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path loss prediction

top height and the building block spacing. Average values were used since these variables change continuously along one route. For the determination of these geometric parameters a map with building database was used [5]. III. THE GRNN ARCHITECTURE The General Regression Neural Network (GRNN) is a neural network architecture that can solve any function approximation problem. The learning process is equivalent to finding a surface in a multidimensional space that provides a best fit to the training data, with the criterion for the “best fit” being measured in some statistical sense. The generalization is equivalent to the use of this multidimensional surface to interpolate the test data. 1 k K i x1 x2 φk (x) xM w1 wk wK Input layer Hidden layer Output layer φ1 (x) xm φK (x) y Figure 1.General regression neural network Figure 1 is the overall network topology implementing the GRNN. As it can be seen from the figure, the GRNN consists of three layers of nodes with entirely different roles: The input layer, where the inputs are applied, The hidden layer, where a nonlinear transformation is applied on the data from the input space to the hidden space; in most applications the hidden space is of high dimensionality. The linear output layer, where the outputs are produced. The most popular choice for the function ϕ is a multivariate Gaussian function with an appropriate mean and autocovariance matrix. The outputs of the hidden layer units are of the form [ ] ( ) ( ) ( ) σ−−−=ϕ 2x k Tx kk 2exp vxvxx (1) when x kv are the corresponding clusters for the inputs and y kv are the corresponding clusters for the outputs obtained by applying a clustering technique of the input/output data that produces K cluster centers [6]. y kv is defined as ( ) ( ) ∑ ∈ = kclusterpy y k pyv (2) Nk is the number of input data in the cluster center k, and ( ) ( ) ( )x k Tx k x k,d vxvxvx −−= (3) with ( ) ( ) ∑ ∈ = kclusterp x k p x xv (4) The outputs of the hidden layer nodes are multiplied with appropriate interconnection weights to produce the output of the GRNN. The weight for the hidden node k (i.e., wk) is equal to ( )∑ = σ − = K 1k 2 2x k k y k k 2 ,d expN v w vx (5) The selection of an adequate set of training examples is very important in order to achieve good generalization properties. The set of all available data is separated in two disjoint sets: training set and test set. The test set is not involved in the learning phase of the networks and it is used to evaluate the performances of the models [7]. The configuration of the neural network model is determined by the nature of the problem to be solved. The dimension of the input vector used defines the number of inputs neurons. IV. PREDICTION MODEL For the LOS case, the proposed neural network model is trained with physical data, which includes the distance between transmitter and receiver, the width of the streets, the height of the buildings, the building separation and the distance between base station and rooftop height. The GRNN model has a single output which represents the normalized propagation path loss. The dimension of the training set is 1013 and the rest of 2026 examples were used to test the model and to compare it with the Walfisch-Bertoni model (WB) [8], the single slope model (SSM) [9] and the modified COST231-Walfisch-Ikegami model (CWI) [4]. Table 1 represents the performance achieved by each of the above-mentioned models for the entire test data. Between the empirical algorithms, in LOS case, the single regression model achieves the best performance. However, this model is based on the distance between transmitter and receiver, the frequency and the propagation factor. It was found that in the LOS paths the power decay factor ranges from minimum 1.56 up to 3.05. Table 1.Comparison between the NN approach and the other empirical models in LOS case µ [dB] Std [dB] RMS [dB] GRNN 5.04 4.54 6.78 SSM 5.43 4.80 7.24 CWI 7.04 4.06 8.23 WB 9.09 4.51 10.24 The better performance achieved by the neural network model is due to the various inputs parameters used to predict the propagation path loss and due to the generalization properties of the network.

Figure 2 represents the measured and predicted propagation path loss by the generalized RBF-NN model, the SSM model and the CWI model for a specific route, in LOS case. -115 -105 -95 -85 -75 -65 -55 13 21 30 39 48 57 67 76 86 95 105 114 124 133 143 152 155 159 164 169 Distance from transmitter [m] Propagationpathloss[dB] Measurements RBF-NN CWI SSM Figure 2.Measured and predicted path loss for LOS case, urban environment For the NLOS case, we have built three neural network models: 1. The first one, called NN1, is trained with four parameters: the distance between transmitter and receiver (d), the width of the street (w), the building separation (b) and the building height (h). 2. The second model, called NN2, in addition to the above-mentioned data we also included the distance between base station height and building height (dhbs). 3. In the third neural model, named NN3, we have included the street orientation (ψ) in addition to the parameters used for the training of NN2. A set of 420 examples was used for training purpose while the rest of 1680 was used for test purpose. Table 2 presents the results obtained by the three different GRNN models, for the training set and for the test set. R represents the correlation between predicted values and the measurement data. Table 2. GRNN models for NLOS case Training set Test set [dB] µ Std RMS µ Std RMS R NN1 4.35 4.64 6.36 5.50 5.27 7.62 0.91 NN2 4.11 4.38 6.00 5.31 5.07 7.34 0.92 NN3 1.47 2.52 2.92 3.67 3.88 5.35 0.96 Table 3 Comparison between the proposed NN3 model and the other empirical models in NLOS case µ [dB] Std [dB] RMS [dB] NN3 3.67 3.88 5.35 SSM 6.35 4.37 7.75 WB 6.08 4.14 7.40 CWI 6.96 4.62 8.38 Table 3 represents the comparison of the performance achieved by the NN3 model and the Single Slope model (SSM) [9], the Walfisch-Bertoni model [8] and the modified COST231-Walfisch-Ikegami model [4] for the entire test patterns. Figure 3 represents the measured and predicted propagation path loss by the GRNN model, CWI model and measurements for one route characterized by a base station antenna located below rooftop -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 19 29 40 54 71 89 108 127 145 164 183 202 221 240 259 278 297 316 Distance [m]Pathloss[dB] Measurements Neural Model CWI Model Figure 3. The comparison between the measured and predicted path loss, in case of a particular route, NLOS case, urban environment V. ERROR CORRECTION MODEL Our purpose is to build an error correction model: the neural network is used to compensate for the errors obtained by applying COST-Walfisch-Ikegami (CWI) model [4]. In Figure 4 is represented the training phase of the neural network structure and in Figure 5 is represented the schematic diagram of the prediction phase. The inputs of the neural network consist of the available physical parameters and the output is represented by the difference error between the path loss computed by CWI model and measurements. Physical data Computed PL Measured PL Neural Network Training Algorithm Neural Network PLcomputed PLmeasured E - + Epredicted PLcomputed PLcorrected = PLcomputed - Epredicted + - Figure 4. The schematic diagram of the training process of the neural network Since our purpose is to train the neural network to perform well for all the routes, we should build the training set including points from the entire set of measurements data. For training and test purpose we have

