Parseval Frames and the Discrete Walsh Transform

Abstract

Suppose that N = 2ⁿ and N1 = 2^n-1, where n is a natural number. Denote by ℂ_N the space of complex N-periodic sequences with standard inner product. For any N-dimensional complex nonzero vector (b₀, b₁,..., b_N-1) satisfying the condition

\({\left| {{b_l}} \right|^2} + {\left| {{b_{l + {N_1}}}} \right|^2} \leq \frac{2}{{{N^2}}},\;\;\;l = 0,1,...,{N_1} - 1,\)

we find sequences u₀, u₁,...., u_r ∈ ℂ_N such that the system of their binary shifts is a Parseval frame for ℂ_N. It is noted that the vector (b₀, b₁,..., b_N-1) specifies the discrete Walsh transform of the sequence u₀, and the choice of this vector makes it possible to adapt the proposed construction to the signal being processed according to the entropy, mean-square, or some other criterion.