Ferkans — Interactive Telecom Tutor

Beyond Store-and-Forward

Traditional relay networks use store-and-forward: the relay receives a packet, buffers it, and retransmits it unchanged. Network coding breaks this paradigm by allowing intermediate nodes to combine information from multiple flows before forwarding. In the classic two-way relay scenario, where nodes $A$ and $B$ exchange messages via relay $R$ , conventional relaying requires 4 time slots ( $A \to R$ , $R \to B$ , $B \to R$ , $R \to A$ ). With network coding, the relay XORs the two messages and broadcasts the combination in a single slot, reducing the exchange to 3 or even 2 time slots. This physical-layer network coding (PNC) doubles the spectral efficiency of two-way relay communication and has profound implications for multi-hop network design.

Definition:
Two-Way Relay Channel

In the two-way relay channel (TWRC), nodes $A$ and $B$ wish to exchange messages via a relay $R$ . Three strategies have increasing spectral efficiency:

Conventional relaying (4 phases): Phase 1: $A \to R$ ( $m_A$ ). Phase 2: $R \to B$ ( $m_A$ ). Phase 3: $B \to R$ ( $m_B$ ). Phase 4: $R \to A$ ( $m_B$ ). Total: 4 time slots for one exchange.

Digital network coding (DNCE, 3 phases): Phase 1: $A \to R$ ( $m_A$ ). Phase 2: $B \to R$ ( $m_B$ ). Phase 3: $R$ broadcasts $m_A \oplus m_B$ . $A$ recovers $m_B = (m_A \oplus m_B) \oplus m_A$ ; similarly $B$ . Total: 3 time slots.

Physical-layer network coding (PNC, 2 phases): Phase 1: $A$ and $B$ transmit simultaneously to $R$ . $R$ receives $y_R = h_A x_A + h_B x_B + n$ . Phase 2: $R$ maps $y_R$ to a network-coded symbol and broadcasts. Total: 2 time slots.

PNC achieves the theoretical minimum of 2 time slots for a bidirectional exchange and doubles the spectral efficiency relative to conventional relaying. The key challenge is reliable decoding at the relay when two signals interfere, which requires careful design of the mapping function and can exploit the structure of lattice codes.

Definition:
Physical-Layer Network Coding (PNC)

In physical-layer network coding, the relay exploits the natural superposition of wireless signals to directly decode a function of the transmitted messages. For binary messages $m_A, m_B \in \{0, 1\}$ with BPSK modulation:

Phase 1: $y_R = h_A(2m_A - 1) + h_B(2m_B - 1) + n_R$

The relay decodes the XOR $m_A \oplus m_B$ (not the individual messages) using a modified decision rule. With equal channel gains ( $|h_A| = |h_B|$ ), the received constellation has 3 distinct points: the sum and difference of the two signals.

Phase 2: The relay broadcasts $m_A \oplus m_B$ . Each end node applies its own message as side information to recover the partner's message.

The achievable sum-rate of PNC is:

$R_{\text{PNC}} = \frac{1}{2}\min\!\left\{ \log_2(1 + |h_A|^2 + |h_B|^2),\; \log_2(1 + \text{SNR}_{RA}) + \log_2(1 + \text{SNR}_{RB}) \right\}$

where the factor $1/2$ reflects the 2-phase protocol (each direction gets half the resources).

Theorem: Spectral Efficiency Gain of PNC

For a symmetric two-way relay channel where all links have the same average SNR $\gamma$ , the sum spectral efficiencies of the three relaying strategies are:

Conventional (4-phase): $R_{\text{conv}} = \frac{1}{2}\log_2(1 + \gamma)$

Digital network coding (3-phase): $R_{\text{DNCE}} = \frac{2}{3}\log_2(1 + \gamma)$

Physical-layer network coding (2-phase): $R_{\text{PNC}} = \log_2(1 + \gamma)$

The spectral efficiency ratios are: $R_{\text{PNC}} : R_{\text{DNCE}} : R_{\text{conv}} = 1 : 2/3 : 1/2$ .

PNC achieves 2 $\times$ the spectral efficiency of conventional relaying.

