Studio Intent

1. Create a document for your studio group’s work.
2. List everyone’s name at the top.
3. Review the overview above carefully. Each person in the group should add a comment to the document indicating they are aware of goals of studio time.

Chapter 1: Binary Basics

Binary: What’s it all mean?

A set of bits, like 10101100 has no inherent meaning. This is one reason we have data types in programming languages like Java and Python — so there are cues about how people want bits to be interpreted.

Take a look at the C Programming Language’s Data Types and indicate at least two that could perfectly fit the binary 10101100 (at least two are directly relevant to the reading). Indicate the meaning of 10101100 for each.

The Unsigned Number Line

In class we looked at a diagram of the segment of the integer number line that can be used for unsigned, 3-bit binary:

4-bits

Sketch out the number line for 4-bit unsigned numbers.

The Signed Number Line

In class we drew the signed number line as a single, long line and as a circle/ring. In some ways it may help to think of “bending” that line so that half is above the bend and half is below the bend:

Notice that the line starts with 0 in the top left (all 0’s) and ends with the maximum value, all 1’s, in the bottom left. Exactly half of the values should be on the top part and half on the bottom part.

Draw both the 3-bit and 4-bit number lines in this bent form. Label the binary and decimal values of each point on the lines assuming it’s using two’s complement numbers.

Signs of negativity

One way to identify a negative 2’s complement number is via the leading (leftmost) bit. Explain how it indicates negative values and what it means in terms of the bent number line.

Bitwise Inversion & the bent number line

Inverting a single bit is flipping it to the complementary value: a 0 becomes a 1 and a 1 becomes a zero. When working with equations we represent bit-wise inversion of a value $A$ by placing a bar over it: $\overline{A}$, or in plain text, with a slash in front of it to indicate the bar: /A. Bitwise inversion of an $n$-bit number flips each individual bit. If $A$ is an $n$-bit number, $\overline{A}$ would represent the bitwise inversion of each individual bit. For example, if $A$ is $010$, $\overline{A}$ would be $101$.

Bitwise inversion simply uses an invertor on each bit of a multi-bit value.

Look carefully your number 3-bit and 4-bit two’s complement number lines you’ve just drawn. Develop an equation that represents the mathematical relationship between $A$’s value and $\overline{A}$’s value.

Mathematical Negation

We will be emulating subtraction by adding the mathematical negation of a value. For example, we will transform $4-5$ into $4 + (-5)$, which can be thought of as $4 + (-1 \times 5)$. Another example: $4 - (-5)$ would be thought of as $4 + (-1 \times -5)$. Consequently, we need to be able to handle the concept of “$-1 \times$”.

Using your work in the prior part, how can we do mathematical negation with just the two operations we already have available, negation and addition? Provide an equation of operations that emulates “$-1 \times$”.

Chapter 2: Gates

Tees

Although you may not find Bill’s shirt today to be a witty take on Shakespeare, he feels that it’s eventually true, no matter what (a logical tautology).

Tee’s Truth via Table

Create a truth table showing that the result of the circuit shown on the shirt will always be true. Include columns for the input, the negation of the input, and the result.

Tee’s Truth via Axioms

Which boolean Algebra axio (from class) is represented by the tee shirt? Axioms are given on slides 19-21 of Tuesday’s class)

Simulation of the Tee

Download this starter file: studio_2a_tee.jls and open it in JLS.

Open the file in JLS and look it over. Note that:

1. Throughout the class “ports” will be a key part of breaking complex circuits into manageable parts. Moreover, JLS requires “ports” to interact with circuits. The only two elements in the circuit are two ports.
2. One port is highlighted because it’s being “watched”. Discuss what this means.

Now:

1. Add in a Signal Generator () and, using it, a description of a test case for this circuit. The description can be just 3 lines… or even just 10 characters on one line!
2. Simulate the circuit (Simulator > Show Simulator Window and then the Start button). Does it clearly show that the circuit is “eventually true”?
3. Update the test case to show that it’s “eventually true, no matter what”. I.e., that it becomes true for any initial condition that could be given.

JLS Simulations

JLS is a simulator and makes assumptions about values at the start of the simulations ($t=0$). Based on the last part, what assumptions does it appear to be making at $t=0$? And why is the “eventually true” a relevant statement?

More Simple Circuits

Now download this starter file: studio_2a_circuit.jls (it’s empty, but please use the starter file anyway).

1. Write out the full truth table for Figure 2.25 of the text: $Y = \overline{B} \cdot \overline{C} + A \cdot \overline{B}$ (or, in text: Y=/B*/C + A*/B).
   1. Be sure to go in a “counting order”, with the bits in the order $A$, $B$, $C$. Initially they should all be zero (false), then $C$ should be come a one (001), then $B$ should become a 1 and C should be a zero (010), etc.
   2. How many rows are needed for the table? Why? (There are three inputs. Consider the 3-bit number line)
2. Create a circuit for the equation using JLS and the provided starter file.
3. Create a simulation that cycles through all possible values of $A$, $B$, and $C$. Be sure your simulation accounts for the computation time (propagation delay) needed for the circuit. Again, use a “Counting Order”, with the bits in the order $A$, $B$, $C$.
4. How does the simulation compare to your truth table? Explain any instants in time where it does not precisely match the table.
5. What is the propagation delay (total time that may be needed) for your circuit?

Looking It Up

As has been mentioned, the concept of a “Look-Up Table” (LUT) is vital in digital logic. Explain how one can use your truth table to “lookup” the output needed.

Challenge: Simple Selection

Another (empty) starter file: studio_2a_lut.jls

We’ll soon be using Multiplexers in larger circuits. Multiplexers can also have some unique behaviors that correspond to LUTs. Can you use a multiplexer () to implement the above equation? You may need to use other parts, like a bundle of wires or constants (hover over parts in JLS for the help text). You should be able to use the same test-case you used before.