Ferkans — Interactive Telecom Tutor

Connecting Billions of Devices

Massive Machine-Type Communication (mMTC) addresses a fundamentally different traffic pattern than eMBB or URLLC. The defining characteristics are:

Massive number of devices: $K_{\mathrm{total}} \sim 10^4$ -- $10^6$ devices per cell, each transmitting infrequently.
Sporadic activity: in any given slot, only a small fraction $p_a \ll 1$ of devices are active, so $K_a = p_a K_{\mathrm{total}} \ll K_{\mathrm{total}}$ .
Small payloads: typically 20--200 bytes per transmission (sensor readings, status updates).
Relaxed latency: delays of seconds to minutes are acceptable.
Energy constraints: devices are battery-powered and must minimise transmission energy.

The classical grant-based access (4G LTE) requires a four-step handshake (scheduling request, grant, data, ACK) that is prohibitively expensive for mMTC: the signalling overhead exceeds the payload, and the random access preamble space limits the number of simultaneously active devices.

This section develops the grant-free access paradigm and the compressed sensing framework for joint activity detection and data recovery.

Definition:
Grant-Free Random Access

In grant-free (also called schedule-free or configured-grant) access, each device is pre-assigned a pilot sequence (preamble) and transmits without waiting for an explicit scheduling grant from the base station. The protocol operates in two phases:

Pilot phase: active device $k$ transmits its assigned pilot sequence $\mathbf{a}_k \in \mathbb{C}^{L_p}$ over $L_p$ resource elements.
Data phase: active device $k$ transmits its data payload using the remaining resources.

The base station must jointly: (a) detect which devices are active (activity detection), and (b) decode the data from active devices (data recovery).

The key advantage is the elimination of the grant handshake, reducing latency and signalling overhead.

Grant-Free vs Grant-Based Access — Comparison of grant-based (left) and grant-free (right) access protocols. Grant-based access requires a four-step handshake (SR $\to$ Grant $\to$ Data $\to$ ACK), consuming significant time and signalling resources. Grant-free access eliminates the handshake: the device transmits pilot + data immediately upon packet arrival, and the base station performs joint activity detection and data decoding.

Definition:
Compressed Sensing Activity Detection Model

Consider $K_{\mathrm{total}}$ devices, each assigned a unique pilot sequence $\mathbf{a}_k \in \mathbb{C}^{L_p}$ . The received pilot signal at the base station (with $M$ antennas) is

$\mathbf{Y} = \mathbf{A} \mathbf{X} + \mathbf{W},$

where:

$\mathbf{A} = [\mathbf{a}_1, \ldots, \mathbf{a}_{K_{\mathrm{total}}}] \in \mathbb{C}^{L_p \times K_{\mathrm{total}}}$ is the pilot matrix,
$\mathbf{X} = [\mathbf{x}_1, \ldots, \mathbf{x}_{K_{\mathrm{total}}}]^T \in \mathbb{C}^{K_{\mathrm{total}} \times M}$ , with $\mathbf{x}_k = \alpha_k \mathbf{h}_k^T$ if device $k$ is active ( $\alpha_k \neq 0$ ) and $\mathbf{x}_k = \mathbf{0}$ otherwise,
$\mathbf{W} \in \mathbb{C}^{L_p \times M}$ is AWGN noise.

Since only $K_a \ll K_{\mathrm{total}}$ devices are active, $\mathbf{X}$ is row-sparse (has only $K_a$ nonzero rows). Activity detection reduces to recovering the support of $\mathbf{X}$ from the compressed measurement $\mathbf{Y}$ .

This is a multiple measurement vector (MMV) compressed sensing problem when $M > 1$ . The $M$ receive antennas provide independent observations of the same sparsity pattern, improving detection performance.

