Problem 1. The behavior of a 1-input, 1-output device is measured by hooking a voltage source to its input and measuring the voltage at the output for several different input voltages:

We're interested in whether this device can serve as a legal combinational device that obeys the static discipline. For this device, obeying the static discipline means that

_IN

_IL

_OUT

_OH

_IN

_IH

_OUT

_OL

When answering the questions below, assume that all voltages are constrained to be in the range 0V to 5V.

Can one chose a V_OL of 0V for this device? Explain.

No, since the static discipline requires that V_OUT <= V_OL for a valid "0" output and V_OUT for this device never falls below 0.5V.

What's the smallest V_OL one can choose and still have the device obey the static discipline? Explain.

The smallest V_OL one can choose is 0.5V, the lowest output voltage produced by the device.

Assuming that we want to have 0.5V noise margins for both "0" and "1" values, what are appropriate voltage levels for V_OL, V_IL, V_IH, and V_OH so that the device obeys the static discipline. Hint: there are many possible choices, just choose one that obeys the constraints listed above.
Here are the rules of operation imposed on us by the static discipline:
1. V_IN >= V_IH implies V_OUT <= V_OL
2. V_IN <= V_IL implies V_OUT >= V_OH
3. V_OL + noise margin = V_IL
4. V_OH - noise margin = V_IH
So if V_OL = 0.5V and assuming 0.5V noise margins, then
and finally we can check that rule 1 is obeyed by these choices.

Assuming that we want to have 0.5V noise margins for both "0" and "1" values, what is the largest possible voltage level for V_OL that still results in a device that obeys the static discipline?
To determine our maximum V_OL, let's set it equal to N. Then
which is simply the noise margin added to V_OL. If we let M be the size of the forbidden region in volts,
Finally,
We'll use the function VTC(v) as the name of the function represented graphically by the voltage transfer characteristic. So, we want to find N and M such that V_OH = VTC(V_IL) and V_OL = VTC(V_IH). If we assume that V_IL and V_IH lie in the range [1,2.5] (we'll check that assumption in a minute) then
Solving the two equations we get N =1 and M = 1 giving us V_OL = 1, V_IL = 1.5, V_IH = 2.5, V_OH = 3. V_IL and V_IH lie in the range we assumed above, so we're all set.

Assuming that we want to have equal noise margins for both "0" and "1" values, what is the largest noise margin we can achieve with this device and still obey the static discipline?
It's clear that to maximize our noise margins, V_OL should be as small as possible and V_OH should be as large as possible. By inspecting the VTC we see that V_OL = 0.5V is the best we can do. From earlier parts we know we can achieve a noise margin N of at least 0.5V.
We can use the following construction to figure out the largest noise margin N. Consider two of these devices hooked in series:

The letters "A" through "E" mark points of interest where we'll want to determine the voltage. Assume we apply 0.5V at point "A". Then:
In order to get 0.5V at point "E", the VTC tells us that the input voltage (voltage at "D") must be greater than or equal to 3V. So 5 - 3N >= 3 which we achieve if N <= 2/3.

Problem 2. Inverter madness.

The following graph plots the voltage transfer characteristic for a device with one input and one output.
Can this device be used as a combinational device in a logic family with 0.75V noise margins?

No. The device doesn't have any region in which its gain is greater than 1, so there are no choices for V_IL, V_OL, V_IH, V_OH which provide for non-zero noise margins.

You are designing a new logic family and trying to decide on values of the four parameters V_IL, V_OL, V_IH, V_OH that lead to non-zero noise margins for various possible inverter designs. Four proposed inverter designs exhibit the voltage transfer characteristics shown in the diagrams below. For each design, either (1) specify suitable values of V_IL, V_OL, V_IH, V_OH. or (2) explain why no values for these parameters satisfy the static discipline.

Device A: V_OL = 1V, V_IL = 3.5V, V_IH = 5V, V_OH = 6V
Device B: no values exist, the gain is never greater than one
Device C: no values exist, the gain is never greater than one
Device D: V_OL = .5V, V_IL = 1V, V_IH = 3.5V, V_OH = 6V

Problem 3. Static discipline.

Consider a combinational buffer with one input and one output. Suppose we set its input to some voltage (V_IN), wait for the device to reach a steady state, then measure the voltage on its output (V_OUT) and find V_OUT < V_OL. What can we say about V_IN?

A combinational device guarantees that valid inputs lead to valid outputs if we wait longer than the propagation delay. But the opposite inference isn't true: a valid output doesn't imply a valid input -- for example, the buffer is allowed a valid "1" before the input has crossed V_IH.
So the only statement that we can make is that V_IN < V_IH since if V_IN >= V_IH the buffer is required to produce V_OUT >= V_OH.

Now consider an inverter. Suppose we set its inputs to some voltage (V_IN), wait for the device to reach a steady state, then measure the voltage on its output (V_OUT) and find V_OUT > V_OH. What can we say about V_IN?

