Ferkans — Interactive Telecom Tutor

When Does Relaying Pay Off?

Relay-assisted communication incurs a fundamental cost: the half-duplex penalty. A half-duplex relay can at best double the effective SNR at the destination while halving the time available for transmission. Whether this trade-off is favourable depends on the operating SNR, the relay position, and the path-loss exponent. In this section, we perform a systematic rate analysis, comparing relay protocols against direct transmission across the SNR range to identify the relay-beneficial region — the set of operating conditions where relaying provides a genuine throughput gain.

Definition:
Relay-Beneficial Region

The relay-beneficial region is the set of parameter values $(d_{SD}, f, \alpha, \text{SNR})$ for which the relay-assisted rate exceeds the direct transmission rate:

$\mathcal{B} = \{(d_{SD}, f, \alpha, \text{SNR}) : C_{\text{relay}} > C_{\text{direct}}\}$

For half-duplex DF relaying at the optimal relay position:

$C_{\text{DF}} > C_{\text{direct}} \iff \frac{1}{2}\log_2(1 + \text{SNR}_{SR}) > \log_2(1 + \text{SNR}_{SD})$

This requires $\text{SNR}_{SR} > (1 + \text{SNR}_{SD})^2 - 1$ , i.e., the relay link must be quadratically better than the direct link to overcome the half-duplex loss.

For path-loss model $\text{SNR}_{SR}/\text{SNR}_{SD} = (d_{SD}/d_{SR})^{\alpha}$ :

$f^{-\alpha} > (1 + \text{SNR}_{SD})^2/\text{SNR}_{SD}$

At high SNR: $f^{-\alpha} > \text{SNR}_{SD}$ , which is satisfied only for very small $f$ (relay very close to source) — relaying provides diminishing gains at high SNR.

The relay-beneficial region shrinks as SNR increases because the direct link already operates at high spectral efficiency, and the half-duplex penalty (factor of 1/2) becomes harder to overcome. Relaying is most valuable in the coverage extension regime (low SNR, long distances) rather than the capacity enhancement regime.

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Definition:
Compress-and-Forward (CF) Protocol

In compress-and-forward (CF), the relay does not decode the source message. Instead, it quantises its received signal $y_R$ to a compressed representation $\hat{y}_R$ at rate $I(\hat{Y}_R; Y_R \mid Y_D)$ and forwards it to the destination. The destination jointly decodes using both its direct observation $y_D$ and the compressed relay observation $\hat{y}_R$ .

The CF achievable rate is:

$C_{\text{CF}} = \frac{1}{2}\min\!\left\{ I(X_S; \hat{Y}_R, Y_D),\; I(X_R; Y_D) \cdot \mathbf{1}_{\text{link}}\right\}$

CF is optimal when:

The relay cannot decode (weak $S \to R$ link), and
The $R \to D$ link is strong (high-capacity forwarding).

CF approaches the cut-set bound when the relay is close to the destination, complementing DF which is optimal when the relay is close to the source.

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Theorem: DF vs. Direct Crossover Condition

For half-duplex DF relaying with relay at fraction $f$ of the $S \to D$ distance, with path-loss $\ell(d) = d^{-\alpha}$ and transmit SNR $\gamma_0$ at unit distance:

The DF rate exceeds the direct rate if and only if:

$\gamma_0 d_{SD}^{-\alpha}(f^{-\alpha} - 1) > (\gamma_0 d_{SD}^{-\alpha} + 1)^2 - (\gamma_0 d_{SD}^{-\alpha})^2 + (\gamma_0 d_{SD}^{-\alpha})^2 \cdot((1-f)^{-\alpha} - 1)/f^{-\alpha}$

In the low-SNR regime ( $\gamma_0 d_{SD}^{-\alpha} \ll 1$ ):

$C_{\text{DF}} > C_{\text{direct}} \iff f^{-\alpha} + (1-f)^{-\alpha} > 2$

This is satisfied for any $f \in (0,1)$ and $\alpha > 1$ : relaying always helps at low SNR.

