Spectrum efficient, localized, orthogonal waveforms: closing the gap with the Balian-low theorem

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    Abstract

    The Balian-Low theorem (BLT) states the fundamental impossibility to design waveforms for L2(ℿ), which 1) form an orthogonal set, 2) are time-frequency localized, and 3) attain a critical waveform density such that they form an orthogonal basis. This article closes the gap between existing waveform designs and the BLT. The main contribution is the design of orthogonal, time-frequency localized, spectrum efficient waveforms for hexagonal lattices. The waveform design is adaptive by a single design parameter, which trades off time-frequency localization with the waveform density. As the orthogonalization procedure is based on employing the minimum number of most time-frequency localized waveforms (Hermite functions) it is argued that the results may be optimal in terms of combined spectrum efficiency and time-frequency localization. An example is provided for waveforms for a hexagonal lattice, which are quasi-orthogonal, time-frequency localized, and up to 99% of the critical waveform density. Although the designed waveforms are not strictly orthogonal, their cross-correlation can be made arbitrarily small. The robustness in doubly dispersive channels and the efficiency for multiuser scenarios are discussed and compared to conventional orthogonal frequency division multiplexing (OFDM).
    Original languageEnglish
    Pages (from-to)2155-2165
    Number of pages11
    JournalIEEE Transactions on Communications
    Volume64
    Issue number5
    DOIs
    Publication statusPublished - May 2016

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    Orthogonal frequency division multiplexing

    Keywords

    • Hermite functions
    • Pulse shaping hexagonal lattice communications
    • Balian-low theorem
    • Orthogonal frequency division multiplexing (OFDM)

    Cite this

    @article{d850c150adbc4d4999bc80caa8e96559,
    title = "Spectrum efficient, localized, orthogonal waveforms: closing the gap with the Balian-low theorem",
    abstract = "The Balian-Low theorem (BLT) states the fundamental impossibility to design waveforms for L2(ℿ), which 1) form an orthogonal set, 2) are time-frequency localized, and 3) attain a critical waveform density such that they form an orthogonal basis. This article closes the gap between existing waveform designs and the BLT. The main contribution is the design of orthogonal, time-frequency localized, spectrum efficient waveforms for hexagonal lattices. The waveform design is adaptive by a single design parameter, which trades off time-frequency localization with the waveform density. As the orthogonalization procedure is based on employing the minimum number of most time-frequency localized waveforms (Hermite functions) it is argued that the results may be optimal in terms of combined spectrum efficiency and time-frequency localization. An example is provided for waveforms for a hexagonal lattice, which are quasi-orthogonal, time-frequency localized, and up to 99{\%} of the critical waveform density. Although the designed waveforms are not strictly orthogonal, their cross-correlation can be made arbitrarily small. The robustness in doubly dispersive channels and the efficiency for multiuser scenarios are discussed and compared to conventional orthogonal frequency division multiplexing (OFDM).",
    keywords = "Hermite functions, Pulse shaping hexagonal lattice communications, Balian-low theorem, Orthogonal frequency division multiplexing (OFDM)",
    author = "Korevaar, {C. Willem} and Kokkeler, {Andre B.J.} and {de Boer}, Pieter-Tjerk and Smit, {Gerard J.M.}",
    year = "2016",
    month = "5",
    doi = "10.1109/TCOMM.2016.2535333",
    language = "English",
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    pages = "2155--2165",
    journal = "IEEE Transactions on Communications",
    issn = "0090-6778",
    publisher = "IEEE",
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    }

    Spectrum efficient, localized, orthogonal waveforms : closing the gap with the Balian-low theorem. / Korevaar, C. Willem; Kokkeler, Andre B.J.; de Boer, Pieter-Tjerk; Smit, Gerard J.M.

    In: IEEE Transactions on Communications, Vol. 64, No. 5, 05.2016, p. 2155-2165.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - Spectrum efficient, localized, orthogonal waveforms

    T2 - closing the gap with the Balian-low theorem

    AU - Korevaar, C. Willem

    AU - Kokkeler, Andre B.J.

    AU - de Boer, Pieter-Tjerk

    AU - Smit, Gerard J.M.

    PY - 2016/5

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    N2 - The Balian-Low theorem (BLT) states the fundamental impossibility to design waveforms for L2(ℿ), which 1) form an orthogonal set, 2) are time-frequency localized, and 3) attain a critical waveform density such that they form an orthogonal basis. This article closes the gap between existing waveform designs and the BLT. The main contribution is the design of orthogonal, time-frequency localized, spectrum efficient waveforms for hexagonal lattices. The waveform design is adaptive by a single design parameter, which trades off time-frequency localization with the waveform density. As the orthogonalization procedure is based on employing the minimum number of most time-frequency localized waveforms (Hermite functions) it is argued that the results may be optimal in terms of combined spectrum efficiency and time-frequency localization. An example is provided for waveforms for a hexagonal lattice, which are quasi-orthogonal, time-frequency localized, and up to 99% of the critical waveform density. Although the designed waveforms are not strictly orthogonal, their cross-correlation can be made arbitrarily small. The robustness in doubly dispersive channels and the efficiency for multiuser scenarios are discussed and compared to conventional orthogonal frequency division multiplexing (OFDM).

    AB - The Balian-Low theorem (BLT) states the fundamental impossibility to design waveforms for L2(ℿ), which 1) form an orthogonal set, 2) are time-frequency localized, and 3) attain a critical waveform density such that they form an orthogonal basis. This article closes the gap between existing waveform designs and the BLT. The main contribution is the design of orthogonal, time-frequency localized, spectrum efficient waveforms for hexagonal lattices. The waveform design is adaptive by a single design parameter, which trades off time-frequency localization with the waveform density. As the orthogonalization procedure is based on employing the minimum number of most time-frequency localized waveforms (Hermite functions) it is argued that the results may be optimal in terms of combined spectrum efficiency and time-frequency localization. An example is provided for waveforms for a hexagonal lattice, which are quasi-orthogonal, time-frequency localized, and up to 99% of the critical waveform density. Although the designed waveforms are not strictly orthogonal, their cross-correlation can be made arbitrarily small. The robustness in doubly dispersive channels and the efficiency for multiuser scenarios are discussed and compared to conventional orthogonal frequency division multiplexing (OFDM).

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    KW - Pulse shaping hexagonal lattice communications

    KW - Balian-low theorem

    KW - Orthogonal frequency division multiplexing (OFDM)

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