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A ROM-less direct digital frequency synthesizer employing trigonometric quadruple angle formula is present in this paper. The worse case spectral purity is better than -130 dBc. The amplitude resolution is up to 13 bits, while the phase resolution is 12 bits. Neither any scaling table nor error correction tables are required. The maximum error is mathematically analyzed. The word length of each multiplier is carefully selected in the digital implementation such that the error range is circumscribed and the resolution is preserved.