Qpsk Symbol Rate Calculator

QPSK Symbol Rate Calculator

In digital telecommunications and radio frequency engineering, calculating the Symbol Rate (also known as Baud Rate) is critical for determining the bandwidth requirements of a transmission channel. This calculator is specifically designed for QPSK (Quadrature Phase Shift Keying) modulation schemes.

What is QPSK?

QPSK is a digital modulation technique that conveys data by changing the phase of a carrier wave. It uses four distinct phases (0°, 90°, 180°, 270°) to encode information. Because there are four possible states ($M=4$), each symbol represents 2 bits of data.

This is derived from the formula $\log_2(M)$, where $M$ is the number of symbols. For QPSK: $\log_2(4) = 2$ bits per symbol.

The Symbol Rate Formula

The relationship between Bit Rate ($R_b$) and Symbol Rate ($R_s$) depends on the modulation efficiency ($m$). For QPSK, $m = 2$.

$$ R_s = \frac{R_b}{m} $$
Where:
Rs = Symbol Rate (Baud)
Rb = Gross Bit Rate (bits per second)
m = Bits per Symbol (2 for QPSK)

Accounting for Forward Error Correction (FEC)

In real-world systems like satellite communications (DVB-S/S2), the useful data (Information Rate) is padded with error correction bits. If you input your "Info Rate" and a coding rate (e.g., 3/4), the Gross Bit Rate increases:

Gross Rate = Info Rate / FEC Ratio

Example: If you want 10 Mbps of data with 1/2 FEC, you must transmit 20 Mbps over the air. The Symbol rate is then calculated from this gross 20 Mbps.

Bandwidth Calculation

Once the Symbol Rate is known, the approximate occupied bandwidth ($BW$) can be calculated using the roll-off factor ($\alpha$) of the filter used (typically Root Raised Cosine):

$$ BW = R_s \times (1 + \alpha) $$

Common roll-off factors range from 0.20 to 0.35 depending on the standard (DVB-S, DVB-S2, etc.).

Example Calculation

Suppose you are setting up a satellite link with the following parameters:

• Target Data Rate: 10 Mbps
• FEC Rate: 3/4
• Modulation: QPSK
• Roll-off: 0.20

Step 1: Determine Gross Bit Rate
$10 \text{ Mbps} / 0.75 = 13.33 \text{ Mbps}$

Step 2: Determine Symbol Rate
Since QPSK carries 2 bits/symbol: $13.33 \text{ Mbps} / 2 = 6.67 \text{ MBaud}$

Step 3: Determine Bandwidth
$6.67 \text{ MBaud} \times (1 + 0.20) = 8.0 \text{ MHz}$