,

Theorem: Pilot Length Scaling Law for Activity Detection

Approximate Message Passing (AMP) for Activity Detection

Input: Pilot matrix

\mathbf{A} \in \mathbb{C}^{L_p \times K}

;

received signal

\mathbf{y} \in \mathbb{C}^{L_p}

(single antenna

for clarity); activity probability

p_a

; noise variance

\sigma^2

;

max iterations

T

Output: Activity indicators

\hat{\alpha}_k \in \{0, 1\}

,

k = 1, \ldots, K

1. Initialise:

\hat{\mathbf{x}}^{(0)} = \mathbf{0}

,

\mathbf{r}^{(0)} = \mathbf{y}

2. For

t = 0, 1, \ldots, T-1

:

a. Effective observation for each device:

\mathbf{z}^{(t)} = \mathbf{A}^H \mathbf{r}^{(t)} + \hat{\mathbf{x}}^{(t)}

b. State evolution: compute effective noise variance

\tau^{(t)} = \sigma^2 + \frac{1}{L_p}\|\hat{\mathbf{x}}^{(t)}\|_0 \cdot \sigma_x^2

(or use the state evolution recursion)

c. Denoising (MMSE under Bernoulli-Gaussian prior):

\hat{x}_k^{(t+1)} = \eta_t(z_k^{(t)})

, where

\eta_t(z) = \frac{p_a \cdot \mathcal{CN}(z; 0, \tau^{(t)} + \sigma_x^2)}{p_a \cdot \mathcal{CN}(z; 0, \tau^{(t)} + \sigma_x^2) + (1-p_a) \cdot \mathcal{CN}(z; 0, \tau^{(t)})} \cdot \frac{\sigma_x^2}{\tau^{(t)} + \sigma_x^2} \cdot z

d. Onsager correction and residual update:

\mathbf{r}^{(t+1)} = \mathbf{y} - \mathbf{A}\hat{\mathbf{x}}^{(t+1)} + \frac{1}{L_p}\mathbf{r}^{(t)} \cdot \sum_k \eta_t'(z_k^{(t)})

3. Threshold: declare device

k

active if

|\hat{x}_k^{(T)}| > \gamma_{\mathrm{th}}

(set to achieve

desired false alarm rate)

CS-Based Activity Detection Performance

Visualise the detection probability and false alarm rate as a function of SNR for compressed sensing activity detection. Adjust the total number of devices, the active fraction, and the pilot length to observe the fundamental trade-offs. The plot shows that (i) longer pilots improve detection, (ii) smaller active fractions are easier to detect, and (iii) there is a sharp phase transition in SNR below which detection fails.

Parameters

Total devices

K_{\mathrm{total}}

500

Active fraction

p_a

0.05

Pilot length

L_p

(0 = auto)0

From ALOHA to Modern Random Access

The classical ALOHA protocol and its slotted variant achieve a maximum throughput of $1/(2e)$ and $1/e$ packets per slot, respectively. Modern mMTC demands much higher throughput from grant-free access. Key advances include:

Irregular Repetition Slotted ALOHA (IRSA): each user transmits multiple replicas in randomly chosen slots. Pointers between replicas enable successive interference cancellation (SIC), analogous to iterative decoding of LDPC codes on a bipartite graph. Throughput exceeds 0.8 packets/slot.
Coded Slotted ALOHA (CSA): generalises IRSA by replacing repetition with local coding across slots, approaching the theoretical limit of 1 packet/slot.
Non-Orthogonal Multiple Access (NOMA): allows multiple users to share the same resource element with different power levels or spreading codes, resolved via SIC at the receiver.

The connection to codes on graphs provides a rigorous framework for optimising the replica distribution via density evolution.

Definition:
Unsourced Random Access

In the unsourced random access (URA) model introduced by Polyanskiy (2017), all $K_a$ active users share a common codebook $\mathcal{C} = \{\mathbf{c}(w) : w \in \{1,\ldots,2^B\}\}$ , where $B$ is the message size in bits. The receiver outputs a list $\hat{\mathcal{L}}$ of decoded messages (without user identities). The per-user probability of error (PUPE) is

$P_e = \frac{1}{K_a} \sum_{k=1}^{K_a} \Pr\!\big[W_k \notin \hat{\mathcal{L}}\big].$

This formulation is appropriate for mMTC because:

Device identity is typically embedded in the message (MAC address), not in the codebook.
A common codebook eliminates the need for per-device pilot assignment, avoiding the pilot exhaustion problem.
The performance metric $P_e$ is symmetric across users.

