Optimizing distance, transmit power, and allocation time for reliable multi-hop relay system

In multi-hop relay systems, the end-to-end channel capacity is restricted by bottleneck node. In order to prevent some relay nodes from being the bottleneck of system and to guarantee the end-to-end channel capacity, the method of optimizing transmit power, distance and allocation time is proposed in this article. We show that the optimizing distance has more end-to-end channel capacity than the optimizing transmit power in case that both the distance and the transmit power are changeable. However, the optimizing transmit power can let the system reach high end-to-end channel capacity when the relay nodes have to shift from the desired location. We also propose the Markov Chain Monte Carlo method to optimize all transmit power, distance and allocation time simultaneously. The optimizing all transmit power, distance and allocation time is the most effective and achieves the highest channel capacity. Based on the average signal-to-noise-ratio, the average channel capacity is evaluated in this article.


Introduction
In the future, it is believed that the MIMO service area will become popular. Therefore, a MIMO relay system is considered. However, in a relay system with one relay, when the number of relay antenna elements is less than the number of transmitter and receiver antenna elements, the capacity of MIMO relay system is lower than that of the original MIMO system. In addition, when the number of relay antenna elements is equal to or more than the number of the transmitter and receiver antenna elements, a MIMO relay system can provide the same average capacity as an original MIMO system. In other words, although the number of relay antenna elements is larger than the transmitter and receiver antenna elements, the capacity of MIMO relay system cannot exceed the channel capacity of original MIMO system [1][2][3]. Therefore, a system with multi relays called multi-hop relay system was proposed and have been discussed in several literatures. The Gaussian MIMO relay channel with fixed channel condition has derived upper bounds and lower bounds that can be obtained numerically by convex programming [4][5][6]. Moreover, the capacity of a particular large Gaussian relay network is determined by the limit as the number of relays tends to infinity [7]. In addition, a multi-hop relay network with multi antenna terminals in a quasi-static slow fading environment also has been considered [8]. However, these researches assumed the signal-to-noise-ratio (SNR) at receiver(s) is fixed, the distance between the transceivers and the transmit power of transmitter(s) are not considered.
In multi-hop MIMO relay systems, when the distance between the base station (Tx) and the final receiver (Rx) is fixed, the distance between the Tx to a relay node (RS), an RS to an RS, an RS to the Rx called the distances between transceivers, is shorten. Consequently, the SNR and the channel capacity are increased. However, according to the number of the relay nodes, the location and the transmit power of each relay node; the channel capacity of each relay node is changed. In addition, the endto-end channel capacity is limited by bottleneck node. Therefore, to obtain the upper bound of end-to-end channel capacity, the location of each relay node meaning the distance between the transceivers and the transmit power of each relay node need to be optimized. We have analyzed performance of multi-hop MIMO relay system with amplify-and-forward (AF) [9]. The distance between the transceivers is optimized when the transmit power of each relay node is assumed to be equal. However, the location of the relay nodes is not always changeable. Consequently, in order to obtain a certain value of endto-end channel capacity, the distance and the transmit power need to be optimized. In this article, the distance and the transmit power are optimized separately or simultaneously to guarantee the end-to-end channel capacity in decode-and-forward scheme multi-hop relay systems. In addition, allocation time is optimized to guarantee and/or to obtain the higher end-to-end channel capacity. Moreover, the channel capacity that is mentioned in this article is average channel capacity. The rest of the article is as follows. After the introduction of the system model in Section 2, we propose the optimizing method of transmit power and distance in Section 3 and the optimizing method of allocation time in Section 4. The optimizing of all transmit power, distance and allocation time simultaneously is described in Section 5. Finally, Section 6 concludes the article.
2 Multi-hop MIMO relay system 2.1 System model Figure 1 shows m relays intervened multi-hop MIMO relay system. Here, K i (i = 0, . . . , m + 1) denotes the number of the antenna elements at the Tx, the Rx and each relay node. d i (i = 0, . . . , m) represents the distance between the transceivers. The distance between the Tx and the Rx is fixed as d. The signal is transmitted from the Tx to the RS 1 . At the RS 1 , the signal is decoded, encoded and transmitted to the RS 2 . Similarly, the signal is transmitted over and over until the signal reaches to the final receiver. We assumed the transmit power of the Tx (E Tx ) and the total transmit powers of relay node (E RS ) are fixed regardless of the number of the relay nodes and the number of antenna elements at each relay. The transmit power of each relay node is denoted by E i . Moreover, the transmit power of each relay is equally divided into each antenna element. On the other hand, as described in Section 1, if the number of antenna elements at one relay is smaller than the other, this relay will be a bottleneck of the system and the end-to-end channel capacity will be restricted by this relay. Since in this article we consider the distance, transmit power and allocation time, the number of antenna elements at each relay is assumed to be the same as that of the Tx and the Rx and denoted by M. Moreover, we assume that the time-division-multipleaccess (TDMA) algorithm is applied to control the transmission of each relay node. The allocation time of RS i is denoted as t i .
Let H ii+1 denotes a K i+1 × K i channel matrix between the RS i and the RS i+1 . Since the path loss is taken into consideration, H ii+1 is the composite matrix. We model H ii+1 as where H wii+1 is a matrix with independent and identically distributed (i.i.d.), zero mean, unit variance, circularly symmetric complex Gaussian entries, and l ii+1

