On the Zeros of Ramanujan Filters

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.