OFDM BER vs SNR (with Simulation)

Theoretical BER vs SNR for OFDM

Fundamental Rule: OFDM is a multiplexing scheme, not a modulation scheme. In an ideal AWGN channel, the BER of OFDM is identical to the BER of the underlying modulation used on its sub-carriers.

If you use QPSK on your OFDM sub-carriers, your theoretical BER curve is simply the QPSK curve.

1. Standard Formulas (AWGN Channel)

Assuming \(N\) sub-carriers and let \(\gamma_b = \frac{E_b}{N_0}\) (SNR per bit):

• BPSK / QPSK OFDM:
  \[P_b = Q\left(\sqrt{2 \gamma_b}\right) = \frac{1}{2}\text{erfc}\left(\sqrt{\gamma_b}\right)\]
• 16-QAM OFDM (Gray Coded):
  \[P_b \approx \frac{3}{4} Q\left(\sqrt{\frac{4}{5} \gamma_b}\right)\]
• 64-QAM OFDM:
  \[P_b \approx \frac{7}{12} Q\left(\sqrt{\frac{2}{7} \gamma_b}\right)\]

Simulation of BER vs. SNR for OFDM Modulation with BPSK

# OFDM Symbols:

Subcarriers (N):

Cyclic Prefix (Ncp):

SNR (dB):

BER Results Log

SNR (dB)	BER

Transmitted Bits (Partial)

Received Bits (Partial)

Compare the BER with Original BPSK →

2. The Cyclic Prefix (CP) Penalty

In practice, OFDM performs slightly worse than single-carrier systems because the Cyclic Prefix consumes transmitter power without carrying new data. This shifts the curve to the right.

If \(T_u\) is the useful duration and \(T_{cp}\) is the prefix duration, the power penalty in dB is:

\[\text{CP Penalty (dB)} = 10 \log_{10} \left( \frac{T_u + T_{cp}}{T_u} \right)\]

Example: In 802.11a Wi-Fi, this penalty is roughly 0.97 dB.

3. Converting raw SNR to \(E_b/N_0\)

To map a physical RF SNR to the theoretical formulas above, use the following conversion:

\[\frac{E_b}{N_0} = \text{SNR} \times \left( \frac{N_{FFT}}{N_{used} \cdot \log_2(M)} \cdot \frac{T_u + T_{cp}}{T_u} \right)\]

Where:

• The ratio \(B/R_b\) acts as a normalization factor between raw signal power and bit energy:

• \(R_b\) (Net Bit Rate): The speed of actual data transmission in bits per second.
• \(N_{FFT}\): Total IFFT size.
• \(N_{used}\): Number of active data sub-carriers.
• \(M\): Modulation order (e.g., 16 for 16-QAM).

Variable Definitions

The ratio \(B/R_b\) acts as a normalization factor between raw signal power and bit energy:

• \(B\) (Transmission Bandwidth): The total frequency span of the signal in Hz.
  In OFDM: \(B \approx N_{FFT} \times \text{subcarrier spacing}\)
• \(R_b\) (Net Bit Rate): The speed of actual data transmission in bits per second.
  \[R_b = \frac{N_{used} \times \log_2(M)}{T_u + T_{cp}}\]

The Logic: If \(B > R_b\), your energy is spread out (robust). If \(R_b > B\), you are packing many bits into a small space (spectral efficiency), which requires a higher SNR to maintain the same BER.

Summary: While the BER curves are theoretically the same as single-carrier, OFDM is preferred because it maintains these curves in Multipath Frequency-Selective channels where single-carrier systems would fail due to Inter-Symbol Interference (ISI).

Return to BER vs SNR Main Page →

Online Simulations Main Page →

Further Reading

OFDM BER vs SNR using 16-QAM, 4-QAM, QPSK and BPSK