Ferkans — Interactive Telecom Tutor

From Fixed MCS to Adaptive Link

All the previous sections have treated one MCS at a time: given a $(Q_m, R)$ pair, here is the BER curve, here is the peak rate. Real wireless links do not operate at one fixed MCS — they switch continuously, adapting to the instantaneous channel quality. This is adaptive modulation and coding (AMC), and its optimisation is the last missing piece of the BICM-standards picture.

The operational question is: given a channel with time-varying SNR $\gamma(t)$ , which MCS should the scheduler pick at each time $t$ ? The answer is cleanest in the block-fading model — the channel SNR is constant within one transmission block, then changes independently from block to block. In that model the optimal AMC policy picks, for each possible SNR $\gamma$ , the highest-rate MCS whose BLER at $\gamma$ is below target. The throughput envelope then traces a staircase that approaches the Shannon curve as the MCS set becomes dense.

AMC pairs with HARQ: any MCS the scheduler picks may still fail, at which point HARQ-IR retransmissions extend the effective time-diversity at a rate cost. Section 9.1 handled the HARQ throughput formula; this section closes the loop by deriving the AMC throughput-optimality theorem and explaining how the CQI-to-MCS mapping is designed.

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Definition:
Adaptive Modulation and Coding (AMC)

An adaptive modulation and coding (AMC) scheme is a transmission protocol in which the transmitter selects, at each time slot $t$ , an MCS index $i(t) \in \{0, 1, \ldots, N_{\rm MCS} - 1\}$ as a function of a channel-state measurement $\hat\gamma(t)$ received from the receiver via a feedback channel. The transmitted rate at time $t$ is $\eta(i(t)) = Q_{m,i(t)} R_{i(t)}$ bits per 2D symbol.

The feedback mechanism is implementation-specific:

5G NR: the UE measures channel-state information (CSI) reference signals, maps the measurement to one of 16 CQI values (4-bit scalar), and feeds back via PUCCH or PUSCH. The gNB translates CQI to MCS index via a proprietary lookup (typically augmented by outer-loop link adaptation, see below).
Wi-Fi 6/7: the station reports MCS capability and per-MCS acknowledgement history; the AP infers channel quality from ACK/NAK patterns.
DVB-S2 ACM: forward feedback via a separate return channel; the gateway selects MODCOD per terminal.

The feedback channel introduces delay: by the time the transmitter uses CQI $\hat\gamma(t - \tau)$ , the true SNR $\gamma(t)$ may have moved. In fast-fading channels (Doppler $> 50$ Hz) the delay forces conservative MCS selection; this is the "margin" built into the AMC design that we quantify in the pitfall block below.

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Theorem: AMC Throughput Optimality via Rate-Bin Decomposition

Consider a block-fading channel with SNR distribution $p_\Gamma(\gamma)$ , and a finite MCS set $\mathcal{M} = \{(Q_{m,i}, R_i)\}_{i=0}^{N-1}$ with per-MCS BLER function $p_b^{(i)}(\gamma)$ . Let $\mathcal{A}_i \subset \mathbb{R}_+$ be the SNR region assigned to MCS $i$ by an AMC policy, partitioning $\mathbb{R}_+ = \bigcup_i \mathcal{A}_i$ . The policy's throughput (subject to BLER target $\varepsilon$ per MCS) is $\eta_{\rm AMC} = \sum_{i=0}^{N-1} \eta_i \int_{\mathcal{A}_i} (1 - p_b^{(i)}(\gamma)) p_\Gamma(\gamma) \,d\gamma,$ where $\eta_i = Q_{m,i} R_i$ . The throughput-optimal policy picks, for each $\gamma$ , the MCS $i^\star(\gamma) = \arg\max_i \eta_i (1 - p_b^{(i)}(\gamma))$ among those satisfying $p_b^{(i)}(\gamma) \le \varepsilon$ , i.e., one MCS per SNR bin defined by the envelope of the $\eta_i (1 - p_b^{(i)}(\cdot))$ curves.

This is the "greedy per-block" principle: because throughput in each block depends only on that block's SNR, the globally optimal policy is the pointwise-greedy one. There is no "coupling" across blocks that would force joint optimisation. The envelope of the $\eta_i (1 - p_b)$ curves is the Pareto frontier in (throughput, SNR) space; the optimal AMC policy traces the upper envelope. As the MCS set becomes denser, the envelope approaches the ergodic capacity curve.

