Usually a wavelet transform is based on dilated–translated wavelets. We propose a symplectic-transformed–translated wavelet family ${\psi }_{r,s}^{*}(z-\kappa )$ ( $r,s$ are the symplectic transform parameters, ${\mid s\mid }^2-{\mid r\mid }^2=1$ , κ is a translation parameter) generated from the mother wavelet ψ and the corresponding wavelet transformation $W_{\psi }f(r,s;\kappa )={\int }_{-\infty }^{\infty }({\mathrm{d}}^{2}z∕\pi )f\left(z\right){\psi }_{r,s}^{*}(z-\kappa )$ . This new transform possesses well-behaved properties and is related to the optical Fresnel transform in quantum mechanical version.