# Computer Science 237 Computer Organization

### Williams College Fall 2006

Lab 4: Digital Design Lab I
Due: Wednesday, October 11, 2006 at 9:00 AM

Lab meets this week in TCL 312b - the digital logic lab. Due to equipment and space limitations, we will break into smaller lab sections. Please arrive promptly at your assigned time. Homework Problems

Please complete each of the following problems, and turn them in on this sheet.

1. Gray code is an alternative binary representation of integers. Interestingly, incrementing a number in gray code causes the representation to change by exactly one bit. We have seen one context where this representation is useful when we labeled our Karnaugh maps. Consider the following table used to convert 3-bit binary integers into their gray code equivalents:

 binary 000 001 010 011 100 101 110 111 gray code 000 001 011 010 110 111 101 100

1. Construct three combinational circuits that compute the 1's digit, the 2's digit, and 4's digit of gray code. Please use the general or-of-ands network (the "sum-of-products" we discussed in class). Do not apply any simplification techniques.
2. Construct the simplest, most elegant circuit you can to convert a binary number (on three inputs) to a gray code number (on three outputs).
2. In class, we saw how any boolean function can be expressed in disjunctive form -- as a disjunction (or) of a set of terms (often called min-terms), each of which is a conjunction (and) of inputs or their negations. This is a handy if you happen to be a digital circuit designer that has a very large pile of and and not gates, and one big or gate. But what if instead you have large piles of or and not gates, but only a single big and? Show that it is possible to express any boolean function as the conjunction of a set of terms, each of which is a disjunction of inputs.
1. Prove that deMorgan's law for converting conjunctions to disjunctions (with negations) holds for n>2 inputs.
2. Prove that deMorgan's law for converting disjunctions to conjunctions (with negations) holds for n>2 inputs.
3. Use these to prove the conjecture.
3. Suppose you are interested in constructing a circuit that is high precisely when four input lines DCBA represent a prime in 4-bit unsigned binary.
1. Use a Karnaugh map to generate a logical expression with the smallest number of terms that computes this function. Do not optimize the expression further.
2. Suppose we didn't care if the function worked on the range 12..15. Use another Karnaugh map to generate a logical expression with the smallest number of terms that computes this function.
Digital Design Lab

Hand in this lab worksheet along with the written homework problems.

Objective: To familiarize yourself with the workings of the

digital lab and various simple TTL circuits.

When you enter lab, you will be handed a collection of TTL chips in antistatic foam. When you are not using the chips, please remember to return them to their antistatic containers. When the environment is dry, static electricity is a significant problem. The antistatic foam helps to dissipate any charges that otherwise might damage the chips.

You should have a number of different chips, including (but not limited to):