Convert between bit rate (bps) and symbol rate (baud) for any digital modulation. Select modulation type or set bits per symbol manually. Visualize the ideal constellation diagram (PSK/QAM) and explore key metrics like Nyquist bandwidth and spectral efficiency.
In digital communications, the baud rate (or symbol rate) is the number of symbol changes per second. Each symbol can carry multiple bits depending on the modulation order M. The fundamental relation is:
Rb (bps) = Rs (baud) × log2(M)
where M = 2m and m is the number of bits per symbol.
This formula is the bedrock of information theory and was formalized by Nyquist and Shannon. For example, QPSK modulation (M=4) carries 2 bits per symbol, so 1 Mbaud yields 2 Mbps.
The term “baud” honors Émile Baudot (1845–1903), a French telegraph engineer who invented the Baudot code, a forerunner of character encoding. In the 20th century, baud rate became crucial for modems, and today it is essential in everything from UART serial links to 5G and Wi‑Fi 6. Understanding the relationship between bit rate and baud rate is fundamental for physical layer design.
From the Nyquist first criterion, the maximum symbol rate in an ideal low‑pass channel is twice the bandwidth: Rs ≤ 2B. With practical pulse shaping (raised‑cosine filter, roll‑off factor α), the occupied bandwidth is Bact = Rs(1+α)/2. The spectral efficiency then becomes η = Rb / Bact = log2(M) / (1+α). This calculator shows the Nyquist minimum bandwidth (α=0), i.e., Bmin = Rs/2.
Real systems often add forward error correction (FEC). For example, 64QAM with a 5/6 code rate would have an effective bit rate multiplied by 5/6. This tool provides the raw modulation rate; you can apply coding factors separately.
| Modulation | Bits/sym m | Order M | Spectral Eff. (bps/Hz) | Typical Applications |
|---|---|---|---|---|
| BPSK | 1 | 2 | 1.0 | GPS, deep‑space comm. |
| QPSK | 2 | 4 | 2.0 | Satellite TV, 4G LTE |
| 8PSK | 3 | 8 | 3.0 | EDGE, some satellite systems |
| 16QAM | 4 | 16 | 4.0 | Wi‑Fi 4/5, cable TV |
| 64QAM | 6 | 64 | 6.0 | DVB‑T, Wi‑Fi 5/6 |
| 256QAM | 8 | 256 | 8.0 | DOCSIS 3.1, 5G NR |
A satellite transponder has a bandwidth of 36 MHz. Using 16QAM (m=4) and a roll‑off factor α=0.2, the maximum symbol rate is Rs = 36×106 / (1+0.2) = 30 Mbaud. The corresponding bit rate is 30 × 4 = 120 Mbps. If we switch to 64QAM (m=6), the bit rate becomes 180 Mbps, but a higher signal‑to‑noise ratio is required. This calculator lets you explore such trade‑offs instantly.
function updateFromBitrate() {
let Rb = parseFloat(bitrateInput.value);
let m = parseFloat(bitsPerSymInput.value);
if (isNaN(Rb) || isNaN(m) || m <= 0) return;
let Rs = Rb / m;
baudrateInput.value = Rs.toFixed(4);
refreshDerived();
}
function updateFromBaudrate() {
let Rs = parseFloat(baudrateInput.value);
let m = parseFloat(bitsPerSymInput.value);
if (isNaN(Rs) || isNaN(m) || m <= 0) return;
let Rb = Rs * m;
bitrateInput.value = Rb.toFixed(4);
refreshDerived();
}