Ferkans — Interactive Telecom Tutor

The Art of Partial Decoding

The strong interference regime says: decode all of the interference. The TIN strategy says: decode none of it. The brilliant insight of Han and Kobayashi (1981) is that in the intermediate regime, you should decode part of the interference.

Each user splits its message into a common part (decodable by both receivers) and a private part (decodable only by the intended receiver). The common part of the interfering user is decoded and removed; the private part is treated as noise. The power split between common and private messages is a design parameter that allows the scheme to interpolate smoothly between TIN (all private) and full interference decoding (all common).

This is the same rate-splitting idea used in the MAC (time-sharing via rate splitting), but here it serves a fundamentally different purpose: managing interference rather than coordinating access.

Definition:
Han-Kobayashi Rate Splitting

Each user $k \in \{1, 2\}$ splits its message $M_k$ into:

A common message $M_{0k}$ at rate $R_{0k}$ , intended for both receivers
A private message $M_{pk}$ at rate $R_{pk}$ , intended only for Rx $k$

Total rate: $R_{k} = R_{0k} + R_{pk}$ .

The transmitted signal is $X_k = U_k + V_k$ , where:

$U_k$ carries the common message $M_{0k}$ (decoded by both Rx 1 and Rx 2)
$V_k$ carries the private message $M_{pk}$ (decoded only by Rx $k$ )
Power allocation: $\mathbb{E}[|U_k|^2] = \alpha_k P_k$ , $\mathbb{E}[|V_k|^2] = (1-\alpha_k) P_k$

The parameter $\alpha_k \in [0,1]$ controls the power split: $\alpha_k = 0$ is pure TIN, $\alpha_k = 1$ is full common (strong interference strategy).

Theorem: The Han-Kobayashi Achievable Region

For the two-user interference channel, the following rate region is achievable. For each user $k$ , split the rate as $R_{k} = R_{0k} + R_{pk}$ . At receiver 1, jointly decode $(M_{01}, M_{p1}, M_{02})$ — the common messages of both users and user 1's private message — treating $V_2$ (user 2's private signal) as noise. The rate constraints at Rx 1 are:

$R_{p1} \leq I(V_1; Y_1 | U_1, U_2)$ $R_{02} \leq I(U_2; Y_1 | U_1, V_1)$ $R_{01} + R_{p1} \leq I(U_1, V_1; Y_1 | U_2)$ $R_{p1} + R_{02} \leq I(V_1, U_2; Y_1 | U_1)$ $R_{01} + R_{p1} + R_{02} \leq I(U_1, V_1, U_2; Y_1)$

Symmetric constraints apply at Rx 2. The full HK region is the union over all power splits $(\alpha_1, \alpha_2)$ and input distributions.

Receiver 1 treats $(U_1, V_1, U_2)$ as a three-user MAC: it jointly decodes its own common and private messages plus user 2's common message. User 2's private signal $V_2$ is treated as noise (with power $(1-\alpha_2) P_2$ ). The more power user 2 puts into the common message (larger $\alpha_2$ ), the less noise Rx 1 sees — but user 2's private rate decreases. This tradeoff is what makes the HK bound powerful.

Proof

Codebook generation

For each user $k$ , generate:

$2^{nR_{0k}}$ common codewords $U_k^n(m_{0k}) \sim \prod P_{U_k}$
$2^{nR_{pk}}$ private codewords $V_k^n(m_{pk}) \sim \prod P_{V_k}$
Transmitted signal: $X_k^n = U_k^n(m_{0k}) + V_k^n(m_{pk})$

Joint decoding at Rx 1

Receiver 1 finds the unique $(\hat{m}_{01}, \hat{m}_{p1}, \hat{m}_{02})$ such that $(U_1^n(\hat{m}_{01}), V_1^n(\hat{m}_{p1}), U_2^n(\hat{m}_{02}), Y_1^n)$ is jointly typical. By the packing lemma for the MAC, decoding succeeds with high probability if the rate constraints are satisfied.

