### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Neural Network–Based Direction Finding
- 3. Modified Neural Multiple-Source Tracking (NM-MUST) Algorithm
- 4. Simulation Results
- 5. Conclusion
- Appendix A
- References

[1] A modified neural network–based algorithm (modified neural multiple-source tracking algorithm (MN-MUST)) is proposed for real-time multiple-source tracking problem. The proposed approach reduced the input size of the neural network without any degradation of the accuracy of the system for uncorrelated sources. In addition, a spatial filtering stage that considerably improves the performance of the system is proposed to be inserted. It is observed that the MN-MUST algorithm provides an accurate and efficient solution to the target-tracking problem in real time.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Neural Network–Based Direction Finding
- 3. Modified Neural Multiple-Source Tracking (NM-MUST) Algorithm
- 4. Simulation Results
- 5. Conclusion
- Appendix A
- References

[2] Direction of arrival (DOA) estimation and target-tracking problems have been studied for years, and several different adaptive algorithms have been developed within the last two decades [*Godara*, 1997]. However, since adaptive algorithms require extensive computation time, it is difficult to implement them in real time in general [*Southall et al.*, 1995]. In recent years, application of neural network (NN) algorithms in both target-tracking problem and DOA estimation [*Southall et al.*, 1995; *Long and Da*, 1993; *Goryn and Kaveh*, 1988; *El Zooghby et al.*, 1997, 1998] have become popular because of the increased computational efficiency.

[3] In a recently published work, a NN algorithm, namely, the neural multiple-source tracking (N-MUST) algorithm, was presented for locating and tracking angles of arrival from multiple sources [*El Zooghby et al.*, 2000]. The algorithm uses a neural network operating in two stages and is based on dividing the field of view of the antenna array into angular sectors. Each network in the first stage of the algorithm is trained to detect signals generated within its sector. Depending on the output of the first stage, one or more networks of the second stage can be activated to estimate the exact location of the sources. Main advantages of the N-MUST algorithm were presented as significant reduction in the size of the training set and the ability to locate more sources than there are array elements.

[4] The algorithm presented in this paper, namely, the modified neural multiple-source tracking algorithm (MN-MUST), consists of three stages that are classified as the detection, filtering, and DOA estimation stages. Similarly to *El Zooghby et al.* [2000], a number of radial basis function neural networks (RBFNN) are trained for detection of the angular sectors which have a source or sources. A spatial filter stage is applied individually to every angular sector which is classified in the first stage as having a source or sources. Each individual spatial filter is designed to filter out the signals coming from all the other angular sectors outside the particular source detected. This stage considerably improves the performance of the algorithm in the case where more than one angular sector has a source or sources at the same time. Insertion of this spatial filtering stage is the main contribution of this paper. The third stage consists of a neural network trained for DOA estimation. In all three stages, the neural network's size and the training data are considerably reduced as compared to the previous approach [*El Zooghby et al.*, 2000] for uncorrelated sources, without loss of accuracy. The reduced-size neural network approach is also applicable to beam forming and direction of arrival estimation algorithms presented by *El Zooghby et al.* [1997, 1998].

[5] The organization of the paper is as follows: In section 2, problem formulation is given, and the neural networks for the problem of DOA estimation are established. In section 3, the MN-MUST algorithm is discussed. Simulation results are given in section 4, and concluding remarks are summarized in section 5.

### 2. Neural Network–Based Direction Finding

- Top of page
- Abstract
- 1. Introduction
- 2. Neural Network–Based Direction Finding
- 3. Modified Neural Multiple-Source Tracking (NM-MUST) Algorithm
- 4. Simulation Results
- 5. Conclusion
- Appendix A
- References

[6] The notations and symbols used in this study are parallel to those of *El Zooghby et al.* [2000], where the NN-based multiple-source tracking algorithm is introduced. The problem is formulated as follows: *M* isotropic antenna elements are placed along a line and are separated by a uniform distance *d* as shown in Figure 1. The number of sources (targets) is *K*, where *K* is not known, and it is allowed to exceed *M*. The antenna elements are assumed to be omnidirectional point sources. The angle of incidence of each source is Θ_{i}, *i* = 1, 2, …, *K*, which are required to be determined.

