Subharmonics and other quantization effects in periodically excited recursive digital filters

Periodic state variations in digital filters have been extensively studied in the context of zero-input limit cycles. Besides such spontaneous oscillations, forced oscillations under periodic excitations also deserve consideration, among which are those for constant input (with unity period). It is observed that, in contrast with its ideal linear counterpart, an actual filter with limited signal wordlength reaches the steady (periodic) state after a finite time with a waveform that depends on the initial conditions, potentially applied in an infinitely remote past. Moreover, the period need not be the same as that of the excitation; subharmonics can occur with periods equal to integer multiples of those of the excitation. In this contribution, the above phenomena are considered as instabilities induced by the nonlinearities due to finite wordlength restrictions, with emphasis on quantization effects. Compared with overflow effects, these effects are relatively small in amplitude and decrease with decreasing quantization steps.