An Application of GA for Symbol Detection in MIMO Communication Systems

An Application of GA for Symbol Detection in MIMO Communication Systems Sajid Bashir Centre for Advanced Studies in Engineering Islamabad, Pakistan Adnan Ahmed Khan Centre for Advanced Studies in Engineering Islamabad, Pakistan Abstract Multi-Input Multi-Output (MIMO) based communication system architecture promises increased capacity and high data rates. Increase in the number of transmit antennas and using higher order complex modulation schemes achieves even higher performance but with exponentially increasing complexity at the receiver end. This paper explores the application of genetic algorithm (GA) for reducing complexity in solving this NP Hard problem. This approach is particularly attractive as GA is well suited for physically realizable, real-time applications, where low complexity and fast convergence is of absolute importance. While an optimal Maximum Likelihood (ML) detection using an exhaustive search method is prohibitively complex, simulation results show that the GA optimized MIMO detection algorithm results in near optimal Bit Error Rate (BER) performance, with significantly reduced complexity. Results also suggest that the GA based MIMO detection out-performs the Vertical Bell labs Layered Space Time (V-BLAST) detector in BER performance without severely increasing the systems complexity 1. Introduction Wireless communication using MIMO antenna architecture also referred to as spatial multiplexed system is one of the most significant technological developments of last decade, which promises to play a key role in realizing the tremendous growth in the field of communication. MIMO systems achieve transmit and receive diversity by employing a Multi-Element Antenna (MEA) structures at both the transmitter and receiver. The challenge is to design signals to be sent by the transmit array and algorithms for processing those seen at the receive array, so that the quality of the transmission (i.e., bit error rate) and/or its data rate are improved. These gains lead to increased reliability, lower power requirements (per transmit antenna) and higher composite data rates (either higher rates per user or more users per link). The most substantial benefit offered by MIMO technology is attainment of these gains without the need Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007 Muhammad Naeem Engineering Sciences Department, Simnon Fraser University, Canada Syed Ismail Shah IQRA Univeristy Islamabad, Pakistan for additional spectral resources. An information theoretic analysis on the capacity of MIMO systems was presented in [1] assuming flat, quasi static and spatially independent Rayleigh fading environment with perfect Channel State Information (CSI) at the receiver. The study concluded that the capacity of multiple antenna channels increases linearly with the smaller of the number of transmit and receive antennas. On the other hand MIMO systems demand tremendous signal processing power at receiver side. Simultaneous transmission of signals from multiple transmitters complicates detection of composite signal at the receiver. Many MIMO detection algorithms for spatial multiplexing [2] are proposed and can be divided into linear and non-linear categories. Linear detectors are found to be computationally less complex with degraded performance when compared to non-linear methods. Maximum Likelihood (ML) and Vertical Bell labs Layered Space Time (V-BLAST) detectors [3] are two famous non-linear MIMO detection methods. The ML detector outperforms VBLAST in terms of BER performance, while the later has significant advantage in terms of complexity. Worst-case complexity of computing exact ML solution is generically exponential, due to NP-hardness [1] and cannot be solved in polynomial time. This paper focuses on non-linear detectors and makes effort to improve their performance using well known heuristic approach of GA. An Algorithm which works “reasonably well” on many cases, but for which there is no proof that it is always fast (e.g., evolutionary techniques) are classified as Heuristic Algorithms. GA is also member of this family of algorithms that has found numerous applications as a search tool in various fields, [8][9] are example of GA being applied in field of communications. Excessively high complexity increasing exponentially with increase in number