### Abstract

For a binary sequence S_{n} = {s_{i} : i = 1, 2,...,n} ∈ {±1}^{n}, n > 1, the peak sidelobe level (PSL) is defined as M(S_{n}) = max_{k=1,2,n-1} |Σ _{i=1}^{n-k} s_{i}s_{i}+k|. It is shown that the distribution of M (S_{n}) is strongly concentrated, and asymptotically almost surely γ(S_{n}) = M(S_{n})/√n In n ∈ [1 - o(1),√2]. Explicit bounds for the number of sequences outside this range are provided. This improves on the best earlier known result due to Moon and Moser that the typical γ(S_{n}) ∈ [o(1/√In n), 2], and settles to the affirmative the conjecture of Dmitriev and Jedwab on the growth rate of the typical peak sidelobe. Finally, it is shown that modulo some natural conjecture, the typical γ(S_{n}) equals √2.

Original language | English (US) |
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Article number | 5361502 |

Pages (from-to) | 545-554 |

Number of pages | 10 |

Journal | IEEE Transactions on Information Theory |

Volume | 56 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2010 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Keywords

- Aperiodic autocorrelation
- Concentration
- Peak sidelobe level (PSL)
- Random binary sequences autocorrelation
- Second moment method

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## Cite this

*IEEE Transactions on Information Theory*,

*56*(1), 545-554. [5361502]. https://doi.org/10.1109/TIT.2009.2034803