  Fourier representation (FR) is an indispensable mathematical formulation for
modeling and analysis of physical phenomenon, engineering systems and signals
in numerous applications. In this study, we present the generalized Fourier
representation (GFR) that is completely based on the FR of a signal, and
introduce the phase transform (PT) which is a special case of the GFR and a
true generalization of the Hilbert transform. We derive the PT kernel to obtain
any constant phase shift, discuss the various properties of the PT, and
demonstrate that (i) a constant phase shift in a signal corresponds to variable
time-delays in all harmonics, (ii) to obtain a constant time-delay in a signal,
one need to provide variable phase shift in all harmonics, (iii) a constant
phase shift is same as the constant time-delay only for single frequency
sinusoid. The time derivative and time integral, including fractional order, of
a signal can be obtained using the GFR. We propose to use discrete cosine
transform (DCT) based implementation to avoid end artifacts due to
discontinuities present in both end of the signal. We introduce fractional
delay of a discrete time signal using the FR, and present the fast Fourier
transform (FFT) implementation of all the above proposed representations. Using
the analytic wavelet transform (AWT), we propose wavelet phase transform (WPT)
to obtain a desired phase-shift in a signal under-analysis, and propose the two
representations of wavelet quadrature transform (WQT) which is special case of
the WPT where phase-shift is $\pi/2$ radians.
