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50.002 Computation Structures
Information Systems Technology and Design
Singapore University of Technology and Design

Digital Abstraction

You can find the lecture video here. You can also click on each header to bring you to the section of the video covering the subtopic.

Detailed Learning Objectives

  1. Explain the Purpose of Digital Abstraction:
    • Explain how digital circuits encode information using voltage levels to represent binary values.
    • Explain the concept of digital abstraction for transforming continuous analog signals into discrete digital values.
  2. Apply the Static Discipline in Digital Systems:
    • Explain how static discipline is used as a contract ensuring predictable behavior in digital systems.
    • Discuss how the static discipline guarantees that valid inputs lead to valid outputs, ensuring system reliability.
  3. Discover Combinational Digital Systems:
    • Define combinational devices and systems, detailing their properties and operational criteria.
    • Differentiate between combinational and sequential logic devices, emphasizing the memory-less nature of combinational systems.
  4. Discover Voltage Encoding and Noise Margin:
    • Explain the concept of using voltage levels to encode binary data, including defining thresholds for ‘0’ and ‘1’.
    • Defend the importance of noise margins in maintaining signal integrity across digital devices.
  5. Examine Voltage Specifications and Their Impact:
    • Justify the need for specifications for valid voltage levels and how they are used to prevent errors due to noise.
    • Discuss how noise margins are established to enhance the robustness of digital systems against external disturbances.
  6. Analyze the Voltage Transfer Characteristic (VTC) Functions:
    • Analyze the VTC function to determine the behavior of digital systems under various input conditions.
    • Evaluate the VTC to ensure that digital devices comply with the static discipline and effectively handle noise.
  7. Set up for Practical Applications:
    • Integrate the theoretical knowledge of voltage levels, static discipline, and combinational logic into designing and evaluating digital circuits.

These objectives aim to equip students with a solid foundation in digital systems, emphasizing the translation of theoretical concepts into practical applications in digital electronics and circuit design.

Overview

One of the cheapest ways to encode information in terms of 0s and 1s is using voltage levels, illustrated in the diagram below.

This is how digital circuits work. This method of determining discrete values out of analog voltage (which value is originally made up of real number, and therefore continuous and infinite) is called the digital abstraction. We need to find a way to determine digital signal values out of analog voltage values.

What are the benefits of the digital abstraction?

  1. The digital abstraction is where one interprets voltage values as binary values, thus allowing us to encode information using voltages.
  2. Using voltages to encode bits of 0’s and 1’s provides a cheap and stable way for us to exchange information through digital devices.
  3. We can also manipulate or change information encoded using voltages very easily.

The voltages that represent digital bits are generated by semiconductor devices (MOSFET) – something that we will learn in the next chapter. The benefit of using semiconductors the ease of generation, and that they require zero power in steady-state. The drawbacks however, is that the voltages generated by these semiconductors are easily affected by external disturbances, and hence they may be unstable.

To preserve the integrity of information encoded in digital devices made of semiconductor materials, we need to set some contracts between these interconnected digital devices. In this notes, we are going to learn how we can use voltages to encode information in a stable way that follows a particular contract called the static discipline to guarantee the behavior of each processing block in the system.

A Digital Processing Element: Combinational Device and Combinational Digital System

A digital device is any device that uses voltages to encode information in terms of “low voltage” (bit 0) and “high voltage” (bit 1). Its output is a pure function of the present input only (there’s no memory of past inputs), and it has the following criteria.

A combinational device is a specific type of digital device that has the following criteria:

  1. One or more digital inputs
  2. One or more digital outputs
  3. A functional specification that details the value of each output for each possible combination of inputs (can be illustrated in terms of truth table / boolean expression)
  4. A timing specification consisting of an upper bound required propagation time for the device to compute the specified output values given a set of valid and stable input value(s)

You have seen this in our previous chapter: the \(A>B\) 2-bit comparator device is a combinational device, consisted of pure logic gates and behave like a pure function.

