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Fourier transforms are used everyday for solving single functions and combinations of functions found in radar and signal processing. Still, many problems that could have been solved by using Fourier transforms have gone unsolved because they require integration that is too computationally difficult. This manual demonstrates how you can solve those integration-intensive problems with an approach to carrying out Fourier transforms. By building upon Woodward's well-known "Rules and Pairs" method and related concepts and procedures, the text establishes a unified system that makes implicit the integration required for performing Fourier transforms on a wide variety of functions. It details how complex functions can be broken down to their constituent parts for analysis. This approach to applying Fourier transforms is illustrated with many specific examples from digital signal processing as well as radar and antenna operation.