Each strategy requires a different number of time slots for one bidirectional exchange. Since the per-phase rate is the same ( $\log_2(1+\gamma)$ per link), the spectral efficiency is inversely proportional to the number of phases: 4, 3, or 2 phases divide into the two directions' worth of information. PNC achieves the minimum possible 2 phases by exploiting the simultaneous transmission in Phase 1.

Proof

Conventional relaying

In 4 phases, each direction ( $A \to B$ and $B \to A$ ) uses 2 phases. The rate per direction is:

$R_{A \to B} = R_{B \to A} = \frac{1}{4}\log_2(1+\gamma) \times 2$

Wait — more precisely, each of the 4 phases carries $\log_2(1+\gamma)$ bits. Two phases serve $A \to B$ , two serve $B \to A$ . But each direction is bottlenecked by the weaker link:

$R_{A \to B} = \frac{1}{4}\min\{\log_2(1+\gamma_{AR}), \log_2(1+\gamma_{RB})\} \times 2 = \frac{1}{4}\log_2(1+\gamma) \times 2$

Sum rate: $R_{\text{conv}} = \frac{2}{4}\log_2(1+\gamma) = \frac{1}{2}\log_2(1+\gamma)$ .

Digital network coding

In 3 phases: Phase 1 ( $A \to R$ ), Phase 2 ( $B \to R$ ), Phase 3 ( $R$ broadcasts $m_A \oplus m_B$ ).

Each direction uses effectively 1.5 phases: Sum rate: $R_{\text{DNCE}} = \frac{2}{3}\log_2(1+\gamma)$ .

Physical-layer network coding

In 2 phases: Phase 1 (simultaneous $A, B \to R$ ), Phase 2 ( $R$ broadcasts network-coded symbol).

The relay decodes $m_A \oplus m_B$ from the superimposed signal. At high SNR, the multiple-access phase supports rate $\log_2(1+2\gamma)$ for the XOR function, which exceeds $\log_2(1+\gamma)$ .

Each direction uses exactly 1 phase: Sum rate: $R_{\text{PNC}} = \frac{2}{2}\log_2(1+\gamma) = \log_2(1+\gamma)$ . $\blacksquare$

Network Coding Animation

Visualise the message exchange in a two-way relay scenario under conventional, digital network coding, and physical-layer network coding protocols. The animation shows the time-slot structure and the information flow at each phase. Adjust the SNR to compare the achievable sum rates of the three strategies. Observe that PNC consistently achieves 2 $\times$ the spectral efficiency of conventional relaying.

Parameters

SNR (dB)15

Number of exchange frames6

Example: Two-Way Relay Rate Comparison

Two nodes $A$ and $B$ exchange data via a relay $R$ . All links have SNR $= 15$ dB.

(a) Compute the sum spectral efficiency for conventional relaying, DNCE, and PNC. (b) If the system bandwidth is 10 MHz, compute the sum throughput for each strategy. (c) How much additional bandwidth would conventional relaying need to match PNC's throughput?

Solution

Spectral efficiencies

(a) $\gamma = 10^{1.5} = 31.62$ .

Conventional: $R_{\text{conv}} = 0.5 \times \log_2(32.62) = 0.5 \times 5.03 = 2.51$ bits/s/Hz.

DNCE: $R_{\text{DNCE}} = \frac{2}{3} \times 5.03 = 3.35$ bits/s/Hz.

PNC: $R_{\text{PNC}} = 5.03$ bits/s/Hz.

Throughput

(b) Conv: $2.51 \times 10 = 25.1$ Mbps. DNCE: $3.35 \times 10 = 33.5$ Mbps. PNC: $5.03 \times 10 = 50.3$ Mbps.

Bandwidth equivalence

(c) Conv needs $50.3/2.51 = 20$ MHz to match PNC at 10 MHz.

PNC saves 50% of the bandwidth — or equivalently, doubles the throughput for the same bandwidth. $\blacksquare$

Quick Check

In physical-layer network coding (PNC) for the two-way relay channel, what does the relay decode in Phase 1?

