代写EE 2301 Introduction to Digital Systems Design Fall 2019 Midterm 1调试数据库编程

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EE 2301

Introduction to Digital Systems Design

Fall 2019

1. Number Systems, Fractional Numbers, and the Complement Method for Negative Numbers [25 points]

(a) Number Systems

In what base is the following operation performed:

DEF + DAD = 1A8B

[10 points]

(b) Fractional Numbers

Consider the number

2 + 10 + 16 + 2/1 + 10/1 + 16/1

(here calculated in base 10). Express this number in base 2, base 10, and in base

16. [5 points]

In class,we learned how to compute the two’s complement of a number in base 2. Consider now a generalization of that concept to other bases. Consider base b. With n digits, represent the negative of a number v as bn - v. (Call this “b’s complement.”)

i. For base 7, what range of numbers can you represent with 2 digits? Express your answer in decimal. [5 points]

ii. Show how to perform. the operation 267 - 157 in 7’s complement arithmetic. [5 points]

2. Truth Tables, Boolean Expressions, and Gates [50 points]

Consider a function f (W, X, Y, Z) that is one if exactly three of its inputs are 1.

(a) Fill in the following truth table. [5 points] inputs

(b) Write a minimal AND-OR two-level expression for this function. [10 points]

(d) Write a minimal NAND-NAND two-level expression for this function. [5 points] (e) Write a minimal NOR-NOR two-level expression for this function. [5 points] (f) Write an XNF expression for this function, that is, an expression consisting of

the XOR of AND terms, with no negations. [10 points]

(g) Draw a network of logic gates that computes this function. [5 points]

3. Circuits for XOR [25 points]

Draw a circuit to compute the XOR function of 31 variables with only AND gates and inverters. Use as few AND gates as possible. (Note that you can deﬁne smaller circuits, put these in “boxes”, and use these boxes to build larger circuits. However, you must specify everything in the smallest box.)