The random coding achievability bound shows that PUPE $\leq \varepsilon$ is achievable with energy-per-bit

$\frac{E_b}{N_0} \geq \frac{2^{B/n} - 1}{B/n} + O\!\left(\frac{K_a}{n}\right)$

where the second term captures the multi-user penalty.

,

Unsourced Random Access: PUPE vs $E_b/N_0$

Compare the per-user probability of error (PUPE) for unsourced random access as a function of $E_b/N_0$ . Adjust the number of active users $K_a$ , the blocklength $n$ , and the message size $B$ . The plot shows both the achievability bound and the performance of practical coded compressed sensing schemes. Observe that the gap to the bound increases with $K_a$ and decreases with $n$ .

Parameters

Active users

K_a

50

Blocklength

n

30000

Message bits

B

100

Quick Check

An mMTC cell has $K_{\mathrm{total}} = 10{,}000$ devices with activity probability $p_a = 0.01$ (so $K_a = 100$ expected active devices). According to the compressed sensing scaling law, the pilot length $L_p$ must scale as $\Omega(K_a \log(K_{\mathrm{total}}/K_a))$ . What is the approximate minimum pilot length?

$L_p \approx 100 \cdot \log_2(100) \approx 664$

$L_p \approx K_{\mathrm{total}} = 10{,}000$ (one pilot per device)

$L_p \approx K_a = 100$

$L_p \approx K_a^2 = 10{,}000$

Correction:

L_p \approx 100 \cdot \log_2(100) \approx 664

$K_a = 100$ and $K_{\mathrm{total}} / K_a = 100$ , so $L_p = \Omega(100 \cdot \log_2(100)) = \Omega(100 \cdot 6.64) \approx 664$ . In practice, a constant factor $C_0 \approx 2$ -- $5$ is needed for robust RIP, so $L_p \approx 1300$ -- $3300$ pilot symbols may be required. With $M$ receive antennas, this can be reduced by up to a factor of $M$ .

Example: Pilot Design for mMTC Activity Detection

A massive MIMO base station with $M = 64$ antennas serves $K_{\mathrm{total}} = 1000$ mMTC devices with activity probability $p_a = 0.05$ . The coherence interval is $T_c = 200$ symbols (time-frequency resource elements within which the channel is approximately constant).

(a) Determine the minimum pilot length $L_p$ based on the compressed sensing scaling law. (b) How many resource elements remain for data after pilot transmission? (c) If each active device transmits $B = 100$ bits, what minimum spectral efficiency (bits per resource element) is needed for the data phase?

Solution

Expected active devices

$K_a = p_a K_{\mathrm{total}} = 0.05 \times 1000 = 50.$ $

Minimum pilot length

The scaling law gives

$L_p = \Omega\!\left(K_a \log_2 \frac{K_{\mathrm{total}}}{K_a}\right) = \Omega\!\left(50 \cdot \log_2(20)\right) = \Omega(50 \cdot 4.32) = \Omega(216).$

With a practical constant factor $C_0 \approx 2$ , we need $L_p \approx 432$ pilot symbols. However, the $M = 64$ receive antennas provide an effective measurement dimension of $L_p \cdot M$ . Using a conservative gain factor of $\sqrt{M}$ (due to noise averaging), we can reduce $L_p$ to approximately $216 / \sqrt{64} \cdot C_0 = 216/8 \cdot 2 = 54$ . Taking $L_p = 60$ as a practical value.

Remaining data resources

$T_{\mathrm{data}} = T_c - L_p = 200 - 60 = 140 \text{ resource elements}.$ $

Required spectral efficiency

Each active device uses 140 REs for data. To transmit $B = 100$ bits:

$\eta = \frac{B}{T_{\mathrm{data}}} = \frac{100}{140} \approx 0.71 \text{ bits/RE}.$

This is a modest spectral efficiency, achievable with QPSK and rate- $1/3$ coding ( $\eta = 2/3 \approx 0.67$ bits/RE), confirming feasibility.

