Linear canonical transforms (LCTs) are important in several areas of signal processing; in particular, they were extended to complex-valued parameters to describe optical systems. A special case of these complex LCTs is the Bargmann transform. Recently, Pei and Huang [J. Opt. Soc. Am. A 34, 18 (2017) [CrossRef] ] presented a normalization of the Bargmann transform so that it becomes possible to delimit it near infinity. In this paper, we follow the Pei–Huang algorithm to introduce the discrete normalized Bargmann transform by the relationship between Bargmann and gyrator transforms in the SU(2) finite harmonic oscillator model, and we compare it with the discrete Bargmann transform based on coherent states of the SU(2) oscillator model. This transformation is invertible and unitary. We show that, as in the continuous analog, the discrete normalized Bargmann transform converts the Hermite–Kravchuk functions into Laguerre–Kravchuk functions. In addition, we demonstrate that the discrete su(1,1) repulsive oscillator functions self-reproduce under this discrete transform with little error. Finally, in the space spanned by the wave functions of the SU(2) harmonic oscillator, we find that the discrete normalized Bargmann transform commutes with the fractional Fourier–Kravchuk transform.

中文翻译：

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离散归一化Bargmann变换通过gyrator变换。

线性规范变换（LCT）在信号处理的多个领域都很重要。特别是，它们被扩展为复数值参数以描述光学系统。这些复杂LCT的特例是Bargmann变换。最近，裴和黄[J．选择。Soc。上午。甲34，18（2017）[交叉引用] 提出对Bargmann变换进行归一化，从而有可能将其定界为无穷大。在本文中，我们遵循Pei-Huang算法，通过SU（2）有限谐振荡器模型中Bargmann和回转器之间的关系引入离散归一化Bargmann变换，并将其与基于相干态的离散Bargmann变换进行比较SU（2）振荡器模型的模型。这种转变是可逆的和单一的。我们证明，如同在连续模拟中一样，离散归一化的Bargmann变换将Hermite–Kravchuk函数转换为Laguerre–Kravchuk函数。另外，我们证明了离散su（1,1）排斥振荡器功能在这种离散变换下具有很小的误差，能够自我再现。最后，