used the same number of patterns as in the prediction model. The test set was used to test the models and to compare them to each other. Also, a comparison between the “best” neural model, measurements values and the CWI model is presented. Physical data Neural Network PLcomputed Epredicted - PL = PLcomputed - Epredicted + - COST-Walfisch- Ikegami Model Figure 5. The schematic diagram of the prediction We have studied the following GRNN models: NN4 with four inputs: distance between transmitter and receiver (d), the width of the street (w), the height of the building (h) and the building separation (b). NN5 with 5 inputs: distance between transmitter and receiver (d), the width of the street (w), the height of the building (h), the building separation (b) and the distance between base station height and the building height (dhbs). NN6 with 5 inputs: distance between transmitter and receiver (d), the width of the street (w), the height of the building (h), the building separation (b) and the street orientation (ψ). NN7 with 6 inputs: distance between transmitter and receiver (d), the width of the street (w), the height of the building (h), the building separation (b), the difference between base station height and building height (dhbs), and the street orientation (ψ) and the difference between base station height. Table 4.Hybrid neural network models Training set Test set [dB] µ Std RMS µ Std RMS R NN4 4.79 5.50 7.30 6.12 6.13 8.66 0.77 NN5 4.35 4.74 6.44 5.68 5.59 7.97 0.81 NN6 1.57 2.66 3.10 3.79 4.07 5.57 0.91 NN7 1.47 2.49 2.90 3.65 3.85 5.30 0.92 The comparison between the results obtained by the GRNN models for the training and test patterns are presented in Table 4. Table 5 represents the comparison between the hybrid model NN7 and CWI model for the entire test set. Table 5. Comparison between the proposed neural model and the CWI model [dB] µ Std RMS NN7 3.65 3.85 5.30 CWI 6.97 4.62 8.38 -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 19 29 40 55 71 89 108 127 145 164 183 202 221 240 259 278 297 316 Distance [m] Pathloss[dB] Measurements Neural Model CWI Model Figure 6. Comparison between the prediction made by the hybrid model, CWI model and measurements, in case of a particular route, urban environment VI. CONCLUSIONS In this paper we have developed applications of the General Regression Neural Networks for the prediction of propagation path loss and we have compared them with measurements and with the prediction made by different empirical models. It is noticed a significant improvement in the prediction made by neural models due to their generalization property. Another advantage of the use of neural networks is the fact that they are trained with measurements, so the included propagation effects are more realistic. REFERENCES [1] B. E. Gschwendtner and F. M. Landstorfer, “Adaptive Propagation Modeling using a Hybrid Neural Technique”, Electronics Letters, Feb. 1996, vol. 32, No. 3, pp. 162-164. [2] T. Balandier, A. Caminada, V. Lemoine and F. Alexandre, “170 MHz Field Strength Prediction in Urban Environments Using Neural Nets”, Proc. IEEE Inter. Symp. Personal, Indoor and Mobile Radio Comm., vol. 1, pp. 120-124, Sept. 1995. [3] P-R. Chang, W-H Yang, “Environment-Adaptation Mobile Radio Propagation Prediction Using Radial Basis Function Neural Networks”, IEEE Trans. Vech. Technol., vol. 46, No. 1, pp. 155-160, Feb. 1997. [4] H. Har, A. M. Watson and A. G. Chadney, “Comment on diffraction loss of roof-to-street in COST231-Walfish-Ikegami model”, IEEE Trans. On Vehicular Technology, vol. 48, no. 5, pp.1451-1452, Sept. 1999 [5] A. Kanatas, N. Moraitis, C. Steriadis, N. Papadakis, E. Angelou, P. Constantinou, D. G. Xenikos and A. Vorvolakos, “Measurements and channel characterization at 1.89 GHz in urban and suburban environments”, The Second Internat. Symp. on Wireless Personal Multimedia Commun., WPMC 99, Amsterdam, The Netherlands, Sept. 1999, pp. 448 – 454. [6] C. Christodoulou, M. Georgiopoulos, Applications of Neural Networks in Electromagnetics, Artech House, 2001 [7] S. Haykin, Neural Networks. A Comprehensive Foundation, IEEE Press, McMillan College Publishing Co., 1994 [8] J. Walfisch and H. L. Bertoni, “A theoretical model of UHF propagation in urban environments”, IEEE Trans. On Antennas and Propagation, vol. 36, no. 12, Dec. 1988, pp. 1788-1796. [9] T. S. Rappaport, Wireless Communications. Principles and practice, Prentice Hall PTR, 1996

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