Symbol rate

The symbol rate of a digital link is the number of symbols transmitted per second, where a symbol is one of the distinct states (amplitude levels, or points in the complex plane) that the transmitter holds for one unit interval. Its unit is the baud (Bd), after Émile Baudot; optical links run at gigabaud, written GBd or GBaud. Binary NRZ carries one bit per symbol, so its symbol rate and bit rate are numerically equal: a 25 Gb/s Ethernet lane runs at 25.78125 GBd. PAM4 carries two bits per symbol, so a 106.25 Gb/s lane runs at 53.125 GBd. Coherent transceivers in service run from about 30 to 200 GBd.

Symbol rate and bit rate

The gross bit rate is the symbol rate multiplied by the bits per symbol and by the number of polarizations that carry independent data:

$$R_b = R_s \cdot \log_2 M \cdot N_\text{pol},$$

where $M$ is the number of constellation points per polarization. For intensity-modulated formats $N_\text{pol} = 1$. Worked values:

These are line rates, including the forward error correction overhead and any coding the standard adds. The payload is smaller: the 512 Gb/s DP-16QAM line rate carries about 410 Gb/s of data if the FEC and framing overhead together are 25%. Lanes of the 200G per lane generation sit near 106–113 GBd, with the exact value set by which FEC layers are present, and the 400ZR coherent pluggable runs DP-16QAM at 59.84375 GBd for a 478.75 Gb/s line rate.

The time per symbol, the unit interval, is $T = 1/R_s$: 18.82 ps at 53.125 GBd and 9.41 ps at 106.25 GBd. The eye diagram is drawn over one or two unit intervals, which is the most direct way to read the symbol rate off an oscilloscope.

Bandwidth

The bandwidth a signal occupies scales with its symbol rate and is almost independent of the number of bits per symbol. By the Nyquist criterion, a baseband signal at $R_s$ needs at least $R_s/2$ of electrical bandwidth to be received without intersymbol interference; on the optical carrier, which carries both sidebands, the minimum occupied bandwidth is $R_s$. Coherent transmitters shape their pulses with a root-raised-cosine filter of roll-off $\beta$, so the optical spectrum spans

$$B_\text{opt} = R_s (1 + \beta).$$

At 64 GBd and $\beta = 0.1$ this is 70.4 GHz, which fits inside a 75 GHz DWDM grid slot. Unshaped NRZ receivers are usually built with an electrical bandwidth of about 0.75 $R_s$, a compromise between noise and intersymbol interference.

For this reason higher-order modulation formats are adopted when components run out of bandwidth: moving from NRZ to PAM4 halves the symbol rate for a given bit rate, and moving from QPSK to 16QAM in a coherent system does the same. The cost is a smaller separation between levels, which raises the required OSNR or received power.

Measurement

In the lab the symbol rate is set by the pattern generator or the transmitter's reference clock, and confirmed in three ways: the period of the eye on a sampling oscilloscope, the clock recovered by the receiver, and the spectrum. On an electrical spectrum analyzer an NRZ or PAM4 signal shows nulls at multiples of $R_s$; on a high-resolution optical spectrum analyzer a Nyquist-shaped coherent signal shows a nearly rectangular spectrum whose width is $R_s(1+\beta)$.

Pitfalls

Baud and bits per second are interchangeable only for binary formats; quoting a PAM4 lane in "Gb/s" when the symbol rate is meant, or the reverse, is a common source of a factor-of-two error. A second is the OSNR scaling: in the usual 0.1 nm reference bandwidth the required OSNR grows in proportion to the symbol rate, 3 dB per doubling, so comparing two transceivers' OSNR figures is meaningful only at equal symbol rates or after rescaling. A third is the overhead: the symbol rate always corresponds to the line rate, so dividing a payload rate by bits per symbol gives a symbol rate that is several percent too low.

Common questions

What is the difference between baud rate and bit rate?

The baud rate counts symbols per second; the bit rate counts bits per second. They are equal for two-level signaling and differ by the factor $\log_2 M \cdot N_\text{pol}$ for multilevel or dual-polarization formats: two for PAM4, eight for DP-16QAM.

Why is the symbol rate not a round number?

Line rates include FEC parity and line-coding overhead on top of the payload. A 100 Gb/s Ethernet payload with RS(544,514) FEC and its transcoding becomes 106.25 Gb/s on the line, hence 53.125 GBd in PAM4.

How much bandwidth does a given symbol rate need?

At least half the symbol rate electrically, and at least the symbol rate in optical occupied bandwidth, increased by the pulse-shaping roll-off. Components with a 3 dB bandwidth near half the symbol rate are typical when equalization is used.

References: J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008); G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010); K. Kikuchi, "Fundamentals of coherent optical fiber communications," Journal of Lightwave Technology 34, 157 (2016); IEEE Std 802.3.