of antennas is a major drawback restricting ML detector’s practical implementation in spatial multiplexed systems. Performance analysis at higher signal to noise ratios (SNR) rules out the choice of VBLAST because of significant comparative degradation in BER performance. Seeking the optimal solution, we report a GA assisted MIMO detection algorithm with a reasonable performance complexity tradeoff. GA based detector operates on an initial population of potential solutions and produces better approximations to the solution using ML equation as an objective function. Each iteration results in a new set of possible solutions created by selecting individuals based on fitness level by applying the so-called genetic operators; crossover and mutation. Direct application of GA, i.e., full power (or pure) GA, requires fine-tuning of its related parameters, and extensive computation, yet as a MIMO detector it might have lower computational complexity than optimum detector based. The rest of the paper is organized as follows. Section-2 provides the wideband spatial multiplexing system model and sets-up MIMO detection problem for flat, quasi static and spatially independent Rayleigh fading environment. A brief overview of existing MIMO detectors is given in section-3. Section-4 details the proposed detection algorithm. Performance of the proposed detector is reported in next sections followed by conclusions. X1 INPUT DATA STREAM DEMULTIPLXER Xn XN r H X × + P N n1 × + P N nm × + P N r1 rm DETECTOR DETECTOR OUTPUT rM Eq.1 illustrates the time selective fading case with one transmit antenna and one receive antenna. Because the channel under consideration is slowly fading, the coefficients will be independent of time, i.e., h[l] = h. In MIMO case MEAs are deployed at both the transmitter and the receiver. Consider the MIMO system shown in fig.1 where N different signals are transmitted and arrive at an array of M (N ≤ M) receivers via a rich-scattering flat-fading environment. The block transmission is assumed to contain one symbol i.e. L=1. Baseband equivalent model of received signal vector at each sampling instant can be represented as: r= 2. MIMO detection in flat fading channels 2.1 System model The channel model assumed is that of quasi-static fading. In this case the path attenuations do not vary over the duration of the block transmission with length ‘L’, thus the channel becomes an LTI system within this block. The attenuations also do not vary over the spectrum of the transmitted signal, and its impulse response is a scaled Dirac delta function. Thus the process can be modeled mathematically as a single-tap filter with complex coefficient h[l], or equivalently by real path attenuation α[l] and phase shift θ[l]: r [l ] = α [l ]e jθ [ l ] x[l ] + n[l ] (1) = h[l ] x[ l ] + n[l ] Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007 (2) Where r is an Mx1 vector of received symbols at each antenna, x is an Nx1 vector of symbols transmitted by each antenna, and n is an Mx1 vector of complex Additive White Gaussian Noise (AWGN) random variables seen at each receive antenna. The channel matrix H is an MxN matrix, whose elements hij represent the complex fading coefficients experienced by a signal transmitted from transmit antenna ‘j’ to receive antenna ‘i’. P is total transmit energy for one transmit antenna system and is normalized for N transmit antenna system. Each element of x is determined from the same set S comprised of C=2b constellation points where b is the number of bits per symbol. The receiver is also assumed to have perfect knowledge of the channel coefficients. This is a reasonable assumption when the fading is slow enough to allow estimation of the CSI with negligible error, as in the case of fixed wireless systems. nM Figure 1. MIMO system model with N transmit and M receive antennas P Hx + n N n r x x̂ Detector H Figure 2. Linear system model for MIMO communication system 2.2 Problem formulation Transmitted symbols from a known finite alphabet χ = {x1,…,xΩ}of size Ω = CN are passed to the channel. The detector chooses one of the possible transmitted symbol vectors from ‘χ’. The optimal detector for a MIMO system returns xˆ = x* , the symbol vector whose (aposteriori) probability of having been sent, given observed vector r, is the largest: x * = arg max P(x r) x∈ χ (3) x * = arg max P ( r x ) P ( x ) x∈χ P (r ) This is known