In signal processing, the analysis of frequency composition of signals plays a key role. The frequecy response is computed with the DFT, and more effectively with the FFT. We present the classic theory of DSP and generalize it to algebraic signal processing theory. We develop the algebraic signal model as the triple, consisting of an algebra, a module and a bijective linear mapping. We define the spectrum and Fourier transform of the model. We present the polynomial signal model and its properties. We show the characterization of group delay. We derive the PDFT algorithm as well as a coarse-fine response algorithm and a group delay estimation algorithm. We determine the precision and computational complexity of the PDFT algorithm. We give an example of the use of the algorithms in localization.
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