In this paper, two second-order state-space digital filter structures are derived based on an input balanced realization, and these two filter structures are transposes of each other under an \(l_2\) -norm dynamic range constraint. Each of the new second-order realizations has only 8 multipliers, which is one multiplier less than the standard state-space realization. The expressions of the roundoff noise gain for these structures are provided. Simulation results are presented to demonstrate the excellent finite wordlength performance of the proposed two sparse structures. The results show that the two proposed structures surpass both the classical optimal state-space realization and the normalized structure in minimizing roundoff noise, especially the transposed one produces the lowest roundoff noise.
