AMP-A-AC: Joint Activity Detection & Channel Estimation

• The paper introduces a per-device MAP activity detection method combined with a conditional MMSE estimator for actual channel estimation.
• AMP-A-AC uses a factor graph with Bernoulli and Gaussian approximations to coordinate estimates across channel taps and multiple antennas.
• Empirical results demonstrate up to 94% error reduction in activity detection, 33% lower MSE, and 96% reduced computation time compared to benchmarks.

AMP-A-AC is an approximate-message-passing algorithm for joint device activity detection and actual time-domain channel estimation in OFDM-based wideband massive grant-free access under frequency-selective fading (Li et al., 5 Aug 2025). It is derived from an exact time-domain probabilistic model and a factor graph that couples device activity with all channel taps and antennas, and it is designed to approximate a per-device MAP activity detector together with a conditional MMSE estimator of the actual channel given activity (Li et al., 5 Aug 2025). The term should not be conflated with AMAPMT, the “Advanced Multiple Access Protocol for Multimedia Transmission” proposed for wireless multimedia MAC scheduling, which is an unrelated data-link-layer protocol (Yu, 2012).

1. Definition and inferential objective

AMP-A-AC is introduced alongside AMP-A-EC in “AMP-based Joint Activity Detection and Channel Estimation for Massive Grant-Free Access in OFDM-based Wideband Systems” (Li et al., 5 Aug 2025). The suffix “A-AC” denotes activity + actual channel, in contrast to AMP-A-EC, which targets activity + effective channel estimation. That distinction is central: AMP-A-AC does not estimate the Bernoulli-masked variable $x_{n,p,m}=a_n h_{n,p,m}$ as its primary output, but instead targets the conditional MMSE estimator

$$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$

The inferential program consists of two coupled tasks. The first is per-device MAP activity detection through the posterior $p(a_n\mid \mathbf Y)$, or equivalently the log-likelihood ratio

$$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$

The second is MMSE actual-channel estimation conditioned on $a_n=1$. The paper states that the MAP-based device activity detection problem and one of the MMSE-based channel estimation problems are formulated for the first time (Li et al., 5 Aug 2025).

This framing implies a stricter inferential target than standard sparse recovery on effective channels. The activity variable is device-level, while the channel unknowns are tap- and antenna-resolved. AMP-A-AC therefore uses a device-level activity belief to coordinate estimates across all $P$ taps and $M$ antennas of the same device, rather than treating those coefficients as independent latent variables.

2. System model and statistical structure

The algorithm is posed for a single-cell uplink with a base station carrying $M$ antennas and $N$ single-antenna IoT devices in a wideband OFDM grant-free access system under frequency-selective fading (Li et al., 5 Aug 2025). The channel for device $n$, tap $$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$0, and BS antenna $$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$1 is

$$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$2

with known large-scale fading power $$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$3. Device activity is modeled as

$$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$4

and the effective channel taps are

$$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$5

The OFDM pilot structure uses $$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$6 subcarriers and pilot length $$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$7, with $$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$8 OFDM pilot symbols. After cyclic-prefix removal and Fourier-domain manipulation, the received pilot observations are written in exact time-domain linear form as

$$\hat h_{n,p,m}^\star(\mathbf Y)=\int h_{n,p,m}\,p(h_{n,p,m}\mid \mathbf Y,a_n=1)\,dh_{n,p,m}.$$9

Here $p(a_n\mid \mathbf Y)$0, $p(a_n\mid \mathbf Y)$1, $p(a_n\mid \mathbf Y)$2, and $p(a_n\mid \mathbf Y)$3 is AWGN (Li et al., 5 Aug 2025). Element-wise,

$p(a_n\mid \mathbf Y)$4

The prior structure is Bernoulli–Gaussian at the effective-channel level: $p(a_n\mid \mathbf Y)$5 Conditioned on $p(a_n\mid \mathbf Y)$6, the variables $p(a_n\mid \mathbf Y)$7 are independent, but marginally they are dependent across taps and antennas because they share the same activity indicator $p(a_n\mid \mathbf Y)$8 (Li et al., 5 Aug 2025). This shared-activity coupling is the main statistical feature exploited by AMP-A-AC.

3. Factor graph and AMP approximation

The joint distribution factors as

$p(a_n\mid \mathbf Y)$9

which induces a tripartite factor graph $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$0 (Li et al., 5 Aug 2025). The exact posterior marginals required for MAP detection and MMSE channel estimation are high-dimensional and computationally intractable.

AMP-A-AC arises by replacing exact sum–product messages with tractable parametric approximations. Messages associated with $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$1 are approximated as complex Gaussian, while messages associated with $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$2 are approximated as Bernoulli. Likelihood-to-variable messages are approximated through central-limit arguments, and the product of Gaussian likelihood messages yields Gaussian messages of the form

$$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$3

The estimator that differentiates AMP-A-AC from AMP-A-EC is the conditional MMSE estimator for the actual channel: $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$4 AMP-A-EC instead denoises the effective channel $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$5 directly. AMP-A-AC compresses the activity inference from $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$6 activity parameters to a single device-level activity parameter $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$7, while retaining per-tap, per-antenna channel estimates $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$8 (Li et al., 5 Aug 2025). This suggests a more explicit exploitation of the shared device activity structure than an effective-channel formulation.

