Linear Array Transmit Beamforming and Sampling RequirementsContext:You are designing the transmit beamformer for a linear ultrasound array. The system needs to focus energy at specific depths along the central axis of the array. You are tasked with calculating the required time delays for the elements and determining the necessary sampling parameters for the system.System Parameters:Array Type: Linear ArrayNumber of Elements ($N$): 16Element Pitch ($d$): $300 \mu m$Speed of Sound ($c$): $1540$ m/sCenter Frequency ($f_c$): 5 MHzFractional Bandwidth: 80%Tasks:Delay Calculation (10 cm Focus):Calculate the transmit time delays required to focus the beam at a depth of 10 cm on the array's central axis ($x = 0$).Assume the delays are calculated relative to the array center.Ensure all delays are positive values (the element closest to the focal point should have the maximum delay, or conversely, the reference time is set such that the first firing element has $t=0$).Plot the Time Delay (ns) vs. Element Number.Effect of Focal Depth:Repeat the delay calculation for focal depths of 8 cm (shallower) and 12 cm (deeper).Generate a single comparative plot showing the delay profiles for all three depths (8 cm, 10 cm, and 12 cm) on the same axes.Observe how the curvature of the delay profile changes as the focal depth increases.Sampling Frequency Requirements:To accurately synthesize these delays digitally, the system requires a sufficient sampling clock.Calculate the upper cutoff frequency ($f_{max}$) of the transducer based on the center frequency and bandwidth.Determine the minimum sampling frequency ($F_s$) required to satisfy the Nyquist criterion for the signal band.