Parseval Frames and the Discrete Walsh Transform

Suppose that N = 2n and N1 = 2n-1, where n is a natural number. Denote by ℂN the space of complex N-periodic sequences with standard inner product. For any N-dimensional complex nonzero vector (b0, b1,..., bN-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,...., ur ∈ ℂN such that the system of their binary shifts is a Parseval frame for ℂN. It is noted that the vector (b0, b1,..., bN-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.