Uriostegui Kenan
J Opt Soc Am A Opt Image Sci Vis. 2020 Jun 1;37(6):951-958. doi: 10.1364/JOSAA.387945.
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. A34, 18 (2017)JOAOD60740-323210.1364/JOSAA.34.000018] 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.
线性规范变换（LCTs）在信号处理的多个领域中都很重要；特别是，它们被扩展到复数值参数以描述光学系统。这些复LCTs的一个特殊情况是巴尔格曼变换。最近，裴和黄[《美国光学学会杂志》A34, 18 (2017)JOAOD60740 - 323210.1364/JOSAA.34.000018]提出了巴尔格曼变换的一种归一化方法，使得在无穷远处对其进行界定成为可能。在本文中，我们遵循裴 - 黄算法，通过SU(2)有限谐振子模型中巴尔格曼变换与旋量变换之间的关系引入离散归一化巴尔格曼变换，并将其与基于SU(2)振子模型相干态的离散巴尔格曼变换进行比较。这种变换是可逆且酉的。我们表明，与连续情形类似，离散归一化巴尔格曼变换将厄米 - 克拉夫丘克函数转换为拉盖尔 - 克拉夫丘克函数。此外，我们证明了离散的su(1,1)排斥振子函数在这种离散变换下能以很小的误差自我再生。最后，在由SU(2)谐振子的波函数所张成的空间中，我们发现离散归一化巴尔格曼变换与分数傅里叶 - 克拉夫丘克变换对易。
