How does FEC provide coding gain?

Coding gain is the reduction in required SNR to achieve a target BER, compared to uncoded transmission. Without FEC: BPSK requires Eb/N0 = 10.5 dB for BER = 10 to the minus 6. With rate 1/2 convolutional code (K=7, Viterbi): requires Eb/N0 = 5.5 dB for the same BER. Coding gain = 10.5 minus 5.5 = 5 dB. However, the rate 1/2 code doubles the bandwidth (or halves the data rate). The net coding gain accounts for the rate penalty: useful when power is more constrained than bandwidth. Shannon proved that reliable communication is possible at any rate below channel capacity C = B log2(1 + SNR), provided sufficiently sophisticated coding is used. The gap to Shannon limit measures how close a code comes to the theoretical maximum: turbo codes (1993): 0.7 dB from Shannon at BER = 10 to the minus 5. LDPC codes (Gallager, 1962, rediscovered 1990s): 0.5 dB from Shannon. Polar codes (Arikan, 2009): theoretically capacity-achieving (0 dB gap) as block length approaches infinity. Soft-decision decoding provides an additional 2 to 3 dB gain over hard-decision by using reliability information from the demodulator.

What FEC codes are used in modern wireless?

5G NR data channel (PDSCH/PUSCH): LDPC codes with two base graphs. BG1 for large blocks and high code rates (transport blocks up to 8448 bits, rates 1/3 to 8/9). BG2 for small blocks and low rates (up to 3840 bits, rates 1/5 to 2/3). Quasi-cyclic structure enables high-throughput parallel decoding. 5G NR control channel (PDCCH/PUCCH): polar codes with CRC-aided successive cancellation list (CA-SCL) decoding. Block sizes 12 to 1706 bits. Excellent performance for short blocks. Wi-Fi 6/6E/7 (802.11ax/be): LDPC codes, mandatory for MCS 10 and 11 (1024QAM). Code rates 1/2, 2/3, 3/4, 5/6. DVB-S2X (satellite): LDPC plus BCH concatenation. Code rates 1/4 to 9/10. Block length 16200 or 64800 bits. Achieves within 0.5 to 1 dB of Shannon limit. Optical (400G/800G): soft-decision LDPC with 15 to 25 percent overhead, providing net coding gain of 11 to 12 dB. Enables transmission over thousands of km of fiber.

Hybrid ARQ (HARQ) combines FEC with retransmission. The first transmission includes enough redundancy for the receiver to correct most errors. If decoding fails, the receiver stores the corrupted data and requests a retransmission. The retransmitted data is combined with the stored first attempt before decoding again. Chase combining (Type I HARQ): the retransmission is identical to the first transmission. The receiver combines both copies (soft combining), doubling the effective SNR (3 dB gain). Incremental Redundancy (Type II HARQ, used in LTE and 5G): the retransmission sends different parity bits (not the same data). This effectively lowers the code rate with each retransmission. First attempt: rate 3/4 (high throughput). Second attempt: rate 1/2 (more redundancy). Third attempt: rate 1/3 (maximum redundancy). This is much more efficient than Chase combining because each retransmission adds new information. 5G NR uses LDPC with incremental redundancy HARQ, with up to 4 retransmissions. The initial transmission targets a 10 percent BLER (Block Error Rate), allowing aggressive MCS selection for high throughput. HARQ corrects the 10 percent errors with minimal overhead.

Channel Coding

FEC

/eff-ee-see/ — Forward Error Correction

Structured redundancy for error correction. R = k/n (code rate). Shannon: C = Blog₂(1+SNR). Convolutional (Viterbi): 3-5 dB gain. Turbo: 0.7 dB from Shannon. LDPC (5G data): 0.5 dB. Polar (5G control): capacity-achieving. Soft decision: +2-3 dB over hard. HARQ: FEC + retransmission (incremental redundancy). Coding gain = less TX power for same BER.

LDPC: 0.5 dB

Turbo: 0.7 dB

5G: LDPC+Polar

Understanding FEC

FEC is the bridge between Shannon's theoretical channel capacity and practical wireless systems. Without FEC, achieving reliable communication requires enormous SNR. With modern FEC (LDPC, polar), reliable communication is possible within 0.5 dB of the theoretical limit, meaning we are extracting nearly 100% of the available channel capacity.

From the RF engineer's perspective, every dB of coding gain is a dB that does not need to come from higher TX power, bigger antennas, or lower noise figures. FEC is the most cost-effective way to improve link reliability.

FEC Equations

Shannon capacity:
C = B × log₂(1+SNR) bits/s
SNR = 0 dB: C = B (1 bit/s/Hz)
SNR = 10 dB: C = 3.46 B

Code rate:
R = k/n (info bits / total bits)
R=1/2: double BW, max 3 dB penalty
R=3/4: 33% overhead, 1.25 dB penalty

Coding gain (uncoded vs coded):
Uncoded BPSK @BER 10⁻⁶: 10.5 dB
Conv R=1/2 K=7: 5.5 dB (gain: 5 dB)
LDPC R=1/2: 1.5 dB (gain: 9 dB)

FEC Code Comparison

Code	Gap to Shannon	Complexity	Latency	Standard
Convolutional	3-5 dB	Low	Low	GSM, 802.11a
Turbo	0.7 dB	High	High	3G, LTE
LDPC	0.5 dB	Medium	Medium	5G data, Wi-Fi
Polar	0 dB*	Medium	Medium	5G control
Reed-Solomon	3-4 dB	Low	Low	DVB, optical

Common Questions

Frequently Asked Questions

Coding gain?

Reduction in required SNR for target BER. Uncoded BPSK @10⁻⁶: 10.5 dB. Conv R=1/2 K=7: 5.5 dB (5 dB gain). LDPC: 1.5 dB (9 dB gain). Rate penalty: R=1/2 costs 3 dB BW. Net gain still positive. Soft decision: +2-3 dB over hard by using demodulator reliability info.

Modern codes?

5G NR data: LDPC (BG1/BG2, quasi-cyclic, R=1/5-8/9). 5G control: polar (CA-SCL decoding). Wi-Fi 6/7: LDPC (mandatory for 1024QAM). DVB-S2X: LDPC+BCH. Optical 400G: soft-decision LDPC (11-12 dB NCG). Turbo: legacy 3G/LTE. Modern = within 1 dB of Shannon.

HARQ?

FEC + retransmission. Chase: resend same, soft-combine (+3 dB). Incremental redundancy (IR): send new parity, lower effective rate. 5G: target 10% BLER first TX (aggressive MCS), HARQ fixes errors. Rate: 3/4 → 1/2 → 1/3 per retransmission. Max 4 HARQ attempts. Efficient: high throughput + reliability.

Related Terms

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