4. Iterative recursion and decision mechanism

The operational AMP-A-AC recursion is given in simplified form by a sequence of residual, activity, and channel updates (Li et al., 5 Aug 2025). For each iteration $$\theta_n=\log \frac{p(a_n=1\mid \mathbf Y)}{p(a_n=0\mid \mathbf Y)}.$$9, the per-antenna residual variance is

$a_n=1$0

Device activity is represented through a device-level probability

$a_n=1$1

where $a_n=1$2 is an approximate LLR assembled from all taps and antennas of device $a_n=1$3: $a_n=1$4

The actual channel update is

$a_n=1$5

The residual is then updated with an Onsager term: $a_n=1$6

The activity hard decision is made through

$a_n=1$7

The paper also introduces best-iterate tracking through the surrogate objective

$a_n=1$8

with $a_n=1$9 built from $P$0, and retains the iterate with minimal $P$1 so far (Li et al., 5 Aug 2025). This is used to mitigate non-monotonic MSE behavior and the lack of convergence guarantees.

5. Computational profile, state evolution, and empirical operating regions

AMP-A-AC has dominant per-iteration complexity

$P$2

and asymptotic order $P$3 (Li et al., 5 Aug 2025). The corresponding dominant term for AMP-A-EC is

$P$4

so AMP-A-AC has lower per-iteration complexity and fewer scalar activity parameters than AMP-A-EC.

A notable theoretical distinction is that AMP-A-EC admits state-evolution analysis, whereas AMP-A-AC does not fit the standard SE framework because of its modified residual definition and the Onsager term tied to $P$5 (Li et al., 5 Aug 2025). The paper therefore evaluates AMP-A-AC empirically rather than through a closed-form SE characterization.

The numerical study compares against AMP-FL-ext, AMP-FS, OMP-based sparse recovery, and ML-MMSE, using activity detection error probability, false alarm and missed detection, channel estimation MSE, and computation time (Li et al., 5 Aug 2025). The reported findings are:

• AMP-A-EC and AMP-A-AC significantly outperform AMP-FL-ext, AMP-FS, and OMP-ext in activity detection and channel MSE.
• The paper reports reductions up to 94% in error probability and 33% in MSE relative to existing AMP-based and OMP-based schemes.
• Compared to ML-MMSE, AMP-A-AC achieves essentially comparable accuracy in both detection and channel estimation, but with much lower complexity, approximately 96% less computation time.

The preferred operating regions are differentiated rather than universal. AMP-A-EC is stated to be slightly better in accuracy when pilot length $P$6 is short or slightly above $P$7, or when the number of antennas $P$8 is small. AMP-A-AC is stated to be slightly better in accuracy and clearly better in complexity when $P$9 is larger and $M$0 is larger, which makes it more favorable in massive MIMO settings with longer pilot budgets (Li et al., 5 Aug 2025).

6. Relation to AMP theory, assumptions, and recurrent sources of confusion

AMP-A-AC inherits the standard AMP reliance on Gaussian approximations and Onsager-style residual correction. In the broader AMP literature, the Onsager term is used to cancel first-order correlations induced by repeated application of the same random matrix, so that the denoiser input behaves like an effective Gaussian observation under large i.i.d. matrix assumptions (Schniter, 2019). AMP-A-AC follows that general design pattern but departs from classical AMP in a way that prevents standard Bayati–Montanari-style state evolution, which is why its analysis remains simulation-based rather than SE-based (Li et al., 5 Aug 2025).

This places AMP-A-AC within a broader line of AMP-based grant-free access and sparse activity inference. Earlier work used AMP to detect device activity and embedded information bits in massive MIMO random access (Senel et al., 2017), and later work incorporated spatially correlated channels into AMP-based joint user identification and channel estimation for mMTC (Djelouat et al., 2021). This suggests that AMP-A-AC extends the same methodological family to an exact time-domain OFDM model under frequency-selective fading, with the additional distinction between effective-channel and actual-channel estimation.

Several interpretive issues recur in discussions of the name. First, AMP-A-AC is not the multimedia MAC protocol of the 2012 paper “A Multiple Access Protocol for Multimedia Transmission over Wireless Networks”; that work concerns AMAPMT, a data-link-layer scheduling protocol based on TTL, traffic priority, and CSI, and the query string “AMP-A-AC” is explicitly stated not to appear there (Yu, 2012). Second, AMP-A-AC is not the effective-channel algorithm AMP-A-EC; the two are paired in the 2025 wideband grant-free access paper but target different MMSE estimators (Li et al., 5 Aug 2025).

The main stated limitations are also specific. AMP-A-AC relies on large-system approximations with large $M$1 and $M$2, small i.i.d. entries $M$3, Gaussian channel priors, and Bernoulli activity. It assumes known $M$4 and $M$5. No rigorous SE-based performance guarantees are available, and although the algorithm targets MAP/MMSE quantities, its optimality is approximate rather than exact (Li et al., 5 Aug 2025).