Frequency selective surface design based on iterative inversion of neural networks

This paper proposes a novel approach to solve a constrained inverse problem encountered in the design of frequency selective surfaces (FSS's). Due to the many-to-one nonlinear functional relationship between an FSS and its frequency response, there is no closed form solution directly from the given desired frequency response to the corresponding surface. Therefore, to design an FSS for a given response, one has to search in the knowledge base through a trial-and-error procedure. This procedure can be a very laborious and tedious process. Our approach adopts an iterative regularized inversion technique, which starts with an inversion algorithm for multilayer perceptrons to generate the corresponding 2-D surface for the given desired frequency response, a constraint satisfaction mechanism is then used to reshape the 2-D surface to satisfy the constraints, and the resulting surface is used as the initial point for the next inversion algorithm. This procedure is mathematically similar to the projection onto convex set algorithm for constrained optimization problems. 1 Frequency Selective Surface Design Frequency selective surfaces (FSS7s) have widespread applications over much of the electromagnetic spectrum [4]. In the microwave region, they are used as reflector antenna dichroic surfaces and antenna radomes. In the far-infrared region, they are used as polarizers, beam splitters, and mirrors for laser applications. Another application of the FSS in this frequency range is in infrared sensors. In the near-infrared and visible portions of the spectrum, they are used t o aid in the collection of solar energy. FSS7s usually comprise periodically arranged identical metallic patch or aperture elements supported by dielectric layers. They exhibit total reflection or transmission in the neighborhood of the element resonance. To model an arbitrarily-shaped patch or aperture elements, the unit cell is uniformly divided into an N x N array of subcells. The geometrical description of the unit-cell is given as follows. Subcells that correspond to the conductor region of the unit cell are represented by 1's whereas subcells outside the conductor region are represented by 0's (one example is shown in Figure l (a) ) . The frequency response of the FSS's are then calculated for a given incident plane wave (see Figure l(b)). There is no closed form solution directly from the given desired frequency response to the corresponding surface. Therefore, to design an FSS for a given response, one has to search in the knowledge base to look for the surface that gives the closest response to the desired one. Unit-cell geometry of the chosen surface is then perturbed (through a trial-and-error procedure) until its response matches all the design criteria. The process is too laborious and human dependent. We propose an alternate design procedure that does not have these negative features. J.N. Hwang, J.J. Choi, S. Oh, R.J. Marks II, "Query learning based on boundary search and gradient computation of trained multilayer perceptrons", Proceedings of the International Joint Conference on Neural Networks, San Diego, June, 1990, 17-21 June 1990, vol. III, pp.III57-III62. Frequency Repom of the Cross-Shaped Patch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 800 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 'O0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 ,0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 5 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ ~ ~ 0 0 0 0 0 0 0 3 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 =O0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 lo[ 0 0 5 10 15 20 25 30 (4 (b) Figure 1: (a) In an FSS, subcells that correspond to the conductor region of the unit cell are represented by 1's whereas subcells outside the conductor region are represented by 0's. (b) The corresponding frequency response of the FSS. 2 Learning and Inversion of Multilayer Perceptrons Multilayer perceptrons are feed-forward neural networks which have one or more layers of hidden neurons between the input and output layers. When the net is trained, it is used to generate response given the input test data. The converse problem of generating input vectors corresponding to a given output vector is referred to as inversion (see Figure 2). The system dynamics in the retrieving phase of an L-layer neural net can be described by the following equations: where aj(l) denotes the activation value of the jth neuron a t the lth layer and f is the nonlinear activation function. Network Learning The learning phase of a multilayer perceptron uses the back propagation learning rule, an iterative gradient descent algorithm designed to minimize the mean squared error between the the desired target values and the actual output values [I]: where E = $ ~2~ (ti u ~ ( L ) ) ~ . Inversion of a Network The inversion of a network will generate the input {aj(0)) (or inputs) that can produce a desired output vector. By taking advantage of the duality between the weights and the activation in minimizing the mean squared error between the the desired target values and the actual output values, the iterative gradient descent algorithm can also be applied to obtain the desired input.