Order statistic (OS) processors have been widely used in the field of signal and image processing [1-3]. OS results can be obtained by sorting the elements of an input vector according to the rank of each element. Ranked outputs such as minimum, median and maximum have been used for target detection with applications in radar, sonar and ultrasonic nondestructive testing [4,5]. The problem of sorting has already been solved by sequential and iterative methods such as the bubble sort, selection sort, insertion sort, and quick sort with computational efficiency ranging between O(NlogN) and O(N²) comparisons and swapping operations [6]. As an alternative to conventional sorting techniques, a neural network design resulting from the harmony theory has been proposed for the sorting operation [7]. Neural network hardware can be implemented with parallel architecture using VLSI and FPGA technology, and this is highly desirable for high-speed computation [8-11].

In this paper, feed-forward neural network models [12] are introduced to find the minimum, the median, and the maximum of the input vectors consisting of real numbers. The back-propagation learning algorithm [13] is utilized in the training phase of the order statistic neural network filters (OSNNF). If the size of the input data is n, there is n! different input vectors including the same real numbers which give the same sorted output. Furthermore, the input vectors with real numbers demand an unlimited number of input vectors for training. Therefore, it is impractical to train an OSNNF with that many input data. Consequently, the trained OSNNF filter might not provide exact sorted results. In spite of this drawback, neural network filters can be trained to provide good approximation for the sorted results, and perhaps this might be sufficient for sorting the random processes in certain applications [4,5].

In practice, it is desirable to develop an efficient neural network model that can be used in finding the estimates of minimum, median, and maximum of the input vectors. To achieve this, simulation data is used in the training phase. The training set of data consists of random numbers with uniform distribution scaled between zero and one. Then, the neural network is trained to yield the ranked output (e.g., the minimum, the median or the maximum value of the input vector with real numbers). In the next section we present the design techniques for the neural network OS filters. Section 3 discusses an improved neural network solution that finds the rank of each input in order to reveal the exact sorted result. Section 4 utilizes these neural network filters to enhance the visibility of target echoes in high scattering clutter using split-spectrum processing (SSP).

Figure 1 displays the structure for the neural network OS filter where the number of inputs is 8 (M = 8). This filter is a fully connected feed-forward neural network. When unordered uniformly distributed random numbers, x(i), are presented as an input vector of real numbers to the neural network, each output of the hidden neurons is the weighted sum of the input nodes and bias node passes