Ferkans — Interactive Telecom Tutor

Beyond Orthogonal Partitioning

Section 19.1 showed that orthogonal multiple access incurs a sum-rate penalty relative to the MAC capacity. A natural question arises: can we close this gap without the exponential complexity of joint maximum-likelihood decoding? The answer is yes, via non-orthogonal multiple access (NOMA). In power-domain NOMA, all users transmit simultaneously over the same time-frequency resource, and the receiver resolves them through successive interference cancellation (SIC). This achieves any point on the boundary of the MAC capacity region with linear decoding complexity. NOMA has attracted enormous interest in 5G research as a path to improved spectral efficiency and massive connectivity.

Definition:
Power-Domain NOMA and SIC Decoding

Consider a two-user Gaussian MAC with received signal:

$y = \sqrt{P_1}\,h_1\,x_1 + \sqrt{P_2}\,h_2\,x_2 + n$

where $h_k$ is user $k$ 's channel, $P_k$ is user $k$ 's transmit power, and $n \sim \mathcal{CN}(0, \sigma^2)$ .

Power-domain NOMA assigns different power levels to users based on their channel conditions. Without loss of generality, assume $|h_1|^2 > |h_2|^2$ (user 1 has the stronger channel).

SIC decoding order: The base station first decodes user 2's signal (the weaker user), treating user 1's signal as noise:

$R_2 \leq \log_2\!\left(1 + \frac{P_2 |h_2|^2}{P_1 |h_1|^2 + \sigma^2}\right)$

After successfully decoding and subtracting user 2's contribution, the base station decodes user 1 interference-free:

$R_1 \leq \log_2\!\left(1 + \frac{P_1 |h_1|^2}{\sigma^2}\right)$

The decoding order can be reversed to achieve a different rate pair on the MAC capacity boundary.

The SIC decoding order determines which corner point of the MAC capacity region is achieved. Decoding the weaker user first (treating the stronger as noise) and then decoding the stronger user interference-free yields the rate pair that maximises the stronger user's rate. Time-sharing between the two decoding orders traces the entire dominant face of the capacity region.

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Definition:
Successive Interference Cancellation (General K-User)

For the $K$ -user Gaussian MAC, SIC decodes users sequentially in some permutation $\pi$ . User $\pi(k)$ is decoded at step $k$ , treating users $\pi(k+1), \ldots, \pi(K)$ as noise (users $\pi(1), \ldots, \pi(k-1)$ have already been cancelled):

$R_{\pi(k)} \leq \log_2\!\left(1 + \frac{P_{\pi(k)} |h_{\pi(k)}|^2} {\sum_{j=k+1}^{K} P_{\pi(j)} |h_{\pi(j)}|^2 + \sigma^2}\right)$

The achievable rate region is the convex hull over all $K!$ possible decoding orders, which equals the MAC capacity region.

Theorem: NOMA Achieves the MAC Capacity Boundary

For the $K$ -user Gaussian MAC with total power constraint $\sum_k P_k \leq P$ and channel gains $\{|h_k|^2\}$ , the capacity region is the set of rate tuples $(R_1, \ldots, R_K)$ satisfying:

$\sum_{k \in \mathcal{S}} R_k \leq \log_2\!\left(1 + \frac{\sum_{k \in \mathcal{S}} P_k |h_k|^2}{\sigma^2}\right) \quad \forall\, \mathcal{S} \subseteq \{1, \ldots, K\}$

Superposition coding at the transmitters with SIC at the receiver achieves every point on the boundary of this region. In particular, the sum-rate maximising point is:

$C_{\text{sum}} = \log_2\!\left(1 + \frac{\sum_{k=1}^{K} P_k |h_k|^2}{\sigma^2}\right)$

which requires decoding users in order of increasing $P_k |h_k|^2$ (weakest first) and is strictly greater than the orthogonal sum rate for $K \geq 2$ .

In orthogonal access, each user gets a fraction of the resource. In NOMA, all users share the entire resource simultaneously. The "interference" from other users is not wasted: SIC peels off each user's signal layer by layer, so the total information carried by the superposed signals equals the capacity of the aggregate power. The successive decoding approach converts a $K$ -user interference problem into $K$ single-user problems, each with progressively less residual interference.