Later on you will learn another type of digital logic devices called the sequential logic device, whose output depends not only on the present input but also on the history of the inputs, hence having a memory.

A set of interconnected circuit elements is combinational and can be labeled as a combinational digital system if and only if:

  • Each circuit element is also combinational with no directed cycles (no feedback loop), and
  • That very device’s input is connected to exactly one output of another device or to some vast supply of 0s and 1s.

Voltage to encode information

The most naive way to use voltage to encode information is to use ‘low’ voltage to encode valid ‘0’ and ‘high’ voltage to encode valid ‘1’, and define the low and high threshold for each valid ‘0’ and ‘1’.

Anything that is between the low and high threshold value is called the invalid zone, as shown in the figure below:

The values of operating voltage in practice is commonly set to be 0.3V for low voltage and 3.3V for high voltage.

The Static Discipline

The static discipline is one of the contracts bound for all logical elements making up a digital system.

The Static Discipline

A digital system must be able to produce a valid output (for the next device connected at its output terminal) according to its specification if it is given a valid input.

This contract guarantees the behavior for each processing block in a system, so that a set of such interconnected devices may work properly (are able to pass and compute valid information at the end of the chain of connections). This is necessary so that the system has a predictable behavior.

Therefore, one can say that a combinational logic device always obeys the static discipline.

However this doesn’t mean that the opposite is true.

A device receiving invalid input does not necessarily produce invalid output. The output from a combinational logic device given invalid input remains unpredictable.

An unconnected terminal or a dangling wire in a digital circuit is not considered a valid low digital signal. In digital electronics, each signal line or terminal must have a definite state: high (1), low (0), or sometimes a high-impedance state.

Voltage Specifications and Noise Margin

A case without noise margin

Consider two digital devices connected in series as shown in the figure below. These devices are called a buffer, meaning that they pass the same bit over (if it receives a low voltage, it will produce a low voltage and vice versa). If we were to naively decide that any voltage below \(V_{low}\)=0.5V as digital bit 0, and any voltage above \(V_{high}\)=2.5V as digital bit 1, then our device may violate the static discipline.

Why?

This explanation can be made clear with the following example. Suppose we supply 0.5V and Device 1 is able to produce also 0.5V, which means digital bit .

  • However, the problem is that a wire, that connects two or more combinational devices together is susceptible to noise.
  • The voltage value that is received at Device 2 may be slightly higher than 0.5V, for example: 0.55V instead, and therefore according to our specification, it is no longer a valid bit 0.

Noise can knock the voltage down as well (not just up, it’s basically random disturbance). The above is just an example that’s detrimental to the function of the devices in this example.

Device 1 in the figure above violates static discipline because given a valid input, it may be unable to produce a valid output (to reach the next device 2), because the 0.5V produced at the output of Device 1 may meet some disturbances that caused it to be slightly off, e.g: 0.55V.

The importance of noise margin

We need to account for the presence of some light noise. Instead of naively setting some voltage \(V_{high}\) and \(V_{low}\) as we did above, we need to set a range of Voltages as valid bit 1 and 0 respectively and need to have something called the noise margin to tolerate noise. It is illustrated as the yellow region in the Figure below. The noise margin is formed by setting four Voltage specifications: \(V_{ol}\), \(V_{oh}\), \(V_{il}\), \(V_{ih}\), where \(V_{ol}\)< \(V_{il}\)< \(V_{ih}\) < \(V_{oh}\) which defines what range of voltage values signifies a valid digital bit 1 and a valid digital bit 0 for any combinational logic component in the system:

Why do we need to have a noise margin?

The noise margin adds as a precaution against external disturbances (noise).

Below are the explanations necessary to understand the figure above:

  1. \(V_{ol}\) (voltage output low) and \(V_{oh}\) (voltage output high) is the voltage that your system outputs, depending on whether your system is outputting bit 0 or 1. The output of this system is going to be received by another system after traversing through some wire.
  2. \(V_{il}\) (voltage input low) or \(V_{ih}\) (voltage input high) is the voltage that your system receives as input from another system.
  3. The absolute difference between \(V_{ol}\) and \(V_{il}\) is called the low bit noise margin, and the absolute difference between \(V_{oh}\) and \(V_{ih}\) is called the high bit noise margin.

