Ferkans — Interactive Telecom Tutor

Why User Cooperation?

In classical point-to-point communication, a single transmitter sends to a single receiver. The relay channel (Chapter 22) introduced a dedicated helper node. But what if there is no dedicated relay? What if two users, each with their own data to send, can also help each other?

The idea of cooperative diversity is precisely this: users relay for each other to achieve macro-diversity, even though no single user has multiple antennas. Each user "borrows" a virtual antenna from its partner. The cost is bandwidth: part of each user's transmission is spent forwarding the partner's message. The question that information theory answers is: when does the diversity gain outweigh the bandwidth cost?

Two-Phase Cooperative Diversity Protocol

Animation of the two-phase cooperative diversity protocol: Phase 1 (broadcast to partner and destination) and Phase 2 (coherent cooperative transmission), creating a virtual MIMO link.

Definition:
The Cooperative Multiple-Access Channel

Consider $K = 2$ users, each with an independent message $W_k \in [1:2^{nR_k}]$ , communicating to a common destination over a memoryless channel $p(y|x_1, x_2)$ . In the cooperative MAC:

Phase 1 (broadcast): User $k$ transmits $X_k^{(1)}$ based on $W_k$ . Both users and the destination receive noisy versions of both signals.
Phase 2 (cooperation): User $k$ transmits $X_k^{(2)}$ based on $(W_k, \hat{W}_{\bar{k}})$ , where $\hat{W}_{\bar{k}}$ is its estimate of the partner's message from Phase 1.

The channel model for the two-phase protocol over $n$ channel uses (with fraction $\alpha$ allocated to Phase 1) is:

Phase 1 ( $\alpha n$ uses): $Y_{12} = h_{12} X_1 + Z_{12}, \qquad Y_{21} = h_{21} X_2 + Z_{21}, \qquad Y^{(1)} = h_1 X_1 + h_2 X_2 + Z^{(1)}$

Phase 2 ( $(1-\alpha)n$ uses): $Y^{(2)} = h_1 X_1^{(2)} + h_2 X_2^{(2)} + Z^{(2)}$

where $h_{12}, h_{21}$ are the inter-user channels, $h_1, h_2$ are the user-destination channels, and all noise terms are $\mathcal{CN}(0, \sigma^2)$ .

The two-phase structure is a simplification. More sophisticated protocols use block-Markov encoding across multiple blocks, as in the relay channel. We start with the two-phase model for clarity.

Cooperative diversity

A technique where users in a multi-user network relay each other's messages to create virtual MIMO links, achieving spatial diversity without requiring multiple antennas at any single terminal.

Theorem: Achievable Rate Region for Cooperative DF

For the two-user cooperative MAC with decode-and-forward at each user, the following rate region is achievable:

$R_1 \le \min\!\Big\{\alpha C(|h_{12}|^2 P_1 / \sigma^2),\; C\!\big(\alpha |h_1|^2 P_1 / \sigma^2\big) + C\!\big((1-\alpha)(|h_1|^2 P_{1c} + |h_2|^2 P_{2c,1} + 2\rho_1 |h_1||h_2|\sqrt{P_{1c} P_{2c,1}}) / \sigma^2\big)\Big\}$

and symmetrically for $R_2$ , where $C(x) = \frac{1}{2}\log(1+x)$ , $P_{kc}$ is the power user $k$ allocates to cooperative transmission, $P_{2c,1}$ is the power user 2 allocates to relay user 1's message, and $\rho_1 \in [0,1]$ is the correlation coefficient for coherent combining.

The first term in the min is the inter-user link constraint: user 2 must decode user 1's message over the inter-user channel. The second term is the effective rate to the destination, which benefits from coherent combining of both users' transmissions in Phase 2. The point is that cooperation creates a virtual MISO channel in Phase 2, where the two users act as a distributed antenna array.

Proof

Codebook generation

Fix a power allocation $(P_1, P_{1c}, P_{2c,1})$ and correlation $\rho_1$ . Generate $2^{nR_1}$ i.i.d. codewords for Phase 1 from $\mathcal{CN}(0, P_1)$ . For each pair $(w_1, w_2)$ , generate Phase 2 codewords with the prescribed correlation structure using the standard technique from the relay channel (Chapter 22, Section 3).

Encoding at user 1

In Phase 1, user 1 transmits the codeword $\mathbf{x}_1^{(1)}(w_1)$ . User 2 decodes $\hat{w}_1$ from $\mathbf{y}_{21}$ , which succeeds if $R_1 < \alpha C(|h_{12}|^2 P_1 / \sigma^2)$ . In Phase 2, both users transmit correlated codewords that coherently combine at the destination.

