for an end-to-end ergodic capacity of multi-hop DF MIMO-OFDM wireless communication system has been presented. Monte-Carlo simulations are conducted to verify the performance of the presented analysis regarding the capacity (bits/s/Hz) for different SNR-dB values for single, 2 × 2, 4 × 4, and multi-hop DF MIMO-OFDM systems. The presented results provide useful insights for the research on the end-to-end ergodic capacity evaluation. This paper presents the closed-form expressions of the ergodic capacity of a single- and multi-hop DF-based MIMO-OFDM relaying systems. The derived expressions are validated using a numerical method. The presented results provide many useful insights, such as the end-to-end ergodic capacity of the considered system model, which degrades with the increasing number of hops. It is also shown that the ergodic capacity of a higher-order MIMO is reduced when the number of hops increases. Moreover, the presented results reveal that the capacity loss of relay-based MIMO-OFDM is approximately constant for all the OFDM subcarriers. A future research direction would be to extend our derived results to a massive MIMO multi-hop relaying system based on the findings presented in this study. present study.


Materials and Methods
This section analyzes the derived closed-form expressions of the ergodic capacity for an endto-end system using a multi-hop DF MIMO-OFDM relaying network.

System Model
This paper considers a K-hop wireless relaying network, as shown in Fig. 1. In the considered network, source node S transmits data to a destination node D via R relay nodes. Also, this work considers a MIMO-OFDM communication model that employs n sub-carriers per OFDM symbol. The number of transmitting antennas is denoted as M, whereas the number of receiving antennas is denoted as L. It is also considered that all M transmitting antennas transmit different symbols simultaneously and operate at the same frequency band. Hence, the considered model is a conventional L× M MIMO-multiplexing relaying model. Moreover, it is assumed that at each time slot, only one node transmits, and that only the receiver end has the channel state information (CSI) available.
represents the fading of the kth OFDM subcarrier, which is modeled as an independent and identically distributed (IID) random variable that obeys complex zero-mean Gaussian random distribution. The gains of relay channels H 0 , H 1 , H 2 , . . . , H R are independent of each other and follow the DF relaying scheme. The signal received at the ith relay node is given by: where x i−1 denotes an mM × 1 vector that represents the transmitted signal, while n i is an nL × 1 complex additive noise vector at the ith node modeled as an IID with a zero-mean, circular symmetry, and complex Gaussian random vector with the variance σ 2 k . The transmitted signal x i from the ith relay node in a DF multi-hop relaying communication system is taken from the transmitted signal x o from the source node.

Multi-Hop DF Relaying System Capacity
The ergodic capacity of a multi-hop DF MIMO-OFDM relaying system is given by: where C i = 1, 2, . . . , R + 1 represents the ith hop's capacity and is given by: where E [.] denotes the expectation operator, I L is an L × L identity matrix, ρ i = P i nMσ 2 0 , and P i , In this work, P i denotes the total transmitted power of the nodes, including both the source node and relay nodes, at the transmitter end. Let us define N min = min (M, L), N max = max (M, L), and Δ = N max − N min . To express the ergodic capacity, it is necessary to use the expressions in terms of eigenvalues λ k, i . . , R, which can be written as: The above-mentioned equation can be simplified as follows: where λ k, i is obtained by a random selection from the set of eigenvalues of The probability density function (PDF) of λ k, i can be expressed as: Then, the cumulative distribution function (CDF) of λ k, i can be calculated by integrating the PDF, which is expressed as: Using n! m! u m μ n−m+1 , the following expression can be obtained: To the best of the authors' knowledge, the closed-form expression of Eq. (8) has not been reported in the related literature, even for a two-hop relaying system. Therefore, this paper presents the closed-form expressions for the CDF in Eq. (8) and the ergodic capacity of a multi-hop DF relaying system. Using Eq. (3), the ergodic capacity can be expressed as: where Similarly, by expressing the ergodic capacity using . . , R, we get: where λ k is selected from the eigenvalue set of λ k , it is assumed that the fading per hop is IID. Let X j n j=1 denote the IID random variables using the order statistics (X 1:n < X 2:n . . . < X n:n ), then the PDF of the ranked random variable (in ascending order) X j:n (1 ≤ j ≤ n) is given by: where F (x) represents the CDF, f (x) represents the PDF, and B (a, b) = 1 0 t a−1 (1 − t) b−1 dt denotes a beta function, and it can be expressed as: Using Eqs. (11) and (12), the PDF of the smallest random variable (i.e., f 1:n (x)) is given by: Thus, the PDF of λ k can be expressed as a combination of CDF and PDF of λ k, i , which is given by: where f λ k, i (λ) and F λ k, i (λ) represent the PDF and CDF of λ k, i given by Eqs. (6) and (7), respectively, and R represents the number of relay stations. Using a binomial expansion, a general expression of Eq. (14) valid for any of relay hops R can be written as: For instance, for R = 1, Eq. (15) becomes: By substituting Eqs. (6) and (7) into Eq. (16), the PDF of λ k is obtained as: After extensive simplification, the ergodic capacity of a multi-hop DF protocol-based MIMO-OFDM relaying system given by Eq. (10) can be obtained as: where The integrals in Eqs. (19)-(21) can be solved using the following approximation: where δ ≈ 0.577. Using the above approximation, Eqs. (19)-(21) can be further expressed in the approximate closed-form, which is given by: Finally, by substituting Eqs. (23)-(25) into Eq. (18) and conducting some mathematical manipulations, the expression for the anticipated ergodic capacity in its approximate closed-form of a multi-hop DF relay protocol for a MIMO-OFDM system can be derived as: As a special case, when Δ ≈ 0.557 in the above expressions, and R = 0 in Eq. (15), then by solving the integral in Eq. (18), the anticipated ergodic capacity in its approximate closed-form of a single-hop DF relay protocol for a MIMO-OFDM system can be derived as:

