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A digital frequency synthesizer has been designed and constructed based on generating digital samples of at time . The real and imaginary parts of this exponential form samples of quadrature sinusoids where the frequency index is allowed to vary . The digital samples drive digital to analog converters followed by low-pass interpolating filters to produce analog sinusoids. The method is superior to digital difference equations with poles on the unit circle since the noise or numerical inaccuracy remains bounded. The digital technique used consists of factoring the exponential into two table look-ups from an efficiently organized small READ-ONLY memory table and performing a complex multiply to produce the real and imaginary components. A small array multiplier efficiently organized performs the multiplications. The technique lends itself to the production of phase coherent or phase controlled sinusoids because of the indexing arrangement used. In addition finer frequency steps than the READ-ONLY memory allows are available by expanding the indexing register at no increase in inaccuracy.