Optimizing distance, transmit power, and allocation time for reliable multi-hop relay system
© Hiep et al; licensee Springer. 2012
Received: 13 October 2011
Accepted: 30 April 2012
Published: 30 April 2012
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.
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 [1–3].
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 [4–6]. Moreover, the capacity of a particular large Gaussian relay network is determined by the limit as the number of relays tends to infinity . In addition, a multi-hop relay network with multi antenna terminals in a quasi-static slow fading environment also has been considered . 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 end-to-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) . 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 end-to-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
where Hwii+1 is a matrix with independent and identically distributed (i.i.d.), zero mean, unit variance, circularly symmetric complex Gaussian entries, and lii+1 represents the path loss between the RS i and the RSi+1. The path loss is described in detail in the following section.
2.2 Path loss
2.3 Channel capacity
Let the channel capacity between the RS i and the RSi+1is Ci, and C denotes the end-to-end channel capacity of the multi-hop MIMO relay system (Figure 2).
3 Optimizing transmit power and distance
3.1 Optimizing transmit power
3.1.1 Optimization method
3.1.2 Numerical evaluation of optimizing transmit power
Antenna elements at Tx, Rx, RS
Transmit power of Tx (mW)
Total transmit power of RS (mW)
Noise power (mW)
Distance between Tx-Rx (m)
Average LOS W (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 RS1 is fixed. Therefore RS1 becomes a bottleneck node if d0 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.
3.2 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.
- (15)is rewritten as(16)
As a result, the optimized di can be obtained by with the condition that di is a real number within (0,d). For the system parameter described in the next section, only is satisfied.
In analyzing the performance of the system that has all channel models between the transceivers which are different, (Wi) is similar. The system in this case is indicated for scheme 2. The Taylor expansion is approximately used for a term . Then, the partial differential equation with respect to each di is obtained, and each di 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 di.
3.2.2 Numerical evaluation of optimizing distance
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-to-end 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.
4 Optimizing allocation time
4.1 Optimization method of allocation time
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.
4.2 Numerical evaluation of optimizing allocation time
5 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.
5.1 Mathematical method
Thus, when W i and d i , i = 1, . . . , m are equal, respectively (the optimum distance of scheme 1), SNRi 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.
5.2 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 and maximal channel capacity Cmax (optimize transmit power, distance and allocation time at scheme 1).
here, rand1, rand2, rand3 are random value within (0,1). Iterate step 2 until standard deviation of all channel capacities is smaller than sigma (σ).
Step 3: If end-to-end channel capacity of scheme 2 is close to maximal channel capacity , 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 σ and α, if σ is not tight enough, the algorithm doesn't converge. On the other hand, if σ is too tight, the algorithm takes a long time for convergence. Hence, for each α, the suitable σ needs to be considered.
In this article, we examined the performance of multi-hop 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 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.
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