### Abstract

Original language | Undefined |
---|---|

Pages (from-to) | 1383 -1390 |

Number of pages | 8 |

Journal | IEEE transactions on signal processing |

Volume | 64 |

Issue number | 6 |

DOIs | |

Publication status | Published - 16 Mar 2016 |

### Keywords

- EWI-26994
- METIS-316916
- Correlation
- Correlation functions
- Hermite functions
- Wigner distribution function
- Signal analysis
- Signal detection
- Eigenvalues and eigenfunctions
- Polynomials
- IR-100339
- Closed-form solutions
- Ambiguity function
- Fourier transforms
- Convolution
- Time-frequency analysis

### Cite this

}

*IEEE transactions on signal processing*, vol. 64, no. 6, pp. 1383 -1390. https://doi.org/10.1109/TSP.2015.2488580

**Closed-form expressions for time-frequency operations involving Hermite functions.** / Korevaar, C.W.; Oude Alink, M.S.; de Boer, Pieter-Tjerk; Kokkeler, Andre B.J.; Smit, Gerardus Johannes Maria.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Closed-form expressions for time-frequency operations involving Hermite functions

AU - Korevaar, C.W.

AU - Oude Alink, M.S.

AU - de Boer, Pieter-Tjerk

AU - Kokkeler, Andre B.J.

AU - Smit, Gerardus Johannes Maria

N1 - eemcs-eprint-26994

PY - 2016/3/16

Y1 - 2016/3/16

N2 - The product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF) of two Hermite functions of arbitrary order n and m are derived and expressed as a bounded, weighted sum of n+m Hermite functions. It was already known that these mathematical operations performed on Gaussians (Hermite functions of the zeroth-order) lead to a result which can be expressed as a Gaussian function again. We generalize this reciprocity to Hermite functions of arbitrary order. The product, convolution, correlation, WDF, and AF operations performed on two Hermite functions of arbitrary order lead to remarkably similar closed-form expressions, where the difference between the operations is primarily determined by distinct phase changes of the weights of the Hermite functions in the result. The closed-form expressions are generalized to the class of square-integrable functions. A key insight from the closed-form expressions is applied to the design of orthogonal, time-frequency localized communication signals which are characterized by an AF with rotational symmetry. In addition to this application, the theoretical expressions may prove useful for signal analysis in fields ranging from communications, radar and image processing to quantum mechanics.

AB - The product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF) of two Hermite functions of arbitrary order n and m are derived and expressed as a bounded, weighted sum of n+m Hermite functions. It was already known that these mathematical operations performed on Gaussians (Hermite functions of the zeroth-order) lead to a result which can be expressed as a Gaussian function again. We generalize this reciprocity to Hermite functions of arbitrary order. The product, convolution, correlation, WDF, and AF operations performed on two Hermite functions of arbitrary order lead to remarkably similar closed-form expressions, where the difference between the operations is primarily determined by distinct phase changes of the weights of the Hermite functions in the result. The closed-form expressions are generalized to the class of square-integrable functions. A key insight from the closed-form expressions is applied to the design of orthogonal, time-frequency localized communication signals which are characterized by an AF with rotational symmetry. In addition to this application, the theoretical expressions may prove useful for signal analysis in fields ranging from communications, radar and image processing to quantum mechanics.

KW - EWI-26994

KW - METIS-316916

KW - Correlation

KW - Correlation functions

KW - Hermite functions

KW - Wigner distribution function

KW - Signal analysis

KW - Signal detection

KW - Eigenvalues and eigenfunctions

KW - Polynomials

KW - IR-100339

KW - Closed-form solutions

KW - Ambiguity function

KW - Fourier transforms

KW - Convolution

KW - Time-frequency analysis

U2 - 10.1109/TSP.2015.2488580

DO - 10.1109/TSP.2015.2488580

M3 - Article

VL - 64

SP - 1383

EP - 1390

JO - IEEE transactions on signal processing

JF - IEEE transactions on signal processing

SN - 1053-587X

IS - 6

ER -