AMP-A-EC: AMP for Grant-Free OFDM Access

• AMP-A-EC is an approximate-message-passing algorithm that performs MAP-based device activity detection and MMSE effective channel estimation using an exact Bayesian time-domain model.
• It employs a factor graph to capture the Bernoulli-Gaussian coupling between device activity and channel taps in OFDM-based panels under frequency-selective fading.
• The algorithm uses best-iterate tracking to counter non-monotonic convergence, yielding improved error probability and MSE performance in pilot-limited regimes.

AMP-A-EC is an approximate-message-passing algorithm for OFDM-based massive grant-free access in wideband systems under frequency-selective fading. Introduced as AMP-based Activity detection and Effective Channel estimation, it is derived from an exact time-domain Bayesian model and a corresponding factor graph that preserves the Bernoulli-Gaussian coupling between device activity and channel coefficients. Its purpose is joint MAP-based device activity detection and MMSE-based effective-channel estimation, with an additional best-iterate tracking mechanism to mitigate the non-monotonic convergence behavior that can arise when pilot length is smaller than or comparable to the number of active devices (Li et al., 5 Aug 2025).

1. Definition and problem setting

AMP-A-EC is formulated for a single-cell uplink in which a base station has $M$ antennas and $N$ single-antenna devices. The channel is block fading and frequency-selective with $P$ taps. Device activity is binary, $a_n\in\{0,1\}$, with sparse activity satisfying $\sum_{n\in\mathcal N} a_n=N_a\ll N$. Large-scale fading coefficients $\beta_n>0$ are assumed known at the base station, and the small-scale fading taps satisfy

$$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$

The effective channel variable is defined by

$x_{n,p,m}=a_n h_{n,p,m},$

so that $x_{n,p,m}$ follows a Bernoulli-Gaussian law conditioned on $a_n$ (Li et al., 5 Aug 2025).

The pilot structure is OFDM-based. The number of subcarriers is $N$0, the pilot length is $N$1, and each device has a unique length-$N$2 pilot split across $N$3 OFDM pilot symbols. After DFT-domain manipulation and stacking across OFDM symbols and receive antennas, the received signal is written in compact form as

$N$4

with $N$5, $N$6, and $N$7. In this model, the block rows of $N$8 encode the activity-weighted channel taps of each device. For large $N$9, the sensing matrix obeys

$P$0

which is the asymptotic scaling used by the algorithmic derivation (Li et al., 5 Aug 2025).

The paper places AMP-A-EC alongside a second algorithm, AMP-A-AC, but the two solve different estimation targets. AMP-A-EC is specifically the method for activity detection and effective channel estimation, whereas AMP-A-AC addresses activity detection and actual-channel estimation of active devices.

2. Bayesian formulation and factor-graph structure

The method is built from three Bayesian estimation problems. The first is MAP activity detection: $P$1 with decision rule

$P$2

The second is MMSE estimation of the effective channel: $P$3

The third, introduced for comparison in the same framework, is MMSE estimation of the actual channel of active devices: $P$4 AMP-A-EC targets the first two of these (Li et al., 5 Aug 2025).

The joint distribution factorizes as

$P$5

with Gaussian likelihood

$P$6

This factorization induces a factor graph that explicitly captures the likelihood factors $P$7, the Bernoulli priors $P$8, and the Bernoulli-Gaussian coupling $P$9. A key structural property is that, for fixed $a_n\in\{0,1\}$0, all $a_n\in\{0,1\}$1 share the same $a_n\in\{0,1\}$2, so they are dependent unconditionally but independent conditioned on $a_n\in\{0,1\}$3. Rows of $a_n\in\{0,1\}$4 are independent across $a_n\in\{0,1\}$5. This graph is the basis for message passing toward $a_n\in\{0,1\}$6 and $a_n\in\{0,1\}$7 (Li et al., 5 Aug 2025).

3. Algorithmic structure of AMP-A-EC

AMP-A-EC applies AMP-style Gaussian approximations to the factor-graph messages and then performs parallel updates of residuals, activity scores, effective-channel estimates, and variance surrogates. The paper introduces the denoising function

$a_n\in\{0,1\}$8

together with its derivative $a_n\in\{0,1\}$9. This function couples the current pseudo-observation, noise level, and activity probability proxy (Li et al., 5 Aug 2025).

