  The special affine Fourier transform (SAFT) is a promising tool for analyzing
non-stationary signals with more degrees of freedom. However, the SAFT fails in
obtaining the local features of non-transient signals due to its global kernel
and thereby make SAFT incompetent in situations demanding joint information of
time and frequency. To circumvent this limitation, we propose a highly flexible
time-frequency transform namely, the special affine wavelet transform (SAWT)
and investigate the associated constant $Q$-property in the joint
time-frequency domain. The basic properties of the proposed transform such as
Rayleigh's theorem, inversion formula and characterization of the range are
discussed using the machinery of special affine Fourier transforms and operator
theory. Besides this, the discrete counterpart of SAWT is also discussed and
the corresponding reconstruction formula is obtained. Moreover, we also drive a
direct relationship between the well known special affine Wigner distribution
and the proposed transform. This is followed by introducing a new kind of wave
packet transform associated with the special affine Fourier transform. Towards
the end, an analogue of the Poisson summation formula for the proposed special
affine wavelet transform is derived.