Proof

Achievability via SIC (two-user case)

For $K = 2$ with $|h_1|^2 \geq |h_2|^2$ , decode user 2 first:

$R_2 = \log_2\!\left(1 + \frac{P_2 |h_2|^2}{P_1 |h_1|^2 + \sigma^2}\right)$

After cancelling user 2, decode user 1:

$R_1 = \log_2\!\left(1 + \frac{P_1 |h_1|^2}{\sigma^2}\right)$

The sum rate is:

$R_1 + R_2 = \log_2\!\left(1 + \frac{P_1 |h_1|^2}{\sigma^2}\right) + \log_2\!\left(1 + \frac{P_2 |h_2|^2}{P_1 |h_1|^2 + \sigma^2}\right)$

$= \log_2\!\left(\frac{\sigma^2 + P_1|h_1|^2}{\sigma^2}\right) + \log_2\!\left(\frac{P_1|h_1|^2 + P_2|h_2|^2 + \sigma^2} {P_1|h_1|^2 + \sigma^2}\right)$

$= \log_2\!\left(1 + \frac{P_1|h_1|^2 + P_2|h_2|^2}{\sigma^2}\right) = C_{\text{sum}}$

Converse

The converse follows from the cut-set bound. For any subset $\mathcal{S}$ , the sum rate of users in $\mathcal{S}$ cannot exceed the capacity of the MAC formed by those users (treating others as known interference that can be cancelled):

$\sum_{k \in \mathcal{S}} R_k \leq \log_2\!\left(1 + \frac{\sum_{k \in \mathcal{S}} P_k |h_k|^2}{\sigma^2}\right)$

These $2^K - 1$ constraints define a polymatroid, and its dominant face is achievable by SIC with appropriate decoding orders and time-sharing. $\blacksquare$

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MAC Capacity Region vs Orthogonal Access

Watch the two-user MAC capacity region (blue pentagon) emerge alongside the orthogonal access region (red). The MAC region strictly contains the OMA region, and the yellow line highlights the sum-rate dominant face achievable only with NOMA/SIC.

The MAC capacity region (NOMA) strictly contains the orthogonal access region for

K \geq 2

users.

NOMA Rate Region vs. Orthogonal Access

Compare the two-user MAC capacity region (achievable with NOMA/SIC) against the orthogonal (TDMA/FDMA) rate region. The pentagon-shaped MAC region strictly contains the triangular orthogonal region. Adjust the total power $P$ and the channel gain ratio to observe how the NOMA advantage grows with asymmetry between users.

Parameters

P

total power [dB]20

|h_1|^2/|h_2|^2

channel gain ratio [dB]10

Example: Two-User NOMA Rate Computation

Two users share a 1 MHz channel. User 1 has channel gain $|h_1|^2 = 10$ (near user) and user 2 has $|h_2|^2 = 1$ (far user). Total power $P = 1$ W, noise PSD $\sigma^2 = 10^{-6}$ W/Hz (so $N_0 W = 1$ W). The power is split as $P_1 = 0.2$ W and $P_2 = 0.8$ W (more power to the weaker user).

(a) Compute the NOMA rates with SIC (decode user 2 first). (b) Compute the orthogonal (TDMA) rates with equal time sharing. (c) Compare the sum rates.

Solution

NOMA rates

Decode user 2 first (treating user 1 as noise):

$R_2 = \log_2\!\left(1 + \frac{0.8 \times 1}{0.2 \times 10 + 1}\right) = \log_2\!\left(1 + \frac{0.8}{3}\right) = \log_2(1.267) = 0.34 \;\text{bits/s/Hz}$

After cancelling user 2, decode user 1:

$R_1 = \log_2\!\left(1 + \frac{0.2 \times 10}{1}\right) = \log_2(3) = 1.585 \;\text{bits/s/Hz}$

NOMA sum rate: $R_1 + R_2 = 1.925$ bits/s/Hz.

TDMA rates (equal sharing)

Each user gets half the time, with full bandwidth and burst power $2P_k$ :

$R_1^{\text{TDMA}} = \frac{1}{2}\log_2(1 + 2 \times 0.2 \times 10) = \frac{1}{2}\log_2(5) = 1.161 \;\text{bits/s/Hz}$

$R_2^{\text{TDMA}} = \frac{1}{2}\log_2(1 + 2 \times 0.8 \times 1) = \frac{1}{2}\log_2(2.6) = 0.689 \;\text{bits/s/Hz}$

TDMA sum rate: $1.161 + 0.689 = 1.850$ bits/s/Hz.

