Joint Estimation of Amplitude and Direction of Arrival for Far Field Sources using Intelligent Hybrid Computing

: In this study, an intelligent hybrid computing technique is presented to estimate jointly the amplitude and Direction of Arrival (Elevation angle) of far field sources. In this intelligent hybrid scheme, Genetic Algorithm (GA) is hybridized with Pattern Search (PS), in which GA is working as a global optimizer while PS is used as local optimizer for further improvement of the results. GA and PS techniques are also applied independently to compare with GA hybridized with pattern search. The fitness evaluation function is formed by the Mean Square Error (MSE) of the desired response with the estimated one. This function is simple and requires a single snapshot to reach the optimum solution. A sufficient number of Monte-Carlo Simulations is used to evaluate the convergence rate, MSE and estimation accuracy of each scheme


INTRODUCTION
Efficient estimation of DOA is one of the most challenging and important concern in the field of array signal processing and wireless communication.It has applications in sonar, radar, seismic exploration and biomedical engineering, etc., Gavan and Ishay (2001) and Khan et al. (2012).Heaps of work has been done in this area by using classical methods as well as heuristic computational approaches (Krim and Viberg, 1996).
The importance and applicability of meta-heuristic techniques is growing rapidly with the passage of time among the researchers because these techniques can be coped easily and performs well even in the presence of local minima and low Signal to Noise Ration (SNR).These techniques include Genetic Algorithm (GA), Differential Evolution (DE), Ant Colony Optimization (ACO), Bee Colony Optimization (BCO) and Particle Swarm Optimization (PSO), etc.It has been observed that the reliability and applicability of meta-heuristic techniques increases drastically if these are combined with any other capable approaches such as Interior Point Algorithm (IPA), Active Set (AS), Pattern Search (PS), etc., Junaid et al. (2011a, b), Fawad et al. (2012a, b, c and d).In Fawad et al. (2012a), PSO is combined with PS while in Fawad et al. (2012b), DE and PSO are hybridized with PS to jointly estimate the parameters (DOA, range, amplitude) of far field sources and near field sources respectively which are impinging on Uniform Linear Array (ULA).In Fawad et al. (2012c); GA and Simulated Annealing (SA) techniques have been used in combination with PS to estimate the 3-D parameter (DOA, elevation angle and Azimuth angle) of far field sources using L shape array.Similarly GA and SA are hybridized with IPA in Fawad et al. (2012d) to estimate the 3-D parameters (range, amplitude and elevation angle) of near field sources.In Fawad et al. (2012a, b, c and d), MSE is used as an objective function and it has been proved that whenever the hybrid approach is used it performed well as compared to GA, PS, DE, PSO, IPA and SA alone.The same fitness function has been used in Fawad et al. (2012e).
In this study, Intelligent Hybrid Computational (IHC) approach based on (GA-PS) is presented for joint estimation of amplitude and DOA for far field sources.The results of hybrid GA-PS technique are compared with GA and PS alone.Different cases are discussed on the basis of various numbers of sources, different values of SNR and MSE.The objective evaluation function is formed by the Mean Square Error (MSE) of the desired response with the estimated one.This function is simple and requires a single snapshot to reach the optimum solution.The applicability as well as the reliability of the given scheme is tested on the basis of large number of Monte-Carlo simulations.Moreover, the robustness of given schemes are examined in the presence of Low Signal to Noise Ratio (SNR) methodology.(Khan et al., 2011).All sources are considered to be narrow band and having known frequency ߱ where each source has different elevation angle (α) and amplitude (s).For L≤M, the output at m-th element for l-th source can be written as: where, m = 1, 2,… M ݊ = The additive white Gaussian noise added at the output of m-th element in the ULA The output of complete array in matrix-vector form can be written as: where in Eq. ( 2 where, Ψ = kd (m-1) cosߙ for l = 1, 2, …, L. In Eq. ( 2), C (α) is called steering matrix composed of steering vectors for sources as defined in Eq. ( 6).It is obvious from Eq. ( 1), the unknown parameters are the amplitudes ܵ and the angle of Arrival ߙ Hence, the problem which has to be solved is they jointly estimation of amplitudes i.e., ܵ ଵ , ܵ ଶ , …, ܵ and DOA of sources i.e., ߙ ଵ , ߙ ଶ , …, ߙ at the output of ULA.

