A water desalination plant has two processors that operate in parallel. Processor A is new and uses a single stage for processing the water to remove salt and Processor B is older, using two stages for processing the water to remove salt. Consider the input to the desalination plant on day n is a volume of water, x[n], and this water is placed in a main holding tank. At the end of day n, just before midnight the contents of the main holding tank, a[n], are piped into the inputs for the two separate desalination processors, with the Processor A input, f[n], getting 40% of the contents of the main tank and the Processor B input, g[n], getting 60% of the contents of the main tank. Just after midnight, Processor A begins processing the water and during that day makes 80% of what it has taken in ready for delivery to the output, y[n] and returns the remaining 20 % to the main holding tank. Also just after midnight, Processor B begins processing the water it has received in two steps: at the end of the first day of processing, it has made 55% of its water available for the output, and passes the remaining 45% through to its second stage just before midnight. After midnight, the second stage processor works on what it received so that by the end of the day it can deliver 65 % of what it has received to the output and it returns the remaining 35% to the main holding tank. Beginning on day n = 0, every day 50,000 cubic meters of salt water are delivered to the plant. a.) Write a difference equation in standard form, where y[n] the volume of desali- nated water produced on day n and x[n] is the volume of new water delivered to the salination plant on day n
b.) Sketch a flow diagram that represents the zero-state system in the 2-domain.
c.) Find the transfer function, Ĥ(z), for the system. Find the values of the poles and zeros for the system.
d.) Specify a closed form formula for x[n] and Ê (2).
e.) If the system can reach steady-state, specify the formula for the steady-state fish population in both the z-domain, Ys(2), and time domain, Yst[n], including values for any constants in the formula. If the system cannot reach steady-state, state the reason it cannot. Note that you do not need to solve for the transient solution output signal.