Sequences with identical autocorrelation spectra

Aperiodic autocorrelation measures the similarity between a finite-length sequence of complex numbers and translates of itself. Autocorrelation is important in communications, remote sensing, and scientific instrumentation. The autocorrelation function reports the aperiodic autocorrelation at every possible translation. Knowing the autocorrelation function of a sequence is equivalent to knowing the magnitude of its Fourier transform. Resolving the lack of phase information is called the phase problem. We say that two sequences are isospectral to mean that they have the same aperiodic autocorrelation function. Sequences used in technological applications often have restrictions on their terms: they are not arbitrary complex numbers, but come from an alphabet that may reside in a proper subring of the complex field or may come from a finite set of values. For example, binary sequences involve terms equal to only $+1$ and $-1$. In this paper, we investigate the necessary and sufficient conditions for two sequences to be isospectral, where we take their alphabet into consideration. There are trivial forms of isospectrality arising from modifications that predictably preserve the autocorrelation, for example, negating sequences or both conjugating their terms and writing them in reverse order. By an exhaustive search of binary sequences up to length $34$, we find that nontrivial isospectrality among binary sequences does occur, but is rare. We say that a positive integer $n$ is barren to mean that there are no nontrivially isospectral binary sequences of length $n$. For integers $n \leq 34$, we found that the barren ones are $1$--$8$, $10$, $11$, $13$, $14$, $19$, $22$, $23$, $26$, and $29$. We prove that any multiple of a non-barren number is also not barren, and pose an open question as to whether there are finitely or infinitely many barren numbers.

Limiting Moments of Autocorrelation Demerit Factors of Binary Sequences

An aperiodic binary sequence of length $\ell$ is written as $f=\ldots,f_{-1},f_0,f_1,\ldots$ with $f_j \in \{-1,1\}$ when $0 \leq j < \ell$ and and $f_j=0$ otherwise. Various problems in engineering and natural science demand binary sequences that do not resemble translates of themselves. The autocorrelation of $f$ at shift $s$ is the inner product of $f$ with the sequence obtained by translating $f$ by $s$ places. The demerit factor of $f$ is the sum of the squares of the autocorrelations at all nonzero shifts for the sequence obtained by normalizing $f$ to unit Euclidean norm. Low demerit factor therefore indicates low self-similarity under translation. We endow the $2^\ell$ binary sequences of length $\ell$ with uniform probability measure and consider the distribution of their demerit factors. Earlier works used combinatorial techniques to find exact formulas for the mean, variance, and skewness of the distribution as a function of $\ell$. These revealed that for $\ell \geq 4$, the $p$th central moment of this distribution is positive for every $p \geq 2$. This article shows that every $p$th central moment is a quasi-polynomial function of $\ell$ with rational coefficients divided by $\ell^{2 p}$. It also shows that, in the limit as $\ell$ tends to infinity, the $p$th standardized moment is the same as that of the standard normal distribution.

Moments of Autocorrelation Demerit Factors of Binary Sequences

Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor. We consider the uniform probability measure on the $2^\ell$ binary sequences of length $\ell$ and investigate the distribution of the demerit factors of these sequences. Previous researchers have calculated the mean and variance of this distribution. We develop new combinatorial techniques to calculate the $p$th central moment of the demerit factor for binary sequences of length $\ell$. These techniques prove that for $p\geq 2$ and $\ell \geq 4$, all the central moments are strictly positive. For any given $p$, one may use the technique to obtain an exact formula for the $p$th central moment of the demerit factor as a function of the length $\ell$. The previously obtained formula for variance is confirmed by our technique with a short calculation, and we demonstrate that our techniques go beyond this by also deriving an exact formula for the skewness.

Sets of Low Correlation Sequences from Cyclotomy

Low correlation (finite length) sequences are used in communications and remote sensing. One seeks codebooks of sequences in which each sequence has low aperiodic autocorrelation at all nonzero shifts, and each pair of distinct sequences has low aperiodic crosscorrelation at all shifts. An overall criterion of codebook quality is the demerit factor, which normalizes all sequences to unit Euclidean norm, sums the squared magnitudes of all the correlations between every pair of sequences in the codebook (including sequences with themselves to cover autocorrelations), and divides by the square of the number of sequences in the codebook. This demerit factor is expected to be $1+1/N-1/(\ell N)$ for a codebook of $N$ randomly selected binary sequences of length $\ell$, but we want demerit factors much closer to the absolute minimum value of $1$. For each $N$ such that there is an $N\times N$ Hadamard matrix, we use cyclotomy to construct an infinite family of codebooks of binary sequences, in which each codebook has $N-1$ sequences of length $p$, where $p$ runs through the primes with $N\mid p-1$. As $p$ tends to infinity, the demerit factor of the codebooks tends to $1+1/(6(N-1))$, and the maximum magnitude of the undesirable correlations (crosscorrelations between distinct sequences and off-peak autocorrelations) is less than a small constant times $\sqrt{p}\log(p)$. This construction also generalizes to nonbinary sequences.

Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro 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 sequence's 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 $A(1.842626\ldots)^n$, where $1.842626\ldots$ is the positive real root of $X^4-3 X-6$. We show that the peak sidelobe level is bounded above by $5(1.658967\ldots)^{n-4}$, where $1.658967\ldots$ 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.