In the high-SNR regime ( $\gamma_0 d_{SD}^{-\alpha} \gg 1$ ):

$C_{\text{DF}} > C_{\text{direct}} \iff f^{-\alpha} > 1 + (1-f)^{-\alpha} + \gamma_0 d_{SD}^{-\alpha}$

This requires exponentially small $f$ as SNR grows: relaying provides negligible gain at high SNR with DF.

At low SNR, the half-duplex penalty is mild (halving a small rate is a small loss) and the relay's proximity advantage (shorter paths) dominates. At high SNR, the direct link is already near capacity, and the half-duplex factor of 1/2 is devastating — the relay must provide a quadratic SNR improvement just to break even.

Proof

Low-SNR expansion

For small $x$ : $\log_2(1+x) \approx x/\ln 2$ .

$C_{\text{direct}} \approx \gamma_{SD}/\ln 2$ .

$C_{\text{DF}} \approx \frac{1}{2}\min\{\gamma_{SR}, \gamma_{SD} + \gamma_{RD}\}/\ln 2$ .

At the optimal relay position where $\gamma_{SR} = \gamma_{SD} + \gamma_{RD}$ :

$C_{\text{DF}} \approx (\gamma_{SD} + \gamma_{RD})/(2\ln 2)$ .

$C_{\text{DF}} > C_{\text{direct}} \iff \gamma_{SD} + \gamma_{RD} > 2\gamma_{SD} \iff \gamma_{RD} > \gamma_{SD}$ .

Since $(1-f)^{-\alpha} > 1$ for $f > 0$ : always true.

High-SNR expansion

For large $x$ : $\log_2(1+x) \approx \log_2(x)$ .

$C_{\text{direct}} \approx \log_2(\gamma_{SD})$ .

$C_{\text{DF}} \approx \frac{1}{2}\log_2(\gamma_{SR})$ .

$C_{\text{DF}} > C_{\text{direct}} \iff \frac{1}{2}\log_2(\gamma_{SR}) > \log_2(\gamma_{SD})$ $\iff \gamma_{SR} > \gamma_{SD}^2$ $\iff f^{-\alpha} > \gamma_{SD}$ .

This requires $f < \gamma_{SD}^{-1/\alpha}$ , which approaches 0 as $\gamma_{SD} \to \infty$ .

Crossover SNR

The crossover SNR $\gamma^{\star}$ (above which direct transmission wins) satisfies:

$\frac{1}{2}\log_2(1 + \gamma^{\star} f^{-\alpha}) = \log_2(1 + \gamma^{\star})$

For $f = 0.5$ , $\alpha = 4$ : $\gamma^{\star} \cdot 16 = (1 + \gamma^{\star})^2 - 1$ , giving $\gamma^{\star} \approx 13$ (11 dB). $\blacksquare$

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Relay Rates vs. SNR

Compare the achievable rates of DF, AF, direct transmission, and the cut-set bound as a function of SNR. Adjust the source-destination distance, relay position, and path-loss exponent. Identify the crossover SNR where relaying becomes beneficial and where it ceases to provide gain. The shaded region shows where the relay rate exceeds the direct rate.

Parameters

d_{SD}

(source-destination distance)2

Relay position fraction

f

0.5

Path-loss exponent

\alpha

3

Example: Relay Crossover SNR

A relay is at the midpoint ( $f = 0.5$ ) between source and destination separated by $d_{SD} = 2$ km with $\alpha = 4$ .

(a) Write the DF and direct rates as functions of the transmit SNR $\gamma_0$ (at 1 km). (b) Find the crossover SNR $\gamma_0^{\star}$ below which DF outperforms direct transmission. (c) Plot the rate advantage $\Delta C = C_{\text{DF}} - C_{\text{direct}}$ as a function of $\gamma_0$ and identify the maximum gain. (d) Explain the practical significance of the crossover.

Solution

Rate expressions

(a) $\gamma_{SD} = \gamma_0 \times 2^{-4} = \gamma_0/16$ . $\gamma_{SR} = \gamma_0 \times 1^{-4} = \gamma_0$ . $\gamma_{RD} = \gamma_0 \times 1^{-4} = \gamma_0$ .

