Comments on ‘Autocorrelation functions for Hermite-polynomial ultra-wideband pulses’

Nadarajah, Saralees

doi:10.1049/el:20070267

Cited by 5 publications

(4 citation statements)

References 18 publications

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“…The Hermite polynomials in the right-hand side of (12) are not orthogonal when multiplied by the weighting function of (11). Therefore, as we aim for a set of orthogonal Hermite functions, the polynomials are reformulated in terms of polynomials , which is achieved by the following identity [23]: (13) The product of two Hermite polynomials as in (11) can now be written as a vector-matrix-vector product as shown in (14) with elements corresponding to ( 12) and ( 13), respectively. .…”

Section: The Product Convolution and Correlation Of Hermite Functionsmentioning

confidence: 99%

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Closed-Form Expressions for Time-Frequency Operations Involving Hermite Functions

Korevaar

Alink²,

Boer

et al. 2016

IEEE Trans. Signal Process.

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The product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF) of two Hermite functions of arbitrary order and are derived and expressed as a bounded, weighted sum of 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.

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Section: The Product Convolution and Correlation Of Hermite Functionsmentioning

confidence: 99%

“…Autocorrelation functions of Hermite functions have first been addressed by Miller in 2003 [12]. In response to that article, Nadarajah derived the autocorrelation functions for Hermite functions [13]. Cross-correlation functions for Hermite functions have also been derived in terms of associated Laguerre polynomials [14].…”

mentioning

confidence: 99%

Closed-Form Expressions for Time-Frequency Operations Involving Hermite Functions

Korevaar

Alink²,

Boer

et al. 2016

IEEE Trans. Signal Process.

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“…In response to that article, Nadarajah derived the autocorrelation functions for Hermite functions [58]. Cross-correlation functions for Hermite functions have also been derived in terms of associated Laguerre polynomials [59].…”

Section: Related Work and Outlinementioning

confidence: 99%

Hermite-based communications

Korevaar

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“…The frequency domain representation of MHPs of order 0, 1,2 and 3 can be expressed as (2.7) The autocorrelation functions of nth order MHPs are defined as [53,54] (2.8)…”

Section: Chapter 2 Ultra Wideband Radio Systems: a Reviewmentioning

confidence: 99%

Orthogonal pulse based modulation schemes for time hopping ultra wideband radio systems

Sudhan¹

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Firstly, I would like to express my utmost thanks and gratitude to my supervisor and career role model, Assoc Prof. A. S. Madhukumar, for his invaluable guidance and constant encouragement during my PhD study in NTU. Without his help and patience, this dissertation could not have reached the current stage. It has been a genuine privilege to have the opportunity to do research as a member of his research group. I am grateful to him because he has taught me not only numerous topics in wireless communication but also many different aspects of real life. I have learned a lot from him and I believe he has prepared me well for the future. I also thank Dr. A. B. Premkumar for serving on my committee, offering valuable suggestions and reviewing my works. I hope to be able to benefit from his profound knowledge and experience in the coming years, too. I would like to thank all the faculty members and technicians in Center for Multimedia and Network Technology (CeMNet) for their kind help and generous encouragement. I would also like to thank the School of Computer Engineering for the financial support provided during my research.

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Comments on ‘Autocorrelation functions for Hermite-polynomial ultra-wideband pulses’

Cited by 5 publications

References 18 publications

Closed-Form Expressions for Time-Frequency Operations Involving Hermite Functions

Closed-Form Expressions for Time-Frequency Operations Involving Hermite Functions

Hermite-based communications

Orthogonal pulse based modulation schemes for time hopping ultra wideband radio systems

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