Decoding at destination

The destination uses joint typicality decoding across both phases. The Phase 1 contribution provides rate $\alpha C(|h_1|^2 P_1 / \sigma^2)$ . The Phase 2 contribution, with coherent combining, provides $(1-\alpha)C\!\big((|h_1|^2 P_{1c} + |h_2|^2 P_{2c,1} + 2\rho_1|h_1||h_2|\sqrt{P_{1c}P_{2c,1}}) / \sigma^2\big)$ . The total rate is the sum of these two terms.

Example: Cooperative Gain in Symmetric Channels

Consider a symmetric cooperative MAC where $|h_1| = |h_2| = 1$ , $|h_{12}| = |h_{21}| = h_c$ , $P_1 = P_2 = P$ , $\sigma^2 = 1$ , and $\alpha = 1/2$ . Compare the sum rate of cooperative DF with the non-cooperative MAC sum rate $C(P) + C(P)$ when $P = 10$ dB and $h_c = 2$ (strong inter-user link).

Solution

Non-cooperative sum rate

Without cooperation, each user achieves $C(P) = \frac{1}{2}\log(1 + 10) \approx 1.73$ bits/use. The sum rate is $R_{\text{sum}}^{\text{nc}} \approx 3.46$ bits/use.

Inter-user link constraint

With $\alpha = 1/2$ and $h_c = 2$ , the inter-user link supports $\frac{1}{2}C(4 \cdot 10) = \frac{1}{4}\log(41) \approx 1.34$ bits/use per user. This exceeds the target rate, so the inter-user link is not the bottleneck.

Cooperative sum rate with equal power split

With equal power allocation $P_{1c} = P_{2c,1} = P/2 = 5$ and $\rho_1 = 1$ (perfect coherent combining), the Phase 2 contribution per user is: $\frac{1}{2}C\!\Big(\frac{5 + 5 + 2\sqrt{25}}{1}\Big) = \frac{1}{4}\log(1+20) \approx 1.09 \text{ bits/use}.$ Adding the Phase 1 contribution: $\frac{1}{2}C(P/2) = \frac{1}{4}\log(6) \approx 0.65$ bits/use. Total per user: $\approx 1.74$ bits/use. Sum rate: $R_{\text{sum}}^{\text{coop}} \approx 3.48$ bits/use.

Interpretation

In this symmetric high-SNR scenario, the sum-rate gain from cooperation is modest. The real benefit of cooperation is diversity: even if one user-destination link fades, the partner provides an alternative path. The diversity gain becomes dramatic in fading channels, which we quantify next via the DMT.

Definition:
Diversity-Multiplexing Tradeoff (DMT)

For a family of codes indexed by $\text{SNR}$ , the multiplexing gain and diversity gain are defined as:

$r = \lim_{\text{SNR} \to \infty} \frac{R(\text{SNR})}{\log \text{SNR}}, \qquad d(r) = -\lim_{\text{SNR} \to \infty} \frac{\log P_e(\text{SNR})}{\log \text{SNR}}.$

The optimal DMT $d^*(r)$ is the supremum of $d(r)$ over all coding schemes at multiplexing gain $r$ . Intuitively, $r$ measures what fraction of the AWGN capacity we achieve, and $d(r)$ measures how fast the error probability decays with SNR.

The DMT was introduced by Zheng and Tse (2003) for point-to-point MIMO. The key insight is that there is a fundamental tradeoff: you cannot simultaneously achieve maximum diversity (steepest error decay) and maximum multiplexing (highest rate scaling).

Diversity-multiplexing tradeoff

The fundamental tradeoff between the rate at which a code's throughput grows with SNR (multiplexing gain $r$ ) and the rate at which its error probability decays (diversity gain $d(r)$ ) in the high-SNR regime.

Related: Cooperative diversity

Theorem: DMT of the Cooperative MAC

Consider the two-user cooperative MAC with i.i.d. Rayleigh fading on all links. Let $d_{\text{nc}}^*(r) = 1 - r$ be the non-cooperative DMT (single-antenna point-to-point). The cooperative DF protocol achieves:

$d_{\text{coop-DF}}^*(r) = 2(1 - 2r), \qquad 0 \le r \le 1/2.$

The cooperative protocol with dynamic decode-and-forward (where the relay listens for a random duration determined by the channel realization) achieves:

$d_{\text{dyn-DF}}^*(r) = 2(1 - r), \qquad 0 \le r \le 1.$

This matches the $2 \times 1$ MISO DMT, confirming that cooperation creates a virtual two-antenna system.

