Noise gain expressions for low noise second-order digital filter structures

The quantization noise of a fixed-point digital filter is commonly expressed in terms of its noise gain, i.e., the factor by which the noise power q²/12 of a single quantizer is amplified to the output of the filter. In this brief, first a closed-form expression for the optimal second-order noise gain in terms of the coefficients of the numerator and denominator polynomials of the transfer function is derived. It is then shown, by deriving a similar expression for its noise gain, that the second-order direct form structure has an arbitrarily larger noise gain the closer the filter poles are to the unit circle. The main result, however, is that the wave digital form and the normal form structures have noise gains which are only marginally larger than the minimum gain. For these forms, the expressions for their noise gain in terms of the transfer function are given as well. The importance of these forms lies in the fact that they use less multipliers than the optimal structure and that they are much easier to design: properly scaled forms are given requiring no design tools.