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.
"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, gradient-index medium system analysis, filter design, time–frequency analysis, radar system analysis, pattern recognition, communications, and many others             . The continuous-time LCT of a signal or function, f ðtÞ A L 2 ðRÞ, is defined as  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 . "
[Show abstract][Hide abstract] 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 band-limited signal viewpoint. However, in the real world, many analog signals encountered in practical engineering applications are non-bandlimited. The purpose of this paper is to derive sampling theorems of the LCT in function spaces for frames without band-limiting constraints. We extend the notion of shift-invariant 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  and  can be reduced to the uncertainty principle (1.1). Although the lower-bounds given in  and  are not as the same as that for the complex signals case given in , all those correspond to the uncertainty principle (1.1).  and , with different conditions and proofs, provide a lower-bound for uncertainty principle for LCT corresponding to the sharper uncertainty principle (1.2). "
[Show abstract][Hide abstract] ABSTRACT: This study devotes to uncertainty principles under the linear canonical transform (LCT) of a complex signal. A lower-bound 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.
IEEE Transactions on Signal Processing 11/2013; 61(21):5153-5164. DOI:10.1109/TSP.2013.2273440 · 2.79 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Recently many uncertainty principles involving the product of signal spreads in two linear canonical transform (LCT) and fractional Fourier transform (FRFT) domains have been presented in the literature. In this paper we derive some new equalities/inequalities involving the sum and product of signal spreads in two FRFT domains. Some equalities involving the sum of signal spreads in two LCT domains are also presented.
Signal Processing 03/2010; 90(3):880-884. DOI:10.1016/j.sigpro.2009.09.010 · 2.21 Impact Factor
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