Abstract: An inexpensive numerically-controlled fast frequency-hopping microwave synthesizer. A voltage-controlled oscillator (VCO) output phase remains locked to an internal direct digital synthesizer (DDS) reference signal over the entire output frequency band, which is an order of magnitude larger than the internal sampling clock frequency. A “Nyquist-boundary hopping” scheme compares signals from an alias band of the DDS with signals from an alias band of the sampled VCO output to derive an output phase error signal, which is forced to zero in a manner that locks the VCO output phase to the DDS output phase over a frequency hop-distance greater than the DDS bandwidth. Accordingly, in a single second, the synthesizer can hop phase-continuously in a single clock cycle to each of hundreds of thousands of different microwave output frequencies with relatively low clock rates (up to 100 MHz) commensurate with silicon application-specific integrated circuits (ASICs).