[7] The sources are assumed to be located in the far field of the antenna array, so the difference in the viewing angle of a given source by the different antenna element is neglected. The signal received on each antenna element can be written as

where *n*_{i}(*t*) is the noise signal received by the *i*th antenna element and *k*_{m} = sin (Θ_{m}), *d* is the spacing between the elements of the array, *c* is the speed of light in free space, and *ω*_{0 }is the angular center frequency of the signal.

[8] In matrix form,

where

and each entry of *A *is defined as

The noise signals {*n*_{i}(*t*), *i* = 1:*M*} received at different antenna elements of the array are assumed to be statistically independent zero mean white noise signals with variance *σ*^{2 }independent of *S*(*t*). The spatial correlation matrix *R *of the received signal is given by

where *H* denotes the conjugate transpose.

[9] It is shown in Appendix A that all entries of correlation matrix *R *starting from the second row are arithmetic combinations of the entries in the first row. Therefore it will be sufficient to calculate only the first row to represent the overall correlation matrix.

[10] On the basis of equation (2), the array can be considered as mapping *G*:*R*^{K} *C*^{M} from the space of the DOA {[θ_{1}, θ_{2}, …, θ_{K}]^{T}} to the space of antenna element output {[*X*_{1}(*t*), *X*_{2}(*t*), …, *X*_{M}(*t*)]^{T}}. In order to construct the inverse mapping *F*:*C*^{M} *R*^{K}, a multistage architecture using NNs is employed. The block diagrams of DOA estimation and MN-MUST algorithm architecture are given in Figures 2 and 3, respectively.

[11] In general, array-processing algorithms utilize a correlation matrix for direction of arrival estimation instead of the actual array output *X*(*t*). In this paper a similar approach is followed with an improvement in the DOA estimation stage by using a three-stage algorithm consisting of the preprocessing, neural network, and postprocessing stages (Figure 2). In the preprocessing stage, the input to the spatial filtering stage is obtained from the antenna element signals *X*(*t*) given by equation (2).

[12] *El Zooghby et al.* [1997, 1998, 2000] use the upper triangular part of the correlation matrix *R* in the DOA estimation applications. On the basis of equation (A9), in this study only the first row of the correlation matrix is used to represent the signals in the antenna elements for the case of uncorrelated sources. Then the input of neural networks is given as

The first entry of *Z*, *R*_{11 }is real, and all the other entries are complex. Therefore the size of *Z *is 2**M* − 1, where *M* is the number of elements of the array structure. On the other hand, in the N-MUST algorithm [*El Zooghby et al.*, 1997, 1998, 2000], the size of *Z *is given to be *M** (*M* + 1). For example, for an 8*8 = 64 element planner smart array structure, the NN input sizes of MN-MUST and N-MUST are 2*64 − 1 = 127 and 64*65 = 4160, respectively, while the output sizes of both NNs are the same. Obviously, a considerable reduction in the training time is obtained with the use of the MN-MUST algorithm.

### 3. Modified Neural Multiple-Source Tracking (NM-MUST) Algorithm

- Top of page
- Abstract
- 1. Introduction
- 2. Neural Network–Based Direction Finding
- 3. Modified Neural Multiple-Source Tracking (NM-MUST) Algorithm
- 4. Simulation Results
- 5. Conclusion
- Appendix A
- References

[13] The MN-MUST algorithm consists of three stages: the detection stage, the filtering stage, and the DOA estimation stage as shown in Figure 3. In the first stage, a number of radial basis function neural networks (RBFNNs) are trained for detection similar to the N-MUST algorithm [*El Zooghby et al.*, 2000] but with reduced input size as discussed in section 2. In the spatial filtering stage, the angular sector of interest is isolated from the others to improve the detection accuracy. The third stage is dedicated to DOA estimation by a NN whose input size is much smaller than the input size of the N-MUST algorithm. The third stage is also similar to the N-MUST algorithm [*El Zooghby et al.* 2000] except for the size of NN.