as Maximum Aposteriori Probability (MAP) detection rule. Making the standard assumption that symbol vectors x ∈ χ are equiprobable i.e. P(x) was sent is constant, the MAP detector rule can be written as: max x* = arg P(r x) x∈χ (4) (5) A detector that always returns an optimal solution satisfying eq.5 is called Maximum Likelihood (ML) detector. Assuming the additive noise n to be white and Gaussian, the ML detection problem can be expressed as minimization of the squared Euclidean distance to a target vector r over N-dimensional finite discrete search set: min P r− H.x̂ x̂ML = arg x̂ N 3.2 Non-Linear MIMO detectors 3.2.1. VBLAST A popular nonlinear combining approach is the vertical Bell labs layered space time algorithm (VBLAST) [3] This detection method is also called Ordered Successive Interference Cancellation (OSIC). It uses the detect-and-cancel strategy similar to that of decision-feedback equalizer. Either ZF or MMSE can be used for detecting the strongest signal component used for interference cancellation. The performance of this procedure is generally better than ZF and MMSE. VBLAST provides a suboptimal solution with lower computational complexity than ML. However, the performance of VBLAST is degraded due to error propagation. (6) Thus conventional ML detection scheme needs to examine CN symbol combinations resulting in an exponentially increasing computational complexity with increases in C and/or N. High speed processing requirements of real time applications demand a comparatively simplified detection scheme. We report a GA assisted spatial multiplexing systems symbol detector that views MIMO symbol detection issue as a combinatorial optimization problem and try to approximate near optimal solution iteratively to counter ML complexity. 3.2.2. ML detector Maximum Likelihood detector is optimal but computationally very expansive. ML detection is not practical in large MIMO systems. Define cost function, cost, variables Select parameters START GENERATE INITIAL POPULATION DECODE CHROMOSOME FITNESS EVALUATION OPITMALITY TEST 3. Existing MIMO detectors 3.1 Linear MIMO detectors YES SOLUTION FOUND WITH BEST INDIVIDUAL S NO MUTATION CROSSOVER SELECTION GENERATE NEW POPULATION A straightforward approach to recover x from r is to use an N x M weight matrix W to linearly combine the elements of r to estimate x, i.e. x̂ = Wr. Figure 3. The GA flow diagram 3.1.1. 4. GA based detection for MIMO systems Zero-Forcing (ZF) The ZF algorithm attempts to null out the interference introduced from the matrix channel by directly inverting the channel with the weight matrix [1]. 3.1.2. Minimum mean squared error (MMSE) A drawback of ZF is that nulling out the interference without considering the noise can boost up the noise power significantly, which in turn results in performance degradation. To solve this, MMSE minimizes the mean squared-error, i.e. J(W) = E{(x- x̂ )*(x- x̂ )}, with respect to W [4], [5]. Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007 GA is an inspiration based on principles of natural genetics and selection. Algorithm starts by defining optimization variables, optimization cost and the cost function. Convergence/Fitness test follows different components of algorithm, as explained. 4.1 Initialization of GA All potential solutions of a problem are level to simplify the following GA operations. In MIMO detection ∀x ∈ χ ML search space are coded as binary encoded in bit recombination that form the strings called chromosomes. It is common to select the initial population randomly from solution space or derive from a linear/non-linear detector output. Each chromosome is a combination of the probable solution for all transmit antennas. Normally, the population size Npop is taken as the product between N, the number of transmit antennas, and C, the number of all possible solutions of each transmit antenna. (7) N pop = 2 b × N by cost. This new population of Nkeep chromosomes forms mating pool of good parents. 