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Using the principle of number theory, periodicity property of the output phase sequences of an M-bit Non-truncation DDS phase-accumulator have been analyzed, and the linear relationships between them have been deduced as well. Combining the symmetry property of the Discrete Fourier Transform Matrix, linear relationships between spectrums of different output signals generated by different frequency control words with the same Lowest Nonzero Bit Position have been deduced. The analysis results show that the output signals can be grouped into M groups, according to the lowest nonzero bit position of their frequency control words. Signals in the same group have the identical Signal-to-Noise-Ratio(SNR) and Spurious Free Dynamic Range(SFDR). As a result, only M DFTs are necessary for the analysis of all output signals generated by an M-bit Non-truncation DDS. Simulation results showed that the academic analysis is correct.