Parameterized transform domain computation of the Hilbert Transform applied to separation of channels in Doppler spectra

The Hilbert Transform is a well known analytical technique which is widely used in demodulation of signals, shift invariant wavelet analysis, and many other domains. Since computation of the Hilbert transform through convolution with the Hilbert kernel is cumbersome, the convolution is usually mapped to multiplication with the frequency response of the Hilbert kernel. However, computation of the Fourier transform is computationally intensive compared to other transforms, such as the Haar wavelet transform and others. This has led to the use of other transforms in computing the Hilbert transform. We demonstrate parameterized methods of computing the Hilbert transform using classical transforms like DCT and Haar, and so called Haar- like transforms. The efficacy of these methods in separating the positive and negative channels is demonstrated on Doppler spectra.