### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. HNN Basic Principles
- 3. HNN Design
- 4. Application and Comparison
- 5. Conclusions
- Acknowledgments
- References

[1] This paper proposes the method of a Hopfield-type neural network (HNN) for extracting Doppler spectrum from ocean echo. First, it introduces the basic principle of HNN for optimized processing. Second, expanding the principle of utilizing autoregression (AR) to estimate frequency spectrum, we point out how to apply HNN in spectrum estimation. Last, the three methods are utilized to process actual data, that is, the conventional fast Fourier transform method, modern spectrum estimation–AR method, and the spectrum estimation method based on HNN. The results obtained by the three methods prove that the spectrum estimation method based on HNN is feasible for extracting the Doppler spectrum from ocean echo.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. HNN Basic Principles
- 3. HNN Design
- 4. Application and Comparison
- 5. Conclusions
- Acknowledgments
- References

[2] High-frequency ground wave radar is widely exploited to detect ocean dynamic parameters and targets such as icebergs and vessels. At present, we are utilizing the system of frequency-modulated continuous wave (FMCW) in which we process ocean echo with dual fast Fourier transform (FFT). Range information is extracted through the first FFT. We acquire Doppler information of certain range cells with a second FFT which is used to process the results from the first FFT during certain coherent integration times. Thus information of ocean dynamic parameters and targets can be extracted. Generally, it takes an accumulation time about 13 min to sense ocean dynamic parameters, and it takes an accumulation time about 3 min to detect the target [*Barrick et al.*, 1994; *Knan and Mitchell*, 1991]. The modern spectrum estimation–autoregression (AR) method takes less accumulation time than conventional FFT because it requires less data. However, its computation is complex and consumes more time [*Vizinho and Wyatt*, 1996]. This paper proposes the spectrum estimation method based on a Hopfield-type neural network (HNN) that can overcome the disadvantage of modern spectrum estimation while preserving its advantage of taking less accumulation time. Moreover, because the spectrum estimation method based on HNN has the capabilities of parallel processing and computation by iterative algorithm, its computation is not complex and does not consume much time [*Ham and Kostanic*, 2003; *Hopfield*, 1982].

### 2. HNN Basic Principles

- Top of page
- Abstract
- 1. Introduction
- 2. HNN Basic Principles
- 3. HNN Design
- 4. Application and Comparison
- 5. Conclusions
- Acknowledgments
- References

[3] The American physicist J. J. Hopfield published two influential papers concerning neural networks [*Hopfield*, 1982; *Hopfield and Tank*, 1985]. He proposed a design which consists of interconnected devices and defined energy function which concerns the state and joining weight of neurons. HNN composed of three neurons is shown in Figure 1 for illuminating the structure of HNN. The first layer is treated as the input of the network; there are no neurons in the first layer which do not have the function of computation. However, the second layer, composed of true neurons, executes the function of accumulating the results which are produced by utilizing input information to multiply weight coefficients. Then the nonlinear transfer function of neurons is exploited to process accumulating results. Thus the output of HNN is acquired. Generally speaking, the model of HNN is a circular neural network because output is connected with input. Thus the input of HNN produces continuous state change. When the input of HNN is selected, the output of HNN can be acquired. Moreover, its feedback can change the value of the input. Thus new output is produced. The feedback can run continuously. If HNN is the network of stabilization, the change caused by feedback and iterative computation is smaller and smaller. HNN can reach the state of balance. At this time, the output of HNN is fixed.

[4] Then HNN is widely applied in many fields such as the traveling salesman problem, pattern recognition, and optimization control. It can be utilized to solve the problems of association memory and optimization computation. The output of neural networks can be either continuous or discrete. Correspondingly, HNN can be divided into two kinds, that is, continuous and discrete. Continuous HNN can solve the problem of optimization computation. Discrete HNN can solve the problem of association memory. Continuous HNN is a single-layered feedback network. The operation mode of each neuron is shown as the following formula:

where *X*_{i} = *F*(*u*_{i}), *i* = 1, 2, ⋯, *N*, *W*_{ij} = *W*_{ji}, *N* is the number of neurons, *u*_{i} is the middle output of the *i*th neuron, *I*_{i} is the threshold value of the *i*th neuron, and *F*(·) is the transfer function, which is usually a sigmoid function. In fact, continuous HNN is the continuous system of nonlinear dynamics; it can be described with one group of nonlinear equations. When the initial state is determined, we can compute the trace of the network by solving the nonlinear differential equations. If the system is steady, it will converge to a certain steady point. We can define one Lyapunov energy function for the description of continuous HNN:

