# Ferreira-2006a

### From IEETA

## Article

Title | Superoscillations: Faster than the Nyquist Rate |
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Author | Paulo J S G Ferreira, Achim Kempf |

Journal | IEEE Transactions on Signal Processing |

Volume | 54 |

Number | 10 |

Pages | 3732-3740 |

Month | October |

Year | 2006 |

DOI | [1] |

Group | |

Group (before 2015) | Signal Processing Laboratory |

Indexed by ISI | Yes |

### Abstract

For any fixed bandwidth there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients or superoscillations can only occur in signals which possess amplitudes of widely different scales. This paper investigates the required dynamical range and energy (squared *L*^{2} norm) as a function of the superoscillation's frequency, number, and maximum derivative. It briefly discusses some of the implications of superoscillating signals, in reference to information theory and time-frequency analysis, for example. It shows, among other things, that the required energy grows exponentially with the number of superoscillations, and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations' period.