Ferkans — Interactive Telecom Tutor

The Gaussian BC: Where Theory Meets Wireless

The Gaussian broadcast channel is the most important specialization of the degraded BC for wireless communications. It directly models the downlink of a single-antenna base station serving two users at different distances (and therefore different received SNRs).

The capacity region takes a particularly clean form: the auxiliary variable $U$ is Gaussian, and the tradeoff between the two users' rates is controlled by a single power-splitting parameter $\alpha$ . This makes the Gaussian BC one of the few multiuser channels where the capacity region can be computed in closed form and has a transparent operational interpretation.

Definition:
The Scalar Gaussian Broadcast Channel

The scalar Gaussian broadcast channel consists of a single transmitter and two receivers:

$Y_1 = X + Z_{1}, \qquad Y_2 = X + Z_{2},$

where $Z_{1} \sim \mathcal{N}(0, N_1)$ and $Z_{2} \sim \mathcal{N}(0, N_2)$ are independent, with $N_1 < N_2$ . The transmit power constraint is $\frac{1}{n}\sum_{i=1}^n x_i^2 \leq P$ .

User 1 is the strong user (lower noise variance) and user 2 is the weak user. As shown earlier, this channel is stochastically degraded: $X \to Y_1 \to Y_2$ .

Definition:
Power Splitting for the Gaussian BC

In the optimal superposition coding scheme for the Gaussian BC, the transmitted signal is:

$X = X_c + X_p,$

where:

$X_c \sim \mathcal{N}(0, \alpha P)$ carries the weak user's message (the cloud center), with power fraction $\alpha \in [0,1]$ ,
$X_p \sim \mathcal{N}(0, (1-\alpha)P)$ carries the strong user's message (the satellite), with the remaining power,
$X_c$ and $X_p$ are independent.

The parameter $\alpha$ is the power-splitting ratio: it controls the tradeoff between the two users' rates. More power to $X_c$ (larger $\alpha$ ) means higher rate for the weak user but lower rate for the strong user, and vice versa.

This corresponds to the general superposition code with $U = X_c$ and $X = U + X_p$ , where $p(x|u)$ is $\mathcal{N}(u, (1-\alpha)P)$ .

Theorem: Capacity Region of the Gaussian Broadcast Channel

The capacity region of the scalar Gaussian BC with noise variances $N_1 < N_2$ and transmit power $P$ is the set of all rate pairs $(R_{1}, R_{2})$ satisfying

$R_{2} \leq \frac{1}{2}\log\!\left(1 + \frac{\alpha P}{(1-\alpha)P + N_2}\right),$

$R_{1} \leq \frac{1}{2}\log\!\left(1 + \frac{(1-\alpha) P}{N_1}\right),$

for some $\alpha \in [0, 1]$ .

The weak user sees its signal $X_c$ corrupted by both the strong user's signal $X_p$ (treated as noise) and the additive noise $Z_{2}$ , giving an effective noise power of $(1-\alpha)P + N_2$ . The strong user first decodes and subtracts the cloud center $X_c$ , then sees only its satellite $X_p$ plus noise $Z_{1}$ , giving effective noise $N_1$ .

Proof

Achievability — Gaussian superposition coding

Apply the general degraded BC capacity region theorem with $U \sim \mathcal{N}(0, \alpha P)$ and $X | U = u \sim \mathcal{N}(u, (1-\alpha)P)$ . Then $X = U + X_p$ with $X_p \sim \mathcal{N}(0, (1-\alpha)P)$ independent of $U$ .

Weak user's rate:

$I(U; Y_2) = h(Y_2) - h(Y_2 | U).$

Since $Y_2 = U + X_p + Z_{2}$ :

$Y_2 \sim \mathcal{N}(0, P + N_2)$ , so $h(Y_2) = \frac{1}{2}\log(2\pi e(P + N_2))$ .
$Y_2 | U = X_p + Z_{2} \sim \mathcal{N}(U, (1-\alpha)P + N_2)$ , so $h(Y_2 | U) = \frac{1}{2}\log(2\pi e((1-\alpha)P + N_2))$ .

Therefore $I(U; Y_2) = \frac{1}{2}\log\frac{P + N_2}{(1-\alpha)P + N_2} = \frac{1}{2}\log\left(1 + \frac{\alpha P}{(1-\alpha)P + N_2}\right)$ .