Static DF pays a factor-of-two rate penalty because it uses half the time for the inter-user phase, limiting the multiplexing gain to $r \le 1/2$ . Dynamic DF removes this penalty by adapting the listening duration. Intuitively, when the direct link is strong, the relay barely needs to help; when it is weak, the relay listens longer and provides full diversity.

Proof

Outage probability analysis

The outage event for cooperative DF at rate $R = r\log\text{SNR}$ requires both the direct link and the cooperative link to be in outage simultaneously. Since the fading coefficients are independent, the probability of this joint event scales as $\text{SNR}^{-2(1-2r)}$ for static DF and $\text{SNR}^{-2(1-r)}$ for dynamic DF.

Converse via cut-set

The cut-set bound for a $2 \times 1$ MISO system gives $d^*(r) \le 2(1-r)$ . Since the two users have independent codebooks (unlike a true MISO transmitter), static DF cannot achieve this bound for $r > 1/2$ . Dynamic DF matches the cut-set bound by adapting the protocol to the channel realization.

Achievability of dynamic DF

The key idea (Azarian, El Gamal, and Yu, 2005) is that the relay decodes after a random number of symbols $n_{\text{listen}}$ determined by a threshold on the accumulated mutual information. When the inter-user link is strong, $n_{\text{listen}} \ll n$ , leaving most of the block for cooperative transmission. This adaptive strategy achieves the full $2(1-r)$ tradeoff.

Diversity-Multiplexing Tradeoff: Non-Cooperative vs Cooperative

Compare the DMT curves for non-cooperative transmission, static cooperative DF, dynamic cooperative DF, and the $2 \times 1$ MISO upper bound. Adjust the number of cooperating users to see how the DMT improves.

Parameters

Number of cooperating users

K

2

Cooperation protocol

Definition:
Coded Cooperation

In coded cooperation, the channel code itself is designed for the cooperative protocol. Each user's codeword is split into two parts:

Systematic part: User $k$ transmits its own information bits in Phase 1.
Parity part: In Phase 2, user $k$ transmits additional parity bits for its partner's message (decoded from Phase 1).

Formally, let $\mathcal{C}_k$ be a rate- $R$ code for user $k$ . The codeword $\mathbf{c}_k(w_k)$ is partitioned as $[\mathbf{c}_k^{(1)}, \mathbf{c}_k^{(2)}]$ . User $k$ transmits $\mathbf{c}_k^{(1)}$ in Phase 1 and $\mathbf{c}_{\bar{k}}^{(2)}$ (partner's parity) in Phase 2.

The effective code at the destination is a distributed code with codewords $[\mathbf{c}_k^{(1)}, \mathbf{c}_k^{(2)}]$ , where the two parts arrive through independent fading channels, providing diversity order 2.

Coded cooperation elegantly integrates cooperation into the code design. The code rate partitioning determines the cooperation level: a higher fraction of parity bits in Phase 2 means more cooperation (more diversity) at the cost of reduced individual rate. Punctured convolutional and turbo codes are natural choices for this framework.

Coded cooperation

A cooperative protocol where the channel code is designed to span both the direct and relay paths, with each user transmitting parity information for its partner's message.

Related: Cooperative diversity

Historical Note: The Origins of Cooperative Communication

2000-2006

The idea that users in a network can help each other communicate is as old as network information theory itself. Cover and El Gamal's 1979 relay channel paper planted the seed. But the modern cooperative diversity framework emerged around 2000-2003, driven independently by Sendonaris, Erkip, and Aazhang (who proposed user cooperation diversity) and by Laneman, Tse, and Wornell (who connected it to the diversity-multiplexing tradeoff). Hunter and Nosratinia then showed that the cooperation could be designed into the code rather than handled at the protocol level. The field exploded: by 2006, cooperative communication was one of the hottest topics in wireless research, with hundreds of papers exploring relaying protocols, partner selection, and resource allocation. The practical impact was eventually realized not through user-to-user cooperation (which requires inter-user channels) but through infrastructure cooperation: CoMP and C-RAN, which we study in Sections 25.2 and 25.3.

Common Mistake: Forgetting the Half-Duplex Constraint

Mistake:

Assuming that a cooperating user can transmit and receive simultaneously on the same frequency band, leading to artificially high cooperative rates.

Correction:

Most practical systems are half-duplex: a node cannot transmit and receive at the same time on the same frequency. The cooperative protocol must account for the fraction $\alpha$ of time spent listening versus transmitting. The DMT analysis above already incorporates this. Full-duplex operation is possible with self-interference cancellation, but requires sophisticated hardware and is limited by residual self-interference, which is typically 50-80 dB below the transmitted power even with state-of-the-art cancellation.

