Part I: Multistage Multirate Digital Filtering. A theorem is introduced which is useful in deriving equivalent multirate filter structures. Frequency responses of multistage multirate filters are derived and defined in terms of their equivalent one-stage filters. A new design principle is proposed to reduce filtering requirements at each stage and move the filter operations to low sampling rate stages and thus results in lower arithmetic rate. New optimum FIR and IIR multistage, multirate filter designs are developed based on this principle. The new design has a one-point passband specification for each decimator and/or interpolator stage resulting in a wider transition region and lower filter order. Efficient comb filter structures as decimators or interpolators are exploited using the theorem to save not only arithmetic but also storage. A new multistage multirate filter structure is proposed to use comb filters for decimation and/or interpolation and an IIR or FIR filter for fine filtering at the lowest sampling rate. This results in an extremely low multiplication rate, very few filter coefficients, low storage requirements, high accuracy arithmetic and simple filter structures. Examples are given to explain the design procedure and comparisons are made to show the superiority of the new filters. A general analysis of the dynamic range and roundoff noise problems with multistage multirate filters is presented, and a cure to reduce roundoff noise is given. Comb filter operation and finite wordlength effects with a comb decimator and interpolator is examined. Part II: New Efficient Recursive FIR Filter Structures and Design. New filter structures have been found to generate piecewise-polynomial and piecewise-(polynomial(.)sinusoid) impulse responses. The filter efficiency depends on the number of piecewise sections, not on the length, of the filter impulse response. When using piecewise-polynomial to represent the impulse response of Hilbert transformer or differentiator, or using piecewise-(polynomial(.)sinusoid) to represent the impulse response of low-pass or high-pass filters, the number of piecewise sections is proportional to log N where N is the filter length. In these cases, new filter structure will be much more efficient for long length filters than the conventional filter structures. Filter efficiencies can also be achieved by using piecewise-polynomial to represent the impulse response of narrow-band low-pass or wide-band high-pass filters or using piecewise-(polynomial sinusoid) to represent the impulse response of narrow-band band-pass or band-stop filter. Minimum 1(,p) error design methods which minimize the frequency domain error are given. Examples illustrated the efficiencies of the new filter structures.