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.
BER Curves by Modulation
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.
Coding Gain
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.
Practical Link Budget Application
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).
C/N = Eb/No + 10log₁₀(Rb/BW)
Shannon: Eb/No_min = -1.59 dB
LDPC Gain: 8-10 dB coding gain
Spectral Eff: QPSK=2, 16QAM=4, 64QAM=6 bps/Hz
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.