Low-bit/high-bit noise margin is formally defined as the maximum voltage amplitude of extraneous (erroneous) signal that can be added to the noise-free input level without causing a drastic change in the output voltage and that it is still within the valid logic level.

Noise Immunity

The noise immunity (like an “overall” or “effective” noise margin) is the minimum between the high bit noise margin and the low bit noise margin.

Note that the value of \(V_{ol}\) is less than \(V_{il}\), because we would want to have some buffer (margin) against noise. A device always outputs a lower voltage value to signify digital bit 00 and accepts a slightly higher low-voltage value as digital bit 00. The same logic applies for the higher region as well, as \(V_{oh}\) is greater than \(V_{ih}\)

In our previous case earlier, if \(V_{ol}\) is set to be 0.5V, and \(V_{il}\) is set to be 0.6V, then Device 2 will be able to tolerate up to 0.1V of noise (if any). Therefore, 0.55V in our example above is still ‘seen’ as a valid bit 0 when it arrives at the input terminal of Device 2, thus making Device 1 obeys the static discipline.

Once set and chosen, these four voltage specifications: \(V_{ol}\), \(V_{oh}\), \(V_{il}\), and \(V_{ih}\) are to be obeyed by every digital device in an entire combinational logic circuit, e.g: your computer. Think of it like some sort of operating standard across all components in your digital system.

Voltage Transfer Characteristic Function (VTC)

The VTC is a plot between the input voltage (\(V_{in}\)) to a digital system/device vs the output voltage (\(V_{out}\)) of this digital system.

VTC does not tell us how fast the device is. It just captures the static behavior of the device and tells us what kind of device it is.

The image below shows the VTC of a buffer: a low \(V_{in}\) gives a low \(V_{out}\) and vice versa.

Forbidden zone is not equal to invalid zone. The latter is the zone where a voltage value does not correspond to digital bit 0 or 1 while the former is the zone whereby static discipline is violated because a valid input voltage does not produce a valid output voltage.

Think!

What will the VTC of an inverter look like?

The purpose of plotting a VTC (typically obtained from device measurements, i.e: we supply input voltages at intervals and measure the output) is to help us to determine whether or not a digital device can be used as a combinational logic device. In other words, we obtain the VTC so that we can find a set of four voltage specifications: \(V_{ol}\), \(V_{oh}\), \(V_{il}\), and \(V_{ih}\) for the device so that the device obeys the static discipline.

Explanation of the VTC figure above:

  • The red zone is called the forbidden zone. It is formed by the four voltage specifications: \(V_{ol}\), \(V_{oh}\), \(V_{il}\), and \(V_{ih}\) that we set for the entire system.
  • The name ‘forbidden zone’ comes from the fact that any value within this zone means that the device receives valid input but is unable to produce a valid output hence violating the static discipline and cannot be used as a combinational logic device.

Can a given device be used as a combinational logic device?

You can quickly tell if a digital device can be potentially be used as a combinational logic device iff: you can find a set of these four voltage specifications: \(V_{ol}\), \(V_{oh}\), \(V_{il}\), and \(V_{ih}\) whereby its VTC curve does not cross the forbidden zone and that \(V_{ol}\)< \(V_{il}\) < \(V_{ih}\) < \(V_{oh}\).