Achievability — Strong user's rate

Strong user's rate:

$I(X; Y_1 | U) = h(Y_1 | U) - h(Y_1 | X).$

Since $Y_1 | U = X_p + Z_{1} \sim \mathcal{N}(U, (1-\alpha)P + N_1)$ :

$h(Y_1 | U) = \frac{1}{2}\log(2\pi e((1-\alpha)P + N_1))$ .
$h(Y_1 | X) = h(Z_{1}) = \frac{1}{2}\log(2\pi e N_1)$ .

Therefore $I(X; Y_1 | U) = \frac{1}{2}\log\frac{(1-\alpha)P + N_1}{N_1} = \frac{1}{2}\log\left(1 + \frac{(1-\alpha)P}{N_1}\right)$ .

Converse — Gaussian maximizes entropy

The converse requires showing that Gaussian $(U, X)$ is optimal. This follows from the entropy power inequality (Section 15.4) or from the maximum-entropy argument: among all distributions with a given covariance, the Gaussian maximizes differential entropy. Since both rate bounds involve differences of differential entropies, the Gaussian achieves the largest region. The full converse is Bergmans' argument, which we develop in Section 15.4.

Corner points and boundary

$\alpha = 1$ : All power to the weak user. $R_{1} = 0$ and $R_{2} = \frac{1}{2}\log(1 + P/N_2)$ .
$\alpha = 0$ : All power to the strong user. $R_{1} = \frac{1}{2}\log(1 + P/N_1)$ and $R_{2} = 0$ .
Varying $\alpha \in [0,1]$ traces the boundary of the capacity region — a curve that lies strictly above the TDMA line connecting the two corner points (whenever $N_1 \neq N_2$ ).

,

Example: Numerical Example: Gaussian BC Capacity Region

Consider a Gaussian BC with $P = 10$ , $N_1 = 1$ , $N_2 = 5$ . Compute the rate pairs at $\alpha = 0, 0.5, 0.8, 1$ and compare with the TDMA achievable rates.

Solution

Superposition coding rates

For each $\alpha$ :

$\alpha$	$R_{1}$	$R_{2}$
0	$\frac{1}{2}\log(1 + 10) = 1.73$	0
0.5	$\frac{1}{2}\log(1 + 5) = 1.29$	$\frac{1}{2}\log(1 + 5/10) = 0.29$
0.8	$\frac{1}{2}\log(1 + 2) = 0.79$	$\frac{1}{2}\log(1 + 8/7) = 0.47$
1	0	$\frac{1}{2}\log(1 + 10/5) = 0.79$

All rates in bits per channel use.

TDMA comparison

With TDMA, user 1 gets capacity $C_{1} = \frac{1}{2}\log(1 + P/N_1) = 1.73$ and user 2 gets $C_{2} = \frac{1}{2}\log(1 + P/N_2) = 0.79$ . Time-sharing with fraction $\theta$ gives $(\theta \cdot 1.73, (1-\theta) \cdot 0.79)$ .

At $\alpha = 0.8$ for superposition coding: $(0.79, 0.47)$ . For TDMA to achieve $R_{2} = 0.47$ , we need $(1-\theta) \cdot 0.79 = 0.47$ , giving $\theta = 0.41$ , and $R_{1} = 0.41 \times 1.73 = 0.70$ .

Superposition coding achieves $(0.79, 0.47)$ while TDMA achieves $(0.70, 0.47)$ — a 13% improvement in $R_{1}$ at the same $R_{2}$ .

Key observation

The superposition coding boundary is a convex curve that lies strictly above the TDMA straight line. The gain is largest when the users have very different channel qualities (large $N_2/N_1$ ), because there is more "room" for layering.

Theorem: Superposition Coding Strictly Outperforms Orthogonal Access

For the Gaussian BC with $N_1 < N_2$ , the capacity region achieved by superposition coding strictly contains the TDMA/FDMA achievable region (except at the two corner points where a single user is served).

TDMA wastes resources because when user 1 is being served, user 2 receives no information, and vice versa. Superposition coding serves both users simultaneously: the weak user decodes its cloud center in every time slot, while the strong user decodes both layers. The strong user effectively "recycles" the power allocated to the weak user's message — it decodes and subtracts it, losing nothing.