The AMP residual and residual-energy estimator are

$\sum_{n\in\mathcal N} a_n=N_a\ll N$0

The activity-related soft factor is

$\sum_{n\in\mathcal N} a_n=N_a\ll N$1

where

$\sum_{n\in\mathcal N} a_n=N_a\ll N$2

Under the Gaussian-message approximation, the approximate posteriors are written as

$\sum_{n\in\mathcal N} a_n=N_a\ll N$3

$\sum_{n\in\mathcal N} a_n=N_a\ll N$4

and

$\sum_{n\in\mathcal N} a_n=N_a\ll N$5

The associated estimators are

$\sum_{n\in\mathcal N} a_n=N_a\ll N$6

with

$\sum_{n\in\mathcal N} a_n=N_a\ll N$7

For large $\sum_{n\in\mathcal N} a_n=N_a\ll N$8 and $\sum_{n\in\mathcal N} a_n=N_a\ll N$9, the algorithm further uses

$\beta_n>0$0

This leads to a fully explicit AMP recursion, including

$\beta_n>0$1

and the Onsager-corrected residual update

$\beta_n>0$2

This places AMP-A-EC in the standard AMP family in the sense that it uses a residual update with an Onsager-style correction, although its denoising and activity-coupling terms are tailored to the Bernoulli-Gaussian wideband access model (Li et al., 5 Aug 2025).

4. Best-iterate tracking and convergence behavior

A distinctive feature of AMP-A-EC is that it does not simply return the final iterate. The paper emphasizes that AMP may fail to converge monotonically when the pilot length $\beta_n>0$3 is smaller than or comparable to the number of active devices. In that regime, the error probability or MSE can decrease first and then increase. AMP-A-EC addresses this by tracking the iterate that minimizes the GROUP-LASSO objective

$\beta_n>0$4

and defining

$\beta_n>0$5

The corresponding best iterate $\beta_n>0$6 and activity scores $\beta_n>0$7 are used for the final output: $\beta_n>0$8

The paper attributes the improvement to three elements: retention of AMP-style low-complexity parallel updates, best-iterate tracking using the GROUP-LASSO objective, and outputting the best past estimate rather than the last one. Reported qualitative consequences are that error probability and MSE first decrease and then remain unchanged, and that the gain is especially visible at $\beta_n>0$9 and $$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$0, where $$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$1 or only slightly exceeds it (Li et al., 5 Aug 2025).

A common misconception is to treat AMP-A-EC as a generic name for any expectation-consistent or orthogonalized AMP method. In the current arXiv literature, AMP-A-EC is a specific algorithm for grant-free wideband access, whereas broader AMP papers on first-order cancellation, EP/GAMP equivalence, or rotationally invariant models describe related methodology but do not define this algorithm by name (Schniter, 2019, Liu et al., 2019, Liu et al., 2024). This suggests that the suffix “A-EC” should be read operationally within this communications context rather than as a universal AMP taxonomy.

5. State evolution and asymptotic characterization

The paper analyzes AMP-A-EC through a scalar state-evolution formalism. The effective observation model is

$$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$2

where $$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$3, $$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$4, and $$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$5 and $$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$6 are independent. The recursion is

$$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$7

The term $$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$8 is

$$g_{n,p,m}\sim \mathcal{CN}(0,1), \qquad h_{n,p,m}=\sqrt{\beta_n}\,g_{n,p,m}.$$9

with

$x_{n,p,m}=a_n h_{n,p,m},$0

For activity detection, the error probability at iteration $x_{n,p,m}=a_n h_{n,p,m},$1 is

$x_{n,p,m}=a_n h_{n,p,m},$2

The paper gives a closed form involving incomplete Gamma functions,

$x_{n,p,m}=a_n h_{n,p,m},$3

where the thresholds are explicit functions of $x_{n,p,m}=a_n h_{n,p,m},$4. The qualitative consequences stated in the paper are that error probability decreases with $x_{n,p,m}=a_n h_{n,p,m},$5 and $x_{n,p,m}=a_n h_{n,p,m},$6, and that $x_{n,p,m}=a_n h_{n,p,m},$7 as $x_{n,p,m}=a_n h_{n,p,m},$8 (Li et al., 5 Aug 2025).

