Hardware IIR Filters: Direct Form I Computing Just Right

Linear Time Invariant (LTI) filters are often specified and simulated using high-precision software, before being implemented in low-precision hardware. A problem is that the hardware does not behave exactly as the simulation due to quantization and rounding issues. This article advocates the construction of LTI architectures that behave as if the computation was performed with infinite accuracy, then rounded only once to the low-precision output format. From this minimalist specification, it is possible to deduce the optimal values of many architectural parameters, including all the internal data formats. This requires a detailed error analysis that captures the rounding errors, but also their infinite accumulation in infinite impulse response filters. This error analysis then guides the design of hardware satisfying the accuracy specification at the minimal hardware cost. This is illustrated on the case of low-precision LTI filters implemented in FPGA logic. This approach is fully automated in a generic, open-source architecture generator tool built upon the FloPoCo framework, and evaluated on a range of Finite and Infinite Impulse Response filters.