The necessary and sufficient conditions for minimum roundoff noise in second-order state-space digital filters and their optimal synthesis

For second-order state-space digital filters, new expressions for the covariance matrix
K
and the unit noise matrix
W
are derived. Using these, the necessary and sufficient conditions for minimum roundoff noise in scaled filters are given in terms of the state parameters. Simple expressions for the second-order modes of the filter and for the attainable lower bound of the unit noise gain are provided. The scaled state realizations having minimum roundoff noise are evaluated for all types of digital filter transfer functions using only their coefficients.