#### 3.1. Detection Stage

[14] In this stage, the entire angular spectrum is first divided into *P *subsectors. For each subsector *p* (1 ≤ *p* ≤ *P*), an RBFNN is trained to determine if any source exists within the sector. Depending on whether or not sources are present in the corresponding sector, the sectorial NN will produce “1” or “0” as its output, respectively. This information is then transferred to the “filtering” stage. In the detection stage, the input vector *Z *is supposed to have size 2**M* − 1 as mentioned in section 2. Network training and test (generalization) phase in training are similar to those of the N-MUST algorithm except for the input size of the network. The test phase produces the input vector *Z*, presents it to the RBFNNs of the detection stage, and then sends the output of each subsector, that is, “1”or “0,” to the next stage.

#### 3.2. Spatial Filtering Stage

[15] The spatial filtering stage aims to isolate each one of the sectors where sources are present from the other sectors. A separate nonlinear (band-pass) filter is designed for each sector and is activated depending on whether or not a target exists in that sector. All these filters are multilayer perceptron-type NNs with input size of 2**M* − 1 and output size of 2**M* − 1.

[16] The spatial filtering stage is basically a NN system. The inputs for the spatial filter network are similar to the vector *Z *of the detection stage of *El Zooghby et al.* [2000]. The output of the spatial filter network for sector *i* is *Z*_{fi}, which does not include the signals from other sectors. In the training phase, *Z*_{fi }pairs are processed, discarding the signals outside the *i*th angular sector. Let the {*S*_{1}(*t*), *S*_{2}(*t*), …, *S*_{K}(*t*)} signal combination for the training phase and {*S*_{t}(*t*), *S*_{q}(*t*), *S*_{l}(*t*)} be the only signal pair coming from direction angular sector *i*, where *t*, *q*, *l* ≤ *K*. The input of the NN, *Z*, is computed through equation (6) having {*S*_{1}(*t*), *S*_{2}(*t*), …, *S*_{K}(*t*)} signal pair while the output of NN, *Z*_{fi}, is computed through equation (6) again, but the signal pair is {*S*_{t}(*t*), *S*_{q}(*t*), *S*_{l}(*t*)} this time. The filtering stage filters out the {*S*_{1}(*t*), *S*_{2}(*t*), …, *S*_{K}(*t*)} − {*S*_{t}(*t*), *S*_{q}(*t*), *S*_{l}(*t*)} signal pair for the subject training example. In the training phase, NN generalizes the input-output (*Z* *Z*_{fi}) relations in the training phase. In the testing phase, one can have a generalized response to inputs that it has never seen before.

[17] The filtered *Z*, which is *Z*_{fi}, will be the input DOA estimation network for sector *i*. In order to understand the spatial filtering stage's function, let us assume it was not used and there were sources in other angular sectors as well as in sector *i*. The algorithm described by *El Zooghby et al.* [2000] can be used, in other words, the algorithm with detection stage first and then the DOA estimation stage. After the detection stage, the angular sectors having the sources would have been identified correctly, but in DOA estimation stages for the related angular sectors, the correct source angles would not be found because the NN was trained in such a way that sources were present only in a specific angular sector. At first glance it may seem that the filtering stage inserted in this study makes the algorithm more complicated with respect to the one of *El Zooghby et al.* [2000], but in reality, it is not so. A single NN is trained for a sector, and the others have been obtained by spatial shifting. The selection of the number of the angular sector basically depends on the total angular range we are looking for and the angular resolution. We have studied 1° angular resolution. One can divide 180° into nine angular sectors with 2° angular resolution. This stage results in a huge performance increase at the cost of a slight increase in the computational complexity of the overall system.