Tx
Rx RS represents the path loss between the RS i and the RS i+1 . The path loss is described in detail in the following section.

Path loss
Since there are a lot of obstacles, such as huge buildings, in propagation environment, the path loss is necessary to consider being attenuated by the reflection. The power of the signal is reduced when the reflection occurs. An amount of the reduction by one time of reflection is called reflection factor. It is natural that the reflection factor is changed according to the matter of the obstacles, the angle of reflections and so on. However, in this article, the reflection factor of all reflections is assumed to be the same and denoted by a. The path loss between the transceivers is described in Figure 2. The path loss in this case is expressed as [10] where ref i is the number of reflection while a signal is transmitted between RS i and RS i+1 . In addition, in order to obtain the number of reflection, the propagation environment coefficient W i is defined as the average distance from a reflection point to the next reflection point. In other words, it is the average of line-of-sight (LOS) distance between RS i and RS i+1 . Therefore, the number of reflection between the transceivers can be expressed as ref i = d i W i . Consequently, the path loss in (2) can be rewritten as (3)

Channel capacity
Let the channel capacity between the RS i and the RS i+1 is C i , and C denotes the end-to-end channel capacity of the multi-hop MIMO relay system ( Figure 2).
The transmission in multi-hop relay system is assumed to be controlled accurately. There-fore, when the signal S i-1 is transmitted from the RS i-1 , the received signal at the RS i is expressed as cite9, Here, P i = diag(p i1 , p i2 , . . . , p iK i ) is the transmit power matrix and is assumed to be subject to a constraint where Tr(·) and p ij denote the trace and transmit power of j th antenna element of RS i , respectively. n i is the noise vector with i.i.d., zero mean, s 2 variance. The transmit power of every antenna in the same relay node was assumed to be equal.
Moreover, the channel capacity is represented as follows. where Since the channel capacity of each relay node is independent from each other, the end-to-end channel capacity is equal to channel capacity of bottleneck relay node.

