This paper investigates a class of weighted-type fractional Fourier transform (WFRFT), which is mainly used in signal processing and image encryption. To date, studies have primarily focused on the application of WFRFT, and few studies have examined its properties in detail. We propose a new reformulation of WFRFT, whose properties can be readily proven. Multiweighted-type fractional Fourier transform (M-WFRFT) has attracted researchers' attention as a generalized form of WFRFT. It is very difficult to prove the properties of M-WFRFT in weighted form, and researchers have assumed that M-WFRFT has the properties of WFRFT in application. The properties of M-WFRFT are proved by applying the new reformulation proposed. The results show that M-WFRFT has boundary and unitary properties, but in few cases. We analyze and discuss the conditions in which that M-WFRFT satisfies the boundary and unitary conditions, providing a theoretical foundation for application of M-WFRFT.

本文研究了一类加权类型的分数阶傅里叶变换（WFRFT），它主要用于信号处理和图像加密。迄今为止，研究主要集中在WFRFT的应用上，很少有研究详细研究WFRFT的特性。我们提出了一种新的WFRFT格式，其性质可以很容易地证明。作为WFRFT的广义形式，多加权分数阶傅里叶变换（M-WFRFT）引起了研究人员的关注。以加权形式证明M-WFRFT的特性非常困难，研究人员已经假定M-WFRFT在应用中具有WFRFT的特性。应用所提出的新公式证明了M-WFRFT的性质。结果表明，M-WFRFT具有边界和单一性质，但极少数情况。