Ramanujan filter banks have been used for identifying periodicity structure in streaming data. This letter studies the locations of zeros of Ramanujan filters. All the zeros of Ramanujan filters are shown to lie on or inside the unit circle in the z-plane. A convenient factorization appears as a corollary of this result, which is useful to identify common factors between different Ramanujan filters in a filter bank. For certain families of Ramanujan filters, further structure is identified in the locations of zeros of those filters. It is shown that increasing the number of periods of Ramanujan sums in the filter definition only increases zeros on the unit circle in z-plane. A potential application of these results is that by identifying common factors between Ramanujan filters, one can obtain efficient implementations of Ramanujan filter banks (RFB) as demonstrated here.

中文翻译：

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拉马努金滤波器的零点


拉马努金滤波器组已用于识别流数据中的周期性结构。这封信研究了拉马努金滤波器的零点位置。拉马努金滤波器的所有零点均位于 z 平面的单位圆之上或之内。方便的因式分解作为该结果的推论出现，这对于识别滤波器组中不同拉马努金滤波器之间的公共因子很有用。对于拉马努金滤波器的某些系列，在这些滤波器的零位置处识别出进一步的结构。结果表明，增加滤波器定义中拉马努金和的周期数只会增加 z 平面单位圆上的零。这些结果的一个潜在应用是，通过识别拉马努金滤波器之间的共同因素，人们可以获得拉马努金滤波器组（RFB）的有效实现，如此处所示。