Phase-derivative imaging. III: Theoretical derivation of first and second order statistics

The echo signal obtained from a homogenous and isotropically scattering medium can be described as a Poisson time series which is convolved with the transmission pulse of the transducer. The probability density function (pdf) of this signal approximates to a Gaussian pdf for narrowband pulse waveform. Methods to derive the phase-derivative (PD) signal from the complex envelope and the preenvelope of the echo signal are described. The first order pdf of the PD asymptotically becomes a Gaussian pdf by smoothing. Since the rectified PD is employed to obtain 2-dimensional grey scale images, the first order pdf as well as the signal-to-noise ratio (SNR) of this signal are also derived. The rectified PD is further smoothed by a cosine time window prior to the imaging. The SNR and the autocorrelation function (in the axial direction) of this latter signal can be derived under the assumption of a Gaussian spectrum of the transmission pulse. These first and second order characteristics of the PD images are calculated for the conditions employed in simulations and experiments reported previously and are quantitatively compared to the values obtained from these.