Using the same argument as above, the only statement we can make is that V_IN < V_IH.

Problem 4. Ternary Logic.

Ternary is a term referring to the number system in base 3. Consider a convention in which a ternary digit is presented as an electric voltage between 0 and 10 V. Let 0-1 V represent a valid "0" output, 4-6 V a valid "1" output and 9-10 V a valid "2" output.

Assuming noise margins 1 V wide, show the mapping of logic levels to voltages for this ternary system. Include valid logic-level outputs, noise margins and forbidden zones. Your chart should resemble the following diagram, except of course it will incorporate 3 valid signal levels:

Graph the transfer characteristic for a device capable of acting as a ternary logic buffer, i.e., a device that produces at its output the same logic level present at its input, as shown below:

We need three "stable" regions, one for each valid output, and a high gain between valid levels to help restore poor inputs to an output valid level.

Can a device with the following transfer characteristic be used as a ternary logic buffer? Why or why not?

No, it can't be used as a ternary logic buffer. The device has only two "stable" regions.

Problem 5. Bread and Circuits, Inc. has discovered an interesting electronic device (a Z-module) which is made using a single yeast cell, a speck of flour, and a grain of salt. The Z-module has two inputs carrying voltages VA and VB and a single output carrying VC. The output VC settles, after a 10ns period of stable input voltages, to the product of the input voltages restricted to range of 0V to 2V. In other words,

V_C =	V_A * V_B when 0 <= V_A * V_B <= 2
	2 when V_A * V_B > 2

Assume that the Z-module treats negative input voltages as if they were 0.

B&C is trying to make a logic family using Z-modules. As a starting point, they claim that

constitutes a valid combinational buffer under the proper voltage-to-logic representation conventions and will yield reasonable noise margins.

Which, if any, of the following proposed voltage thresholds yields a valid buffer with positive noise margins?
```
A. VOL = 1.5V  VIL = 1.6V  VIH = 1.7V  VOH = 1.8V
B. VOL = 0.7V  VIL = 0.9V  VIH = 1.1V  VOH = 1.3V
C. VOL = 0.1V  VIL = 0.3V  VIH = 1.7V  VOH = 1.9V
D. VOL = 0.5V  VIL = 0.5V  VIH = 1.5V  VOH = 1.5V
```
Choice A would allow two "low" inputs (e.g., 1.6V) to generate a "high" output (1.6 * 1.6 = 2.56 which is above VOH) => not okay.
Choice B would allow two "high" inputs (e.g., 1.1V) to generate an underdetermined output (1.1 * 1.1 = 1.21 which is above VOL but below VOH) => not okay.
Choice C has positive noise margins and the allowed input voltages result in the desired behavior => okay
Choice D doesn't have positive noise margins => not okay.

Determine the voltage thresholds that maximize the noise margins of B&C's buffer. If the noise margins are not independent, maximize the smaller of the two.

To maximize the noise margins of the buffer, we consider the voltage transfer characteristic of the Z-module. Since VIN² >= 2 will result in a VOH of 2, then we have VIH >= sqrt(2). Thus, VIH = sqrt(2), and VOH = 2.
Since VOUT = VIN² in the lower region of the voltage transfer characteristic, then let us represent VIN with "x" and VOUT with "x²". Then the lower noise margin is represented by the function f(x) = x - x².
To maximize the noise margin, we take the first derivative of f(x) and set it to zero: f'(x) = 1 - 2x = 0. Solving for x gives 0.5. Thus, VIL = x = 0.5V, and VOL = x² = 0.25V.

Ivan Idea, chief logician at B&C, is exploring the use of a single Z-module as a two-input logic gate, again with positive noise margins. He suspects that a Z-module, under the appropriate logic conventions, can be used for an AND or OR gate which obeys the static discipline. Ivan's been at it for several weeks and needs your help.
Can you find a way to use a Z-module for an AND or OR combinational device? If so, give the appropriate input and output voltage thresholds and the function performed. If not, carefully explain why the Z-module can't be used as AND or OR.

No, the Z-module can't be used as either an AND gate or an OR gate.
In order to implement an OR gate, we must have the behavior that a logic "1" input and a logic "0" input produces a logic "1" output. In the case of a single Z-module, if we had 2V as the first input and 0V as the second input (noiseless, perfect input voltages) the output is 0V, instead of a logic "1" voltage.
In order to implement an AND gate, we must have the behavior that a logic "1" input and a logic "0" input produces a logic "0" output. According to the static discipline, a "0" output means VOUT should be less than equal to VOL. If we substitute 2V for the logic "1" input and consider the "0" input, we have VOUT = VIL * 2 <= VOL. But we can't have VOL > VIL, so the Z-module can't function as an AND gate.

