PAPR Problem for Walsh Systems and Related Problems

High peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and degradation of several transmission performances. We continue the theoretical contributions made by Boche and Farell toward the understanding of peak value reduction, using the strategy known as tone reservation for orthogonal transmission schemes. There it was shown that for orthogonal frequency-division multiplexing (OFDM) systems, the combinatorial object called arithmetic progression plays an important role in setting limitations for the applicability of the tone reservation method. In this paper, we show that the combinatorial object introduced as perfect Walsh sum (PWS) plays a similar role for code-division multiple access (CDMA) systems as arithmetic progression for OFDM systems. By specific construction, we show that for a chosen numbers <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, all subsets <inline-formula> <tex-math notation="LaTeX">$\mathcal {I} $ </tex-math></inline-formula> of the set <inline-formula> <tex-math notation="LaTeX">$[N]$ </tex-math></inline-formula> of the first <inline-formula> <tex-math notation="LaTeX">$N=2^{n}$ </tex-math></inline-formula> natural numbers, which has the density in <inline-formula> <tex-math notation="LaTeX">$[N]$ </tex-math></inline-formula> larger than a given <inline-formula> <tex-math notation="LaTeX">$\delta \in (0,1)$ </tex-math></inline-formula>, i.e., <inline-formula> <tex-math notation="LaTeX">$\left |{ \mathcal {I} }\right |/N\geq \delta $ </tex-math></inline-formula>, and which is sufficiently large enough, in the sense that <inline-formula> <tex-math notation="LaTeX">$\left |{ I }\right |\geq 2(2/\delta)^{2^{m}-1}$ </tex-math></inline-formula>, contains a PWS of size <inline-formula> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula>. By means of this result, and motivated by the previously mentioned connection between arithmetic progression and PWS, we show results for the PWS which are analogous to the famous Szemerédi theorem on arithmetic progressions, Conlon-Gower’s theorem on probabilistic construction of “sparse” sets containing an arithmetic progression, and even a solution of an analogon to the Erdős’ conjecture on arithmetic progressions. Those results give in particular an insight into the asymptotic limitations of tone reservation method for the CDMA systems. Besides, we show that a subset <inline-formula> <tex-math notation="LaTeX">$I$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$[N]$ </tex-math></inline-formula> is a PWS if and only if the embedding inequality of the subspace of <inline-formula> <tex-math notation="LaTeX">$L^{1}([{0,1}])$ </tex-math></inline-formula>, containing linear combinations of Walsh functions indexed by elements of <inline-formula> <tex-math notation="LaTeX">$\mathcal {I} $ </tex-math></inline-formula>, holds with the minimum possible embedding constant <inline-formula> <tex-math notation="LaTeX">$\sqrt { \left |{ \mathcal {I} }\right |}$ </tex-math></inline-formula>. The corresponding approach based in particular by the fact that the PWSs are the only Walsh sums having unit <inline-formula> <tex-math notation="LaTeX">$L^{1}$ </tex-math></inline-formula>-norm, proven in this paper. By means of that results, we show that the minimum possible threshold constant for which the tone reservation method is applicable yields <inline-formula> <tex-math notation="LaTeX">$\sqrt { \left |{ \mathcal {I} }\right |}$ </tex-math></inline-formula> if and only if the information set <inline-formula> <tex-math notation="LaTeX">$\mathcal {I} $ </tex-math></inline-formula> is a PWS.