Doppler power spectrum from a Gaussian sample volume

A closed-form expression for the Doppler power spectrum due solely to the range of blood velocities passing through a Gaussian sample volume placed anywhere in a vessel under conditions of axisymmetric flow, uniform backscatter and negligible intrinsic spectral broadening has been derived. The formulation presented here allows the independent specification of the sample volume position and width, in the three dimensions, and enables simple estimations of spectral shape for pulsed wave Doppler systems. Simpler expressions were derived for the cases of symmetric sample volume projections onto the vessel cross-section and/or sample volumes centred in the vessel. Closed form expressions were derived for mean frequency and spectral width in the case of a symmetric sample volume projection centred in the vessel. The effects of sample volume size and position on the Doppler spectral width and mean frequency are shown for a range of velocity profiles.