Optimizing transmit power and distance
In order to explain the optimizing of the distance and transmit power clearly, the allocation time of each relay node is assumed to be the same, t i = 1 m+1 . In addition, H w ii + 1 H H w ii + 1 (i = 0, . . . , m) is independent from the distance and the transmit power. Consequently, the SNR can be examined instead of a channel capacity. Hence, in order to avoid that some nodes become the bottleneck and to obtain high channel capacity, the channel capacity of each node should be equal. Consequently, the received SNR of each node is necessary to be equal. here,

Optimizing transmit power 3.1.1 Optimization method
When the location of the relay nodes is not changeable, the transmit power of each relay node should be optimized to increase the channel capacity of bottleneck relay node. Consequently, the necessary transmit power can be obtained from (7).
Note that total transmit power of relay node is fixed.
From (8) we have By substituting each E i to (9), it can be rewritten as Consequently, the necessary transmit power of RS 1 is obtained.
Similarly, the transmit power of each relay node is obtained by,

Numerical evaluation of optimizing transmit power
The system parameters summarized in Table 1 are used as an example for evaluating the optimizing method mentioned above. Let the distance between the transceivers be random. The channel capacity, in case the number of the relay nodes are 3, 6, 9, are described in Figure 3, here the average propagation environment coefficient W, , meaning the average of LOS distance between the Tx and Rx, is set as 500 m.
As shown in Figure 3, in case the number of the relay nodes is 9, the channel capacity of bottleneck node is improved and the end-to-end channel capacity is increased. However, since the transmit power of Tx is assumed to be constant, the channel capacity of RS 1 is fixed. Therefore RS 1 becomes a bottleneck node if d 0 is large, such as the number of the relay nodes is 3. In this case, the end-to-end channel capacity can not be improved. Moreover, SNR is increased by transmit power to the 1st power and decreased by distance to the 2nd power. Hence, in order to increase the channel capacity, the huge transmit power needs to be provided when the distance is large. Under the assumption that total transmit power of relay node is fixed; the channel capacity is not considerably improved. It is the reason why the end-to-end channel capacity of the system with 6 relay nodes is low.

Optimizing distance between transceivers 3.2.1 Optimization method
Since the optimizing of transmit power remains some drawbacks as mentioned above, the distance between the transceivers needs to be considered. In order to analyze the distance more easily, the transmit power of each relay node is assumed to be equal. Therefore, the channel capacity only depends on the distance.
By solving (7), the optimized distance becomes as follows.
Firstly, at scheme 1, we assume that all channel models between the transceivers are the same (W ). Therefore, by substituting the path loss of (3) in (7), this equation becomes, The Taylor expansion is approximate to g(d i ), and (14) can be expressed as here, (15) is rewritten as The function y is added to this equation. Hence, Moreover, we can describe as Therefore, From these equations, the resolvent cubic equation can be obtained.
Let c 2 1 = y , the resolvent cubic equation can be rewritten as From this equation, function y is obtained. here Additionally, by applying the resolvent cubic equation to (18), we have Thus, Consequently, the function x can be obtained.
As a result, the optimized d i can be obtained by 4b 4 with the condition that d i is a real number within (0,d). For the system parameter described in the next section, only In analyzing the performance of the system that has all channel models between the transceivers which are different, (W i ) is similar. The system in this case is indicated for scheme 2. The Taylor expansion is approximately used for a term a −2d i W i . Then, the partial differential equation with respect to each d i is obtained, and each d i can be obtained similarly to be mentioned above. However, in this case, we have made the Taylor expansion, solving partial differential m + 1 times to obtain each d i .

