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Register a new userCrosscorrelation of Rudin-Shapiro-Like Polynomials
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Daniel Katz
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We consider the class of Rudin-Shapiro-like polynomials, whose $L^4$ norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial $f(z)=f_0+f_1 z + cdots + f_d z^d$ is identified with the sequence $(f_0,f_1,ldots,f_d)$ of its coefficients. From the $L^4$ norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. In this paper, we study the crosscorrelation properties of pairs of sequences associated to Rudin-Shapiro-like polynomials. We find an explicit formula for the crosscorrelation merit factor. A computer search is then used to find pairs of Rudin-Shapiro-like polynomials whose autocorrelation and crosscorrelation merit factors are simultaneously high. Pursley and Sarwate proved a bound that limits how good this combined autocorrelation and crosscorrelation performance can be. We find infinite families of polynomials whose performance approaches quite close to this fundamental limit.
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Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which are infinite families of binary sequences, usually represented as polynomials. Each family of Rudin-Shapiro-like sequences is obtained from a starting sequence (which we call the seed) by a recursive construction that doubles the length of the sequence at each step, and many sequences produced in this manner have exceptionally low aperiodic autocorrelation. Borwein and Mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most $3$, and found the seeds of length $40$ or less that produce the maximum asymptotic merit factor of $3$. The definition of Rudin-Shapiro-like sequences was generalized by Katz, Lee, and Trunov to include sequences with arbitrary complex coefficients, among which are families of low autocorrelation polyphase sequences. Katz, Lee, and Trunov proved that the maximum asymptotic merit factor is also $3$ for this larger class. Here we show that a family of such Rudin-Shapiro-like sequences achieves asymptotic merit factor $3$ if and only if the seed is either of length $1$ or is the interleaving of a pair of Golay complementary sequences. For small seed lengths where this is not possible, the optimal seeds are interleavings of pairs that are as close as possible to being complementary pairs, and the idea of an almost-complementary pair makes sense of remarkable patterns in previously unexplained data on optimal seeds for binary Rudin-Shapiro-like sequences.
Sequences with low aperiodic autocorrelation and crosscorrelation are used in communications and remote sensing. Golay and Shapiro independently devised a recursive construction that produces families of complementary pairs of binary sequences. In the simplest case, the construction produces the Rudin-Shapiro sequences, and in general it produces what we call Golay-Rudin-Shapiro sequences. Calculations by Littlewood show that the Rudin-Shapiro sequences have low mean square autocorrelation. A sequences peak sidelobe level is its largest magnitude of autocorrelation over all nonzero shifts. H{o}holdt, Jensen, and Justesen showed that there is some undetermined positive constant $A$ such that the peak sidelobe level of a Rudin-Shapiro sequence of length $2^n$ is bounded above by $A(1.842626ldots)^n$, where $1.842626ldots$ is the positive real root of $X^4-3 X-6$. We show that the peak sidelobe level is bounded above by $5(1.658967ldots)^{n-4}$, where $1.658967ldots$ is the real root of $X^3+X^2-2 X-4$. Any exponential bound with lower base will fail to be true for almost all $n$, and any bound with the same base but a lower constant prefactor will fail to be true for at least one $n$. We provide a similar bound on the peak crosscorrelation (largest magnitude of crosscorrelation over all shifts) between the sequences in each Rudin-Shapiro pair. The methods that we use generalize to all families of complementary pairs produced by the Golay-Rudin-Shapiro recursion, for which we obtain bounds on the peak sidelobe level and peak crosscorrelation with the same exponential growth rate as we obtain for the original Rudin-Shapiro sequences.
Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences (i.e., sequences with terms in ${-1,1}$). If $f$ is a nonzero sequence, then its autocorrelation demerit factor, $text{ADF}(f)$, is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing $f$ to have unit Euclidean norm. If $(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demerit factor, $text{CDF}(f,g)$, is the sum of the squared magnitudes of the aperiodic crosscorrelation values over all shifts for the sequences obtained by normalizing both $f$ and $g$ to have unit Euclidean norm. Pursley and Sarwate showed that for binary sequences, the sum of $text{CDF}(f,g)$ and the geometric mean of $text{ADF}(f)$ and $text{ADF}{(g)}$ must be at least $1$. For randomly selected pairs of long binary sequences, this quantity is typically around $2$. In this paper, we show that Pursley and Sarwates bound is met for binary sequences precisely when $(f,g)$ is a Golay complementary pair. We also prove a generalization of this result for sequences whose terms are arbitrary complex numbers. We investigate constructions that produce infinite families of Golay complementary pairs, and compute the asymptotic values of autocorrelation and crosscorrelation demerit factors for such families.
We show that a recently proposed Rudin-Shapiro-like sequence, with balanced weights, has purely singular continuous diffraction spectrum, in contrast to the well-known Rudin-Shapiro sequence whose diffraction is absolutely continuous. This answers a question that had been raised about this new sequence.
In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we show that the Mahler measure of the Rudin-Shapiro polynomials of degree $n=2^k-1$ is asymptotically $(2n/e)^{1/2}$, as it was conjectured by B. Saffari in 1985. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers.
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