Construction of Invertible Sequences for Multipath Estimation

Abstract

J. Ruprecht has proposed coding schemes that allow for multipath estimation. His schemes use sequences a₀…a _nwith a_j = ±1for each j such that the associated polynomial $f(z) = \sum {{a_{j}}{z^{j}}}$ has a large

$${R_{p}}(f) = \frac{{n + 1}}{{\sum\limits_{{k = 0}}^{n} {|f({e^{{2\pi ik/(n + 1)}}}){|^{{ - 2}}}} }}.$$

Most sequences have a small R(f), and those with maximal are hard to find. This note shows for n of the form n =q-1,qa prime, one can construct sequences with R(f)≥n-O(n^1/3).Since R(f) ≥ n+1 for any sequence, this construction is asymptotically close to optimal. It also produces large values of R(f) for small n.

It is also shown that for n =q - 1 qa prime, there exist sequences a₀,..., a_n, such that the associated polynomialf(z)satisfies

$$ \left| {f\left( {e^{2\pi ik/\left( {n + 1} \right)} } \right)} \right| = \left( {1 + o\left( 1 \right)} \right)n^{1/2} as n \to \infty $$

uniformly for 0 ≤k≤n.