Rate region via Fourier-Motzkin

The five constraints at Rx 1 (and five at Rx 2) define a region in the six-dimensional space $(R_{01}, R_{p1}, R_{02}, R_{p2}, R_{01}, R_{02})$ . Fourier-Motzkin elimination of the common rates $R_{01}, R_{02}$ produces the HK region in terms of the operational rates $(R_{1}, R_{2})$ .

Optimization over power split

The full HK region is the union over all $\alpha_1, \alpha_2 \in [0,1]$ of the above regions. The optimal power split depends on the interference parameters $a, b$ and the SNR values. In general, this optimization must be done numerically.

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Theorem: Within One Bit of Capacity (Etkin-Tse-Wang)

For the two-user Gaussian IC, the Han-Kobayashi scheme with a specific choice of power split achieves a rate region that is within 1 bit per user of the capacity region: $R_{k}^{\text{HK}} \geq R_{k}^* - 1 \quad \text{for } k = 1, 2$

where $R_{k}^*$ is the capacity. The specific choice is to set the private message power at the level of the noise floor at the unintended receiver: $(1 - \alpha_k) P_k = \min\left(P_k, \frac{\sigma^2}{b_k^2}\right)$ where $b_1 = b$ and $b_2 = a$ are the cross-channel gains.

The Etkin-Tse-Wang (ETW) result is one of the most elegant in interference channel theory. The idea is to set the private message power just below the noise floor at the unintended receiver — so the private message causes negligible additional interference. The common message uses the remaining power and is decoded by both receivers. This simple power split is within 1 bit of optimal for all parameter regimes.

The "1 bit" gap comes from the fact that TIN incurs at most a 1-bit penalty per user in the "noisy interference" regime (where $\text{INR} < 1 + \text{SNR}$ ), and the HK scheme with the ETW power split handles the transition between regimes smoothly.

Proof

Outer bound (genie-aided)

Provide Rx 1 with a genie that reveals either $S_1 = \sqrt{b^2 P_1 + \sigma^2} Z_1 + b X_1$ (a "noisy" version of the interference) or the interference $X_2^n$ directly. These genie-aided bounds produce outer bounds that can be evaluated in closed form.

Achievable rate with ETW power split

With the ETW private power level $(1-\alpha_k)P_k = \sigma^2/b_k^2$ :

Private rate: $R_{pk} = \frac{1}{2}\log(1 + \sigma^2/(b_k^2 \sigma^2)) \leq \frac{1}{2}$ bit
Common rate: optimized via the MAC constraints at each receiver
Total rate: within 1 bit of the outer bound

The 1-bit gap

The gap arises because the outer bound assumes perfect side information (genie), while the achievable scheme has finite-precision interference management. The 1-bit gap is independent of SNR and INR — it is a universal constant, which is remarkable for such a complex channel.

Historical Note: The Han-Kobayashi Bound: Four Decades and Counting

1981-2008

Te Sun Han and Kingo Kobayashi published their achievable region in 1981, and for over 40 years it has remained the best known inner bound for the general interference channel. The original paper considered the discrete memoryless case; the Gaussian case and the "within 1 bit" result by Etkin, Tse, and Wang (2008) added the crucial quantitative insight.

What makes the HK bound so resilient is that it combines the only two strategies available for dealing with interference: decode it (common message) or tolerate it (private message treated as noise). Any improvement would require a fundamentally new idea — and despite decades of effort, none has been found for the general two-user case.

Example: ETW Power Split for the Symmetric IC

For the symmetric Gaussian IC with $P_1 = P_2 = P = 100$ , $\sigma^2 = 1$ , and $a = b = 0.5$ (moderate interference: $\text{INR} = 25$ ), compute the ETW power split and the resulting sum rate. Compare with TIN.

Solution

ETW power split

Private power: $(1-\alpha)P = \sigma^2/a^2 = 1/0.25 = 4$ . Common power: $\alpha P = 100 - 4 = 96$ . So $\alpha = 0.96$ — almost all power goes to the common message.

