Basic Digital Concepts

admin — Fri, 11 Sep 2009 10:24:22 +0000

By converting continuous analog signals into a nite number of discrete states, a process
called digitization, then to the extent that the states are suciently 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 incred-
ible 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 dening the binary states and for doing binary arithmetic.

Binary Arithmetic

admin — Fri, 11 Sep 2009 10:20:19 +0000

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. An example of a 8-bit number in this case is 010011112 = 027 + 126 + + 120 = 64 + 8 + 4 + 2 + 1 = 7910 (Generally the subscripts will be omitted, since it will be clear from the context.) To convert from base 10 to binary, one can use a decomposition like above, or use the following algorithm illustrated by 79: 79=2 = 39, remainder 1, then 39=2 = 19 r 1, and so forth. Then assemble all the remainders in reverse order. 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 signicant 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 0 = 0, 01 = 0, and 1 1 = 1. Note that this is dierent 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 dierent 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.

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Basic Digital Concepts

Binary Arithmetic