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Ramanujan-sum expansions for finite duration (FIR) sequences

Ramanujan sums have in the past been used to represent arithmetic sequences. It is shown here that for finite duration (FIR) sequences with length N, the traditional representation is not suitable. Two new types of Ramanujan-sum expansions are proposed here for the FIR case, each offering an integer basis. One of these is particularly suited to identify periodicities in the FIR sequence. This representation in fact expresses any FIR sequence as a sum of orthogonal sequences each with a hidden periodicity corresponding to a divisor of N.