Estimation of error variance due to coefficient quantization of digital filters excited by random signals

Expanded state equations are formulated for the time-domain analysis of an arbitrary configuration of digital filters with multiple inputs and outputs. In the equations, the outputs of adders and multipliers are defined as the variables v(k) in addition to the state variables x(k). The permutation of v(k) is introduced to clarify the condition of computability. Based on the obtained equations, a unified analysis is developed for the estimation of errors due to coefficient quantization of digital filters. The variance of output errors is calculated in quadratic form. Moreover, the dynamic range constraint (l2norm) is considered for the fixed-point implementation. An optimal scaling method using scaling matrices is presented. Detailed examples including a new triangular pillar configuration with two inputs and two outputs are presented to show the unified analysis and the optimal scaling method.