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Definition:
Gap to Capacity

The gap to capacity $\Gamma$ (in dB) quantifies how far a practical modulation and coding scheme operates from the Shannon limit at the same spectral efficiency $\eta$ :

$\text{SNR}_{\text{required}} = \Gamma \cdot (2^\eta - 1)$

or equivalently,

$\Gamma = \frac{\text{SNR}_{\text{required}}}{2^\eta - 1}$

A gap of $\Gamma = 0$ dB means the scheme achieves capacity. Uncoded QAM at BER $= 10^{-6}$ has $\Gamma \approx 8.8$ dB. Modern coded modulation (LDPC + QAM) achieves $\Gamma \approx 1$ - $2$ dB.

Practical Code Families Approaching Capacity

Three families of codes have been shown to approach AWGN capacity:

Turbo codes (Berrou, Glavieux, Thitimajshima, 1993): Parallel concatenation of two convolutional codes with a random interleaver. Iterative decoding achieves near-capacity performance. First codes to operate within 0.5 dB of capacity.

LDPC codes (Gallager, 1962; MacKay, 1996): Sparse parity-check matrix codes decoded by belief propagation. Can be designed to achieve capacity on the BEC (irregular LDPC) and approach capacity within 0.04 dB on the AWGN channel.

Polar codes (Arikan, 2009): The first provably capacity-achieving codes with explicit construction and polynomial complexity encoding/decoding. Based on channel polarisation: transforming $N$ copies of a channel into channels that are either near-perfect or near-useless.

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Turbo vs LDPC vs Polar Codes

Property	Turbo codes	LDPC codes	Polar codes
Year introduced	1993	1962 (rediscovered 1996)	2009
Gap to capacity (AWGN)	0.3-0.5 dB	0.04-0.5 dB	0.2-0.5 dB
Decoder	Iterative (BCJR)	Belief propagation	Successive cancellation (list)
Decoding complexity	$O(N)$ per iteration	$O(N)$ per iteration	$O(N \log N)$
Latency	Medium (iterative)	Medium (iterative)	High (sequential)
Error floor	Can be problematic	Design-dependent	No error floor
Standard adoption	3G/4G data	5G data, Wi-Fi 6	5G control
Provably capacity-achieving	No (empirical)	Yes (BEC)	Yes (any B-DMC)

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Example: Gap to Capacity for 16-QAM with LDPC Coding

A system uses 16-QAM ( $\eta = 4$ bits/s/Hz) with a rate-3/4 LDPC code. The effective spectral efficiency is $\eta_{\text{eff}} = 4 \times 0.75 = 3$ bits/s/Hz. The system achieves BER $= 10^{-5}$ at $E_b/N_0 = 4.5$ dB. Compute the gap to capacity.

Solution

Shannon limit at $\eta = 3$

$\text{SNR}_{\text{Shannon}} = 2^3 - 1 = 7 = 8.45$ dB.

$E_b/N_0|_{\text{Shannon}} = \text{SNR}/\eta = 7/3 = 2.33 = 3.68$ dB.

Gap computation

$\text{SNR}_{\text{required}} = \eta \times E_b/N_0 = 3 \times 10^{4.5/10} = 3 \times 2.818 = 8.454$

$\Gamma = \text{SNR}_{\text{required}} / (2^\eta - 1) = 8.454 / 7 = 1.208 = 0.82$ dB.

Alternatively: $\Gamma = E_b/N_0|_{\text{actual}} - E_b/N_0|_{\text{Shannon}} = 4.5 - 3.68 = 0.82$ dB.

This LDPC-coded 16-QAM system operates within 0.82 dB of the Shannon limit — excellent performance. $\blacksquare$

Adaptive Modulation and Coding (AMC)

AMC is the practical bridge between information theory and system design. Instead of continuously adapting rate and power (as water-filling prescribes), AMC selects from a discrete set of modulation and coding scheme (MCS) combinations:

Low SNR: QPSK with rate-1/3 coding ( $\eta_{\text{eff}} \approx 0.67$ bits/s/Hz)
Medium SNR: 16-QAM with rate-1/2 coding ( $\eta_{\text{eff}} = 2$ bits/s/Hz)
High SNR: 256-QAM with rate-5/6 coding ( $\eta_{\text{eff}} \approx 6.67$ bits/s/Hz)

The MCS is selected to maximise throughput while keeping the block error rate below a target (typically 10%). The throughput curve of AMC closely tracks the Shannon capacity curve, with a gap determined by the granularity of the MCS table and the code's distance from capacity.

Quick Check

An uncoded 64-QAM system requires SNR $= 26$ dB for BER $= 10^{-5}$ . The spectral efficiency is $\eta = 6$ bits/s/Hz. What is the gap to capacity?

