# Digital quantities and analog quantities.

1. Digital theory (15 points total)

a. Give two examples of “digital” quantities and two examples of “analog” quantities.
b. One byte is eight bits. How many possible characters can one byte represent (I.E. how many different combinations of these eight bits are possible? Show your reasoning.)

c. Mathematically, what makes a piece of information “digital?” (That is, if it is a “digital” signal or piece of information, what type of numbers must be used to represent it? (Hint: the answer is not “binary” or “hexadecimal”.)
d. Using three NOR gates, show how to make an AND gate. (Q = A * B)

e. A logic gate is called a “universal” gate if any other Boolean logic circuit can be built, using only that type of gate. Specifically, for a two-input gate to be universal, it must be capable of inverting an input under at least some conditions, and its output must depend on each input under some, but not all, conditions.

Name the two standard universal gates:

2. DeMorgan’s Theorem (10 points total)

Simplify the following expressions, using DeMorgan’s Theorem. (Simplified expressions should only have negation on single variables, not negated expressions.
I.E. (A + B) is okay but (A + B) would need to be simplified. )

Watch out for trick questions; just because a variable is in an expression, doesn’t mean it will be a part of the simplified answer! After all, (A + A) == 1 !
(Show your work or reasoning for partial credit.)

a. (A + B) * (A + B)
b. (A * B * C)

c. (A + B) + (C + D)

d. (A + B) + (A * B)
3. Karnaugh map minimization (20 points total)
Use a 4-variable Karnaugh map to minimize the following function. The final Boolean expression should use only ANDs and ORs (for instance, (E + F) etc). Assume that all variables are available as both positive and negative, if needed. You can use a truth table to construct the K-Map, if you want, but this is not required. Show your work (I.E. show how each term in the final expression maps to one or more 1s in the K-Map.)

Q = ABCD + ABCD + ABCD + ABCD + ABCD
4. Practical considerations in Digital Design (20 points total)
a. What is wrong with this circuit? (I.E. why will it not work correctly when turned on?)
Explain what the circuit will do when the switch is closed, and how would you correct it.
b. Draw a circuit using one pull-up resistor (1k) and one SPST switch to provide a logical input to an inverter gate (shown). Show all voltage sources and sinks that you use.
c. Will the following circuit produce a 1 or a 0 output? Show your reasoning by tracing the signals through each gate; no credit will be given if your work is not shown.
(For example, two “1” inputs into an AND gate will produce a “1” output…)

d. What pins (by pin number) are outputs on a 74LS02 Quad 2-input NOR chip?
(See the chip pinout for the 7402 inside the cover of your book.)
5. Digital Arithmetic (10 points total; show your work for all steps):
a. Convert this hexadecimal expression to binary.
(Just convert the numbers; do not perform the subtraction yet): A1hex – 42hex.

b. Take the twos’ complement of the second binary number (the 42hex) from #b above.
c. Add this number to the first binary number from #a (the A1hex) to do the subtraction.
d. Convert the resulting number from binary to decimal.

e. Convert the resulting number from binary to hexadecimal.
6. Circuit design exercise (25 points total):
Create a circuit to implement a “three-of-four” function.
This circuit takes four digital inputs (A, B, C, and D) and produces a single output, Q.

Q should be 1 when at least three inputs are high, and 0 if fewer than three are high.

You may use as many AND, OR, and NOT gates as you like, but you may only use these types of gates for this exercise.

a.) Complete the four-variable truth table for the “three-of-four” function. (Remember, this function should be a 1 if three or four inputs are high, and a 0 otherwise.) (5 points)

b.) Make a 4-variable Karnaugh map of your function. (Use the template below.)
(5 points)

c.) Circle the simplified terms on the Karnaugh map, above (5 points).

d.) From your circled terms, identify the simplified Boolean equation for the function. (5 points)

e.) Draw a schematic of the circuit. Make sure to label A, B, C, D, and Q. Make sure AND and OR gates are clearly distinguishable (AND gates have a straight back; OR gates are concave at the back and have a sharper front.)
Extra credit (worth up to 10 points extra; partial credit given):

A 555 timer IC can be used as an “A-and-not-B” gate. The symbol for this is shown here:

Prove that this gate (although unusual and asymmetric) is universal, by using them to construct any of the following (choose one option):

• AND and NOT gates (you must construct both of these); OR

• OR and NOT gates (you must construct both of these); OR

• A NAND gate (known to be universal); OR

• A NOR gate (also known to be universal).

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