$C_{\text{direct}} = \log_2(1 + \gamma_0/16)$ . $C_{\text{DF}} = \frac{1}{2}\min\{\log_2(1+\gamma_0), \log_2(1+\gamma_0/16+\gamma_0)\} = \frac{1}{2}\log_2(1+\gamma_0)$ .

(The second argument dominates since $\gamma_0/16 + \gamma_0 > \gamma_0$ .)

Crossover SNR

(b) $C_{\text{DF}} = C_{\text{direct}}$ :

$\frac{1}{2}\log_2(1+\gamma_0) = \log_2(1+\gamma_0/16)$

$(1+\gamma_0)^{1/2} = 1+\gamma_0/16$

Squaring: $1+\gamma_0 = 1 + \gamma_0/8 + \gamma_0^2/256$

$\gamma_0(1-1/8) = \gamma_0^2/256$

$\gamma_0 = 256 \times 7/8 = 224$ (23.5 dB).

Maximum gain

(c) The gain peaks around $\gamma_0 \approx 10$ (10 dB): $C_{\text{DF}} = 0.5\log_2(11) = 1.73$ . $C_{\text{direct}} = \log_2(1.625) = 0.70$ . $\Delta C = 1.03$ bits/s/Hz.

At $\gamma_0 = 100$ (20 dB): $C_{\text{DF}} = 0.5\log_2(101) = 3.33$ . $C_{\text{direct}} = \log_2(7.25) = 2.86$ . $\Delta C = 0.47$ bits/s/Hz.

Practical significance

(d) Below 23.5 dB reference SNR, the relay provides a throughput gain. Above this threshold, direct transmission is superior due to the half-duplex loss. In coverage extension scenarios (cell-edge users with low SNR), relaying is highly beneficial. In high-SNR capacity enhancement scenarios, relaying offers diminishing returns. $\blacksquare$

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Quick Check

At very high SNR, why does half-duplex DF relaying typically underperform direct transmission?

Because the relay introduces additional noise

Because the half-duplex factor of $1/2$ costs a full rate halving that the relay cannot compensate at high SNR

Because the relay has limited transmit power

Because path loss is too high for the relay links

Correction:

Because the half-duplex factor of

1/2

costs a full rate halving that the relay cannot compensate at high SNR

At high SNR, the direct link operates at $\log_2(\text{SNR}_{SD})$ bits/s/Hz, which grows logarithmically. The relay provides $\frac{1}{2}\log_2(\text{SNR}_{SR})$ . For the relay to compensate the factor of 2, $\text{SNR}_{SR}$ must be quadratically larger than $\text{SNR}_{SD}$ , which becomes increasingly difficult as SNR grows.

Compress-and-Forward (CF)

A relay protocol where the relay quantises its received signal to a compressed representation and forwards it to the destination without decoding the source message. CF is optimal when the relay is close to the destination and the $S \to R$ link is too weak for decoding. It approaches the cut-set bound in this regime.

Relay-Beneficial Region

The set of operating conditions (SNR, distance, path-loss exponent, relay position) for which relay-assisted communication achieves higher throughput than direct transmission. For half-duplex DF, this region is largest at low-to-moderate SNR and high path-loss exponents, and shrinks at high SNR where the half-duplex penalty dominates.

⚠️Engineering Note

5G NR Integrated Access and Backhaul (IAB) Relay Constraints

5G NR Release 16 introduced Integrated Access and Backhaul (IAB) as the standardised relay architecture. IAB nodes operate in time-division multiplexing between access (serving UEs) and backhaul (connecting to the donor gNB), implementing half-duplex DF relaying at the protocol level.

