Noise reduction strategies for digital filters: Error spectrum shaping versus the optimal linear state-space formulation

The noise reduction performance of error spectrum shaping (ESS) structures and of the optimal linear state-space (LSS) structure are compared for second-order digital filter sections. It is shown that optimal direct form 1 and direct form 2 ESS realizations have a higher signal-to-noise ratio than the optimal LSS structure. In practice, suboptimal ESS structures with simple hardware implementations are of greater interest. Several of these implementations are considered and optimal values for the ESS coefficients in these structures are derived. For filters with zeros at z = -1, it is shown that some of the simple ESS structures can outperform the optimal LSS structure. For elliptic filters, it is shown that several of the suboptimal ESS structures perform poorly, but that others still perform well.