Show Hint

Separate the throughput integral into contributions per MCS region.

Show that the integrand is maximised pointwise by choosing the highest-throughput MCS at each SNR.

There is no constraint coupling the regions (e.g., no average-power constraint), so pointwise optimisation is globally optimal.

Proof

Step 1: Factor the throughput

$\eta_{\rm AMC}$ as given is a sum of integrals over disjoint regions, where the integrand for MCS $i$ is $\eta_i (1 - p_b^{(i)}(\gamma))$ . Rewriting, $\eta_{\rm AMC} = \int_0^\infty f(\gamma) p_\Gamma(\gamma) \, d\gamma,$ where $f(\gamma) = \eta_{i^\star(\gamma)} (1 - p_b^{(i^\star(\gamma))}(\gamma))$ and $i^\star(\gamma)$ is the MCS chosen at SNR $\gamma$ .

Step 2: Pointwise optimisation

At each $\gamma$ , we want to choose $i^\star(\gamma)$ to maximise the integrand $\eta_i (1 - p_b^{(i)}(\gamma))$ subject to the BLER constraint $p_b^{(i)}(\gamma) \le \varepsilon$ . There is no inter- $\gamma$ constraint (no budget like average power being split across SNR bins), so this pointwise choice is the unique optimum.

Step 3: Envelope interpretation

$f^\star(\gamma) = \max_i [\eta_i (1 - p_b^{(i)}(\gamma))]$ over the admissible set. This is the upper envelope of the family of rate-vs-SNR curves. Each MCS contributes to the envelope over some SNR interval; the ordered boundaries between these intervals define the AMC decision thresholds.

Step 4: Asymptotic Shannon match

If the MCS set $\mathcal{M}$ is dense enough that for every $\gamma$ some MCS achieves $\eta_{i^\star} \approx \log_2(1 + \gamma) - \Delta$ (for some fixed gap $\Delta$ , e.g. 1 dB), then $\eta_{\rm AMC} \to \mathbb{E}_\Gamma[\log_2(1 + \Gamma)] - \Delta$ as the set fills. With 28 MCSs (5G NR) the gap is $\sim 1$ -1.5 dB; with 88 MODCODs (DVB-S2X) the gap shrinks to 0.5 dB. $\blacksquare$

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AMC Throughput Staircase vs Shannon Capacity

The AMC throughput envelope as a function of instantaneous SNR for 5G NR (MCS Table 2), LTE (28 MCSs), and Wi-Fi 6 (11 MCSs). The Shannon curve $\log_2(1 + \text{SNR})$ is the dashed grey bound. The visible "staircase" structure is a direct consequence of the discrete MCS set: each step corresponds to switching from one MCS to the next. Notice that NR's 28-MCS set hugs Shannon to within $\sim 1$ dB across the working SNR range, while Wi-Fi 6's 11-MCS set shows visible $\sim 2$ dB gaps between step boundaries. This is the denseness-versus-capacity trade.

Parameters

Standard

CQI-to-MCS Mapping with Outer-Loop Link Adaptation

Complexity:

O(N_{ m MCS})

per block;

O(1)

outer-loop update

Input: CQI feedback

c \in \{0, 1, \ldots, 15\}

, outer-loop offset

\Delta_{\rm OLLA}

(dB)

Output: MCS index

i \in \{0, \ldots, 27\}

1. Look up target

\text{SNR}_{c}

(the SNR at which CQI

c

was defined, BLER

\le 10^{-1}

)

2.

