Introduction to digital logic

While an analog signal is continuous, a digital one is discrete. Specifically, an analog signal represents an infinite continuum of levels while the digital one represents finite, discrete levels.

Our world is inherently analog. We sense continuous signal levels such as the intensity of light or sound. In contrast, if humans were born with digital sensory organs we would only sense binary states, for example brightness or darkness, silence or noise. Not surprisingly, most of us who are fortunate to enjoy functioning analog sensory organs would never want to trade these for digital ones. Based on this observation it appears that in comparison to analog, digital technology is inherently limited and crude. Nevertheless, the prevailing trend in technology clearly favors the digital approach over the 'old' analog one, indicating significant advantages of this 'new' technology. Let's explore the main advantages with the following analogy.

Imagine that you are in a concert hall with two 'musicians' who are hidden from the audience by a curtain. The task for the performers is simple: they are each asked to play a single note on their instrument and then the audience is asked to describe the sound intensity or if they even heard that instrument was played.

After the first performer plays a few identical notes on a violin you probably would get a wide response from the listeners ranging from 'quiet' to 'loud'. After all, the perception is subjective and based on the listener's music preference, hearing capability and distance from the player. Furthermore, depending on the relationship between how loud the note was played and how loud the background noise was in the hall, it may even be hard for the listeners to discern the state of the instrument, i.e. was it played at all or did they only hear noise.

Next imagine the second player, who uses an extremely loud and obnoxious sounding car horn as his instrument. Again he plays one note by turning the horn on or off multiple times. Now, most listeners would clearly agree on the state of the perceived sound intensity, i.e. either it was quiet or loud—the horn was 'on' or 'off'. As long as the car horn is louder than the background noise level in the concert hall (a very likely condition) no ambiguity of the signal state exits.

If our only concern is to make certain that the audience perceives the signal's state, i.e. is the instrument being played or not, then clearly the second musician with his digital approach wins. By using an instrument capable of only producing two clearly discernable states, complete silence and ear piercing noise, he is able to transmit his message across the concert hall without ambiguity or errors. It should come as no surprise why this digital message system works so well; after all, it was designed to work in an extremely noisy environment, such as traffic, to deliver a clear (warning) message.

In summary, by only using clearly identifiable states, digital technology is able to transmit a signal state that is (almost) immune to noise. By the same reasoning, it follows that these states can also be duplicated and stored without degradation by ambient noise. In digital technology, such a binary 'signal' is referred to as a 'bit' and its state as 'on' or 'off'.

Of course, one could make the argument that we are really comparing two completely different systems in our concert hall analogy. It is one thing to convey a clear message by, for example, either remaining completely quiet or blasting an obnoxious car horn. However, it is a completely different matter to convey a more complex, nuanced and probably esthetically pleasing message, such as the violinist may produce with her analog instrument. Instead, the question we should ask is: can we produce a more nuanced message with our simple car horn? At first, this may seem impossible if we are only allowed to use the two clearly discernable states that our car horn emits, silence and noise. However, the digital world gets around this issue by 'bundling' bits into a package and then assigning meaning to this sequence of bits. This is actually not all that different from the way some drivers operate their car horn: no honking means you are ok; one brief honk means you have mildly irritated someone; a long series of honks clearly means you should get off your phone. While this is a bit oversimplified, it shows that we can express a discrete level of emotions by using only silence and no silence.

It also highlights another aspect of digital technology, namely that the more signals, or bits, we agree to use in our sequence, the finer we can express these levels. However, as long as we are forced to use a finite sequence of bits, the number of levels we can express will also be finite. Going back to our traffic analogy, while with one bit we can only express two levels, 'ok' or 'angry', with just two bits we can signal four levels, 'ok', 'mildly irritated', 'somewhat irritated' and 'angry'.

Another advantage that digital technology has over analog is the ease with which the data can be manipulated using very simple 'operators', called gates, to mimic mathematical operations.

