Noise may be reduced or eliminated from a digital sawtooth signal representing the phase of a periodic signal. This may be done precisely, even when inexpensive fixed-point arithmetic is used. In one aspect of the invention, the input signal (noise plus true signal) 12 is filtered to produce, in succession: (a) mod one differentiated noise plus slope of true phase signal 28; (b) mod one differentiated noise plus slope of residual phase signal (true phase signal minus estimated slope of true phase signal) 36; (c) mod one differentiated noise 46; (d) estimated noise 62; and (e) smoothed phase signal 72. In a second aspect, a noisy phase signal 12 is extracted from a first arbitrary periodic signal and the above steps are used to generate a noise-reduced phase signal 72. The noise-reduced phase signal 72 is then used to generate a second arbitrary periodic signal of the same frequency.