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Roundoff noise and limit cycle oscillations due to postmultiplication rounding can seriously limit the application of recursive digital filters in practice, particularly for poles near the unit circle or near z = +1. Rounding is usually performed by zero memory quantizers. "Second-order quantizers" are introduced, which can be used for implementing recursive digital filters with practically no roundoff errors or limit cycle oscillations. Based on the idea of changing the transfer function used to compute roundoff errors, these quantizers save the low-order bits to correct the product at future iterations. For several pole locations, this can be done with minimal amount of extra hardware.