Numerical evaluation of optimizing distance
The system model is the same as mentioned above, and the system parameters are summarized in Table 1. Figure 4 shows the end-to-end channel capacity responded to each number of the relay nodes, i.e., 3, 6, 9. By contrast to the optimizing of the transmit power, the optimizing of the distance can change the channel capacity of RS 1 and improve the channel capacity of all bottleneck nodes. The comparison in channel capacity of optimizing the distance and optimizing the transmit power is shown in Figure 5. It is clear that in the case of optimizing the distance, the channel capacity of all relay nodes is the same and relatively higher than the end-to-end channel capacity in the case of optimizing the transmit power. The optimizing of the distance is effective; however, the relay node can not always be set at the desired location. For example, the desired location is on roads, in rivers, and so on. Thus, the relay node needs to be shifted in the front or the rear of the desired location. As a result, the channel capacity is decreased. In order to remain high channel capacity, the transmit power is adjusted after shifting distance by method of optimizing transmit power mentioned above. Figures 6 and 7 show the end-to-end channel capacity before and after adjusting the transmit power. In this scenario, the channel capacity of all relay nodes after adjusting the transmit power is almost the same, and achieves the end-to-end channel capacity of the system without shifting the location of the relay nodes. The optimizing of the transmit power in this scenario is much more effective.
In order to compare the end-to-end channel capacity when the number of the relay nodes is changed, the average end-to-end channel capacity from 1, 000× of calculation is shown in Figure 8. The end-to-end channel capacity is increased when the number of the relay nodes is small. Moreover, when the number of the relay nodes exceeds a certain value, the end-to-end channel capacity is decreased. The reason is that the SNR increases when the number of the relay nodes increases, however the allocation time of each relay node reduces rapidly. Therefore, although the SNR is high, the endto-end channel capacity is decreased. As a result, there is the optimum number of the relay nodes for maximum end-to-end channel capacity.
As shown in Figure 8, we can confirm that after shifting the location of the relay nodes, the end-to-end channel capacity is rapidly decreased, especially when the shifted distance is large and/or the number of the relay nodes increases. However, the optimizing of the transmit power is quite effective in this case, and the end-toend channel capacity after adjusting the transmit power is approximate to the end-to-end channel capacity of the system without shifting the location of the relay nodes.

Optimization method of allocation time
Similar to transmit power and distance, in order to guarantee the end-to-end channel capacity, the allocation time of each relay needs to be optimized. To explain the optimizing of the allocation time, we assume all distance and transmit power to be fixed. When every relay transmits the signal in its allocation time, the endto-end channel capacity is as follows.
The practicable channel capacity of the system is guaranteed when As a result, Therefore, the optimized allocation time is expressed as Consequently, the channel capacity of each relay node is the same and the end-to-end channel capacity is expressed as follows.
From (30), we have It means that the end-to-end channel capacity of the system after optimizing allocation time is higher than that of the system with equal allocation time.

Numerical evaluation of optimizing allocation time
In this section, the optimizing of the distance, the transmit power and the allocation time by mathematical method at scheme 1 is compared. The channel model is the same as mentioned above. Figure 9 shows the channel capacity of each relay node in case the number of the relay nodes are 3 and 9. In the case of 3 relay nodes, the difference of channel capacity before and after optimizing the allocation time is small. It can be explained that if there is a relay node that has the channel capacity much smaller than that of another, the endto-end channel capacity before and after optimizing allocation time is restricted by this relay node. In the case of optimizing allocation time, let's assume that the channel capacity of RS k is the lowest (C k ≪ C i , i ≠ k,). It means 1 C k 1 C i . Consequently, the end-to-end channel capacity in (30) can be changed as follows.
However, as mentioned in the previous section, the end-to-end channel capacity after optimizing allocation time is higher than that before the optimizing. Additionally, the end-to-end channel capacity of optimizing allocation time is lower than that of optimizing distance, but higher than that of optimizing transmit power (in the case of 9 relay nodes in Figure 9). We can confirm the relation between optimization results to Figure 10. Figure 10 shows the average from 1, 000 × calculation of the end-to-end channel capacity when the number of the relay nodes is changed from 1 to 10.

Optimized transmit power, distance, and allocation time simultaneously
Till now, we explained the method of optimizing transmit power, distance and allocation time separately in Sections 3 and 4. Each method is effective. However, the optimizing of the transmit power, the distance and the allocation time simultaneously is expected to achieve higher channel capacity than optimizing each one separately.

