Basic Digital Concepts
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By converting continuous analog signals into a finite number of discrete states, a process called digitization, then to the extent that the states are sufficiently well separated so that noise does create errors, the resulting digital signals allow the following (slightly idealized):
- storage over arbitrary periods of time
- flawless retrieval and reproduction of the stored information
- flawless transmission of the information
Some information is intrinsically digital, so it is natural to process and manipulate it using purely digital techniques. Examples are numbers and words.
The drawback to digitization is that a single analog signal (e.g. a voltage which is a function of time, like a stereo signal) needs many discrete states, or bits, in order to give a satisfactory reproduction. For example, it requires a minimum of 10 bits to determine a voltage at any given time to an accuracy of » 0:1%. For transmission, one now requires 10 lines instead of the one original analog line.
The explosion in digital techniques and technology has been made possible by the incredible increase in the density of digital circuitry, its robust performance, its relatively low cost, and its speed. The requirement of using many bits in reproduction is no longer an issue. The more the better.
This circuitry is based upon the transistor, which can be operated as a switch with two states. Hence, the digital information is intrinsically binary. So in practice, the terms digital and binary are used interchangeably. In the following sections we summarize some conventions for defining the binary states and for doing binary arithmetic.
Binary Logic States
The following table attempts to make correspondences between conventions for defining binary logic states. In the case of the TTL logic gates we will be using in the lab, the Low voltage state is roughly 0-1 Volt and the High state is roughly 2.5-5 Volts.
The convention for naming these states is illustrated in Fig. 1. The "positive true" case is illustrated. The relationship between the logic state and label (in this case "switch open") at some point in the circuit can be summarized with the following:
The labelled voltage is High (Low) when the label's stated function is True (False). In the Figure, the stated function is certainly true (switch open), and this does correspond to a high voltage at the labelled point. (Recall that with the switch open, Ohm's Law implies that with zero current, the voltage difference across the "pull up" resistor is zero, so that the labelled point is at +5 Volts. With a closed switch, the labelled point is connected to ground, with a 5 Volt drop across the resistor and a current of I = V=R = 5 mA through it.)
Figure 1: Illustration for labelling logic states ("positive true").
With the convention known as "negative true", the label would be changed to "switch
closed" with a bar over it: switch closed. Our statement becomes:The labelled voltage is Low (High) when the label's stated function is True (False). So in the figure, the stated function (switch closed) is true when the voltage is low. The bar is meant to envoke the boolean inversion operation:
Binary Arithmetic
Each digit in binary is a 0 or a 1 and is called a bit, which is an abbreviation of binary digit. There are several common conventions for representation of numbers in binary. The most familiar is unsigned binary.The largest number which can be represented by n bits is 2n - 1. For example, with 4 bits the largest number is 11112 = 15. The most significant bit (MSB) is the bit representing the highest power of 2, and the LSB represents the lowest power of 2.
Arithmetic with unsigned binary is analogous to decimal. For example 1-bit addition and multiplication are as follows: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 0, 0 x 0 = 0, 0 x 1 = 0, and
1 x 1 = 1. Note that this is different from Boolean algebra, as we shall see shortly, where
1 + 1 = 1.
Another convention is called BCD ("binary coded decmal"). In this case each decimal digit is separately converted to binary. Therefore, since 7 = 01112 and 9 = 10012, then 79 = 01111001 (BCD). Note that this is di erent than our previous result. We will use BCD quite often in this course. It is quite convenient, for example, when decimal numerical displays are used.
Hexadecimal Representation
It is very often quite useful to represent blocks of 4 bits by a single digit. Thus in base 16 there is a convention for using one digit for the numbers 0,1,2,: : :,15 which is called hexadecimal. It follows decimal for 0-9, then uses letters A-F.