26 dB

$26 - 10\log_{10}(2^6 - 1) = 26 - 17.8 = 8.2$ dB

6 dB

1.59 dB

Correction:

26 - 10\log_{10}(2^6 - 1) = 26 - 17.8 = 8.2

dB

Correct. $\text{SNR}_{\text{Shannon}} = 2^6 - 1 = 63 = 18.0$ dB (more precisely $17.99$ dB). $\Gamma = 26 - 18.0 = 8.0$ dB. This large gap is typical for uncoded modulation.

Why This Matters: AMC in LTE and 5G NR

In LTE, the MCS table contains 29 entries spanning QPSK (rate 0.12) to 64-QAM (rate 0.93), with spectral efficiencies from 0.15 to 5.55 bits/s/Hz. 5G NR extends this to 256-QAM, reaching 7.41 bits/s/Hz.

The scheduler selects the MCS every 1 ms (LTE) or every slot (5G NR, as short as 0.125 ms), based on CQI feedback from the UE. This rapid adaptation means the system operates near the instantaneous capacity most of the time, with the gap determined by the finite MCS granularity and the LDPC/polar code performance.

The throughput of a well-designed AMC system typically achieves 85-95% of the ergodic capacity, with the loss due to:

Finite MCS table (discrete rate adaptation)
CQI feedback delay and quantisation
HARQ retransmission overhead

See full treatment in Adaptive Modulation and Coding in OFDM

🔧Engineering Note

LDPC Decoder Complexity and Throughput in 5G NR

5G NR uses quasi-cyclic LDPC (QC-LDPC) codes with two base graphs: BG1 for large transport blocks (up to 8448 bits) and BG2 for smaller blocks (up to 3840 bits). Key implementation constraints:

Decoder iterations: Typically 6-15 iterations of min-sum or offset min-sum decoding. Each iteration requires reading and updating the full factor graph, consuming significant memory bandwidth.
Throughput target: 5G NR targets up to 20 Gbps downlink. At code rate 8/9 with 256-QAM, this requires the LDPC decoder to process ~2.5 Gbps of coded bits — demanding highly parallel hardware implementations.
Quantisation: Fixed-point decoders use 5-8 bit message quantisation. Aggressive quantisation (4 bits) saves silicon area but degrades performance by 0.1-0.3 dB.
Early termination: Checking syndrome at each iteration allows early stopping. On average, decoders terminate in 3-6 iterations at operating SNR, saving 50-70% of energy.

Practical Constraints

•
5G NR BG1: max info bits 8448, max code rate 8/9
•
Typical decoder: 6-15 iterations, min-sum algorithm
•
Decoder throughput requirement: up to 2.5 Gbps coded bits
•
Quantisation: 5-8 bits LLR, 0.1-0.3 dB loss at 4 bits

📋 Ref: 3GPP TS 38.212 §5.3.2 (LDPC base graphs)

Key Takeaway

Information theory tells us the limit ( $C$ ); coding theory gets us close to it. The gap to capacity $\Gamma$ has shrunk from 8-10 dB (uncoded modulation) to under 1 dB (modern LDPC and polar codes). In practice, adaptive modulation and coding provides a finite-rate approximation to the theoretically optimal water-filling strategy, achieving 85-95% of capacity in current 4G/5G systems.

Why This Matters: Coded Modulation and BICM in the CM Book

The gap to capacity discussed in this section depends critically on how coding and modulation are combined. The Coded Modulation (CM) book develops this in full depth:

Trellis-coded modulation (TCM): Ungerboeck's approach to joint coding and modulation design
Bit-interleaved coded modulation (BICM): the pragmatic alternative used in all modern wireless standards (CommIT contribution: Caire, Taricco, Biglieri, 1998)
Multilevel coding and multistage decoding: optimal capacity-achieving scheme for PAM/QAM constellations
Constellation shaping: approaching the 1.53 dB shaping gap between uniform QAM and Gaussian signalling

Understanding CM is essential for bridging the gap between the information-theoretic capacity derived here and the achievable rates in practical 5G NR systems.

Gap to Capacity

The ratio (in dB) of the SNR required by a practical scheme to the SNR at the Shannon limit for the same spectral efficiency: $\Gamma = \text{SNR}_{\text{req}} / (2^\eta - 1)$ . Modern coded systems achieve $\Gamma < 2$ dB.

Adaptive Modulation and Coding (AMC)

A link adaptation technique that selects the modulation order and code rate based on the instantaneous channel quality. AMC approximates the water-filling capacity by using a finite set of MCS entries. Employed in LTE, 5G NR, and Wi-Fi.

Operational Implications

Definition: Gap to Capacity

Practical Code Families Approaching Capacity

Turbo vs LDPC vs Polar Codes

Example: Gap to Capacity for 16-QAM with LDPC Coding

Shannon limit at $\eta = 3$

Gap computation

Adaptive Modulation and Coding (AMC)

Quick Check

Why This Matters: AMC in LTE and 5G NR

LDPC Decoder Complexity and Throughput in 5G NR

Key Takeaway

Why This Matters: Coded Modulation and BICM in the CM Book

Gap to Capacity

Adaptive Modulation and Coding (AMC)

Definition:
Gap to Capacity