It is proved that the Window Average Gradient (WAG) scheme designed to detect singularities in the turbulence is approximately equivalent to the Hilbert transform of the "Mexican Hat" wavelet. Several identities are derived between these schemes and their validity are theoretically and experimentally shown through the fluctuating wall shear stress and velocity data taken in the turbulent boundary layer. The instantaneous amplitude-frequency representation of WAG at the large scale is also considered. These results indicate that the wall turbulence is significantly regular. It is shown that the near wall singularities involve in the large scale frequency shift key process and that the corresponding instantaneous phase consists of discontinuous line segments.