Digital Theory
By Steve Jackson
When you go to the supermarket and buy groceries, you count out money using numbering system called base-10. The base-10 system has 10 digits representing the numbers 0 to 9. Each ÒplaceÓ in the system can hold 9 Òunits.Ó When a tenth unit is added, it overflows into the next Òplace.Ó For example, the first place is the ÒonesÓ place. If it holds 9 units and another unit is added (9+1) then the ones has 0 units, and the next place (the ÒtensÓ) has one unit, making a 10.
Computers cannot directly use the base-10 system. The electronics that make digital systems possible are like standard light bulbs. A light bulb has two positions, on and off. These two positions allow only two ÒnumbersÓ to be represented, 0 and on (unlike the 10 available in base-10). These digits are called ÒbinaryÓ digits.
Binary digits can be used to encode larger numbers than just 0 or 1, just like base-10. For example the binary number 10 is the same as the base-10 number 3. 11 is the same as the base-10 number 4, and 100 is the same as the base-10 number 5.
Sidebar: bit depth and how much information can be stored with a binary number
1 bit = 2
2 bit = 4
3 bit = 8
4 bit = 16
5 bit = 32
6 bit = 64
7 bit = 128
8 bit = 256
9 bit = 512
10 bit = 1024
11 bit = 2048
12 bit = 4096
13 bit = 8192
14 bit = 16384
15 bit = 32768
16 bit = 65536
17 bit = 131072
18 bit = 262144
19 bit = 524288
20 bit = 1048576
21 bit = 2097152
22 bit = 4194304
23 bit = 8388608
24 bit = 16777216
The larger the number encoded, the larger the binary digit needed to encode it. This means the more Òlight bulbsÓ needed to store it, which affects how much space you need to store the information and also affects how much ÒbandwidthÓ is needed to move the information from one place to another. The amount of information that a binary number can encode is called Òbit depth.Ó Common bit depths used in media are 8 bit, 16 bit, and 24 bit. 8 bit numbers can encode 256 choices. 16 bits can encode 65536 choices. 24 bits can encode over 16 million choices
Sidebar: terms for bit depths
kilobit kbit
megabit Mbit 1,000 kbits
gigabit Gbit 100,000 kbits
terabit Tbit 100,000 mbits
petabit Pbit 1,000,000 mbits
A single binary number, when used in computers, is called a Òbit.Ó A single bit is the same as saying that you have a single binary number. Because a bit is such a small number, it is often common to use groups of bits, for example kilobits (Kb) which is 1,000 bits, and megabits, which is 1,000,000 bits. Thus IEEE 1394a (Fire Wire 400), a connection standard for moving digital signals, has a capacity of 400 megabits per second, or 400,000,000 bits / second.
Information is also commonly discussed in terms of bytes. A byte is an octet, or 8 bits. While bits are used in most measurements of digital information (especially transmission speed), bytes are primarily used to measure storage. To convert bits to bytes, simply divide by 8. For example IEEE 1394a (Fire Wire 400) has the theoretical maximum ability to move data of 50 MB / second, while Fast Ethernet such as 100BaseT has a theoretical maximum of 12.5 MB / second. It should be noted that while data storage is often spoken about in terms of bytes, data transmission speed rarely is.
Sidebar: terms for storage
kilobyte KB 8 kbits
megabyte MB 1,000 KB
gigabyte GB 100,000 KB
terabyte TB 1,000,000 KB
petabytes PB 1,000,000 MB
One interesting fact about digital theory is that you may multiply digital numbers simply by adding them. For example, 8 bits (256) x 4 bits (16) = 12 bits (4096). Division works in a similar manner. This is an advantage with math using a binary system.
Each time a digital reproduction (called a sample) is being made two factors are important. The first is the Òbit depth,Ó or amount of memory (in term of bits) that the sample is allocated. The second is the number of samples made of the item to be reproduced. The more samples, the higher the Òresolution.Ó Resolution and depth equal quality for any digital media. The higher the resolution and depth also determines how much space and bandwidth a digital file takes up.
Sidebar: Nyquist–Shannon Sampling Thereom
Theory that states a signal can only be accurately reproduced if it is has a frequency (or bit rate) more than twice the signalÕs bandwidth. For example a human ear is capable of hearing sound from 20 to 20,000 hertz (for a bandwidth of slightly less than 20,000 cycles per second). When the standards for audio CDs (called the Red Book of CDDA standard) were developed and released in 1980, a sampling rate of 44,100 hertz was chosen because it was more than twice the frequency of the required bandwidth to reproduce high quality sound.
Resolution in photographs or other digital still images is expressed in terms of pixels. When a digital photograph is printed, the more pixels that are used in a given space means higher quality for a photograph. This quality is expressed in terms of pixels per inch or pixels per centimeter.
Television images and images used on the Internet do not have fixed resolution in terms of pixels per inch. Instead, they have resolutions based on the bandwidth available for transmission and the quality required. Standard definition television uses 640 pixels x 480 pixels. High definition television uses either 1280 pixels x 720 pixels or 1920 x 1080 pixels. Televisions signals have another aspect of quality: frame rate. Standard television uses 30 frames per second. Video shot to conform to the image rate of film uses 24 pictures per second.
Sidebar: vocabulary
Bandwidth: The amount of data that can be passed along a communications channel in a given period of time.
Bitstream: Any digital data stream. A digital data stream can be Òreal timeÓ (meaning that transmission is connection to a time base, such as broadcast television, or not real time, meaning that the transmission of data is not dependent on a time reference (for example, the Internet).
Color Depth: The number of bits used to record color. 24-bit color is considered Òphotorealistic.Ó Lower bit depths reproduce less realistic color.
Hertz: A unit of frequency in terms of cycles or events per second. 1 hertz = 1 event a second. Hertz is abbreviated Hz. 1000 Hz is called a kilohertz (kHz) while 1,000,000 Hz is a MHz.