Comparison

NOMA achieves $1.925$ bits/s/Hz vs. TDMA's $1.850$ bits/s/Hz, a 4% gain. Note that user 1 gains significantly ( $1.585$ vs. $1.161$ ) while user 2 loses ( $0.34$ vs. $0.689$ ). A different power split or SIC order would shift the rate pair along the MAC capacity boundary, enabling a more balanced allocation. The sum capacity is:

$C_{\text{sum}} = \log_2(1 + 0.2 \times 10 + 0.8 \times 1) = \log_2(3.8) = 1.925$

confirming that NOMA achieves the sum capacity. $\blacksquare$

Quick Check

In a two-user NOMA system, user 1 is near the base station ( $|h_1|^2 = 20$ dB) and user 2 is far ( $|h_2|^2 = 0$ dB). To achieve the sum capacity, which user should be decoded first by SIC?

Decode user 2 (far user) first, then user 1 interference-free

Decode user 1 (near user) first since it has higher SNR

Decode both simultaneously using joint ML detection

The decoding order does not matter for NOMA

Correction:

Decode user 2 (far user) first, then user 1 interference-free

To maximise the sum rate, decode users in order of increasing received power. User 2 (weaker) is decoded first, treating user 1 as noise. Then user 1 is decoded without interference. This achieves the sum-rate corner point of the MAC region.

Historical Note: Superposition Coding and the MAC

1970s -- 2013

The capacity region of the Gaussian multiple-access channel was characterised independently by Ahlswede (1971) and Liao (1972). Cover (1972) introduced superposition coding as an achievability technique, showing that layered transmission with SIC could achieve the entire capacity region without joint ML decoding. This result is one of the cornerstones of network information theory. The modern interest in NOMA for 5G was largely driven by Saito et al. (2013), who demonstrated practical gains of power-domain NOMA over OFDMA in LTE-Advanced scenarios, reigniting interest in non-orthogonal techniques that had been known theoretically for decades.

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Why This Matters: MAC Capacity Region in the ITA Book

This section presents the Gaussian MAC capacity region and SIC achievability at a level sufficient for system design. The ITA book (Information Theory and Applications) develops the full multi-user information theory:

Ch 7-8: MAC capacity region with general input distributions, converse via Fano's inequality, polymatroidal structure
Ch 9: Broadcast channel duality and dirty-paper coding
Ch 10: Interference channel (Han-Kobayashi region)
Ch 11: Network information theory (relay, distributed source coding, Slepian-Wolf)

The NOMA/SIC framework here is the simplest instance of multi-user coding; the ITA book shows how these ideas generalise to arbitrarily complex network topologies.

⚠️Engineering Note

SIC Error Propagation in Practice

The theoretical analysis of NOMA assumes perfect SIC: each user's signal is decoded and subtracted without error before decoding the next. In practice, decoding errors propagate through the SIC chain:

If user $k$ 's signal is decoded incorrectly, the residual interference after subtraction is $2P_k|h_k|^2$ (double the original power), catastrophically degrading all subsequent users.
Error propagation is most severe when the first-decoded user (weakest) has marginal SINR. A single error cascades.

Practical mitigations in 5G NR study items (3GPP TR 38.812):

Use channel coding (LDPC/polar) with CRC to detect decoding errors; if CRC fails, skip the subtraction and fall back to treating interference as noise.
Limit SIC depth to 2 users (the 3GPP NOMA study found diminishing returns beyond 2 SIC layers).
Use hybrid HARQ: if the first-decoded user fails, request a retransmission before proceeding with SIC.

The 3GPP NOMA study item (Rel-16) ultimately concluded that the gains of NOMA over OFDMA with MU-MIMO were marginal in practical deployments, and NOMA was not adopted in 5G NR Release 16/17.

Practical Constraints

•
SIC depth limited to 2 layers in practice
•
CRC-aided error detection adds 24 bits per codeblock
•
3GPP NOMA study (TR 38.812) showed < 5% gain over MU-MIMO

📋 Ref: 3GPP TR 38.812 (NOMA study item, Rel-16)

NOMA

Non-Orthogonal Multiple Access: a class of multiple access schemes in which users share the same time-frequency resource and are separated by power level, code, or other non-orthogonal signatures. The receiver employs successive interference cancellation or other advanced techniques to resolve users.

Related: SIC, OFDMA

SIC

Successive Interference Cancellation: a receiver technique that decodes multi-user signals sequentially, subtracting each decoded user's contribution before decoding the next. When combined with superposition coding, SIC achieves the MAC capacity region with linear per-user complexity.

Related: NOMA

Non-Orthogonal Multiple Access (NOMA)