4.2 Fitness evaluation using cost function 4.6 Crossover Each member of the population at each generation is evaluated, and according to its fitness value (output of the cost function), it is assigned a probability to be selected for reproduction. Following equation acts as the cost function to evaluate the fitness of each chromosome: Crossover operation is applied to pairs of selected parents and creates offspring. Process of mating by applying crossover operator creates two offspring by combining subparts of bit strings of two selected parents [6]. The crossover point is randomly selected along the chromosome lengths and portions up to that point are exchanged between two parents. Probability of crossover pc is user controlled and usually set to a high value. f = r − Hx 2 (8) The optimal solution of eq.8 should yield a minimum value. ML detector evaluates ∀x ∈ χ results in computation time that varies exponentially with the number of transmit antennas. As will be explained, the use of GA in a MIMO system can reduce the computation times of eq.8 significantly. The best chromosome in a generation should have the least value of the objective function. If the value of the best chromosome in the present generation is larger than its counterpart in the previous generation, the chromosome with the largest value of the objective function in the present generation will be replaced by the best chromosome of the previous generation. This operation ensures that at least the useful information contained in the present generation is passed on to the next. 4.3 Optimality test If the optimal criterion is satisfied, that is, when any one of all f in the population is less than a predetermined threshold, or if the generation number has exceeded a predefined value, which is also commonly taken as the product between N and C, then go to 4.8. Otherwise, go to Step 4.4. 4.5 Reproduction This step is intended to replace the chromosome with largest objective function value by the best chromosome of the same generation. The Offspring to replace the bad parents is created by the processes of ‘Crossover’ and ‘Mutation’. 4.7 Mutation These offspring are mutated through mutation operator and new members of next generation are produced. Mutation operator simply alters each bit of the binary chromosome randomly with a user-controlled probability pm. Generally, the crossover probability pc is close to 1 and the mutation probability pm is close to 0. New individuals replace members of previous populations with worst fitness values. Algorithm iterates until optimization of objective function is achieved. For given predefined number of generations and population size, the computation times of eq.8 vary linearly with (Npop X Ngen) for GA based detector, which is much smaller than the factor of 2NXbin the ML detection. The improvement is clearly significant. 4.8 Decision making If the defined optimality criterion is met the algorithm terminates its execution and output is generated. Otherwise further refining of solution is carried out through step 4.2 by producing next generation. 4.4 Selection This process uses fitness value and serves to provide chromosomes for the subsequent recombination operations. Of Npop chromosomes in a generation, only Nkeep= Npop * psel survive for mating, and bottom Npop – Nkeep are discarded to make room for new offspring. Deciding how many chromosomes to keep is arbitrary and normally 50% (Psel = 0.5) of population survives ‘Selection’ process after chromosomes have been sorted Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007 5. Simulation and numerical results BER performance analysis of reported GA-MIMO detector and its comparison with other MIMO detectors is presented in this section. Improvement in performance of spatial multiplexing system using reported detector is shown with simulation results and a comparison of computational complexity is also presented in last section. 5.1 BER Performance Analysis Performance of GA-MIMO detector is analyzed for a 4x4 (MxN) and a 6x6 MIMO system using 4-QAM and 16-QAM schemes. Population size ‘Npop’ and number of generations ‘Ngen’ are dependent on system configuration i.e. number of transmit antennas used in spatial multiplexing system and the QAM size. For each case the parameters of GA-MIMO detector are different and noted against in the simulation results. For example for a 4x4, 4-QAM system, ‘Npop’ equals ‘14’ and it grows to ‘26’ for 16-QAM system. ‘Ngen’ is kept as 10 and 18 for the two systems respectively. The SNR (Eb/No) is average signal to noise ratio per antenna ρ / σ 2 where average power per antenna is ρ and σ 2 is noise variance. Simulation environment assumes quasi static Rayleigh flat-fading channel with no correlation between sub-channels. Fig.4 presents the bit error rate (BER) versus