Ivan sidesteps the previous enigma by allowing himself the use of several Z-modules as components of a single logic gate. He assumes (and you may too) that noise enters the system only between logic gates, not between the components of a single gate. He notes that each of B&C's proposed buffers (using one Z-module) is an amplifier with gain greater than one. Ivan reasons that by cascading many such devices (as shown below), he can achieve arbitrarily high gain and hence excellent noise margins.

His plan is to use such a high-gain cascade on the output of a Z-module to restore the validity of marginal signal levels. Describe the voltage transfer characteristic (i.e, VIN vs. VOUT) of a cascade of a large number of Z-modules (via an expression or sketch).

For N cascaded Z-modules, the voltage transfer characteristic is:

Is there any way by which many Z-modules can be used to build a 2-input AND gate whose noise margins are both greater than 0.75V? If so, sketch an approach (giving a diagram and calculating the noise margins). If not, give a brief but convincing explanation.

It is possible to build an AND gate whose noise margins greater than 0.75V. Our strategy is to clean up both input signals independently before computing the logic function. A cascade of 5 Z-modules is enough to restore marginal signals, and the last Z-module on the right performs the AND function. The diagram looks like this:

Problem 6. Combinational construction rules

In lecture, we learned two basic principles regarding the class of combinational devices. The first allows us to build a combinational device from, e.g., electronic components:

A combinational device is a circuit element that has
- one or more digital inputs
- one or more digital outputs
- a functional specification that details the value of each output for every possible combination of valid input values
- a timing specification consisting (at minimum) of an upper bound t_pd on the required time for the device to compute the specified output values from an arbitrary set of stable, valid input values.

while the second allows us to construct complex combinational devices from acyclic circuits containing simpler ones:

A set of interconnected elements is a combinational device if
- each circuit element is combinational
- every input is connected to exactly one output or to some vast supply of 0's and 1's
- the circuit contains no directed cycles

In this problem, we ask you to think carefully about why these rules work - in particular, why an acyclic circuit of combinational devices, constructed according to the second principle, is itself a combinational device as defined by the first. You may assume for the following that every input and output is a logical 0 or 1.

Consider the following 2-input acyclic circuit whose two components, A and B, are each combinational devices:

The propagation delay - the upper bound on the output settling time - for each device is specified in nanoseconds. The functional specifications for each component are given as truth tables detailing output values for each combination of inputs:

Give a truth table for the acyclic circuit, i.e. a table that specifies the value of z for each of the possible combinations of input values on x and y.

Describe a general procedure by which a truth table can be computed for each output of an arbitrary acyclic circuit containing only combinational components. [HINT: construct a functional specification to each circuit node].

Since each circuit node is connected to an output of some combinational element, the functional specification for each circuit node is given by the functional specification for the combinational element which drives it. If we start with components that are only connected to inputs to the overall circuit, we can build truth tables for their outputs that only involve those inputs. We can then work on the next tier of components and build truth tables for their outputs that only involve overall circuit inputs. Continuing in this manner, we'll eventually reach components that drive the overall circuit outputs and be able to construct truth tables for each output that only involves overall circuit inputs.
In our example, we'd first build a truth table for the output of the A module and then use that table to build a table for Z:

Specify a propagation delay (the upper bound required for each combinational device) for the circuit.

The propagation delay for Z is 5ns.

Describe a general procedure by which a propagation delay can be computed for an arbitrary acyclic circuit containing only combinational components. [HINT: add a timing specification to each circuit node].
1. Label each INPUT to the circuit with tpd=0.
2. Repeatedly
3. The prop delay spec for the device is the MAX of the prop delay labels on the output nodes.

Do your general procedures for computing functional specifications and propagation delays work if the restriction to acyclic circuits is relaxed? Explain.

No. Without cycles, you're guaranteed to be able to find a new output node to label (i.e., the output of some element E whose inputs are already labeled) until the entire circuit is labeled. If you have cycles, the algorithm breaks down. You can be left with a cycle of elements whose outputs are unlabeled and some of whose inputs are unlabeled.

Problem 7. If you are given the following 2-input and 2-output combinational block:

with the following functional description: The output X is the the logical complement of the input A, and the output Y is the the logical complement of the input B. And valid ouputs are guaranteed after valid inputs have been established for 1 second.

Does this device adhere to the static disipline?

Yes: there's an unambiguous functional specification for each output and a maximum propagation delay has been specified.

Suppose that the output X is connected to the input B, what output would you expect?

If you assume that the circuit was composed of two inverters (one with A as its input, the other with B as its input) then we would expect Y = A after 2 second propagation delay.
However, there are other implementations that have the same functional specification. In particular, X might be implemented with logic that uses both A and B as inputs. In this case, connecting X to B would create a cycle and the value of Y might be undetermined.

Suppose the functional description was changed to the following: The ouput X is a 1 if both A and B are "0", and Y is a 1 if either A or B but not both are "1". Does this change the answer the previous question?

No. Since X and Y are functions of both A and B, it's more obvious that a cycle would be created when X was connected to B.

The digital abstraction