#### 3.3. DOA Estimation Stage

[18] The DOA estimation stage consists of a NN that is trained to perform the actual direction of arrival estimation. When the output of one or more networks of sectors from the first stage is “1,” the corresponding second-stage network(s) are activated. The sectors (associated NNs) that have outputs of 1 are considered as the sectors having targets. Then the second-stage filtering is activated. The input information to each second-stage network is *Z*, while the output is the filtered *Z*_{fi}. For each sector having output 1 from stage 1, the corresponding third-stage network(s) are activated. The input information to each third-stage network is *Z*_{fi}, while the output is the actual DOA of the sources.

[19] For a sector Θ_{W }and a minimum angular resolution of ΔΘ_{min}, the number of output nodes is given by

DOA estimates are obtained by postprocessing the neural network outputs of this stage. *J* output nodes represent bins in a discrete angular sector centered at intervals of width ΔΘ_{min}. For example, if Θ_{W }is a sector of 10° width and the desired accuracy ΔΘ_{min} is 1°, then the number of output nodes is 10. The NN is trained so as to produce “0” or “1” values at the outputs. An output of 1 indicates the presence of a source exactly on that bin, and 0 represents no source.

### 4. Simulation Results

- Top of page
- Abstract
- 1. Introduction
- 2. Neural Network–Based Direction Finding
- 3. Modified Neural Multiple-Source Tracking (NM-MUST) Algorithm
- 4. Simulation Results
- 5. Conclusion
- Appendix A
- References

[20] In this section, simulation results are presented to demonstrate different features of the MN-MUST algorithm discussed in the previous sections.

[21] The use of reduced input size in the NN-based system decreases the computational time while keeping the computational accuracy unaffected (Figures 4, 5, and 6) . In Figure 4, neural network training time versus size of the smart antenna structures in N-MUST and MN-MUST algorithms is compared. As the number of array elements increases, the training times of the two systems become drastically different as expected. Figure 5 shows the total training error of both networks within the training times given in Figure 4. The training error threshold is set to the same (one) for all examples for both N-MUST and MN-MUST algorithms. In Figure 6, comparison of memory requirements of the MN-MUST algorithm and N-MUST algorithm are provided. It is observed that after a certain threshold value in the number of array elements, the MN-MUST algorithm training requires much less memory as well as having high speed for the same training error threshold level.

[22] Performance of the MN-MUST algorithm is demonstrated in Figures 7, 8, and 9for some different cases. For this purpose, the angular spectrum between 1° and 30° is divided into three angular sectors in 10° intervals. The angular separation is 1° within each angular sector. For the first two examples, three stages of the MN-MUST algorithm are trained for the three-target case with a three- and five-element linear array, respectively. Three separate detection neural networks are trained to determine whether or not there exists a target in that sector. Three network filters are then trained to annihilate the targets outside the corresponding angular sector. Finally, three separate DOA networks are trained to find the actual target locations within the corresponding sectors.

[23] In the first scenario there exist three targets in three different angular sectors located at 1°, 15°, and 24°, respectively. The performance of the MN-MUST algorithm is given in Figure 7.

[24] In the second scenario presented in Figure 8, there exist three targets in one angular sector case located at 22°, 24°, and 29°, respectively. This scenario is run using a 5 dB signal-to-noise ratio (SNR) for both MN-MUST and N-MUST algorithms.

[25] In the last scenario, the MN-MUST algorithm is applied to the case where there exist four targets and a three-element antenna array, that is, where the number of targets is greater than the number of antenna elements and 5 dB SNR. The four targets were located at 21°, 24°, 27°, and 29°. The critical point in this application is that all the targets are located within the same angular sector. Simulation results are given in Figure 9.

[26] In all three cases demonstrated in Figures 7–9, it is observed that the MN-MUST algorithm finds the targets correctly no matter whether or not the targets are located within the same angular sector. In addition, as the number of targets exceeds the number of antenna elements, the algorithm can still perform sufficiently well even in a high-SNR condition. The maximum number of targets within an angular sector depends upon the size of the sector and the angular resolution.