Key implementation constraints:

Resource partitioning: IAB nodes share time/frequency resources between access and backhaul. 3GPP defines "soft" and "hard" resource multiplexing modes. In hard mode, access and backhaul use non-overlapping resources (no self-interference); in soft mode, simultaneous operation is allowed with cross-link interference management.
Multi-hop: IAB supports up to 4 hops (3 intermediate relay nodes) in Release 17, with each hop adding latency ( $\sim 0.5$ -- $1$ ms per hop for sub-6 GHz). The end-to-end rate is limited by the bottleneck hop.
Topology adaptation: IAB nodes can dynamically switch their parent node (donor or another IAB node) based on backhaul link quality, enabling network resilience.
Timing: All IAB nodes must maintain tight timing alignment ( $\leq 3\,\mu$ s for FR1, $\leq 1.5\,\mu$ s for FR2) relative to the donor gNB for proper TDD UL/DL switching.

Typical deployment: mmWave small cells with wireless backhaul in urban street canyons. The relay beneficial region analysis from this chapter directly applies: IAB is most effective when the direct gNB-to-UE link is blocked (high path loss exponent) and the IAB node has good backhaul quality (relay near source).

Practical Constraints

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Max 4 hops per IAB topology in Rel-17
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Half-duplex TDM between access and backhaul
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Timing alignment: ≤3 μs (FR1), ≤1.5 μs (FR2)
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Self-interference limits soft resource multiplexing

📋 Ref: 3GPP TS 38.174 (IAB for NR)

⚠️Engineering Note

Self-Interference Cancellation for Full-Duplex Relaying

The half-duplex constraint that dominates this chapter's analysis can be eliminated if the relay can transmit and receive simultaneously on the same frequency (full-duplex operation). The fundamental challenge is self-interference (SI): the relay's own transmitted signal ( $\sim 20$ -- $40$ dBm) must be suppressed to below the receiver noise floor ( $\sim -90$ dBm) — a dynamic range of $110$ -- $130$ dB.

State-of-the-art SI cancellation achieves $\sim 110$ dB through three cascaded stages:

Antenna isolation ( $\sim 40$ -- $60$ dB): Physical separation, directional antennas, or circulators. Advanced designs use auxiliary transmit antennas to create a null at the receive antenna.
Analog cancellation ( $\sim 30$ -- $40$ dB): An analog circuit taps the transmit signal, adjusts its amplitude and phase (multi-tap filter), and subtracts it from the received signal before the ADC. This prevents ADC saturation.
Digital cancellation ( $\sim 20$ -- $30$ dB): After ADC, a digital filter models the residual SI channel (including PA nonlinearities) and subtracts it. Requires precise channel estimation of the SI path.

Practical implications for relay rate analysis:

With perfect SIC ( $\beta = 0$ residual SI), the DF rate becomes $R_{\mathrm{DF}}^{\mathrm{FD}} = \min\{C_{SR}, C_{SD} + C_{RD}\}$ — no half-duplex penalty. With residual SI power $\beta P_R$ , the relay observes $\text{SINR}_{SR} = P_S |h_{SR}|^2 / (\sigma^2 + \beta P_R)$ , reducing the $S \to R$ link capacity and partially restoring the half-duplex regime's advantage at high transmit power.

Current silicon implementations (e.g., Rice University's FlexiCan) achieve $\sim 105$ dB SIC over $80$ MHz bandwidth, sufficient for sub-6 GHz small-cell relays but not yet for high-power macro relays.

Practical Constraints

•
State-of-the-art SIC: ~110 dB (3 cascaded stages)
•
ADC dynamic range limits analog cancellation bandwidth
•
PA nonlinearity modeling needed for digital cancellation
•
Residual SI increases with transmit power and bandwidth

Relay Rate Analysis

When Does Relaying Pay Off?

Definition: Relay-Beneficial Region

Definition: Compress-and-Forward (CF) Protocol

Theorem: DF vs. Direct Crossover Condition

Low-SNR expansion

High-SNR expansion

Crossover SNR

Relay Rates vs. SNR

Parameters

Example: Relay Crossover SNR

Rate expressions

Crossover SNR

Maximum gain

Practical significance

Quick Check

Compress-and-Forward (CF)

Relay-Beneficial Region

5G NR Integrated Access and Backhaul (IAB) Relay Constraints

Self-Interference Cancellation for Full-Duplex Relaying

Definition:
Relay-Beneficial Region

Definition:
Compress-and-Forward (CF) Protocol