Symbol rate

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In digital communications, symbol rate, also known as baud rate or modulation rate; is the number of symbol changes (signalling events) made to the transmission medium per second using a digitally modulated signal or a line code. The Symbol rate is measured in baud (Bd) or symbols/second. In the case of a line code, the symbol rate is the pulse rate in pulses/second. Each symbol can represent or convey one or several bit of data. The symbol rate is related to but should not be confused with gross bitrate expressed in bit/s.

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 \over f_s},

where fs is the symbol rate.

A simple 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.

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[edit] Relationship to gross bitrate

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 bit (a binary bit always represents exactly two states).

Example of use and misuse of "baud rate": It is correct to write "the baud rate of my COM port is 9,600" if we mean 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 Mbaud" or "the baud rate of my modem is 56,000" if we mean bit rate. See below for more details on these techniques.

The difference between baud (or signalling rate) and the data rate (or bit rate) is like a man using a single semaphore flag who can move his arm to a new position once each second, so his signalling rate (baud) is one symbol per second. The flag can be held in one of eight distinct positions: Straight up, 45° left, 90° left, 135° left, straight down (which is the rest state, where he is sending no signal), 135° right, 90° right, and 45° right. Each signal carries three bits of information. It takes three binary digits to encode eight states. The data rate is three bits per second. In the Navy, more than one flag pattern and arm can be used at once, so the combinations of these produce many symbols, each conveying several bits, a higher 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 \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[1] constructed a measure of the gross bitrate R as:

R = f_s \log_2(M), \,

where fs is the baud rate in symbols/second or pulses/second. (See Hartley's law).

[edit] 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 certain frequency, amplitude and phase. The 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 10 bits per symbol.

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:

  • A V.22bis modem transmits 2400 bit/s using 1200 Bd (1200 symbol/s), where each quadrature amplitude modulation symbol carries two bits of information. The modem can generate M=22=4 different symbols. It requires a bandwidth of 2400 Hz (equal to the baud rate). The carrier frequency (the central frequency of the generated spectrum) is 1800 Hz, meaning that the lower cut off frequency is 1800 -2400/2 = 600 Hz, and the upper cutoff frequency is 1800 + 2400/2 = 3000 Hz.
  • A V.34 modem may transmit symbols at a baud rate of 3,420 Bd, and each symbol can carry up to ten bits, resulting in a gross bit rate of 3420 * 10 = 34,200 bit/s. However, the modem is said to operate at a net bit rate of 33,800 bit/s, excluding physical layer overhead.

[edit] Line codes for baseband transmission

In case of a baseband channel such as a telegraph line, a serial cable or a Local Area Network twisted pair cable, data is transferred using line codes, i.e. pulses rather than sinewave tones. In this case the baud rate is synonymous to the pulse rate in pulses/second.

The maximum baud rate or pulse rate for a base band channel is called the Nyquist rate, and is half the bandwidth (half the upper cut-off frequency).

The simplest digital communication links (such as individual wires on a motherboard or the RS-232 serial port/COM port) typically have a symbol rate equal to the gross bit rate.

Common communication links such as 10 Mbit/s Ethernet (10Base-T), USB, and FireWire typically have a symbol rate slightly higher than the data bit rate, due to the overhead of extra non-data symbols used for self-synchronizing code and error detection.

J. M. Emile Baudot (1845–1903) worked out a five-level code (five bits per character) for telegraphs which was standardized internationally and is commonly called Baudot code.

More than two voltage levels are used in advanced techniques such as FDDI and 100/1000 Mbit/s Ethernet LANs, and others, to achieve high data rates.

1000 Mbit/s Ethernet LAN cables use many wire pairs and many bits per symbol to encode their data payloads. 1000BASE-T uses four wire pairs and two data bits per symbol to get a symbol rate of 125 MBd (i.e., 1.25×108 Bd).

[edit] Digital television and OFDM example

In digital television transmission the symbol rate calculation is:

symbol rate in symbols per second = (Data rate in bits per second * 204) / (188 * bits per symbol)

The 204 is the number of bytes in a packet including the 16 trailing Reed-Solomon error checking and correction bytes. The 188 is the number of data bytes (187 bytes) plus the leading packet sync byte (0x47).

The bits per symbol is the (modulation's power of 2)*(Forward Error Correction). So for example in 64-QAM modulation 64 = 26 so the bits per symbol is 6. The Forward Error Correction (FEC) is usually expressed as a fraction, i.e., 1/2, 3/4, etc. In the case of 3/4 FEC, for every 3 bits of data, you are sending out 4 bits, one of which is for error correction.

Example:

given bit rate = 18096263
Modulation type = 64-QAM
FEC = 3/4

then

\textrm{symbol}\;\textrm{rate} = \cfrac{18096263}{(6)*(3/4)} ~ \cfrac{204}{188} = \cfrac{18096263}{6} ~ \cfrac{4}{3} ~ \cfrac{204}{188} = 4363638

In digital terrestrial digital television (DVB-T, DVB-H and similar techniques) OFDM mudulation is used, i.e. multi-carrier modulation. The above symbol rate should then be divided by the number of OFDM sub-carriers in view to achieve the OFDM symbol rate. See the OFDM system comparison table for further numerical details.

[edit] Relationship to chip rate

Some communication links (such as GPS transmissions, CDMA cell phones, and other spread spectrum links) have a symbol rate much higher than the data rate (they transmit many symbols called chips per data bit. In these systems, the baud rate is called chip rate.

Representing one bit by a chip sequence of many symbols overcomes co-channel interference from other transmitters sharing the same frequency channel, including radio jamming, and is common in military radio and cell phones. Despite the fact that using more bandwidth to carry the same bit rate gives low channel spectral efficiency in (bit/s)/Hz, it allows many simultaneous users, which results in high system spectral efficiency in (bit/s)/Hz per unit of area.

[edit] Relationship to bit error rate

The disadvantage of conveying many bits per symbol is that the receiver has to distinguish many signal levels or symbols from each other, which may be difficult and cause bit errors in case of a poor phone line that suffers from low signal-to-noise ratio. In that case, a modem or network adapter may automatically choose a slower and more robust modulation scheme or line code, using fewer bits per symbol, in view to reduce the bit error rate.

An optimal symbol set design takes into account channel bandwidth, desired information rate, noise characteristics of the channel and the receiver, and receiver and decoder complexity.

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