A spectrum analyzer having an improved local oscillator for use in a digital step sweep is capable of minimizing a dynamic spurious response which is inverse proportional to a unit step time in the step sweep. The local oscillator includes a random clock delay which provides a random clock to a direct digital synthesizer to incorporate random timings in a time length of the unit step time for sweeping the local oscillator. In another aspect, the local oscillator includes a sweep step number control which maximizes the number of steps in the step sweep to ultimately decrease the dynamic spurious response.