The individual messages $m_A$ and $m_B$ separately

The XOR (or a function) of the two messages $m_A \oplus m_B$ , not the individual messages

Nothing — the relay simply amplifies and forwards

A compressed version of the superimposed signal

Correction:

The XOR (or a function) of the two messages

m_A \oplus m_B

, not the individual messages

PNC exploits the fact that each end node already knows its own message. The relay only needs to decode the network-coded combination $m_A \oplus m_B$ . Each end node then XORs the broadcast with its own message to recover the partner's message.

Network Coding

A technique where intermediate network nodes combine (encode) data from multiple incoming flows before forwarding, rather than simply routing individual packets. In wireless two-way relay channels, network coding reduces the number of time slots needed for bidirectional exchange from 4 to 3 (digital NC) or 2 (physical-layer NC).

Physical-Layer Network Coding (PNC)

A technique where the relay decodes a function (typically XOR) of simultaneously transmitted messages by exploiting the natural superposition of wireless signals. PNC achieves the minimum 2-phase protocol for two-way relay exchange, doubling the spectral efficiency over conventional 4-phase relaying.

Common Mistake: Assuming PNC Works Without Tight Synchronisation

Mistake:

"Physical-layer network coding simply exploits the natural superposition of wireless signals — no additional synchronisation is needed beyond standard OFDM timing."

Correction:

PNC requires the two end nodes to transmit simultaneously so that their signals arrive at the relay within the cyclic prefix duration. In practice, this means:

Symbol-level synchronisation: The relay must coordinate the two nodes' transmission timing to within the CP duration (e.g., 4.7 $\mu$ s in normal CP for 15 kHz SCS, corresponding to $\sim 1.4$ km of propagation distance mismatch).
Carrier frequency offset (CFO): Residual CFO between the two nodes' oscillators creates inter-carrier interference in the superimposed OFDM signal. PNC performance degrades sharply when the CFO exceeds $\sim 1\%$ of the subcarrier spacing.
Power control: If one node's signal is significantly stronger, the XOR decoding at the relay is unreliable (the near-far problem). Transmit power control or successive interference cancellation is needed.

These practical challenges have limited PNC deployment in standards despite its theoretical appeal.

Comparison of Relay Protocols

Property	DF (Decode-and-Forward)	AF (Amplify-and-Forward)	CF (Compress-and-Forward)	PNC (Physical-Layer NC)
Relay processing	Full decoding + re-encoding	Linear amplification	Quantisation + Wyner-Ziv coding	XOR / lattice decoding of sum
Noise at relay	Removed (if decoded correctly)	Amplified and forwarded	Quantised (distortion-bounded)	Partially cancelled via structure
Best regime	Strong $S \to R$ link (relay near source)	All regimes (simple, suboptimal)	Strong $R \to D$ link (relay near destination)	Two-way exchange (bidirectional)
Half-duplex phases	2 (listen + forward)	2 (listen + forward)	2 (listen + forward)	2 (simultaneous Tx + forward)
Achieves capacity?	Yes, for degraded relay channel	No (suboptimal in general)	Approaches cut-set bound near destination	Optimal for TWRC under certain conditions
Complexity	High (full codec at relay)	Low (analog amplification)	High (source coding at relay)	High (structured codes, synchronisation)
Standards adoption	LTE-A Type 1 relay, 5G IAB	LTE-A Type 2 relay (transparent)	Research stage	Research stage

Two-Way Relay Channel (TWRC)

A three-node network where two end nodes exchange messages via a common relay, with no direct link between the end nodes. The TWRC is the canonical setting for network coding in wireless networks.

Network Coding

Beyond Store-and-Forward

Definition: Two-Way Relay Channel

Definition: Physical-Layer Network Coding (PNC)

Theorem: Spectral Efficiency Gain of PNC

Conventional relaying

Digital network coding

Physical-layer network coding

Network Coding Animation

Parameters

Example: Two-Way Relay Rate Comparison

Spectral efficiencies

Throughput

Bandwidth equivalence

Quick Check

Network Coding

Physical-Layer Network Coding (PNC)

Common Mistake: Assuming PNC Works Without Tight Synchronisation

Comparison of Relay Protocols

Two-Way Relay Channel (TWRC)

Definition:
Two-Way Relay Channel

Definition:
Physical-Layer Network Coding (PNC)