,

🎓CommIT Contribution(2021)

Coded Compressed Sensing for Unsourced Random Access

A. Fengler, S. Haghighatshoar, P. Jung, G. Caire — IEEE Trans. Information Theory, vol. 67, no. 5, pp. 2925--2951

Fengler, Haghighatshoar, Jung, and Caire developed a practical coded compressed sensing (CCS) scheme for unsourced random access that approaches the Polyanskiy (2017) random coding achievability bound within a few dB.

Key innovations:

Divide-and-conquer coding: Each $B$ -bit message is split into $L$ sub-messages. Each sub-message selects a column from a separate compressed sensing sub-codebook. The concatenation creates an effective codebook of size $2^B$ without storing or searching $2^B$ codewords.
Massive MIMO receiver: With $M$ base station antennas, the activity detection problem becomes an MMV (multiple measurement vector) compressed sensing problem. The $M$ antennas provide an $M$ -fold increase in effective measurements, dramatically improving detection reliability.
Non-Bayesian approach: Unlike AMP-based methods that assume a specific prior on the activity pattern, the CCS scheme uses non-Bayesian (covariance-based) detection that is robust to model mismatch and does not require knowledge of the number of active users.

Performance: For $K_a = 300$ active users transmitting $B = 100$ bits over $n = 30{,}000$ channel uses with $M = 64$ antennas, the scheme achieves per-user error probability $P_e \leq 0.05$ at $E_b/N_0 \approx 2$ dB, within 3 dB of the Polyanskiy bound.

unsourced-random-accesscompressed-sensingmassive-mimocommitView Paper →

⚠️Engineering Note

mMTC Device Constraints and Battery Life

mMTC devices (NB-IoT, LTE-M, and 5G NR RedCap) operate under severe hardware and energy constraints that shape the system design:

Device density:

3GPP mMTC target: up to $10^6$ devices per km $^2$ .
Typical activity factor: 0.1%--1% per coherence interval, so $K_a / K_{\mathrm{total}} \approx 10^{-3}$ -- $10^{-2}$ .
This sparsity is what enables compressed sensing approaches.

Battery life:

Target: 10+ years on a single AA battery ( $\sim 2500$ mAh).
At 3.6 V, total energy budget $\approx 9$ Wh $= 32.4$ kJ.
If the device transmits 100-byte packets once per hour with $P_{\mathrm{tx}} = 23$ dBm ( $200$ mW) and Tx duration $\sim 1$ ms, the energy per transmission is $200\,\mu$ J.
With $8760$ transmissions/year over 10 years: total Tx energy $\approx 17.5$ J — negligible compared to the battery.
The battery bottleneck is the control plane: RACH, authentication, and DRX wake-up consume $\sim 100\times$ more energy than data transmission. Grant-free access eliminates the control-plane overhead.

Receiver sensitivity:

NB-IoT achieves $-164$ dBm MCL (maximum coupling loss) using 128 repetitions — enabling coverage in deep indoor/basement scenarios.
The repetition coding trades throughput for coverage and is equivalent to increasing $n$ (blocklength) at fixed rate.

Practical Constraints

•
Target: 10^6 devices/km^2 with 10+ year battery life
•
Activity factor: 0.1-1% per coherence interval
•
NB-IoT MCL: -164 dBm with 128 repetitions
•
Grant-free access eliminates control-plane energy overhead

📋 Ref: 3GPP TS 36.888 (mMTC study), 3GPP TS 38.300 (NR RedCap)

Historical Note: From ALOHA to Coded Random Access: 50 Years of Evolution

1970--2021

The history of random access begins with Norman Abramson's ALOHA protocol (1970) at the University of Hawaii, which demonstrated that uncoordinated transmission could work at the cost of collisions. Pure ALOHA achieves throughput $1/(2e) \approx 18.4\%$ ; slotted ALOHA (Roberts, 1975) doubles this to $1/e \approx 36.8\%$ .

For decades, these limits seemed fundamental. The breakthrough came with Contention Resolution Diversity Slotted ALOHA (CRDSA) (Casini et al., 2007) and its generalisation Irregular Repetition Slotted ALOHA (IRSA) (Liva, 2011), which applied the successive interference cancellation idea from turbo/LDPC decoding to random access. By transmitting replicas across multiple slots and iteratively cancelling decoded packets, IRSA achieves throughput approaching 1 packet/slot — a $3\times$ improvement over slotted ALOHA.

