When computational resources are limited, especially multipliers, distributed arithmetic (DA) is used in lieu of the typical multiplier-based filtering structures. However, DA is not well suited for adaptive applications. The bottleneck is updating the memory table. Several attempts have been done to accelerate updating the memory, but at the expense of additional memory usage and of convergence speed. To develop an adaptive DA filter with an uncompromised convergence rate, the memory table must be fully updated. In this research, an efficient method for fully updating a DA memory table is proposed. The proposed update method is based on exploiting the temporal locality of the stored data and subexpression sharing. The proposed update method reduces the computational workload and requires no additional memory resources. DA using the proposed update method is called conjugate distributed arithmetic. Filters can also be constructed from analog components. Often, for lower precision computations, analog circuits use less power and less chip area than their digital counterparts. However, digital components are often used because of their ease of reprogrammability. Achieving such reprogrammability in analog is possible, but at the expense of additional chip area. A reprogrammable mixed-signal DA finite impulse response (FIR) filter is proposed to address the issues with reprogrammable analog FIR filters that are constructing compact reprogrammable filtering structures, non-symmetric and imprecise filter coefficients, inconsistent sampling of the input data, and input sample data corruption. These issues are successfully addressed using distributed arithmetic, digital registers, and epots. Also, a mixed-signal DA second-order section (SOS), which is used as the building block for higher order infinite impulse response filters, was proposed. The type of issues with an analog SOS filter are similar to those of an analog FIR filter, which are the lack of a compact reprogrammable filtering structure, the imprecise filter coefficients, the inconsistent sampling of the data, and the corruption of the data samples. These issues are successfully addressed using distributed arithmetic and digital registers.