Baud: how symbol rate relates to gross bit rate
Baud measures symbol rate, while gross bit rate measures the number of bits conveyed per second. In telecommunications and electronics, symbol rate is one component determining communication speed over a data channel. Baud expresses symbols or pulses per second, counting distinct signaling events in a digitally modulated signal or a line code. Its symbol is Bd. A bit is a binary digit and always represents one of two states, but a transmitted symbol can have more than two possible states. That difference allows a symbol to represent more information than a single bit, separating the two rates.
The name honors Emile Baudot, creator of the Baudot code for telegraphy. Its spelling follows the convention for SI units: Bd begins with an uppercase letter, while baud is normally lowercase when written out. A sentence beginning with the word or title capitalization can change that presentation. The CCIT, a forerunner of today's ITU-T, took up the unit in 1926. An earlier measure used words per minute, but varying word length made that measure less robust. A rate based on signaling events avoids making the size of a word the basis of the communication-speed measurement.
Time provides a second way to express a signaling pace. Symbol duration, also called the unit interval, is the reciprocal of symbol rate. It can be measured by observing the time between transitions in an eye diagram on an oscilloscope. Communication at 1000 Bd means sending 1000 symbols per second. For a modem, the example corresponds to 1000 tones per second; for a line code, it corresponds to 1000 pulses per second. Each symbol then lasts 1/1000 of a second, or one millisecond. The duration and rate describe the same timing from opposite directions: interval length and events during a second.
Metric prefixes scale the baud unit for different signaling rates. One kilobaud, written kBd, equals 1000 Bd. One megabaud, or MBd, equals 1000 kBd, and one gigabaud, or GBd, equals 1000 MBd. The prefixes change the scale attached to the unit rather than substituting a count of bits for a count of symbols.
The connection to gross bit rate depends on information per symbol. If N bits are conveyed by each symbol and R is the bit rate, including channel-coding overhead, the symbol rate is R divided by N. Reversing the relationship gives R as symbol rate multiplied by N. Systems with precisely two symbols, typically representing binary 0 and 1, have matching baud and bit-per-second values. Old modems and the simplest digital communication links use that one-bit-per-symbol arrangement. More advanced modem and transmission techniques allow symbols with additional states, so the same event count need not imply the same information count.
Hartley's relationship expresses gross bit rate through the number of distinct messages that can be sent. The information per pulse is the base-two logarithm of that message count, without rounding; using a ceiling function would instead give the number of bits needed for a fixed-length binary label. The ceiling of 2.11, for example, is 3. Multiplying information per pulse by pulses per second gives the communication rate in bits per second. In that relationship, the number of different symbols is two raised to the information-per-symbol value N. In a modem, the different symbols may be time-limited sine-wave tones distinguished by combinations of amplitude, phase, or frequency. A line code can instead use different voltage levels, making the distinct signaling states physical alternatives on the transmission medium.
A 64QAM modem gives a specific case with 64 symbols. The base-two logarithm of 64 is 6, making its bit rate six times its baud rate. Ratios between bits and symbols can also be noninteger. In 4B3T coding, the bit rate is 4/3 of the baud rate. A typical basic-rate interface illustrates that relationship with a raw rate of 160 kbit/s and operation at 120 kBd. The QAM example and the line-code example therefore use different relationships between information carried and signaling pace, with neither requiring the two quantities to share a numerical value.
Channel characteristics influence where higher-information symbols are useful. Codes with many symbols, yielding a gross bit rate above the symbol rate, are most useful on channels such as telephone lines with limited bandwidth but a high signal-to-noise ratio within that bandwidth. Other applications can place bit rate below symbol rate. Eight-to-fourteen modulation used on audio CDs has a bit-to-baud ratio of 8/17. That coding requires three merging bits between adjacent fourteen-bit codewords. The additional merging bits contribute to the coded signaling structure, producing a relationship in which the count of signaling events exceeds the rate of the underlying bits.