New Sequences Design From Weil Representation With Low Two-Dimensional Correlation in Both Time and Phase Shifts

Zilong Wang, Guang Gong

Abstract: A new elementary expression of the construction first proposed by Gurevich, Hadani, and Sochen is given, which avoids the explicit use of the Weil representation. The sequences in this signal set are given by both multiplicative character and additive character of finite field <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\BBF_{p}$</tex></formula> . Such a signal set consists of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p^{2}(p-2)$</tex></formula> time-shift distinct sequences, the magnitude of the two-dimensional autocorrelation function (i.e., the ambiguity function) in both time and phase of each sequence is upper bounded by <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$2\sqrt {p}$</tex></formula> at any shift not equal to (0, 0). Furthermore, the magnitude of their Fourier transform spectrum is less than or equal to 2. For a subset consisting of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p(p-2)$</tex> </formula> phase-shift distinct sequences in this signal set, the magnitude of the ambiguity function of any pair is upper bounded by <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$4\sqrt {p}$</tex></formula> . A proof is given through finding a new expression of the sequences in the finite harmonic oscillator system. An open problem for directly establishing these assertions without involving the Weil representation is addressed.