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.