CSCI 2500 —- Computer Organization

Homework 04 —- Due October 25, 2024

●  This homework is due by 11:59pm EDT on the above date via a Submitty gradeable named “Homework 04”.

●  This homework is to be completed individually. Do not share your solutions with anyone else.

Overview

● Answer the questions and provide solutions to all problems given below.

●  You  MUST  type  up  your  answers.  Handwritten   solutions  will not be accepted or graded, even if they are scanned into a PDF file.

●  You are recommended to use LaTeX (https://www.latex-project.org/). If you have never used LaTeX, you might want to follow some tutorials, like this one: https://www.latex-tutorial.com/tutorials/. There is a convenient online LaTeX editor called Overleaf (https://www. overleaf.com/) that has a free plan for students. You can also use Microsoft Word or free Google Doc (in your Google drive) both of which have equation editors.

●  Submit your answers for this homework on Submitty as a single PDF file named hw04.pdf.

●  All  circuit  diagrams  (schematic  diagrams)  must  be  submitted  as  both pictures   in   your   PDF   and   OpenCircuits   files   (https://opencircuits.io/) uploaded to Submitty.

● This homework will be manually graded by our TAs.

● Explicitly denote any assumptions you need to make.

● Show all work! Because the majority of the problems below have online calculators which would make them otherwise trivial to answer correctly, no work=no credit for this assignment.

●  Clearly  label  every  problem.  If  a  problem  has  a   single  answer,  like 0x561a9d36, make it bold, and use a larger font for better visibility, like this: 0x561a9d36 . You still need to show all the work!

● There are 10 problems in this homework.

Boolean Algebra

1.  Simplify the following expressions using Boolean algebraic laws. Give each step ofyour simplification and denote which laws you’re using for each step. Do not skip or combine steps! (Character * means AND operator)

2.  Find all solutions of the following Boolean equations without using the truth tables: (Character * means AND operator)

3.  Simplify  the following expression by first constructing a truth table, using that truth table to construct a K-map, and then using that K-map to simplify.

(Character * means AND operator)

Logical Circuits

4.  Convert the following truth table into its sum ofproducts representation:

5.  Draw  a logical circuit diagram that represents the above sum of products expression  using  OpenCircuits  (https://opencircuits.io/).  Clearly  label  all inputs/outputs and all components. Make sure you connect appropriate input components  (e.g., buttons,  switches,  clocks,  etc.)  and  output  components (e.g., LEDs, displays,  etc.) to  facilitate testing of your circuit. Download your   diagram   using  OpenCircuits ’  “Download”   feature,   rename   it  to hw04_SOP.circuit, and submit on Submitty along with your hw04.pdf file.

6.  Test your circuit by supplying appropriate inputs and observing the expected  values of the output. Explain why your set of tests is sufficient to prove that  your logical circuit does in fact implement the required Boolean function.  For each test, provide a picture (snapshot) of your circuit. Insert all such  pictures  in  the  hw04.pdf PDF  file.  You  can  download  pictures  (PNG,  JPEG, or PDF) of your circuit diagram using OpenCircuits ’ “Export Image” feature.

7.  Given  inputs A and B, show that NOR{(A  +  B)}  is functionally complete by giving logical circuits equivalent to AND {(A  *  B)}, OR {(A  +  B)}, and NOT {A} gates using only NOR gates in their construction.

Numerical Conversions and Arithmetic (Please note that you only need to solve Problems 8 and 9 if we cover Floating Point numbers during the lectures by Thursday, October 24 . Otherwise,just answer Problem 10.)

8.  For  each  of  the  following  numbers,  convert  them  to  their  closest  single precision IEEE  754  floating point representation. First, denote the binary values   of   the   sign,   fraction,   and   exponent.   Then   provide   a   32-bit hexadecimal value. Show your steps.

a.  50. 4375

b.  0. 0

c.  − Infinity

d.  1. 0000001

9.  For  each  of the  following  hexadecimal  values,  convert them from single precision   IEEE   754   floating   point   representation  to   decimal  rational numbers. You may leave large powers of two in the exponential form, and you may express your answer as a ratio (e.g., − 8/5, 264/1, etc.).  Show your steps.

a.  0xc349a000

b.  0x ffe00001

c.  0x80000000

d.  0x00400000

10.  Give  a  reason  why we use 2’s complement representations for negative numbers in computer arithmetic. Give an example of its usage.

Submission and Grading Criteria

For  this  assignment,  you  will  submit  your  answers  to  the  problems  and  your OpenCircuits diagrams into the Submitty gradeable.