,..., 1 , ,..., 1
,n ,k K n N
k = =
α for each set of Rk ,k =1,...,K. We employ (2.1) for the f(c) but set Pe=10−4 and N0 =1. Table 3.3 shows the average of 250 − ×100%
d D
d for each (N, K, APBS), where D and d denote the actual optimal power consumption of (2.3) and the power consumption of good enough solution obtained by our approach, respectively. For each ABPS, the average of the average d−dD×100% of various N and K is shown in the last row of Table 3.3, which indicates the average deviation of d from D is around 1.0% in various congestion condition of the system. This shows that the good enough solutions we obtained are really good enough.
Table3.3
The average d−dD×100% for each set of N,K and ABPS
Average − ×100% d
D d
N K ABPS =3 ABPS =4 ABPS =5
32 4 0.246 0.300 0.283 32 6 0.646 1.260 0.874 32 8 1.606 1.634 1.115 32 10 1.837 1.778 1.568 64 4 0.198 0.114 0.131 64 8 0.557 0.422 0.241 64 12 1.872 1.663 1.295 64 16 2.227 2.328 2.090 128 4 0.056 0.081 0.080 128 8 0.131 0.110 0.144 128 16 0.501 0.863 0.852 128 32 2.572 2.842 2.793 Average 1.037 1.116 0.956
3.4 Concluding Remarks
In this chapter, we have proposed an OO theory based four-stage approach to solve the adaptive subcarrier assignment and bit allocation problem of multiuser OFDM system for a good enough feasible solution. To resolve the computational complexity problem caused by the DPG method in our approach, we propose a hardware architecture to implement the DPG method so as to exploit deepsubmicron technology. Comparing with some existing methods, the quality of the good enough feasible solution obtained by our approach is excellent, and the estimated computation time meets the real-time application requirement.
Chapter 4
A Computationally Efficient Method for Large Dimension Subcarrier Assignment and Bit Allocation Problem of Multiuser OFDM System
In multiuser OFDM communication system, the increasing dimension will (i) adverse the computational complexity of the already time consuming mathematical programming based approaches [3], [5] and (ii) enlarge the discrete solution space, which will degrade the quality of the solutions obtained by the more local-like approaches [6]-[9] as well as the corresponding computation time. Although the method presented in chapter 3 can result a good enough solution and can meet real-time application requirement, however, implementing the first stage in hardware is almost impossible for area concern due to the large-dimension ASABA problem. Thus, dealing with large-dimension ASABA problem based on software-like method is a challenging issue in wireless communication, and the purpose of our proposed second method is proposing a computationally efficient method to solve the considered problem for a good enough solution for large-dimension ASABA problem.
The quality of the solution obtained by the mathematical programming based approach [3]
is considered to be one of the best so far. However, they arbitrarily round the optimal continuous subcarrier assignment pattern off to the closest discrete values may cause infeasibility problem and not guarantee to be a good solution, if feasible. To avoid the undesirable effect caused by rounding off, we will handle the discrete solution space directly and use a global-like approach. However, the global searching techniques [30] such as Genetic Algorithm (GA), Simulated Annealing method, Tabu Search method and Evolutionary Programming are not adequate here because of their tremendous computation time, which is even worse than the mathematical programming based approach due to (i) evaluating the objective value of a feasible subcarrier assignment pattern is time consuming, (ii) handling the constraints is not an easy task and (iii) the size of the discrete solution space is huge.
Evaluating the exact performance (i. e. the objective value) of a feasible subcarrier assignment pattern is a conventional “value” concept. However, it is indicated in OO theory that the performance order of discrete solutions is likely preserved even evaluated by a surrogate
model. In other words, the OO theory claims that there is high probability that we can find the actual good discrete solutions if we limit ourselves to the top n% of the estimated good discrete solutions evaluated by a surrogate model [12]. Thus, to retain the merit of global searching technique while avoiding the cumbersome conventional performance evaluation of a discrete solution, our approach is based on OO theory to solve the considered problem for a good enough solution with high probability using limited computation time.
The approach consists of three OO stages. First of all, we will reformulate the considered problem to separate it into subcarrier assignment and bit allocation problem such that the objective function of a feasible subcarrier assignment pattern is the corresponding optimal bit allocation for minimizing the total consumed power. Then, in the first stage, we will develop an easy-to-evaluate approximate objective function to estimate the objective value of a subcarrier assignment pattern and employ a GA to search through the huge discrete solution space to find the top s subcarrier assignment patterns based on the estimated objective values.
In the meantime, a subtle representation scheme and a repair operator for GA need be designed to handle the constraints of the considered problem. In the second stage, we use an off-line trained ANN to estimate the objective values of the s subcarrier assignment patterns obtained in stage 1 and pick the top l patterns based on the estimated objective value. In the third stage, we use the exact objective function to evaluate the l subcarrier assignment patterns obtained in stage 2, and the best one associated with the corresponding optimal bit allocation is the good enough solution that we seek. In the proposed three-stage approach, the models employed to evaluate a solution are varying from very rough (stage 1) to exact (stage 3). In the meantime, the candidate solution space is reduced from the original huge solution space (stage 1) to only l candidate solutions (stage 3). In general, a more accurate approximate objective function will take more time to evaluate a solution; however as can be seen from our three-stage approach, when a more accurate approximate objective function is used, the search space is already reduced considerably, and the computation time is largely reduced accordingly.
We organize Chapter 4 in the following manner. In Section 4.1, we will reformulate the considered problem. In Section 4.2, we will present our three OO stages to solve the
considered problem. In Section 4.3, we will apply our algorithm to numerous large-dimension ASABA cases and compare with some existing algorithms in the aspects of solution quality and computation time. Finally, we will draw a conclusion in Section 4.4.