  • We typically begin by guessing each value of \(V_{ol}\), \(V_{oh}\), \(V_{il}\), and \(V_{ih}\) and check if the curve crosses the forbidden zone (check if static discipline obeyed) formed by these four values.
  • If static discipline is violated, we either adjust our guess or find another device.
  • Also, we want to choose \(V_{ol}\), \(V_{oh}\), \(V_{il}\), and \(V_{ih}\) that maximises noise immunity. -

If you can satisfy the condition highlighted above, then it means that the device is a combinational logic device. Its VTC curve has to possesses both characteristics below:

  1. There exist some region in the VTC whereby its absolute Gain is \(>1\) . Gain is actually a function of \(V_{in}\) and is formally defined as:

    \[\begin{aligned} \text{Gain}(V_{in}) = \frac{d V_{out}}{d V_{in}} \end{aligned}\]

    In layman terms you can approximate Gain during some transition \(V_{in_i}\) to \(V_{in_j}\) that results in some \(V_{out_k}\) to \(V_{out_l}\) respectively by the simply computing the slope between these two points on the VTC:

    \[\text{Gain} \approx \frac{V_{out_l}-V_{out_k}}{V_{in_j}-V_{in_i}}\]

    If you have found four voltage specifications \(V_{ol}\), \(V_{oh}\), \(V_{il}\), and \(V_{ih}\) for which the device still obeys the static discipline, you can approximate device’s maximum Gain by computing:

    \[\max\text{Gain}\approx \frac{V_{oh} - V_{ol}}{ V_{ih} - V_{il}}\]

    If absolute Gain \(>1\), then there is a finite, positive noise margin. If absolute Gain\(=1\), then there’s zero noise margin. It is impossible to have absolute Gain \(<1\) and still have the four Voltage specifications \(V_{ol}\) < \(V_{il}\) < \(V_{ih}\) < \(V_{oh}\). Also, having absolute Gain > 1 maintains the signal passed through the system as signal loss is inevitable through the system.

  2. The device has a Non-linear Gain, meaning that Gain is a function of Vin and therefore the gradient along the entire curve varies.

    • The VTC curve for a combinational logic device should not be entirely made of a single, constant gradient like the shape of a plot from a basic line equation, but rather more towards an “S” (or mirrored S) shape.

If both characteristics above aren’t satisfied in the VTC curve, then it is not the VTC of a combinational logic device.

Summary

You may want to watch the post lecture videos here.

Here are the key points from this notes:

  1. Static Discipline: An important contract (via setting the four voltage specifications), to ensure that the output of a digital component behaves correctly given specific valid input levels. This contract governs all digital devices in a system (not just a single gate!)
  2. Noise Margin and Noise Immunity: We need to take into account the presence of noise when we establish the voltage specification contract, so that we can still obey the static discipline despite the presence of noise.
  3. VTC Graph: A plot describing the behaviour (Vout vs Vin) of a digital device. This is very useful to determine reasonable voltage specifications that obey the static discipline. We need to ensure that the graph does not cross the forbidden zone.

Smaller logic gates can be assembled to form more complex units and ultimately entire computer systems. We can be confident that the computer system will work predictably by ensuring that each sub-component obey the static discipline. This layered approach not only simplifies the design and development of digital devices but also ensures that even small components adhere to necessary standards to function correctly within the larger system.

Next Steps

In the next chapter, we will learn about the MOSFET (transistor), that is one of the smallest component (building block) that makes up a digital device, and how we can use them to form a proper combinational logic elements we call gates. These gates can be used to form an even larger combinational circuits such as the adder, shifter, etc, and an even larger one such as the Arithmetic Logic Unit (you will build them in Lab 2 and 3).

Each larger device will provide greater level of abstraction:

  • At first we have MOSFETs (transistors) that we can use to make logic gates. Once we are using logic gates, we abstract away the details about our transistors.
  • We can then use logic gates to make more complex logic units like ROMs, microcontrollers, multiplexers, registers, and CPU. Realise that once we have a CPU, we abstract away the details about our little logic gates.
  • Once we have a CPU, we can create an assembler to help us program an operating system in assembly. We can create more useful programs like compilers to help us create programs easily in higher level language. This is how abstraction works: we create smaller components and use them together so that we can do our job (as programmers) easily.

Therefore, it is imperative that each combinational logic device / component, no matter how small, must conform to the static discipline and the established four voltage specifications (that must be chosen such that it fits with their VTC) so that the larger system can work as intended.