What is the Eb/No required for a given modulation scheme and bit error rate?
Eb/No and BER Performance
Eb/No is the standard figure of merit for digital communication system performance because it normalizes out the effects of data rate and bandwidth, allowing direct comparison of different modulation and coding combinations.
| Parameter | Free Space | Urban | Indoor |
|---|---|---|---|
| Path Loss Model | Friis (1/r²) | Okumura-Hata | IEEE 802.11 |
| Fading Margin | 0 dB | 10-30 dB | 5-15 dB |
| Multipath | None | Severe | Moderate-severe |
| Typical Range | Line of sight | 1-30 km | 10-100 m |
| Shadow Fading (σ) | 0 dB | 6-12 dB | 3-8 dB |
Margin Allocation
The BER as a function of Eb/No follows different curves for each modulation scheme: (1) BPSK and QPSK: BER = Q(sqrt(2×Eb/No)), where Q() is the Q-function (complementary cumulative distribution of the Gaussian). BPSK and QPSK have identical BER vs Eb/No performance because QPSK can be decomposed into two independent BPSK channels (I and Q), each carrying one bit per symbol with the same energy allocation. At Eb/No = 10 dB: BER ≈ 4×10^-6. At Eb/No = 12 dB: BER ≈ 10^-8. (2) M-QAM: BER ≈ (4/log2(M)) × (1 - 1/sqrt(M)) × Q(sqrt(3×log2(M)×Eb/No / (M-1))). The penalty for higher-order modulation: 16-QAM requires approximately 4 dB more Eb/No than QPSK for the same BER. 64-QAM requires approximately 8.5 dB more. 256-QAM requires approximately 13.5 dB more. The benefit: higher spectral efficiency (bits/s/Hz): QPSK = 2, 16-QAM = 4, 64-QAM = 6, 256-QAM = 8. (3) M-PSK (M > 4): less power-efficient than M-QAM because the constellation points are on a circle rather than a grid. 8-PSK requires 3.5 dB more than QPSK but provides only 50% more bits per symbol. Generally: M-QAM is preferred over M-PSK for M > 4.
Propagation Modeling
Forward error correction (FEC) reduces the required Eb/No by adding redundancy: (1) Convolutional codes (Viterbi decoding): coding gain = 3-5 dB (depending on constraint length and rate). Rate 1/2, K=7 convolutional code: approximately 5 dB gain at BER = 10^-6. (2) Turbo codes: coding gain = 7-9 dB. Within 0.7-1.0 dB of Shannon limit. Used in 3G/4G cellular (LTE uses turbo codes for data channels). (3) LDPC codes: coding gain = 8-10 dB. Within 0.3-0.5 dB of Shannon limit for long block lengths (>10,000 bits). Used in 5G NR, DVB-S2/S2X, Wi-Fi 6/7, and 802.11ad/ay. (4) Polar codes: within 0.5 dB of Shannon limit with successive cancellation list decoding. Used in 5G NR control channels. The net effect: with modern LDPC coding, QPSK achieves BER = 10^-6 at Eb/No ≈ 1.5 dB (compared to 10.5 dB uncoded). This represents a massive improvement in link budget efficiency.
- Performance verification: confirm specifications against the application requirements before finalizing the design
- Environmental factors: temperature range, humidity, and vibration affect long-term reliability and parameter drift
- Cost vs. performance: evaluate whether the application demands premium components or standard commercial grades
Fade Mitigation
To use Eb/No in a link budget: (1) Determine the required Eb/No from the BER specification and chosen ModCod (modulation and coding combination). Example: DVB-S2 QPSK rate 3/4 LDPC requires Eb/No = 4.03 dB for quasi-error-free (QEF, BER < 10^-7). (2) Calculate the required C/N: C/N = Eb/No + 10×log10(Rb/BW). For 30 Msym/s QPSK (Rb = 60 Mbps) with 36 MHz transponder: Rb/BW = 60e6/36e6 = 1.67. C/N = 4.03 + 10×log10(1.67) = 4.03 + 2.2 = 6.2 dB. (3) Ensure the link budget provides C/N > 6.2 dB plus implementation margin (1-2 dB) and fade margin. (4) Adaptive coding and modulation (ACM): modern systems dynamically switch ModCod based on current link conditions. In clear sky: use 32-APSK rate 9/10 (spectral efficiency = 4.5 bits/s/Hz, requires Eb/No = 12 dB). During rain fade: fall back to QPSK rate 1/4 (0.5 bits/s/Hz, requires Eb/No = -0.8 dB).
Frequently Asked Questions
Why do BPSK and QPSK have the same BER performance?
QPSK transmits 2 bits per symbol by modulating in-phase (I) and quadrature (Q) carriers independently. Each carrier is essentially a BPSK signal. The total symbol energy is twice the bit energy (Es = 2×Eb), but since each symbol carries 2 bits, the energy per bit is the same as BPSK. The I and Q channels are orthogonal (do not interfere with each other), so each channel achieves the same BER as BPSK. Result: QPSK has twice the spectral efficiency of BPSK with no penalty in Eb/No. This is why QPSK is universally preferred over BPSK.
How close are modern codes to the Shannon limit?
State-of-the-art LDPC codes (as used in 5G NR and DVB-S2X) operate within 0.2-0.5 dB of the Shannon limit for block lengths of 10,000-64,000 bits. For shorter blocks (100-1000 bits, as needed for low-latency applications): the gap increases to 1-3 dB. Polar codes with CRC-aided successive cancellation list decoding: within 0.5 dB for medium block lengths. Turbo codes: within 0.7-1.0 dB. In practical implementations: the achievable gap is limited by finite block length, implementation precision (fixed-point quantization in the decoder), and latency constraints (fewer decoder iterations = worse performance).
What implementation margin should I add beyond the theoretical Eb/No?
Implementation margin accounts for real-world imperfections not captured in the theoretical BER curve: (1) Phase noise from local oscillators: 0.5-1.0 dB degradation for 64-QAM (higher for 256-QAM). (2) I/Q imbalance: 0.2-0.5 dB. (3) ADC quantization noise: 0.1-0.5 dB (depends on ADC resolution, typically 8-12 bits). (4) Timing recovery jitter: 0.1-0.3 dB. (5) Filter group delay distortion: 0.2-0.5 dB. Typical total implementation margin: 1.0-2.0 dB for QPSK, 1.5-3.0 dB for 64-QAM, 2.5-4.0 dB for 256-QAM. Always add this margin on top of the theoretical Eb/No and the fade/rain margin.