For active-device channel estimation, the normalized MSE

$x_{n,p,m}=a_n h_{n,p,m},$9

is characterized analytically as a function of $x_{n,p,m}$0, $x_{n,p,m}$1, and $x_{n,p,m}$2. The paper states that this MSE decreases as state evolution improves. It also reports that the analytical state-evolution predictions for AMP-A-EC match simulation closely.

6. Computational complexity, numerical results, and positioning

The dominant per-iteration flop counts for AMP-A-EC are reported stepwise. The dominant term is

$x_{n,p,m}$3

yielding overall complexity

$x_{n,p,m}$4

The paper states that AMP-A-EC is more complex than AMP-A-AC because AMP-A-EC computes $x_{n,p,m}$5, $x_{n,p,m}$6, and the more detailed $x_{n,p,m}$7 for every $x_{n,p,m}$8. Comparative orders given in the paper are: ML-MMSE has $x_{n,p,m}$9; AMP-FS and OMP-ext. have $a_n$0; and AMP-FL-ext., AMP-A-EC, and AMP-A-AC have $a_n$1 (Li et al., 5 Aug 2025).

The simulation setup includes $a_n$2, subcarrier spacing $a_n$3 kHz, noise PSD $a_n$4 dBm/Hz with $a_n$5, activity probability $a_n$6, and typical defaults

$a_n$7

Pilots are i.i.d. $a_n$8 and normalized so that $a_n$9. The experiments use 500 Monte Carlo realizations.

The paper reports that AMP-A-EC and AMP-A-AC significantly outperform AMP-FL-ext, AMP-FS, and OMP-ext, with up to 94% reduction in error probability, up to 33% reduction in MSE, and up to 96% reduction in computation time versus ML-MMSE. It also reports several monotonic trends: increasing $N$00 improves error probability, false alarm, missed detection, and MSE; increasing $N$01 improves all estimation metrics; increasing $N$02 or $N$03 worsens performance; increasing SNR improves performance; and computation time grows with $N$04, $N$05, and $N$06 but is essentially independent of SNR (Li et al., 5 Aug 2025).

The paper distinguishes the preferable operating regions of the two proposed algorithms. AMP-A-EC is preferable at small $N$07 and small $N$08. AMP-A-AC is preferable at large $N$09 and large $N$10, and has lower dominant complexity. AMP-A-EC is described as somewhat more accurate in some small-dimension regimes, while AMP-A-AC can be faster and better in larger regimes. A plausible implication is that the two algorithms are best viewed as complementary solvers within a shared Bayesian and factor-graph framework rather than as strict substitutes.

7. Relation to the broader AMP literature

AMP-A-EC belongs to the family of approximate-message-passing algorithms in that it uses Gaussian approximations, scalar denoising structure, and an Onsager-corrected residual update. In the broader AMP literature, standard linear-regression AMP is often motivated by the Onsager term’s cancellation of leading self-interaction, yielding an asymptotically Gaussian effective noise characterized by state evolution (Schniter, 2019). EP-based derivations likewise connect AMP and GAMP to a unified message-passing rule under large-system approximations, especially for AWGN measurement channels (Liu et al., 2019).

However, AMP-A-EC is not a generic statement about AMP theory. It is a communications-specific construction for OFDM-based grant-free access under frequency-selective fading, based on an exact time-domain signal model, a factor graph that couples activity and channel taps, and a best-iterate strategy added to address practical non-monotonicity. By contrast, recent rotationally invariant AMP frameworks derive long-memory Onsager terms from free cumulants and OAMP reductions, but do not define an algorithm literally named AMP-A-EC (Liu et al., 2024). The nomenclature therefore identifies a particular algorithmic instance rather than a general AMP subclass.

Within that narrower meaning, AMP-A-EC’s significance lies in combining four elements in a single procedure: exact wideband modeling, MAP activity detection, MMSE effective-channel estimation, and best-iterate tracking under challenging pilot-limited regimes. The paper’s central conclusion is that this combination yields strong accuracy-complexity tradeoffs for massive grant-free access, especially when pilot length is limited and conventional AMP behavior is least reliable (Li et al., 5 Aug 2025).