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A function defined on a finite interval may be expressed as the sum of two functions, one of odd and one of even symmetry in the interval. These two functions as defined on the half-interval may each in turn be expressed as the sum of two functions, and so on. Such analysis is called symmetry analysis. For sampled functions, if the number of samples under analysis is a power of 2, symmetry analysis is identical with Walsh analysis. Characteristic basis vectors for all orders of symmetry analysis are derived.