What is coding gain and how is it measured?

Coding gain is the reduction in required SNR to achieve a target BER compared to uncoded transmission: G_c = SNR_uncoded - SNR_coded (in dB) at the same BER. For BER = 10^-6, convolutional codes (K=7, R=1/2) provide about 5 dB coding gain. Turbo codes achieve 8-10 dB gain. LDPC and polar codes achieve 9-11 dB, approaching within 0.5 dB of the Shannon limit. Coding gain comes at the cost of bandwidth expansion by factor 1/R.

Why does 5G NR use both LDPC and polar codes?

5G NR uses LDPC for data channels and polar codes for control channels. LDPC excels at high throughput with parallelizable decoding, supporting rates from 1/5 to 0.95 with block sizes up to 66560 bits. Polar codes are mathematically proven to achieve Shannon capacity and are better suited to the shorter block lengths (up to 1024 bits) and lower latency required by control signaling. This dual approach optimizes performance across both use cases.

What is the difference between hard and soft decision decoding?

Hard-decision decoding quantizes each received symbol to a single bit (0 or 1) before decoding. Soft-decision decoding passes multi-bit reliability information (log-likelihood ratios) to the decoder, preserving the analog confidence level. Soft decoding provides approximately 2-3 dB improvement over hard decoding. All modern FEC systems (turbo, LDPC, polar) use soft-decision decoding. The trade-off is increased decoder complexity and memory requirements.

FEC / Digital Communications

Channel Coding

/CHAN-ul KOH-ding/

Forward error correction (FEC) adds structured redundancy to digital data before RF transmission, enabling the receiver to detect and correct bit errors from noise and fading. The Shannon limit C = B·log₂(1 + SNR) sets the theoretical ceiling. Modern codes (turbo, LDPC, polar) approach within 0.5 dB. Coding gain G_c = SNR_uncoded − SNR_coded at target BER.

Code rate: R = k/n

Gain: 5–11 dB typical

Near Shannon: 0.5 dB gap

Understanding Channel Coding

Every RF communication link contends with noise, interference, and fading that introduce bit errors. Channel coding combats these impairments by adding redundancy in a mathematically structured way, enabling the receiver to reconstruct the original data even when some bits are corrupted. The code rate R = k/n expresses the ratio of useful information bits (k) to total transmitted bits (n). A lower code rate provides more protection but consumes more bandwidth.

The evolution of channel coding spans four generations. First-generation block codes (Hamming, BCH, Reed-Solomon) offer algebraic decoding with modest coding gain. Second-generation convolutional codes with Viterbi decoding became the workhorse of 2G/3G cellular and satellite links. Third-generation turbo codes, discovered in 1993, demonstrated near-Shannon-limit performance and were adopted for 3G/4G. Fourth-generation LDPC and polar codes now dominate 5G NR, Wi-Fi 6/7, and next-generation satellite systems, with polar codes being the first codes mathematically proven to achieve Shannon capacity.

Fundamental Equations

Shannon capacity:
C = B·log₂(1 + SNR) bps

Code rate:
R = k/n (information bits / total bits)

Coding gain:
G_c = SNR_uncoded − SNR_coded dB (at same BER)

Spectral efficiency with coding:
η = R × log₂(M) bps/Hz
M = modulation order

Eb/N0 relationship:
E_b/N₀ = (C/N) × (B / R_b)
R_b = information bit rate

FEC Code Family Comparison

Code Family	Rate Range	Gap to Shannon	Decoder	Latency	Standard
Convolutional	1/2–7/8	2–5 dB	Viterbi / MAP	Low	GSM, DVB-S
Reed-Solomon	Variable	3–6 dB	Berlekamp-Massey	Low	DVB-S, CD/DVD
Turbo	1/3–0.95	~1 dB	Iterative BCJR	High	3G/4G LTE
LDPC	1/5–0.95	0.5–1 dB	Belief propagation	Medium	5G data, Wi-Fi 6
Polar	1/8–0.95	~0.5 dB	SCL + CRC	Low-Med	5G NR control

5G NR Coding Configuration

Channel Type	FEC Code	Max Block	Base Graph	Rate Range	HARQ
PDSCH (data DL)	LDPC	8448 bits	BG1 (high rate)	1/5–0.95	Incremental redundancy
PUSCH (data UL)	LDPC	8448 bits	BG2 (low rate)	1/5–0.95	Incremental redundancy
PDCCH (ctrl DL)	Polar	512 bits	N/A	1/8–0.75	N/A
PUCCH (ctrl UL)	Polar/RM	1706 bits	N/A	Variable	N/A
PBCH (broadcast)	Polar	864 bits	N/A	1/4	N/A

Common Questions

Frequently Asked Questions

What is coding gain?

Coding gain is the SNR reduction to achieve a target BER vs. uncoded: G_c = SNR_uncoded − SNR_coded. Convolutional (K=7): ~5 dB. Turbo: ~8–10 dB. LDPC/Polar: ~9–11 dB, within 0.5 dB of Shannon at BER = 10⁻⁶. The cost is bandwidth expansion by 1/R.

Why LDPC and polar in 5G?

LDPC handles data channels: parallelizable decoding supports multi-Gbps throughput with block sizes up to 66560 bits. Polar handles control channels: proven Shannon-achieving with shorter blocks (up to 1024 bits) and lower latency. This dual architecture optimizes both use cases.

Hard vs. soft decision?

Hard decision: 1-bit quantization per symbol. Soft decision: multi-bit log-likelihood ratios preserving confidence. Soft decoding yields 2–3 dB improvement. All modern FEC (turbo, LDPC, polar) use soft input. The trade-off is increased decoder memory and complexity.

Related Terms

← Channel Capacity Channel Equalization →

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