In a digitally modulated signal or a line code, symbol rate, modulation rate or baud rate is the number of symbol changes, waveform changes, or signaling events across the transmission medium per unit of time. The symbol rate is a kind of aperiodic frequency, measured in baud (Bd) or symbols per second. In the case of a line code, the symbol rate is the pulse rate in pulses per second. Each symbol can represent or convey one or several bits of data. The symbol rate is related to the gross bit rate, expressed in bits per second.
Symbols A symbol may be described as either a pulse in digital baseband transmission or a tone in passband transmission using modems. A symbol is a waveform, a state or a significant condition of the communication channel that persists, for a fixed period of time. A sending device places symbols on the channel at a fixed and known symbol rate, and the receiving device has the job of detecting the sequence of symbols in order to reconstruct the transmitted data. There may be a direct correspondence between a symbol and a small unit of data. For example, each symbol may encode one or several binary digits (bits). The data may also be represented by the transitions between symbols, or even by a sequence of many symbols. The symbol duration time, also known as unit interval, can be directly measured as the time between transitions by looking into an eye diagram of an oscilloscope. The symbol duration time Ts can be calculated as:
T s = 1 f s {\displaystyle T_{s}={1 \over f_{s}}}
where fs is the symbol rate. For example, a baud rate of 1 kBd = 1,000 Bd is synonymous to a symbol rate of 1,000 symbols per second. In case of a modem, this corresponds to 1,000 tones per second, and in case of a line code, this corresponds to 1,000 pulses per second. The symbol duration time is 1/1,000 second = 1 millisecond.
Relationship to gross bit rate The term baud rate has sometimes incorrectly been used to mean bit rate, since these rates are the same in old modems as well as in the simplest digital communication links using only one bit per symbol, such that binary "0" is represented by one symbol, and binary "1" by another symbol. In more advanced modems and data transmission techniques, a symbol may have more than two states, so it may represent more than one binary digit (a binary digit always represents one of exactly two states). For this reason, the baud rate value will often be lower than the gross bit rate. Example of use and misuse of "baud rate": It is correct to write "the baud rate of my COM port is 9,600" if one means that the bit rate is 9,600 bit/s, since there is one bit per symbol in this case. It is not correct to write "the baud rate of Ethernet is 100 megabaud" or "the baud rate of my modem is 56,000" if one means bit rate. See below for more details on these techniques. The difference between symbol rate and bit rate can be understood by considering a person using a single semaphore flag who can move the flag to a new position once each second, so the signaling rate (baud) is one symbol per second. The flag can be held in eight distinct positions: Straight up, 45° left, 90° left, 135° left, straight down (which is the rest state, corresponding to no signal), 135° right, 90° right, and 45° right. Each signal (symbol) carries three bits of information, since it takes three binary digits to encode eight states. The data rate is three bits per second. In practice, two flags are used together increasing the number of symbols and the data rate. If N bits are conveyed per symbol, and the gross bit rate is R, inclusive of channel coding overhead, the symbol rate can be calculated as:
f s = R N {\displaystyle f_{s}={R \over N}}
In that case M = 2N different symbols are used. In a modem, these may be sinewave tones with unique combinations of amplitude, phase and/or frequency. For example, in a 64QAM modem, M = 64. In a line code, these may be M different voltage levels. By taking information per pulse N in bit/pulse to be the base-2-logarithm of the number of distinct messages M that could be sent, Hartley constructed a measure of the gross bit rate R as:
R = f s log 2 ( M ) {\displaystyle R=f_{s}\log _{2}(M)}
where fs is the baud rate in symbols/second or pulses/second. (See Hartley's law).
Modems for passband transmission Modulation is used in passband filtered channels such as telephone lines, radio channels and other frequency division multiplex (FDM) channels. In a digital modulation method provided by a modem, each symbol is typically a sine wave tone with a certain frequency, amplitude and phase. Symbol rate, baud rate, is the number of transmitted tones per second. One symbol can carry one or several bits of information. In voiceband modems for the telephone network, it is common for one symbol to carry up to 7 bits. Conveying more than one bit per symbol or bit per pulse has advantages. It reduces the time required to send a given quantity of data over a limited bandwidth. A high spectral efficiency in (bit/s)/Hz can be achieved; i.e., a high bit rate in bit/s although the bandwidth in hertz may be low. The maximum baud rate for a passband for common modulation methods such as QAM, PSK and OFDM is approximately equal to the passband bandwidth. Voiceband modem examples:
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