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As a paradigm for nonlinear spatial-temporal processing, cellular nonlinear networks (CNN) are biologically inspired systems where computation emerges from a collection of simple locally coupled nonlinear cells. Our investigation is an exploration of an important and difficult aspect of implementing arbitrary Boolean functions by using CNN. A typical class of basic key Boolean functions is the class of linearly separable ones. In this paper, we focus on establishing a complete set of mathematical theories for the linearly separable Boolean functions (LSBF) that are identical to a class of uncoupled CNN. First, we obtain an essential relationship between the template and the offset levels as well as the basis of the binary input vector set in the uncoupled CNN. More precisely, we construct a neat binary input-output truth table and some interesting properties of the offset levels of the uncoupled CNN, and develop a practical design formula for the class of CNN template. Especially, we found a criterion for LSBF, which depends only on symbolic relations between a Boolean function's outputs. Furthermore, we develop a method for representing any linearly nonseparable Boolean function into a logic operation of a sequence of linearly separable ones for a small number of inputs.Index Terms-Binary input-output truth table, cellular nonlinear network (CNN), linearly nonseparable Boolean function, linearly separable Boolean function (LSBF), template design.

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Phased array antennas have played a very important role in many different areas and applications. It requires precise excitation of each antenna element by various synthesis techniques to obtain desired array pattern features. However, due to reasons such as manufacturing imperfections, component aging, and temperature variation, the realistic antenna element excitation inevitably differs from their expected values in practice. This paper presents a tutorial-like review to deal with excitation errors for phased array antennas. Two kinds of analysis methods, probabilistic methods and interval arithmetic (IA) based methods, are presented to evaluate the effects of excitation errors for phased array antennas. State-ofthe-art calibration methods along with various signal processing techniques are reviewed, their advantages and challenges are discussed in a comparative manner. Some other common errors and open research directions are also presented.INDEX TERMS Phased array antenna, excitation error, interval arithmetic, array calibration.