We can prove that the energy function of HNN is limited. The formula *dE*/*dt* ≤ 0 can demonstrate that the system is steady. The state of network always changes with the tendency of making *E* decrease until *E* reaches minimum. Meanwhile, *X*_{i}, namely, the steady value of the *i*th neuron, is constant. Therefore HNN has the function of computing minimum automatically. As has been discussed above, continuous HNN can be utilized to solve the optimization problem [*Ham and Kostanic*, 2003; *Hopfield*, 1982].

### 3. HNN Design

- Top of page
- Abstract
- 1. Introduction
- 2. HNN Basic Principles
- 3. HNN Design
- 4. Application and Comparison
- 5. Conclusions
- Acknowledgments
- References

[5] At first, we will briefly describe AR spectrum estimation. One *p*th-order AR model is presumed to satisfy the difference equation

Here *a*_{1}, *a*_{2}, ⋯, *a*_{p} are constant, *a*_{p} ≠ 0, and ɛ(*n*) indicates the noise sequence whose mean and variance are *e*(*n*) and σ_{ɛ}^{2}, respectively. We define that the predicted value (*n*) is computed from parameters *a*_{1}, *a*_{2}, ⋯, *a*_{p}. Then the difference exists and can be described by the equation

We realize the AR spectrum estimation by estimating parameters *a*_{1}, *a*_{2}, ⋯, *a*_{p}, which are produced by making *E*[ɛ^{2}(*n*)] reach minimum *E*[ɛ^{2} (n)]_{min}. More concrete details about how to solve the Yule-Walker equation are illuminated as follows:

Then we acquire the values of coefficients *a*_{1}, *a*_{2}, ⋯, *a*_{p} and *E*[ɛ^{2}(*n*)]_{min}. Here *r*_{x} (*m*) = *E*[*x*(*n*) *x*(*n* + *m*)]. According to the equation

we can estimate the signal frequency spectrum [*Cristi*, 2003]. Here *E*[ɛ^{2} (*n*)]_{min} is the variance of white noise in the AR model. We pay more attention to coefficients *a*_{1}, *a*_{2}, ⋯, *a*_{p}. *E*[ɛ^{2}(*n*)]_{min} should not be neglected, so we are not normalizing by giving it the value 1. Thus there is the unitary equation

that can be utilized to realize spectrum estimation [*Kay and Marple*, 1981].

[6] In order to utilize the neural network to solve the optimization problem, we must set up the relationship between the neural network and the problem. In this section, we illuminate how to set up the relationship between HNN and the problem of acquiring parameters *a*_{1}, *a*_{2}, , *a*_{p} by minimizing *E*[^{2}(*n*)]. *E*[^{2} (*n*)] can be expressed as the equation

It is not difficult to discover that we can estimate AR model parameters *a*_{1}, *a*_{2}, , *a*_{p} by minimizing the formula {*x*(*n*) − *a*_{i}*x*(*n* − *i*)}^{2}. Here *N* is the number of data that are used to estimate the spectrum. Moreover, the formula {*x*(*n*) − *a*_{i}*x*(*n* − *i*)}^{2} can be expressed as the following matrix norm form:

where *B* = [*x*(*p* + 1), *x*(*p* + 2), ⋯, *x*(*N*)]^{t}, *X* = [*X*_{1}, *X*_{2}, ⋯, *X*_{P}]

