Lower precision arithmetic can improve the throughput of adaptive filters, while requiring less hardware resources and less power. Such benefits are crucial for adaptive filters, especially for IoT and wearable applications. In order to apply lower precision arithmetic to adaptive filters, a clear rounding error analysis framework is required, since lower precision arithmetic can degrade the filter performance. Previously, rounding error analyses of adaptive filters were based on forward error analysis. This limited the descriptiveness of rounding error impact on adaptive filter performance in relation to other external variables such as measurement noise, regularisation, and numerical stability of an algorithm. To overcome such limitations, we first present a new backward error analysis framework for adaptive Recursive Least Squares (RLS) filters. Our framework transforms finite precision arithmetic adaptive filters into exact arithmetic adaptive filters with the input data corrupted by rounding error noise that is additive to measurement noise. Findings throughout our backward error analysis framework can provide a guide on how to apply lower precision arithmetic to adaptive filters: (i) the magnitudes of the rounding error noise depend on the numerical stability of the implementation algorithm, arithmetic precision, and regularisation, (ii) the rounding error noise is independently additive to measurement noise, (iii) a higher regularisation is recommended for lower precision arithmetic adaptive filters, and (iv) adaptive filters using lower precision arithmetic have equivalent filter performance to those using higher precision if the magnitudes of rounding error noise are lower than measurement noise.

中文翻译：

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走向低精度自适应滤波器：RLS 后向误差分析的事实


较低精度的算术可以提高自适应滤波器的吞吐量，同时需要更少的硬件资源和功耗。这些优势对于自适应滤波器至关重要，特别是对于物联网和可穿戴应用。为了将较低精度的算术应用于自适应滤波器，需要一个清晰的舍入误差分析框架，因为较低精度的算术会降低滤波器的性能。以前，自适应滤波器的舍入误差分析基于前向误差分析。这限制了与其他外部变量（例如测量噪声、正则化和算法的数值稳定性）相关的舍入误差对自适应滤波器性能影响的描述性。为了克服这些限制，我们首先提出了一种新的自适应递归最小二乘（RLS）滤波器的后向误差分析框架。我们的框架将有限精度算术自适应滤波器转换为精确算术自适应滤波器，输入数据因舍入误差噪声而损坏，而舍入误差噪声是测量噪声的附加值。我们的后向误差分析框架中的发现可以为如何将较低精度算术应用于自适应滤波器提供指导：（i）舍入误差噪声的大小取决于实现算法的数值稳定性、算术精度和正则化，（ii） ) 舍入误差噪声独立地添加到测量噪声中，(iii) 建议对较低精度算术自适应滤波器采用更高的正则化，并且 (iv) 使用较低精度算术的自适应滤波器与使用较高精度的自适应滤波器具有等效的滤波器性能，如果舍入误差噪声低于测量噪声。