Parseval Frames and the Discrete Walsh Transform

Suppose that $N=2^n$ and $N_1=2^{n-1}$, where $n$ is a natural number. Denote by ${\mathbb C}_N$ the space of complex $N$-periodic sequences with standard inner product. For any $N$-dimensional complex nonzero vector $(b_0,b_1,\dots,b_{N-1})$ satisfying the condition
$$ |b_{l}|^2+|b_{l+N_1}|^2 \le \frac{2}{N^2}\,, \qquad l=0,1,\dots,N_1-1, $$
we find sequences $u_0,u_1,\dots,u_r\in {\mathbb C}_N$ such that the system of their binary shifts is a Parseval frame for ${\mathbb C}_N$. Moreover, the vector $(b_0,b_1,\dots, b_{N-1})$ specifies the discrete Walsh transform of the sequence $u_0$, 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.