TIN sum rate

$R_{\text{TIN}} = 2 \times \frac{1}{2}\log\left(1 + \frac{100}{0.25 \times 100 + 1}\right) = 2 \times \frac{1}{2}\log\left(1 + \frac{100}{26}\right) \approx 2 \times 1.03 = 2.06 \text{ bits}$ $

HK sum rate (approximate)

With the ETW split, Rx 1 decodes $(U_1, V_1, U_2)$ jointly. The effective noise includes only the private part of user 2: $(1-\alpha) a^2 P = 0.04 \times 0.25 \times 100 = 1$ . So the effective noise at Rx 1 for the common messages is approximately $\sigma^2 + 1 = 2$ . The common-message MAC sum rate is approximately $\frac{1}{2}\log(1 + (96 + 0.25 \times 96)/2) \approx \frac{1}{2}\log(61) \approx 2.97$ bits. Adding private rates: total sum rate $\approx 2.97 + 2 \times 0.5 = 3.97$ bits, a significant improvement over TIN's 2.06 bits.

Gap to outer bound

The outer bound (interference-free) gives $2 \times \frac{1}{2}\log(101) \approx 6.66$ bits. The HK sum rate of $\approx 3.97$ bits is well within 2 bits of this bound, consistent with the ETW guarantee.

Rate splitting

A coding technique where each user divides its message into sub-messages (typically "common" and "private") encoded at different power levels. The common message is decodable by unintended receivers; the private message is not. Rate splitting enables partial interference decoding in the interference channel.

Common Mistake: Common Message Does Not Mean Multicast

Mistake:

Interpreting the "common message" in Han-Kobayashi as useful data for the unintended receiver, as in a multicast setting.

Correction:

The "common" message carries data intended only for the corresponding user. It is called "common" because it is encoded at a power level that allows the unintended receiver to decode it as well — but the unintended receiver discards the decoded data. The purpose of decoding it is to subtract interference, not to obtain useful information.

Quick Check

In the ETW power split, the private message power is set to $\sigma^2/a^2$ . What is the operational significance of this choice?

The private signal arrives at the unintended receiver at the same power as the noise, making it negligible

It minimizes the total power consumption

It maximizes the private message rate

Correction:

The private signal arrives at the unintended receiver at the same power as the noise, making it negligible

With private power $(1-\\alpha)P = \\sigma^2/a^2$ , the interference from the private part at the unintended receiver is $a^2 \\cdot \\sigma^2/a^2 = \\sigma^2$ . This is exactly the noise power — so the private signal adds at most 3 dB of additional noise, which costs at most 1 bit in rate.

Key Takeaway

The Han-Kobayashi scheme achieves rates within 1 bit of the two-user Gaussian IC capacity for all parameter regimes. The key idea is rate splitting: divide each message into a common part (decoded and subtracted by the unintended receiver) and a private part (treated as noise). The ETW power split — setting the private power at the noise floor of the unintended receiver — is a near-universal prescription that works across all interference regimes.

The Han-Kobayashi Achievable Region

The Art of Partial Decoding

Definition: Han-Kobayashi Rate Splitting

Theorem: The Han-Kobayashi Achievable Region

Codebook generation

Joint decoding at Rx 1

Rate region via Fourier-Motzkin

Optimization over power split

Theorem: Within One Bit of Capacity (Etkin-Tse-Wang)

Outer bound (genie-aided)

Achievable rate with ETW power split

The 1-bit gap

Historical Note: The Han-Kobayashi Bound: Four Decades and Counting

Example: ETW Power Split for the Symmetric IC

ETW power split

TIN sum rate

HK sum rate (approximate)

Gap to outer bound

Rate splitting

Common Mistake: Common Message Does Not Mean Multicast

Quick Check

Key Takeaway

Definition:
Han-Kobayashi Rate Splitting