Abstract: A frequency synthesizer takes advantage of the simplicity of the phase lock loop and the rapid settling time and spectral purity of direct methods of frequency synthesis. "Approximations" and resulting frequency output errors are controlled and optimized to provide high performance, low noise, rapid frequency acquisition and stabilization, wide frequency range, high frequency resolution, and high accuracy and stability. In one embodiment, phase locked loops provide an output frequency with a resolution considerably less than the reference frequencies due to the application of a unique direct synthesis method. Additional resolution (and correction of error due to the approximations) is obtained by slightly varying the frequency of the master reference frequency. As direct frequency synthesis does not suffer from the long lockup times associated with phase locked loops operating at low reference frequencies, such small frequency changes can be made almost instantaneously.