An L × L ultrasound transducer is centered at the origin and pointing in the +z direction. The narrowband signal generated by the transducer has an envelope

Two point scatterers with reflectivity of R are located at (0, 0, z₀) and (0, 0, z₀ + z). Ignore attenuation and multiple reflections.

(a) Assume the scatterers are within the region where the geometric assumption holds and find the estimated reflectivity Rˆ (x, y, z).

(b) How far should the scatterers be separated in order for the imaging system to distinguish them?

(c) If the scatterers are in the far field where the Fraunhofer approximation holds, what would be the estimated reflectivity Rˆ (x, y, z)?

(d) Derive an expression for the depth resolution in the far field as a function of z₀.