Eb/No performance of reported GA detector compared with ML, VBLAST and ZF detectors in 4x4 spatial multiplexing system using 4-QAM. As shown by simulation results of fig.4, at BER of 10-3 proposed detector has a performance gain of approx 5-dB compared to VBLAST. Fig.5 shows BER performance comparison of two 4x4 MIMO systems using different modulation schemes. Using higher order modulation scheme increases the complexity of optimization problem. To counter this increased complexity Npop and Ngen are increased from 14 to 26 and from 10 to 18 for 16-QAM system respectively. Relative improvement in performance compared to VBLAST for case of 16-QAM seems little but as shown in fig.6 a further tuning of GA parameters enhances relative BER performance of reported detector. This results in a linear increase in computational complexity as discussed in next section. Thus to achieve a balance between BER performance and computational complexity careful selection of values for GA parameters is vital. Performance analysis of MIMO systems with different configurations using identical modulation scheme is presented in fig.7. Systems with 4x4 and 6x6 configurations using 4-QAM are used. Ngen is taken as 10 and 16 whereas Npop is taken as 14 and 20 for 4x4 and 6x6 systems respectively. Relative improvement in performance for the two systems using GA-MIMO detector at 10-4 BER is also shown. It is evident that relative improvement for higher order system is more than for a lower order system. For example at 10-4 BER, the performance gain for 4x4 system is approx 4 dB whereas for 6x6 system it is approx 7 dB. 5.2 Computational complexity comparison Computational complexity of reported GA detector is examined and compared with ML and VBLAST detectors Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007 in this section. As the hardware cost of each algorithm is implementation-specific, we try to provide a rough estimate of complexity in terms of number of complex multiplications. The computational complexity is computed in terms of the N, M and C i.e. constellation size. ML detection requires N(M+1)CN complex multiplications, as seen from (6). Where CN(NM) is for matrix multiplication and CN(N) is for square operation. For VBLAST, pseudo-inverse of matrix (HHH)-1HH requires 4N3+2MN2 computations [7]. The pseudoinverse matrix is calculated N times with decreasing dimension. In addition, the complexity of ordering and interference canceling is ∑ [N (N N −1 i=0 − i) + 2 N ] . Therefore, total complexity of VBLAST (γVBLAST) results in: ∑ (4i N i =0 3 + 2 Mi 2 ) + ∑ ( N ( N − i ) + 2 N ) N −1 i =0 (9) 1 ⎞ ⎛7 ⎛5 2 ⎞ γ VBLAST = N 4 + ⎜ + M ⎟ N 3 + ⎜ + M ⎟ N 2 + NM (10) 3 ⎠ ⎝2 ⎝2 3 ⎠ For proposed detector, before applying GA operators, fitness of each chromosome in initial population (Npop) is assessed. Based on the parameters used for simulation i.e. selection probability ps=0.5, crossover operator with pc=0.9 is applied to pairs of Nkeep chromosomes which undergo logical AND/OR operations. Hence number of crossover operations is equal to 0.5*Npop(1-ps)*pc. Mutation of one bit i.e. logical NOT operation is applied to crossover results with probability pm=0.1 on a population of size Npop-Nkeep and these logical operations are performed Ngen times. However the most expensive computational process of multiplication results in complexity of: (11) γ Genetic = N pop N gen (MN ) Since the Proposed detector takes initial solution guess as the output of ZF-VBLAST therefore, total computational complexity of proposed detector results in: γ GA−MIMO = γ Genetic + γ VBLAST (12) Computation complexity of reported GA-MIMO detector for different system configurations along with the performance gains are mentioned in table-1. The results are derived from simulations results for BER performance analysis discussed in previous section. The complexity estimate in table-1 is only meaningful in the order of magnitude sense since it is based on the number of complex multiplications. 