Article
Uncertainty Principles for Linear Canonical Transform
Dept. of Electron. Eng., Beijing Inst. of Technol., Beijing
IEEE Transactions on Signal Processing (Impact Factor: 2.79). 07/2009; 57(7):28562858. DOI: 10.1109/TSP.2009.2020039 Source: DBLP
ABSTRACT
This correspondence investigates the uncertainty principles under the linear canonical transform (LCT). First, a lower bound on the uncertainty product of signal representations in two LCT domains for complex signals is derived, which can be achieved by a complex chirp signal with Gaussian envelope. Then, the tighter lower bound for real signals in two LCT domains proposed by Sharma and Joshi is also proven to hold for arbitrary LCT parameters based on the properties of moments for the LCT. The uncertainty principle for the fractional Fourier transform is a special case of the achieved results.

 "It can be regarded as a generalization of many mathematical transforms such as the Fourier transform , Laplace transform, the fractional Fourier transform, and the Fresnel transform. Many fundamental properties of this extended transform are already known, including shift, modulation, convolution, and correlation and uncertainty principle, for example, in [1] [2] [3] [4] [5] [6]. "
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ABSTRACT: We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied. 
 "Many signal processing operations, such as the Fourier transform (FT), the fractional Fourier transform (FRFT), the Fresnel transform, and the scaling operations are special cases of this transform. The LCT has proven to be a powerful tool for optical systems, gradientindex medium system analysis, filter design, time–frequency analysis, radar system analysis, pattern recognition, communications, and many others [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]. The continuoustime LCT of a signal or function, f ðtÞ A L 2 ðRÞ, is defined as [2] F M ðuÞ ¼ L M ff ðtÞgðuÞ ¼ R R f ðtÞK M ðu; tÞ dt; b a 0 ffiffiffi d p e ðjcd=2Þu 2 f ðduÞ; b ¼ 0 ( ð1Þ where L M denotes the LCT operator, and kernel K M ðu; tÞ is given by K M ðu; tÞ ¼ A b e ðja=2bÞt 2 þ ðjd=2bÞu 2 À ðj=bÞut ð2Þ where M ¼ ða; b; c; dÞ, a; b; c; d A R satisfying ad À bc ¼ 1, and A b ¼ 1= ffiffiffiffiffiffiffiffiffi ffi j2πb p . "
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ABSTRACT: The linear canonical transform (LCT) has proven to be a powerful tool in optics and signal processing. Most existing sampling theories of this transform were derived from the LCT bandlimited signal viewpoint. However, in the real world, many analog signals encountered in practical engineering applications are nonbandlimited. The purpose of this paper is to derive sampling theorems of the LCT in function spaces for frames without bandlimiting constraints. We extend the notion of shiftinvariant spaces to the LCT domain and then derive a sampling theorem of the LCT for regular sampling in function spaces with frames. Further, the theorem is modified to the shift sampling in function spaces by using the Zak transform. Sampling and reconstructing signals associated with the LCT are also discussed in the case of Riesz bases. Moreover, some examples and applications of the derived theory are presented. The validity of the theoretical derivations is demonstrated via simulations. 
 "However, both [18] and [23] can be reduced to the uncertainty principle (1.1). Although the lowerbounds given in [16] and [21] are not as the same as that for the complex signals case given in [24], all those correspond to the uncertainty principle (1.1). [22] and [25], with different conditions and proofs, provide a lowerbound for uncertainty principle for LCT corresponding to the sharper uncertainty principle (1.2). "
Article: A Tighter Uncertainty Principle for Linear Canonical Transform in Terms of Phase Derivative
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ABSTRACT: This study devotes to uncertainty principles under the linear canonical transform (LCT) of a complex signal. A lowerbound for the uncertainty product of a signal in the two LCT domains is proposed that is sharper than those in the existing literature. We also deduce the conditions that give rise to the equal relation of the new uncertainty principle. The uncertainty principle for the fractional Fourier transform is a particular case of the general result for LCT. Examples, including simulations, are provided to show that the new uncertainty principle is truly sharper than the latest one in the literature, and illustrate when the new and old lower bounds are the same and when different.
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