\gamma_{\rm eff} \leftarrow \text{SNR}_{c} - \Delta_{\rm OLLA}

(adjust for observed ACK/NAK bias)

3.

i^\star \leftarrow \arg\max_i \eta_i

subject to

p_b^{(i)}(\gamma_{\rm eff}) \le \varepsilon

4. return

i^\star

Outer-loop update (after each HARQ process completes):

5. if the block succeeded:

\Delta_{\rm OLLA} \leftarrow \Delta_{\rm OLLA} - \delta_{\rm up}

6. else:

\Delta_{\rm OLLA} \leftarrow \Delta_{\rm OLLA} + \delta_{\rm down}

7. Typical

\delta_{\rm up} / \delta_{\rm down} = \varepsilon / (1 - \varepsilon) \approx 0.1

The outer-loop link-adaptation (OLLA) offset $\Delta_{\rm OLLA}$ is the key piece of engineering that separates a theoretical AMC from a deployable one. CQI feedback is quantised and subject to UE-specific biases (receiver noise figure, implementation margin). Outer-loop ALPHA corrects these by adjusting a per-UE SNR offset so that the long-run BLER matches the target. The Sato (2002) and 3GPP RAN1 tutorials treat this in detail.

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Common Mistake: CQI Feedback Delay in Fast Fading

Mistake:

A naive AMC implementation treats CQI reports as describing the current channel state. In a fast-fading channel (vehicular UEs, Doppler $> 100$ Hz), the channel has moved several decorrelation distances between CQI measurement and MCS use.

Correction:

The CQI delay is typically $5$ - $10$ ms in NR (including measurement, scheduling, and DCI delivery). For pedestrian UEs (Doppler $< 10$ Hz, coherence time $> 100$ ms) this is negligible. For vehicular UEs (60 km/h, Doppler $\approx 300$ Hz at 5 GHz, coherence time $\approx 5$ ms) the CQI is effectively stale — the scheduler should adopt a conservative MCS one or two steps below the CQI-indicated optimum. 3GPP specifies a "CQI offset" parameter per UE class that the gNB applies. Missing this correction causes systematic BLER $> 10\%$ in high-mobility deployments.

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🔧Engineering Note

LTE AMC: The Reference Implementation

LTE introduced the CQI/MCS/TBS framework in Release 8 (2008); NR adopts and extends it. The key design choices that persisted to NR:

CQI is a 4-bit scalar (16 levels). Always a compromise between feedback overhead and AMC resolution. Studies in the 2010s suggested 5-bit CQI would give 0.3 dB more throughput, but the extra PUCCH overhead was deemed not worth it.
Outer-loop link adaptation (OLLA) with a step size of 0.1 dB up / 0.9 dB down for a 10% target BLER. Typical dynamic range of $\Delta_{\rm OLLA}$ is $\pm 3$ dB.
CQI reporting is wideband or sub-band, based on scheduler configuration. Sub-band CQI gives 1-2 dB more throughput in frequency-selective channels but increases feedback.

NR inherits all of the above and adds (i) shorter CQI periodicities, (ii) MCS Table 3 for URLLC (different CQI-MCS mapping at $10^{-5}$ BLER), and (iii) per-layer CQI for rank-adaptive MIMO. The NR technical specification 38.214 §5.2.2 enumerates the CQI tables.

Practical Constraints

•
CQI = 4 bits = 16 levels
•
OLLA step 0.1 up / 0.9 down for 10% BLER target
•
Feedback delay: 4-8 slots in NR numerology
•
Sub-band CQI optional

📋 Ref: 3GPP TS 38.214 §5.2.2

Example: AMC Throughput at SNR 15 dB on NR Table 2

Using NR MCS Table 2, with the following BLER curves (BLER $= 10^{-1}$ thresholds): $\text{SNR}_{7} = 8$ dB for MCS 7 (16-QAM, $R \approx 0.5$ , $\eta = 2.04$ ), $\text{SNR}_{13} = 13.5$ dB for MCS 13 (64-QAM, $R \approx 0.5$ , $\eta = 3$ ), $\text{SNR}_{20} = 19.2$ dB for MCS 20 (64-QAM, $R \approx 0.6$ , $\eta = 3.61$ ), $\text{SNR}_{22} = 22.5$ dB for MCS 22 (256-QAM, $R \approx 0.5$ , $\eta = 3.94$ ). Compute the AMC-selected MCS and its effective throughput at $\gamma = 15$ dB.

Solution

Identify the admissible MCS set at 15 dB

MCS $i$ is admissible at $\gamma$ if $p_b^{(i)}(\gamma) \le 0.1$ , i.e., $\text{SNR}_{i} \le \gamma$ . At $\gamma = 15$ dB:

MCS 7 ( $\text{SNR}_{7} = 8$ dB): admissible, $\eta = 2.04$ .
MCS 13 ( $\text{SNR}_{13} = 13.5$ dB): admissible, $\eta = 3.0$ .
MCS 20 ( $\text{SNR}_{20} = 19.2$ dB): NOT admissible (15 < 19.2).
MCS 22 ( $\text{SNR}_{22} = 22.5$ dB): NOT admissible.