Graphical Results and Analytical Expressions
All figures presented in this paper are obtained using MATLAB software. The analytical work is validated using MC simulations. Analytical expressions defining some symbols with descriptions are provided in Tab. 1.

Simulation Results
This section presents the numerical and simulation results to verify the proposed system's analytical findings of Fig. 1. The comparison of the numerical and theoretical results for the 2 × 2 and 4 × 4 MIMO multiplexing systems conforming to the IEEE 802.16j standard regarding the two-hop DF capacity performance is presented in Fig. 2. The results were obtained using the number of antennas and considering the high value of the SNR region. The results in Fig. 2 show that the capacity of the DF MIMO multiplexing relaying network decreased compared with the network that did not include relays, i.e., when R = 0. The simulation results of the capacity of 2 × 2 and 4 × 4 multi-hop DF MIMO relaying configurations are presented in Figs. 3 and 4, respectively, are compared with the theoretical results. As shown in Figs. 3 and 4, higher losses in capacity were caused by the increase in the order of the MIMO-DF relaying network configuration. Furthermore, the derived expressions were evaluated for different multi-hop DF relaying MIMO-OFDM systems. The results of different OFDM systems having different numbers of carriers, including the single-and multi-carrier configurations, are displayed in Fig. 5. The performances of the single-carrier, 256-OFDM, and 1024-OFDM systems were evaluated with 2 × 2 MIMO multiplexing using the DF protocol. The relaying approach has been envisioned in the sub-plots of Fig. 5 considering two setups, i.e., single-hop and three hops, where R = 0 and R = 2, respectively. As shown in Figs. 5a and 5b, for the same SNR value, the capacity was reduced when relaying was used. For the single-carrier OFDM, the capacity was reduced by 3.82 bits/s/Hz; for the 256-OFDM carrier, the decrease was 3.91 bits/s/Hz; and for 1024-OFDM, the decrease was 3.9 bits/s/Hz. Thus, based on the simulation result, it can be concluded that there is a hopping loss in MIMO-OFDM even if the number of relays exceeds one. Consequently, the presented results highlight the importance of the proposed expressions that have been derived through rigorous analysis and validated by MC simulations.

Conclusions
This paper presents the closed-form expressions of the ergodic capacity of a single-and multihop DF-based MIMO-OFDM relaying systems. The derived expressions are validated using a numerical method. The presented results provide many useful insights, such as the end-to-end ergodic capacity of the considered system model, which degrades with the increasing number of hops. It is also shown that the ergodic capacity of a higher-order MIMO is reduced when the number of hops increases. Moreover, the presented results reveal that the capacity loss of relaybased MIMO-OFDM is approximately constant for all the OFDM subcarriers. A future research direction would be to extend our derived results to a massive MIMO multi-hop relaying system based on the findings presented in this study.

Funding Statement:
The author(s) received no specific funding for this study.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.