Design and analysis of recursive periodically time-varying digital filters with highly quantized coefficients

A design for recursive filters with coefficients ε {1, 0, -1} that can change at each filter update is introduced. This flexibility of allowing the multipliers to be functions of time can be used to offset the reduction in degrees of freedom when such quantization is used. This result will be especially useful in filtering applications in which a controllable or adaptive filter is required and high-speed multipliers are not available. A general transition-matrix state-variable formulation similar to Floquet theory for differential equations is developed that can be used to design IIR (infinite-impulse response) filters characterized by a periodically time-varying feedback matrix. By defining a time-varying similarity transformation and introducing circulant feedback matrices, a significant simplification of the highly quantized time-varying state recursion equation is realized. This leads to the enumeration of the filter transfer function and to a simple eigenvalue structure that facilitates the design of filters with desired pole locations. These analytical results are verified in simulations of lowpass and bandpass filter designs