Select the highest-rate admissible MCS

Maximum $\eta$ over admissible MCSs is $\eta_{13} = 3.0$ . Selected MCS: 13.

Compute effective throughput

$\eta_{\rm AMC}(\gamma = 15 \text{ dB}) = 3.0 \times (1 - 0.1) = 2.7$ info bits per 2D symbol. The 10% BLER bleed is the "channel margin" budgeted into the system.

Compare to Shannon

At $\text{SNR} = 15$ dB = 31.6 linear, $\log_2(1 + 31.6) = 5.03$ bits/2D. The BICM-staircase gap is $5.03 - 2.7 = 2.33$ bits/2D — large because MCS 20 sits just above 15 dB. This SNR is exactly at a staircase edge; 16 dB would jump straight to MCS 20 and close the gap. $\blacksquare$

Why This Matters: AMC and HARQ Form a Two-Layer Link Adaptation

AMC and HARQ are not independent mechanisms — they compose. A rate-1/2 transmission that fails HARQ-IR once has effective rate 1/4 (new bits halve the rate). The AMC-then-HARQ pair can therefore cover an SNR range exceeding the natural range of the MCS set.

Operationally this means: if the scheduler picks MCS 18 (64-QAM 4/5, SNR threshold 18 dB), a single failed transmission followed by a HARQ-IR retransmission gives effective rate (64-QAM 2/5), needing about 12 dB SNR to succeed. So the MCS 18 + HARQ process works at any SNR above 12 dB, with the spectral efficiency decaying smoothly from $\eta_{18}$ at 18 dB down to $\eta_{18}/2$ at 12 dB. HARQ thus "smooths" the AMC staircase, partially filling the gaps visible in the interactive plot above.

In Chapter 14 we analyse the MIMO version of this composition — the ARQ-diversity-multiplexing tradeoff of El Gamal-Caire-Damen (2006) — where each HARQ retransmission adds both diversity and multiplexing gain.

Quick Check

The AMC throughput theorem says the optimal MCS at each SNR is chosen pointwise (independently across SNR bins). Why does this not require joint optimisation across bins?

There is no budget (like average power) that couples the different SNR regions

All MCSs have the same average power

The Lagrangian simplifies to a single envelope

The bin widths are infinitesimal

Correction:

There is no budget (like average power) that couples the different SNR regions

Without a coupling constraint, each SNR bin can be optimised independently — the greedy envelope is optimal. If a power budget were added (e.g., water-filling over SNR), joint optimisation would be required.

Adaptive Modulation and Coding (AMC)

A link-adaptation scheme in which the transmitter selects an MCS based on feedback from the receiver. Optimal under block-fading: pick the highest-rate MCS that meets the BLER target at the current SNR. Implemented in every modern cellular and Wi-Fi standard.

Channel Quality Indicator (CQI)

A 4-bit scalar in LTE/5G NR that the UE feeds back to the gNB to report the highest MCS index that would give BLER $\le 10^{-1}$ at the current channel quality. Used as the primary input to the CQI-to-MCS mapping table and to outer-loop link adaptation.

Block Error Rate (BLER)

The fraction of transport blocks not decoded successfully after all HARQ retransmissions. Target values: $10^{-1}$ for eMBB (throughput- oriented), $10^{-5}$ for URLLC (reliability-oriented), $10^{-11}$ for DVB-S2 (quasi-error-free broadcast).

Key Takeaway

AMC selects the throughput-optimal MCS per SNR bin. The optimality argument is simple pointwise optimisation because there is no cross-bin constraint. The resulting throughput envelope traces a staircase that approaches Shannon as the MCS set becomes dense. HARQ-IR composes with AMC to smooth the staircase — a failed MCS $i$ plus one retransmission approximates MCS $i$ at half the rate. Feedback delay (for CQI) is the main practical limitation in fast-fading channels.

Adaptive Modulation and Coding, HARQ