Quick Check

In the two-user cooperative MAC with static decode-and-forward, the maximum achievable multiplexing gain is $r = 1/2$ , not $r = 1$ . Why?

Because each user has only one antenna

Because the inter-user channel has limited capacity

Because static DF uses half the block for the inter-user phase, limiting the end-to-end rate

Because the destination cannot do successive interference cancellation

Correction:

Because static DF uses half the block for the inter-user phase, limiting the end-to-end rate

Static DF fixes $\alpha = 1/2$ : half the block is used for the inter-user phase, during which the destination receives at a reduced rate. This halves the effective multiplexing gain. Dynamic DF recovers the full $r = 1$ by adapting $\alpha$ to the channel realization.

Common Mistake: Assuming Any Partner Is Beneficial

Mistake:

Cooperating with a partner whose inter-user channel is weak (e.g., $|h_{12}|^2 \ll |h_1|^2$ ), which wastes Phase 1 bandwidth without providing useful relay information.

Correction:

Cooperation is beneficial only when the inter-user link is strong enough that the relay can decode reliably without consuming too much of the block. The condition is roughly $|h_{12}|^2 \gtrsim |h_1|^2$ : the inter-user link should be at least as strong as the direct link. When the inter-user link is weak, non-cooperative transmission is preferable. Optimal partner selection is itself an important research problem with connections to matching theory and graph optimization.

Key Takeaway

Cooperative diversity creates virtual MIMO. Two single-antenna users cooperating via decode-and-forward achieve a $2 \times 1$ MISO diversity order. Dynamic DF matches the full MISO DMT $d^*(r) = 2(1-r)$ by adapting the listening duration to the channel realization. The cost is bandwidth: the inter-user phase consumes resources that could otherwise carry data. Cooperation is most valuable when inter-user links are strong and the diversity gain outweighs the rate loss.

Quick Check

In coded cooperation, what is the role of the parity bits transmitted by the partner in Phase 2?

They repeat the original codeword for power gain

They provide incremental redundancy through an independent fading channel, creating a distributed code with diversity

They enable the destination to perform interference cancellation

Correction:

They provide incremental redundancy through an independent fading channel, creating a distributed code with diversity

The partner's parity bits, arriving through an independently faded channel, create a distributed code whose components experience independent fading. This is formally equivalent to a $2 \times 1$ space-time code, achieving diversity order 2.

⚠️Engineering Note

Practical Overhead of Cooperative Protocols

Cooperative diversity protocols require: (1) a mechanism for partner discovery and pairing, (2) channel estimation of the inter-user link (adding pilot overhead), (3) synchronization between cooperating users (within the cyclic prefix duration in OFDM systems), and (4) signaling to indicate successful/failed decoding at the relay. In LTE-Advanced and 5G NR, these overheads have limited the adoption of user-to-user cooperation. Instead, infrastructure cooperation (CoMP) is preferred, where base stations are connected by high-capacity backhaul and coordination is handled by the network.

Practical Constraints

•
Partner discovery adds latency (10-50 ms in typical protocols)
•
Inter-user CSI estimation requires additional pilot symbols
•
Timing synchronization within CP duration (4.7 us in normal CP, 5G NR)
•
Relay ACK/NACK signaling on control channel

Cooperative Diversity

Why User Cooperation?

Two-Phase Cooperative Diversity Protocol

Definition: The Cooperative Multiple-Access Channel

Cooperative diversity

Theorem: Achievable Rate Region for Cooperative DF

Codebook generation

Encoding at user 1

Decoding at destination

Example: Cooperative Gain in Symmetric Channels

Non-cooperative sum rate

Inter-user link constraint

Cooperative sum rate with equal power split

Interpretation

Definition: Diversity-Multiplexing Tradeoff (DMT)

Diversity-multiplexing tradeoff

Theorem: DMT of the Cooperative MAC

Outage probability analysis

Converse via cut-set

Achievability of dynamic DF

Diversity-Multiplexing Tradeoff: Non-Cooperative vs Cooperative

Parameters

Definition: Coded Cooperation

Coded cooperation

Historical Note: The Origins of Cooperative Communication

Common Mistake: Forgetting the Half-Duplex Constraint

Quick Check

Common Mistake: Assuming Any Partner Is Beneficial

Key Takeaway

Quick Check

Practical Overhead of Cooperative Protocols

Definition:
The Cooperative Multiple-Access Channel

Definition:
Diversity-Multiplexing Tradeoff (DMT)

Definition:
Coded Cooperation