In summary, the high noise immunity of transmitting and storing digital signals and the ease with which they can be manipulated with mathematical operations give digital an advantage over analog. This explains the current trend away from analog and towards digital technology. Nevertheless, you should keep in mind that we started out by saying that our world (including us humans) is inherently analog. Therefore, at least as long as humans have no digital ports, we will always be working with both analog and digital technology and the interface between these two worlds, analog-to-digital and digital-to-analog converters.

When we refer to the binary state of a bit we will use the conventional labels of 'on' or 'off', '1' or '0' or 'high' or 'low' (which is sometimes also written as 'HI' or 'LO'). We will use these labels interchangeably and agree that 'on', '1' and 'high' are considered equivalent and so are 'off', '0' and 'low'.

In a physical environment, such as an electronic circuit, explicit voltage or current level standards exist for a specific digital logic state. Multiple conventions exist and the most common ones are listed in table 1.1. To conserve energy and to increase switching speeds, the 3.3 V low voltage transistor–transistor logic (TTL) standard is increasingly replacing the old, established 5 V TTL level.

Table 1.1 is deliberately kept simple and it does not list the acceptable threshold levels for high and low input and output signals. Typically these are within one volt of the 'ideal' levels listed above. For example, for the 3.3 V CMOS systems, an input voltage between 0 and 0.8 V is acceptable as a low input, signals between 2.0 and 3.3 V are considered as a high input; you want to avoid input signal levels between 0.8 V and 2.0 V because they are ambiguous.

Discrete components require external power and the levels listed above state their supply voltages.

Finally, avoid applying input signals exceeding the high or low levels stated above. Doing so can destroy the components!

As mentioned previously, the power of digital logic comes from the ability to manipulate the signals through logic gates. These logic gates are to digital technology what operators are to mathematics. There are only three fundamental gates: the NOT, AND and OR gates. All other gates and the entirety of digital technology are based on repeatedly applying these three operators in various combinations. While this may be hard to believe, remember that a large part of mathematics is composed of repeated applications of the addition and subtraction operator, resulting essentially in multiplication and division. As you will see later, by combining these gates you will be able to build a simple calculator, a music player and even a microprocessor.

The three fundamental gates of digital logic are the NOT, OR and AND gate.

The NOT gate is an inverter. It negates the current state and switches it to its opposite. For example, by NOT being 'on', something is 'off'. (Note: the NOT operator is sometimes written as $\bar{X}$, ∼X or !X. As you will see later, if X represents a collection of bits then the meanings of ∼X and !X are not equivalent. Avoid the confusion and stick with ∼X.)

Applying a NOT operator successively an even number of times results in no change since every other NOT operation cancels the previous one. For example, applying the NOT operator twice as in: $Q=\bar{(\bar{X})}$ results in $Q={X}$. A schematic representation of this expression is:

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The NOT operator symbol is often drawn in an abbreviated form, omitting the triangle and using only the open circle at its end. The three circuits below illustrate this. All three circuits depict the functionality of a NOR gate, i.e. an OR gate followed by a NOT gate:

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The same abbreviated form of a NOT gate is also used at the gate inputs, as depicted in the three circuits below. The circuits represent a 'NOT A, NOT B, OR-gate' (of course, the functionality of these three circuits is not equivalent to the three circuits shown directly above!):

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The remaining fundamental gates are the OR and the AND gates. Unlike the NOT gate which is a unary, single-input operator, both the OR and AND gates have at least two inputs.

First, note the commutative nature of these gates, and Boolean logic in general, shown below:

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Although the gates shown above are two-input gates, we could easily expand their inputs to any number while maintaining their functionality. An example of a three-input OR gate is shown below. The circuit on the right is displayed in its concise form while the one on the left details its implementation with two-input OR gates. This approach could be extended further and it works for any two-input gate. Note that we implicitly made use of the Boolean associative properties, namely:

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Finally, let's examine the difference between the OR and AND gates with the following two examples.