Proof

TDMA achievable region

With time fraction $\theta$ for user 1:

$R_{1}^{\text{TDMA}} = \theta \cdot \frac{1}{2}\log\!\left(1 + \frac{P}{N_1}\right), \quad R_{2}^{\text{TDMA}} = (1-\theta) \cdot \frac{1}{2}\log\!\left(1 + \frac{P}{N_2}\right).$

This traces a straight line between the corner points $(C_{1}, 0)$ and $(0, C_{2})$ .

Superposition coding achieves more

At any interior point of the TDMA line with $R_{2} > 0$ , set $\alpha$ such that the superposition $R_{2}$ equals the TDMA $R_{2}$ . Then:

$R_{1}^{\text{SC}} = \frac{1}{2}\log\!\left(1 + \frac{(1-\alpha)P}{N_1}\right) > \theta \cdot \frac{1}{2}\log\!\left(1 + \frac{P}{N_1}\right) = R_{1}^{\text{TDMA}}$

whenever $N_1 < N_2$ . The strict inequality follows from the concavity of $\log$ and the fact that superposition coding uses the full bandwidth all the time, while TDMA splits the time.

The intuition

In TDMA, user 1 is silent for a fraction $(1-\theta)$ of the time. In superposition coding, user 1 gets rate $\frac{1}{2}\log(1 + (1-\alpha)P/N_1)$ in every time slot. The price is that user 2 treats user 1's signal as noise — but this cost is borne by the weak user, not the strong one. And the weak user was going to suffer from noise anyway; the marginal noise from user 1's signal is small compared to $N_2$ .

Superposition Coding vs. Orthogonal Multiple Access

Property	Superposition Coding	TDMA/FDMA
Achievable region	Full capacity region	Strict subset (except corner points)
Bandwidth usage	Both users use full bandwidth simultaneously	Users allocated disjoint time/frequency slots
Receiver complexity	Strong user requires SIC	Simple single-user decoder
Power allocation	Power splitting parameter $\alpha$	Each user uses full power in its slot
Gain over orthogonal	Largest when $N_2/N_1 \gg 1$	Baseline
Robustness	Sensitive to SIC errors	Robust — no inter-user interference

Gaussian BC Capacity Region

Visualize the capacity region of the Gaussian broadcast channel. The blue curve shows the superposition coding boundary; the dashed line shows the TDMA region. Adjust power and noise variances.

Parameters

P

(transmit power)10

N_1

(strong user noise)1

N_2

(weak user noise)5

Rate Tradeoff: $\alpha$ Sweeps the Boundary

See how varying the power-splitting parameter $\alpha$ traces the boundary of the Gaussian BC capacity region. The slider controls $\alpha$ , and the plot shows the corresponding rate pair.

Parameters

\alpha

(power fraction for weak user)0.5

P

(transmit power)10

N_1

1

N_2

5

Quick Check

In the Gaussian BC with power $P = 20$ , $N_1 = 1$ , $N_2 = 10$ , which power-splitting $\alpha$ maximizes the weak user's rate?

$\alpha = 0$

$\alpha = 0.5$

$\alpha = 1$

$\alpha = N_1 / N_2 = 0.1$

Correction:

\alpha = 1

With $\alpha = 1$ , all power goes to the weak user's cloud center, giving $R_{2} = \frac{1}{2}\log(1 + P/N_2) = \frac{1}{2}\log(3) \approx 0.79$ bits. The strong user gets rate 0.

⚠️Engineering Note

NOMA in 5G: Superposition Coding in Practice

The superposition coding principle is implemented in modern cellular systems under the name NOMA (Non-Orthogonal Multiple Access). In power-domain NOMA, the base station serves a near user (strong) and a far user (weak) simultaneously by superimposing their signals with different power levels.