PROPOSED METHODOLOGY
In this section, we discussed the proposed methodology structure for the estimation of amplitude and DOA of far field sources.The flow diagram of proposed structure is shown in Fig. 2, in which the solution is initiated with GA and ended up with PS.A MATLAB built-in optimization tool box is used for GA and PS for which the parameters settings are shown in Table 1.PS method is gradient free technique and can be used as global and local optimizers.It plays very important rule for optimization problems like "Bound constrained minimization" and "Globaly Convergent Augmented Lagrangian algorithm" Torczon (1997).PS method is more effective especially in the presence of fewer minima and become more effective when it is hybridized with DE, GA and PSO (Fawad et al., 2012a, b and c).Hence to take advantage of PS, in present study it is not only applied independently but also used as a local search optimizer with GA i.e., the best individual results got through GA is given as an input to PS for further improvement.
In the current problem the length of each chromosome is 2*L where L is the total number of sources.The first L genes in each chromosome represent the elevation angles while the next L genes show amplitudes of sources.The i-th chromosome can be written as: where ‫ܮ‬ ௦ and ܷ ௦ are the lower and upper bound of signals amplitude for i = 1, 2, …, k, j = 1, 2, …, L.
Step 2: In step 2, the error is minimized between desired response and estimated response using the following relation: where, i represents i-th chromosome.In Eq. ( 9), ‫ݕ‬ is defined in Eq. ( 1) while: ˆêxp( ( 1) cos( ) 1 Step 3: Terminate, if any of the following condition is satisfied and go to step 5 else go to step 4: • The fitness value has been achieved i.e., ߦ ≤10 ିଵହ • The total number of iteration has been completed • Predefined value of the TolFun has been achieved Step 4: By using the parameters of elitism, mutation and crossover as shown in Table 1, reproduces the new population and go to step 2.
Step 5: Give the best individual results of GA as an input to PS for further improvement.
Step 6: Store all the values for later discussion and comparison.

RESULTS AND DISCUSSION
In this section, comparison between GA, PS and GA-IPA is made on the basis of convergence rate, MSE and estimation accuracy and proximity effect.All the values of DOA are taken in radians while the enter element spacing between two consecutive elements in the array is same i.e., λ/2.Initially no noise is added from case 1 to case 3.All the results are carried out for 100 independent runs and for a threshold value of MSE is 10 ିଶ .
Case 1: In this case, the convergence rate, MSE and estimation accuracy are evaluated for two far field sources.The ULA is composed of five elements and the desired values of amplitudes and DOA are ‫ݏ({‬ ଵ = 7, ߙ ଵ = 0.5236 rad), ‫ݏ(‬ ଶ = 5, ߙ ଶ = 1.9199)}.As shown in Table 2, all the three techniques produced fairly good results.However, it is quite obvious that GA becomes more accurate when it is combined with PS and hence, it produces better results as compared to GA and PS alone.The second best scheme in this archive is GA alone.
In Fig. 3 and 4, the percentage convergence and MSE of each scheme is shown respectively.One can see that the convergence arte and MSE of each scheme improves with the increase of elements in the array.In this case also, the hybrid GA-PS approach converges more times and maintains a minimum MSE as compared to PS and GA alone for each number of elements in the ULA.3, the performance of all three schemes despoiled a bit due to increase of sources.However, once again the hybrid GA-PS technique proves to be the finest technique as compared to PS and GA alone.The second best is once again GA alone.4.
In Fig. 7 and 8, the convergence rate and MSE are evaluated for four sources respectively.Once again, it is quite evident that GA-PS not only converges more times but also maintained minimum MSE as compared to PS and GA alone.The Second best is again GA.

CONCLUSION AND RECOMMENDATIONS
In this study, intelligent hybrid approach is used for joint estimation of amplitudes and DOA of far field sources.The proposed hybrid scheme produced better results as compare to GA and PS alone.MSE has been used as fitness function which is optimal in nature and only single snapshot is required to attain the optimum solution.A major advantage of the given scheme is its simplicity in concept, ease in implementation and a fewer budgets is required for hardware implementation.All the three schemes fail when the number of sensors in the array is kept less than the number of sources.In future, we will use the same intelligent hybrid approach for null steering and side lobes in the field of adaptive beam forming.

Fig. 1 :
Fig. 1: ULA having M elements Problem formulation for far field sources: In this section, a data model is developed for L sources lying in the far field and impinging on ULA from different directions.The ULA is composed of M elements having same spacing ″d″ between two consecutive elements as shown in Fig. 1(Khan et al., 2011).All sources are considered to be narrow band and having known frequency ߱ where each source has different elevation angle (α) and amplitude (s).For L≤M, the output at m-th element for l-th source can be written as:

Table 1 :
Parameter settings for GA and PS GA -

Table 2 :
Estimation accuracy of two sources Scheme

Table 4 :
Estimation accuracy for four sources