Mathematical method
In order to optimize the transmit power, the distance and the allocation time, firstly let's fix the distance, equalize the allocation time and optimize the transmit power. Therefore, the optimization method and the result in Section 3 can be applied and (7) can be rewritten by To solve this equation, each path loss should be expressed by Taylor expansion. It is relatively complex, especially at scheme 2. Consequently, in order to optimize the distance, the transmit power and the allocation time more easily in any channel model, the Markov Chain Monte Carlo (MCMC) method is proposed in the following section. Moreover, from (13), the SNR i can be expressed as follows.  Thus, when W i and d i , i = 1, . . . , m are equal, respectively (the optimum distance of scheme 1), SNR i becomes maximum. In this case, the equal allocation time is also the optimum solution. In other words, the optimized distance, transmit power and allocation time at scheme 1 is one of the optimal solutions for maximal end-to-end channel capacity of any channel model. Consequently, the optimizing of the distance, the transmit power and the allocation time lets the end-to-end channel capacity reach to this maximum.

Markov chain Monte Carlo method
The MCMC method is constructed to find the optimal state of transmit power, distance and allocation time that has the end-to-end channel capacity close to the maximal channel capacity. The algorithm is explained as follows.
Calculate W = 1 (m+1) m i=0 (W i ) and maximal channel capacity C max (optimize transmit power, distance and allocation time at scheme 1).
Step 1: Create d i , E i and t i randomly. Subject to (t i ) = 1.
Step 2: Calculate all channel capacities C i , and soften them from high to low. Adjust the distance, the transmit power and the allocation time to make all channel capacity to be almost the same.
Step 3: If end-to-end channel capacity of scheme 2 is close to maximal channel capacity C max −C C max ≤ α , the algorithm is finished. Otherwise, return to step 1.
Compare to the mathematical method, MCMC algorithm is easier to optimize the distance, the transmit power and the allocation time simultaneously at any channel model. However, the MCMC algorithm requests running in the computer and the convergence of this algorithm should be discussed. The convergence is dependent on s and a, if s is not tight enough, the  On the other hand, if s is too tight, the algorithm takes a long time for convergence. Hence, for each a, the suitable s needs to be considered.
Let's denote the average of all channel capacities to beC . When the end-to-end channel capacity (C = min(C i )) is approximate to the maximal channel capacity (C max ), we haveC ≈ C max and|max (C i ) −C| ≈ |min (C i ) −C| . Thus, s can be described by As a result, s = C max a is the suitable value. Since C max changes when the number of the relay nodes changes. s is changed for each number of the relay nodes and a. Figure 11 shows the end-to-end channel capacity in case s is changed, i.e., 1%, 5%, and 10%. Here, let W i be random within (0, 1000) and satisfy 1 m+1 m i=0 W i = 500 . According to a, the end-to-end channel capacity of MCMC method is different. However, with small alpha, MCMC method optimizes the transmit power, the distance, the allocation time simultaneously and achieves the maximal channel capacity in any channel model.

Conclusion
In this article, we examined the performance of multihop relay systems with decode-and-forward method. The optimizing of the transmit power, the distance and the allocation time is effective in preventing some relay nodes from becoming the bottleneck of the system and in guaranteeing the end-to-end channel capacity. However, the optimizing of the distance is the most effective and the optimizing of the transmit power is the least effective. The optimizing of the transmit power is effective when the location of the relay nodes is shifted within a short range from the desired location. The    Number of relay nodes End-to-end channel capacity [bit/s/Hz] Optimizing distance, equal trans. power and allocation time Optimizing trans. power, random distance, equal allocation time Optimizing allocation time, random distance, equal trans. power Figure 10 Comparing the end-to-end channel capacities of optimizing the transmit power, the distance and the allocation time. MCMC algorithm was proposed to optimize all transmit power, distance and allocation time simultaneously. MCMC method can achieve the maximal channel capacity.
In this article, in order to simplify the analysis, we have analyzed the system under Gaussian channel model. However, the performance of this system needs to be analyzed under the channel model which is close to the practice. Additionally, in order to apply the optimization method to any channel model, more general path loss functions needs to be considered. We considered the transmit power, the distance and the allocation time to guarantee the end-to-end channel capacity. The other method, such as the changing of modulation and/ or coding is left for future studies.