Key practical considerations:

SIC error propagation: The strong user must decode and subtract the weak user's signal. Decoding errors propagate and degrade performance. In practice, the error rate of SIC is higher than the theoretical prediction.
Channel estimation: Both users need accurate CSI. The strong user needs channel estimates for both the direct channel and the effective channel after SIC.
User pairing: NOMA works best when users have significantly different channel qualities ( $N_2/N_1 \gg 1$ ). Pairing users with similar channels yields negligible gains over OFDMA.
3GPP status: While studied extensively for 5G NR, power-domain NOMA was not adopted in the initial 3GPP Release 15/16 standards, in part due to the complexity of multi-user SIC receivers. Research continues for Release 18+ (6G).

🔧Engineering Note

From Scalar to MIMO: Dirty Paper Coding

The superposition coding result for the scalar Gaussian BC extends to the MIMO case, but with a crucial twist: the MIMO BC is generally not degraded (unless the channel matrices are specially structured).

For the MIMO BC, the capacity-achieving scheme is dirty paper coding (DPC), which can be viewed as a generalization of superposition coding combined with Gel'fand–Pinsker precoding. The transmitter encodes user $k$ 's message while treating the signals of users $1, \ldots, k-1$ as known interference (dirt), using Costa's result to pre-cancel them.

The DPC capacity region of the MIMO BC equals the MAC-BC dual region — a remarkable result that connects the downlink capacity to the easier uplink optimization (Chapter 16).

Historical Note: Bergmans and the Gaussian BC Converse

1970s

Patrick Bergmans proved the capacity region of the degraded Gaussian broadcast channel in 1973, shortly after Cover proposed superposition coding. Bergmans' key insight was to use the entropy power inequality (EPI) to establish the converse.

The challenge was to show that Gaussian inputs are optimal — i.e., that no non-Gaussian choice of $(U, X)$ can enlarge the capacity region. The EPI provides exactly the right tool: it shows that Gaussian noise is the "hardest" noise, which means Gaussian inputs are the "best" inputs.

The full proof for the $K$ -user Gaussian BC came later, with contributions from Gallager (1974) and the culmination in the EPI-based approach. The MIMO extension required entirely different tools (MAC-BC duality and Weingarten, Steinberg, and Shamai's 2006 result).

Common Mistake: Confusing $\alpha$ as the Strong User's Power Fraction

Mistake:

Using $\alpha$ for the strong user's power fraction and $(1-\alpha)$ for the weak user, leading to reversed rate expressions.

Correction:

In the standard convention (Cover & Thomas, El Gamal & Kim), $\alpha$ is the weak user's power fraction. The strong user gets $(1-\alpha)P$ . This convention is natural because the weak user needs more power to maintain a nonzero rate.

Always check which convention is being used when reading papers. Some authors reverse the convention. What matters is the mapping between the user labels and the noise variances.

Gaussian BC Capacity Region vs TDMA

The power-splitting parameter

\alpha

traces the boundary of the Gaussian BC capacity region — a convex curve that lies strictly above the TDMA straight line. The animation shows the operating point sliding along the boundary as

\alpha

varies from 0 to 1.

The Gaussian Broadcast Channel

The Gaussian BC: Where Theory Meets Wireless

Definition: The Scalar Gaussian Broadcast Channel

Definition: Power Splitting for the Gaussian BC

Theorem: Capacity Region of the Gaussian Broadcast Channel

Achievability — Gaussian superposition coding

Achievability — Strong user's rate

Converse — Gaussian maximizes entropy

Corner points and boundary

Example: Numerical Example: Gaussian BC Capacity Region

Superposition coding rates

TDMA comparison

Key observation

Theorem: Superposition Coding Strictly Outperforms Orthogonal Access

TDMA achievable region

Superposition coding achieves more

The intuition

Superposition Coding vs. Orthogonal Multiple Access

Gaussian BC Capacity Region

Parameters

Rate Tradeoff: α\alphaα Sweeps the Boundary

Parameters

Quick Check

NOMA in 5G: Superposition Coding in Practice

From Scalar to MIMO: Dirty Paper Coding

Historical Note: Bergmans and the Gaussian BC Converse

Common Mistake: Confusing α\alphaα as the Strong User's Power Fraction

Gaussian BC Capacity Region vs TDMA

Definition:
The Scalar Gaussian Broadcast Channel

Definition:
Power Splitting for the Gaussian BC

Rate Tradeoff: $\alpha$ Sweeps the Boundary

Common Mistake: Confusing $\alpha$ as the Strong User's Power Fraction