We present a variable-frequency variant of the complex demodulation technique. Complex demodulation is the digital equivalent of heterodyne detection, in which the input signal is multiplied by a carrier oscillation {exp(−i&ohgr;mt)} of modulation frequency &ohgr;mand then low-pass filtered. The time-dependent amplitude and relative phase can then be computed from the filtered demodulate. This technique fails when the frequency of oscillation differs from the modulation frequency by more than the cutoff frequency of the low-pass filter. We modified the standard technique by computing iteratively a time-dependent modulation frequency{&ohgr;m(t)}that tracks the signal frequency to within ±0.1&percent; of the Nyquist frequency. This small difference frequency allows the use of very narrow filter bandwidths for the study of narrow-band oscillations with improved signal-to-noise ratio. We apply this analysis tool to Mirnov oscillations measured by a magnetic probe and show the time variation of the amplitude of the fundamental and harmonic frequencies during a plasma current ramp down. Future work will involve the application of this technique to ECE measurements. ©1997 American Institute of Physics.