ot = [a (t) + jHilbert (a (t))} e -*\" 0*, (A-3) F igure A . 1 : A Modi f i ed N e w t o n search far the es t imat ion of the t ime delay. T h e root of the phase is i terat ively searched. In each step of the i tera t ion, the phase is calculated and approximated by a linear funct ion w i t h a slope equal to the transducers centroid frequency. T h e intercept of this l inear function w i t h the abscissa is the new estimate for the t ime delay [90]. Fur thermore the post-compression signal has to be re-sampled by a sub-sample t ime shift [90]. A m o n g a l l available re-sampling methods, linear interpolation is used for the purpose of real t ime applicat ions since i t is more accurate for base-band da ta [89]. A.2 Combined Autocorrelation Method (CAM) 128 A . 2 Combined Autocorrelation Method (CAM) T h e me thod has the merits of phase domain processing but wi thout al iasing, since i t combines the result of phase correlat ion w i t h that of envelope correlat ion, b o t h of w h i c h are calculated d i rec t ly from the R F signal using complex autocorrela t ion processing [102,103]. Phase correlat ion is used for displacement measurement and envelope correlat ion is used for phase unwrapping . In order to perform the complex autocorrelat ion, phase informat ion is needed. S i m i l a r l y to the P R S method , the H i l b e r t t ransform can be used to convert the t ime-domain signals to analyt ic signals. Af te r this step the complex cross-correlation function is used to detect the phase shift: \/\u2022to\/2 Rab (t- n)= a+(t + v) b*+ (t + nT\/2 + v) dv = Ru (t; r - n T \/ 2 ) e-^o(r-r\u00abr\/2) ^ ( A_ 4) J-to\/2 where T is per iod of ultrasonic wave, UJQ is carrier angular frequency, r is the t ime shift and Ru(t;T) is the autocorrelat ion of the envelope. Fo r the special case of n = 0, the displacement d can be obtained for the phase shift ip = LJQT. If the displacement is less than A \/ 4 i t can be obtained di rect ly and wi thout ambigui ty : < A - 5 > Since large displacements may be necessary for elast ici ty imaging, C A M uses another phase unwrapp ing step by using the envelope normal ized correlat ion coefficient defined by ^ u.^ \\Rab(t;n)\\ Cu{t'n)-\\a(t)\\\\b(t + nT\/2)\\> (A\"6) A c c o r d i n g to E q u a t i o n (A-4) and E q u a t i o n (A-6) , for each t ime t, two sets of Cu and (p = UQT may be obta ined as c u (t) \u2014 {Cu M , . . . , Cu 1 , C \u00b0 , C \u201e , . . . C^ , } v(t) = {