In the first example, assume that you have 1000 individual particle detectors and you want to monitor when at least one of them has been activated. Assume that when an individual detector is hit by a particle, the detector's output goes high. The easiest way to look for such an event is to connect all the particle detector outputs to one giant OR gate (having 1000 inputs) and monitor its (single) output. When the OR gate's output goes high, we know that at least one particle has passed through one of the detectors. In other words, the OR gate goes high when any one of its inputs goes high. This is the reason the OR operator is also known as the 'ANY' operator.

For the second example, consider that we are trying to locate a mysterious source of particles originating somewhere in the universe, far away. We will use a particle 'telescope' to detect their origin. It works similarly to an optical telescope, except that it employs a number of particle detectors in place of lenses, arranged in a straight line. We can pinpoint the particles' origin by sweeping our telescope across the sky and monitoring for an event when all our detectors are activated at the same time. This occurs only when all the detectors are aligned with the mysterious source. (In this example, we assume that the particles travel in a straight line and close to the speed of light.) This technique, often called an n-fold coincidence technique, where n refers to the number of detectors placed in coincidence, can be implemented by connecting all the detector outputs to the input of a giant AND gate. The output of the AND gate will go high only when ALL the inputs are high, meaning that we must have aligned our detectors in the direction of the source. This example illustrates the finicky nature of AND gates: unlike the 'combiner' OR gates, AND gates are very selective and act like a 'filter'. Since all its input conditions must be satisfied for its output to go high, an AND gate is sometimes called an 'ALL' gate.

Finally, a quick word on Boolean notation for these two gates: the OR operator is usually expressed by using the addition symbol, +. However, Verilog does have an addition operator and this can lead to confusion since the behavior is different. This will be discussed in more detail later. In Boolean notation, the AND operator is often omitted and it is implicitly understood that AB really means A · B.

In the preceding sections we derived and manipulated Boolean expressions using truth tables. In this section we convert truth tables to their corresponding Boolean logic expressions.

A formal procedure for implementing truth tables involves Karnaugh maps, detailed information can be found online. However, for our purpose, a simplified version of such a procedure is provided by the algorithm listed below. It will work with any arbitrary truth table and will produce a Boolean logic expression that correctly represents the original table. However, unlike other algorithms, it will not necessarily provide the most optimized expression. This is not really an issue for us since our applications are simple and, if needed, the Verilog compiler will optimize the expressions for us.

Example 1.  Finding the Boolean expression for an XOR gate

Attesting to its usefulness, the XOR gate has its own schematic symbol and operator. However, it is not a fundamental gate and it can be constructed with AND, OR and NOT gates. We will now show how to derive its Boolean expression starting with its truth table:

Although we may be able to figure out a logic expression by trial and error, it can be implemented more systematically by noting the following observations:

Step 1. Note that the output of the entire truth table is represented by the four states labeled 0 to 3 in the table above. In other words, we could express the table with the following general Boolean expression involving four OR operations:

$$\begin{eqnarray}&&Q={Q}_{0}+{Q}_{1}+{Q}_{2}+{Q}_{3},\end{eqnarray} \tag{ 1.1 }$$

where

$$\begin{eqnarray}&&{Q}_{0}=0,{Q}_{1}=1,{Q}_{2}=1\; \text{and}\; {Q}_{3}=0.\end{eqnarray} \tag{ 1.1a }$$

Step 2. We have already shown that X + 0 = X. Therefore, we can eliminate all states with Q_i = 0 since these will not affect our result. In our example, we end up with following simplified expression of Q:

$$\begin{eqnarray}&&Q={Q}_{1}+{Q}_{2}.\end{eqnarray} \tag{ 1.2 }$$

In other words, Q₀ and Q₃ have been eliminated since they are 0.

