Consider a phased array ultrasound system operating at 2 MHz and capable of B-mode and M-mode imaging that is located on the starship Prometheus. We are examining a fetus in utero (in the uterus of a living woman), which is suspected to be an alien.

(a) Calculate the depth of penetration that can be acheived with an 80 dB dynamic range. Assume an absorption coefficient α = af dB cm⁻¹, for a = 1.0 dB cm⁻¹ MHz⁻¹ and f in MHz.

b) What is the maximum frame rate acheivable, for a 256 × 256 B-mode image? Assume a speed of sound c = 1, 480 m/s.

(c) We notice a quickly moving fetal heart, located at a depth of z = 10 cm and lateral position x = 5 cm. We wish to focus the pulse at this location. Assume we have 129 transducer elements, where the ith (i from −64 to 64) element is located at position  cm and range z = 0 cm. Calculate an expression for the delay times, τi, and find the value of τ₆₄. Let τ₀ = 0. Delays may be negative.

(d) An M-mode image is produced from the signal focused on the heart. Calculate maximum the pulse repetition rate for this image. Alien hearts oscillate like a sine wave with frequency 500 beats per minute. Is our M-mode frame rate high enough to sample this periodic signal without introducing aliasing? How about our B-mode frame rate? Explain