5.3 Complexity-Performance trade off For a 4x4 MIMO system BER performance of different detectors shown in fig.4. As shown in table.1 a gain of 5 dB is achieved for proposed detector over VBLAST at a BER of 10-4. The complexity of GAMIMO detector for achieving this performance gain is 57.6% to that of ML detector. The complexity performance trade off is found to be better when the proposed GA-MIMO detector is used with higher order system configurations. As shown with fig.5 and fig.6 for a 4x4 system with 16-QAM the gain in BER performance increases from 1 dB to 6 dB with a corresponding increase in relative complexity from 0.626% to 1.32%. Performance gain of 7 dB is observed for a 6x6 system with 4-QAM with a relative complexity of 8.5%. These numeric figures suggest that GA-MIMO detector achieves a complexity/performance trade off balance for system configurations where practical implementation of ML detector is not realizable and performance of VBLAST is severely degraded. [4] S. Haykin, Adaptive Filter Theory, 3rd ed. Prentice-Hall, 1996. [5] A. Sayed, Fundamentals of Adaptive Filtering. WileyIEEE Press, 2003. [6] Randy L.Haupt and Sue Ellen Haupt, “Practical Genetic Algorithms”, Second Edition. A John Wiley & sons, 2004. [7] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. John Hopkins University Press, 1996. [8] M. Y. Alias, S. Chen and L. Hanzo, “Genetic Algorithm Assisted Minimum Bit Error Rate Multiuser Detection in Multiple Antenna Aided OFDM”, 2004. [9] Kai Yen and Lajos Hanzo, “Genetic Algorithm Assisted Joint Multiuser Symbol Detection and Fading Channel Estimation for Synchronous CDMA Systems”, IEEE Journal on selected areas in communication, vol. 19, No. 6, June 2001. 0 10 6. Conclusion 7. References -1 BER 10 -2 10 -3 10 -4 10 20 Eb/No 30 40 0 10 ML 4-QAM VBLAST 4-QAM GA-MIMO 4-QAM ML 16-QAM VBLAST 16-QAM GA-MIMO 16-QAM -1 10 -2 10 10 [2] H. Bolcskei, D. Gesbert, C. and Papadias , Space-Time Wireless Systems: From Array Processing to MIMO Communications, Cambridge University Press, 2005. 10 Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007 10 System with Ngen=10 and Npop=14 [1] I. Emre Telatar. Capacity of multi-antenna Gaussian channels. Technical report, October 1995. [3] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs Technical Journal, vol. 1, pp. 41-59, Autumn 1996. 0 Figure 4. Bit Error Vs Eb/No for a 4x4 4-QAM MIMO BER In this paper, power of GA is explored when used as a search tool. The algorithm is applied to MIMO detection problem and it achieves reduced computational complexity for complex system configurations. This natural optimization method shows promising results when compared with traditional ML detection for systems with increased number of antennas and using higher order complex modulation schemes. The optimality is however compromised slightly but reduction of computations show a major gain. Although VBLAST detector has a reduced complexity, its BER performance is much inferior to the proposed detector. The simulation results show that the proposed detector significantly out-performs the VBLAST detector without severely compromising the systems complexity. Simulation results also show that a trade off can be made between complexity and optimality. Employing parallel processors for proposed algorithm will further reduces this complexity. GA based solution of MIMO detection problem has proven to be very cheap and shows a promise for Heuristic Algorithms in MIMO detection for complex modulation schemes. ML ZFE VBLAST GA-MIMO -3 -4 0 10 20 Eb/No 30 40 Figure 5. BER Analysis for 4x4 system using 4-QAM and 16-QAM, with values of Npop taken as 14 and 26 & Ngen taken as 10 and 18 respectively. 0 10 -1 ML Detector (4X4 System) ZFE VBLAST GA-MIMO -1 10 10 --------- 6x6 System -2 -2 10 BER BER 10 -3 10 -3 10 -4 10 0 VBLAST POPULATION SIZE = 30 POPULATION SIZE = 40 ZF POPULATION SIZE = 36 10 20 Eb/No -4 10 30 40 0 Figure 6. Effect of Population Size on BER performance for a 4x4 system using 16-QAM with Npop = 26 5 10 15 20 25 Eb/No (dB) 30 35 Figure 7. BER Performance Comparison for a 4x4 and 6x6 System using 4-QAM with Ngen= [10 16] and Npop= [14 20] for the two systems. Table 1. Complexity and performance analysis System Configuratoin Population Size (Npop) Number of Generatoins (Ngen) γML γGA-MIMO = γGenetic+γVBLAST 4x4 MIMO System with 4-QAM 14 10 5120 2240 + 712 = 2952 26 18 1310720 7488 + 712 = 8200 13192 15688 17352 172032 11520 + 3048 = 14568 4X4 MIMO System with 16-QAM 6X6 MIMO System with 4-QAM 30 36 40 16 26 20 Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007 Performance/Complexity Comparison ML & GA-MIMO Complexity Relative Performance Improvement to ML relative to VBLAST γ GA− MIMO (at BER = 10-4) ×100 γ ML 57.6% 5 dB 0.626 % 1 dB 1% 1.19% 1.32% 2 dB 4 dB 6 dB 8.5% 7 dB