Step 3. In each state, Q_i is a function of its inputs, i.e. Q_i = Q_i (A_i , B_i ). For our case this means

$$\begin{eqnarray}&&Q={Q}_{1}({A}_{1},{B}_{1})+{Q}_{2}({A}_{2},{B}_{2}).\end{eqnarray} \tag{ 1.3 }$$

Now we have to find a Boolean expression for the case listed that will result in a high output for its, and only its, specific input. Therefore, we must find a Boolean expression that satisfies the following condition:

$$\begin{eqnarray}&&{Q}_{1}({A}_{1},{B}_{1})={Q}_{1}(A=0,B=1)=1,\end{eqnarray} \tag{ 1.4 }$$

$$\begin{eqnarray}&&{Q}_{2}({A}_{2},{B}_{2})={Q}_{2}(A=1,B=0)=1.\end{eqnarray} \tag{ 1.5 }$$

The only expressions satisfying our requirement for Q₁ and Q₂ are

$$\begin{eqnarray}&&{Q}_{1}=\bar{A}B,\end{eqnarray} \tag{ 1.6 }$$

$$\begin{eqnarray}&&{Q}_{2}=A\bar{B}.\end{eqnarray} \tag{ 1.7 }$$

(It should come as no surprise that such a specific request can only be satisfied with an AND operator.)

Step 4. Putting equations (1.2), (1.6) and (1.7) together results in

$$\begin{eqnarray}&&Q={Q}_{1}+{Q}_{2}=\bar{A}B+A\bar{B}.\end{eqnarray} \tag{ 1.8 }$$

This is the Boolean definition of an XOR gate and we have successfully implemented the truth table.

Example 2.  Three-input truth table to Boolean expression

Simple two-input truth tables can often be evaluated with a trial, error and good luck approach. However, more complex truth tables may resist this approach and may require the algorithm shown above. For example, consider a three-input truth table which means that Q is now composed of eight states:

$$\begin{eqnarray*}&&Q={Q}_{0}+{Q}_{1}+{Q}_{2}+{Q}_{3}+{Q}_{4}+{Q}_{5}+{Q}_{6}+{Q}_{7}.\end{eqnarray*}$$

The rest follows the outline above. See if you can implement the table below and show that it conforms to the Boolean expression listed below the table:

$$\begin{eqnarray*}&&Q=\bar{A}\bar{B}C+\bar{A}B\bar{C}+A\bar{B}\bar{C}+ABC.\end{eqnarray*}$$

Now that we know how to derive a Boolean expression from a truth table, let's see how to represent it with a digital circuit. Start by working your way from the most general expression at the output back to the input and identify its basic operators. For example, starting at the output, the expression derived at the end of the previous section can be rewritten as

$$\begin{eqnarray}&&Q={Q}_{1}+{Q}_{2}+{Q}_{4}+{Q}_{7}.\end{eqnarray}$$

Each of the four terms, Q₁, Q₂, Q₄ and Q₅, are joined by an OR operation which can be represented with the schematic symbol of a four-input OR gate.

Working our way to the inputs, the terms Q_i are composed of three-input AND gates which may contain negation operators at some of their inputs. (Remember: the Boolean symbol for the AND operation, a middle dot ' · ', is usually dropped and, hence, often overlooked.)

$$\begin{eqnarray*}&&{Q}_{1}=\bar{A}\bar{B}C,\end{eqnarray*}$$

$$\begin{eqnarray*}&&{Q}_{2}=\bar{A}B\bar{C}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{Q}_{4}=A\bar{B}\bar{C},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{Q}_{7}=ABC.\end{eqnarray*}$$

Putting this all together results in the schematic shown below:

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We have made it easy on ourselves by using multi-input gates, such as the three-input AND and the four-input OR gate. If we were forced to use only two-input gates we could replace these gates with a series of two-input gates as shown earlier.

To convert a schematic representation to a Boolean expression, start at the inputs and then work through each gate, inserting after each gate the corresponding Boolean expression. (This is the opposite order of the previous example.) See the example above where we have omitted the final Boolean expression for the circuit since it has already been stated in the text.