(Design Problem) Can you think how to dramatically reduce the complexity of implementing the function of Exercise 1.27? (Hint: Rather than starting with the truth table, think through a simple implementation of direct Boolean functions to implement the mappings between the four inputs-d28, d29, d30, and d31-and the five binary outputs).
(a) Write down your equations for simplified implementation of Exercise 1.27.
(b) Characterize its complexity in the same way as in part (c) of Exercise 1.26 and Exercise 1.27.
Exercise 1.27
(Design Problem) Now consider a different way to achieve the same result. Keep the Calendar system exactly as discussed in class. But, add a new component that takes as inputs the four outputs-d28, d29, d30, and d31-and maps these into the five outputs as described in Exercise 1.26.
(a) Develop the truth table for the new portion of the function.
(b) Write down the Boolean equations for each of the five outputs.
(c) Once again, characterize the complexity of the implementation by tabulating the number of gates of various inputs needed to realize the five outputs.
(d) Given the complexity of the original calendar subsystem, and this new subsystem, how does this solution compare with the one you developed for Exercise 1.26? Which is better?
Exercise 1.26
(Design Problem) Consider the Calendar subsystem presented in this chapter. We will change the output specifications slightly while the inputs will remain the same. Directly generate the 5-bit binary number for the number of days in the month: 28 = 111002, 29 = 111012, 30 = 111102, and 31 = 111112.
(a) Develop the truth table for the revised function, with four inputs to represent the month, one input to indicate a leap year, and the five outputs as indicated above.
(b) Writer down the Boolean equations for each of the five outputs.
(c) Characterize the complexity of this implementation by counting the number of AND, OR, and NOT gates of various input sizes needed to realize each of the five outputs (e.g., so many 2-input ANDs, 3-input ANDs, etc.).