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A Comparative Study of the Signal-to-Noise Ratios of Different Representations for Symbolic Sequences

Based on the numerical representations by T basic vectors of a symbolic sequence consisting of T symbols, first, we prove mathematical that the total Fourier spectrum of the sequence is the square of the length of the sequence. In the meantime, we define the indicator sequences vector. Using the orthogonal or row orthogonal transformations of the indicator sequences vector, we construct some special numerical representations of the symbolic sequence and characterize the signal-to-noise ratios of the power spectrum of the numerical representations. After calculating the discrete Fourier transform of those special numerical representations, the signal-to-noise ratios of them can be figured out. Mathematical theorems prove that the signal-to-noise ratio of the Fourier spectrum of those special representations of the symbolic sequence is T/(T-1) times the signal-to-noise ratio of the representation by T base vectors. The results are applied in analyzing the properties of the DNA sequences or protein sequences in the frequency domain, if one uses the signal-to-noise ratios of special representations as the distinguishing criterion, the distinguishing results only depend upon the distribution of the symbols in the symbolic sequence and their mathematical constructions of representations, but do not relate to the chemical or biological meanings of the representations.