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In this paper, a universal mathematical method is proposed to determine the upperbound of the spurious-free dynamic range (SFDR) in direct digital frequency synthesizers (DDFSs) realized by piecewise polynomial interpolation methods. The Fourier series is used to establish a linear matrix relationship between the frequency spectrum of the interpolated sinusoidal signal and the coefficients of the interpolating polynomials. This matrix relationship can be considered as a linear overdetermined system of equations, which can be solved for the ideal spectrum where the fundamental harmonic has an amplitude of one and the other harmonics are zero. It is shown that the Moore-Penrose pseudoinverse and Chebyshev minimax methods find the coefficients corresponding to the largest signal-to-noise ratio and maximum SFDR designs, respectively. The proposed method is used to show that the maximum SFDR of a DDFS based on the even fourth-order polynomial interpolation is 74.35 dBc. A DDFS based on the aforementioned method is designed and its architecture is optimized to obtain an SFDR of 72.2 dBc. A VLSI implementation of the proposed DDFS is also reported.