We unfold equation (9) to acquire the equation

*B*^{t}*B* is neglected because it is irrelevant to parameters *a*_{1}, *a*_{2}, ⋯, *a*_{p}. We can transform the right-hand part of equation (10) into the polynomial *X*_{i}^{t}*X*_{j}*a*_{i}*a*_{j} − 2*B*^{t}*X*_{i}*a*_{i}. By comparing it with the energy function of HNN, we can easily set up the relationship between them. That is to say, the joining weight between the *i*th neuron and the *j*th neuron is expressed as W_{ij} = − *X*_{i}^{t}*X*_{j}, and the threshold value of the *i*th neuron is computed through the formula *I*_{i} = *B*^{t}*X*_{i}. According to the relation between the AR method and the HNN method, the number of neurons in the HNN method corresponds to the number of order in the AR method. A sigmoid function whose output is always positive cannot be selected as the transfer function *F*(·) because the output value of neurons in HNN can be either positive or negative. The rule which can decide the selection of transfer function *F*(·) is that the derivative of transfer function *F*(·) cannot be negative [*Minsky and Papert*, 1988]. Here we select a special purelin function [*Hopfield and Tank*, 1985] *v*_{i} = *u*_{i}/β, which is specially designed for the problem in the paper. The value of β is selected according to the weight and threshold values of HNN. A suitable one can make the network converge to minimum quickly. The value of β is 40 in my simulating experiment.

[7] HNN usually has the problem of local minimum, which may interfere with its converging to global minimum and may produce wrong results. However, the equation in this paper for computing the minimum is a simple quadratic polynomial without any constraint, so it has only one global minimum without local minimum. As a result, it is impossible for the network of HNN to converge to an incorrect local minimum.

### 4. Application and Comparison

- Top of page
- Abstract
- 1. Introduction
- 2. HNN Basic Principles
- 3. HNN Design
- 4. Application and Comparison
- 5. Conclusions
- Acknowledgments
- References

[8] In this section, we want to prove the feasibility of the spectrum estimation method based on HNN. The three methods, that is, the conventional FFT method, the modern spectrum estimation–AR method, and the spectrum estimation method based on HNN, are utilized to process the same actual data. Then we compare processing results.

[9] At first, we select actual data which come from the Radio Propagation Lab in Wuhan University to extract ocean dynamic parameters at 0645 UT on 13 April 2004 on the Zhujiajian Island of Zhejiang province in China. The radar parameters are set as follows. The carrying frequency is 7.958 MHz, and range resolution is 2.5 km. The sweep period is 0.653 s, and coherent integration time is about 13 min. The number of range cells is 28. The number of data is 1024. The same data are processed by the three methods. The FFT method utilizes the whole data, and the AR and HNN methods utilize the data whose number is 256. The length of the slide window is 192. The number of moving slide windows is 64 in the AR and HNN methods. The order of the AR method is 64, and the number of neurons in HNN is also 64. The results of processing with the three methods are shown in Figures 2, 3, and 4, respectively.

[10] From Figures 2–4, it can be seen that the spectrum estimation method based on HNN can produce a satisfactory processing result in the detection of ocean dynamic parameters. The region of the first-order echo spectrum produced by the spectrum estimation method based on HNN is consistent with the regions produced by the other two methods; moreover, the signal noise ratio (SNR) is about 35 dB, which is higher than the SNR of the other two methods, which is 20 dB.

[11] Then we consider the application of the HNN-based spectrum estimation method in the field of detecting the target. Actual data from the Radio Propagation Lab in Wuhan University are selected for extracting information on the target at 0052 UT on 19 April 2004 on the Zhujiajian Island of Zhejiang province in China. The radar parameters are set as follows. The carrying frequency is 7.928 MHz, and range resolution is 1.25 km. The sweep period is 0.653 s, and coherent integration time is about 3 min. The number of range cells is 44. The order of the AR method is 64, and the number of neurons in HNN is also 64. The results of processing by the three methods are shown in Figures 5, 6, and 7, respectively.

[12] The spectrum estimation method based on HNN can be utilized to detect the target as the other two methods can. The SNR is 30 dB, which is higher than the 20 dB of the other two methods. Because the number of neurons is consistent with the order of the AR method, the Akaike information criterion rule that is utilized to ascertain the order of the AR method [*Kashyap*, 1980] can be also exploited to ascertain the number of neurons. However, the drawback-caused spectrum spread in the method based on HNN should be considered in future research. We can select the middle value in the region of the first-order spectrum to overcome the disadvantage.