Unified fractional Fourier transform and sampling theorem

Erseghe, Tomaso; Kraniauskas, P.; Carioraro, G.

doi:10.1109/78.806089

Cited by 146 publications

(73 citation statements)

References 11 publications

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“…Both definitions are of identical form since the value of for is [1]. It is also worth noting that the functions we have just defined are chirp-periodic in the sense of [6], [8].…”

Section: Fundamental Theorem For Lctsmentioning

confidence: 90%

“…The discrete LCT of has been defined as follows for [3], [4]: is not the continuous LCT of . The special case of (3) for the FRT has been defined in [6], but we note that this definition is different than the discrete FRT in [7]. The definition in (3) can be made unitary by including an additional factor .…”

Section: Discrete Linear Canonical Transformsmentioning

confidence: 99%

“…This result is the generalization of the Poisson sum formula [5], and is related to the LCT sampling theorem [9]- [11]. The right-hand side of this expression defines the discrete-time LCT [4] and its special case for the FRT defines the discrete-time FRT [6]. Now, sample in (11) with a sampling interval chosen as with an arbitrary integer.…”

Section: Fundamental Theorem For Lctsmentioning

confidence: 99%

“…In this case, the series will give the periodically replicated and phase modulated extension of outside its finite extent. Unlike the discrete-time LCTs, which take discrete signals to continuous signals [4], [6], [12], linear canonical series, which take continuous signals to discrete signals, do not seem to have received attention in the literature.…”

Section: Fundamental Theorem For Lctsmentioning

confidence: 99%

“…Had the proof of the theorem been carried out by applying the operator to the sampled version of instead of applying to the sampled version of , one would obtain the duals of (7) and (11): (13) (14) Here (13) provides the exact relation for the inverse DLCT and (14) gives the expression for the linear canonical series, which generalizes ordinary Fourier series. The fractional Fourier series in [6], [12], [13] is a special case of this series. Just as periodic functions have Fourier series, a function in the form of (5), which is chirp-periodic, has a linear canonical series.…”

Section: Fundamental Theorem For Lctsmentioning

confidence: 99%

See 4 more Smart Citations

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

Oktem

Özaktaş

2009

IEEE Signal Process. Lett.

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Abstract-Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. We present the exact relation between continuous and discrete LCTs (which generalizes the corresponding relation for Fourier transforms), and also express it in terms of a new definition of the discrete LCT (DLCT), which is independent of the sampling interval. This provides the foundation for approximately computing the samples of the LCT of a continuous signal with the DLCT. The DLCT in this letter is analogous to the DFT and approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We also define the bicanonical width product which is a generalization of the time-bandwidth product.Index Terms-Bicanonical width product, fractional Fourier transform, linear canonical series, linear canonical transform.

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“…Both definitions are of identical form since the value of for is [1]. It is also worth noting that the functions we have just defined are chirp-periodic in the sense of [6], [8].…”

Section: Fundamental Theorem For Lctsmentioning

confidence: 90%

Section: Discrete Linear Canonical Transformsmentioning

confidence: 99%

Section: Fundamental Theorem For Lctsmentioning

confidence: 99%

Section: Fundamental Theorem For Lctsmentioning

confidence: 99%

Section: Fundamental Theorem For Lctsmentioning

confidence: 99%

See 3 more Smart Citations

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

Oktem

Özaktaş

2009

IEEE Signal Process. Lett.

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References

2022

Time‐Frequency Analysis of Seismic Signals

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Generalized convolution theorem associated with fractional Fourier transform

Shi

Sha

Song

et al. 2012

Wireless Communications

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The fractional Fourier transform (FRFT)-a generalization of the well-known Fourier transform (FT)-is a comparatively new and powerful mathematical tool for signal processing. Many results in Fourier analysis have currently been extended to the FRFT, including the ordinary convolution theorem. However, the extension of the ordinary convolution theorem associated with the FRFT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, a generalized convolution theorem for the FRFT is proposed, and the dual of it is also presented. The ordinary convolution theorem and some of its existing extensions related to the FRFT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

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Unified fractional Fourier transform and sampling theorem

Cited by 146 publications

References 11 publications

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

References

Generalized convolution theorem associated with fractional Fourier transform

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