The paradigm shifted again with Polyanskiy's unsourced random access model (2017), which abandoned user identity entirely and asked: what is the minimum energy per bit needed for $K_a$ users to each deliver $B$ bits to a common receiver? Fengler, Jung, and Caire (2021) showed that coded compressed sensing schemes closely approach this fundamental limit.

Common Mistake: Dimensioning mMTC for Simultaneous Activation of All Devices

Mistake:

"Our cell has $K_{\mathrm{total}} = 50{,}000$ IoT devices. We need pilot resources for all 50,000 to avoid collisions."

Correction:

The activity factor in mMTC is extremely small: typically only $K_a = 50$ -- $500$ devices are active in any given coherence interval (activity factor $\sim 10^{-3}$ -- $10^{-2}$ ). The pilot dimension must scale with $K_a$ , not $K_{\mathrm{total}}$ .

The compressed sensing framework exploits this sparsity: with $L_p = O(K_a \log(K_{\mathrm{total}}/K_a))$ pilot symbols, all active devices can be detected. For $K_a = 100$ and $K_{\mathrm{total}} = 50{,}000$ :

$L_p = O(100 \cdot \log_2(500)) \approx O(900).$

This is far more efficient than orthogonal pilot allocation, which would require $L_p = K_{\mathrm{total}} = 50{,}000$ pilot symbols — an impossibly large overhead.

Grant-Free Random Access

A multiple access protocol where devices transmit data without first requesting and receiving a scheduling grant from the base station. Reduces latency by eliminating the 4-step RACH handshake and enables massive connectivity by removing the grant bottleneck.

Unsourced Random Access

A random access model (Polyanskiy, 2017) where all active users share a common codebook and the receiver outputs an unordered list of decoded messages without user identities. The performance metric is the per-user probability of error (PUPE).

Random Access Protocols for Massive IoT

Property	Slotted ALOHA	IRSA (Liva 2011)	Grant-Free CS	Unsourced RA (CCS)
Max throughput	$1/e \approx 0.37$ packets/slot	$\sim 0.97$ packets/slot	Limited by pilot length	Approaches Polyanskiy bound
Collision handling	Discard and retransmit	SIC (iterative cancellation)	CS-based joint detection	Coded CS decoding
User identity	Required (packet header)	Required	Required (pre-assigned pilot)	Not required (anonymous)
Pilot overhead	None (data only)	None	$O(K_a \log(K_{\mathrm{total}}/K_a))$	Embedded in codebook
Scalability	Poor ( $K_a > 100$ problematic)	Good (up to $\sim 1000$ )	Good (CS scales with sparsity)	Excellent (designed for massive $K_a$ )
Standards	LTE RACH, WiFi	DVB-S2/RCS2	5G NR grant-free (Type A)	Research stage

mMTC: Massive Machine-Type Communication

Connecting Billions of Devices

Definition: Grant-Free Random Access

Grant-Free vs Grant-Based Access

Definition: Compressed Sensing Activity Detection Model

Theorem: Pilot Length Scaling Law for Activity Detection

Approximate Message Passing (AMP) for Activity Detection

CS-Based Activity Detection Performance

Parameters

From ALOHA to Modern Random Access

Definition: Unsourced Random Access

Unsourced Random Access: PUPE vs Eb/N0E_b/N_0Eb​/N0​

Parameters

Quick Check

Example: Pilot Design for mMTC Activity Detection

Expected active devices

Minimum pilot length

Remaining data resources

Required spectral efficiency

Coded Compressed Sensing for Unsourced Random Access

mMTC Device Constraints and Battery Life

Historical Note: From ALOHA to Coded Random Access: 50 Years of Evolution

Common Mistake: Dimensioning mMTC for Simultaneous Activation of All Devices

Grant-Free Random Access

Unsourced Random Access

Random Access Protocols for Massive IoT

Definition:
Grant-Free Random Access

Definition:
Compressed Sensing Activity Detection Model

Definition:
Unsourced Random Access

Unsourced Random Access: PUPE vs $E_b/N_0$