The Fourier representations (FRs) are indispensable mathematical formulations for modeling and analysis of physical phenomena and engineering systems. This study presents a new set of generalized Fourier representations (GFRs) and phase transforms (PTs). The PTs are special cases of the GFRs and true generalizations of the Hilbert transforms. In particular, the Fourier transform based kernel of the PT is derived and its various properties are discussed. The time derivative and integral, including fractional order, of a signal are obtained using the GFR. It is demonstrated that the general class of time-invariant and time-variant filtering operations, analog and digital modulations can be obtained from the proposed GFR. A narrowband Fourier representation for the time-frequency analysis of a signal is also presented using the GFR. A discrete cosine transform based implementation, to avoid end artifacts due to discontinuities present in the both ends of a signal, is proposed. A fractional-delay in a discrete-time signal using the FR is introduced. The fast Fourier transform implementation of all the proposed representations is developed. Moreover, using the analytic wavelet transform, a wavelet phase transform (WPT) is proposed to obtain a desired phase-shift in a signal under-analysis. A wavelet quadrature transform (WQT) is also presented which is a special case of the WPT with a phase-shift of $\pi /2$ radians. Thus, a wavelet analytic signal representation is derived from the WQT. Theoretical analysis and numerical experiments are conducted to evaluate effectiveness of the proposed methods.

傅里叶表示（FR）是对物理现象和工程系统进行建模和分析所必不可少的数学公式。这项研究提出了一组新的广义傅立叶表示（GFR）和相变（PT）。PT是GFR的特殊情况，是Hilbert变换的真实概括。特别地，派生了基于傅立叶变换的PT内核，并讨论了其各种特性。使用GFR获得信号的时间导数和积分（包括分数阶）。事实证明，可以从建议的GFR中获得时不变和时变滤波操作，模拟和数字调制的一般类别。使用GFR还提供了用于信号时频分析的窄带傅立叶表示。提出了一种基于离散余弦变换的实现方式，以避免由于信号两端存在不连续性而导致的末端伪像。引入了使用FR的离散时间信号中的小数延迟。开发了所有提议表示的快速傅里叶变换实现。此外，使用解析小波变换，提出了一种小波相位变换（WPT），以在信号欠分析中获得所需的相移。还提出了小波正交变换（WQT），这是WPT的特殊情况，其相移为 开发了所有提议表示的快速傅里叶变换实现。此外，使用解析小波变换，提出了一种小波相位变换（WPT），以在信号欠分析中获得所需的相移。还提出了小波正交变换（WQT），这是WPT的特殊情况，其相移为 开发了所有提议表示的快速傅里叶变换实现。此外，使用解析小波变换，提出了一种小波相位变换（WPT），以在信号欠分析中获得所需的相移。还提出了小波正交变换（WQT），这是WPT的特殊情况，其相移为$\pi /2$弧度。因此，从WQT得出小波分析信号表示。进行了理论分析和数值实验，以评估所提出方法的有效性。