In a degraded Gaussian broadcast channel, user 1 has better SNR than user 2. The sender uses superposition coding, allocating power alpha*P to user 2's message and (1-alpha)*P to user 1's message (0 < alpha < 1). Why does user 2 get more power?
AUser 2 needs more power because they have a weaker channel — their message must be robust enough to decode despite higher noise
BUser 2 always has higher priority in broadcast systems
CThe power allocation is arbitrary and does not affect capacity
DUser 1 gets more power because they have the better channel
In superposition coding, user 2's message is encoded at higher power so that BOTH receivers can decode it — user 2 (weak receiver) decodes only this high-power layer. User 1 (strong receiver) first decodes user 2's message (treating user 1's message as noise), subtracts it (like SIC), then decodes their own lower-power message from the residual. The stronger receiver can afford to decode the weaker receiver's message because it has better channel quality. The power split alpha controls the rate tradeoff between users.
Question 2 True / False
The broadcast channel is simply the multiple access channel with the communication direction reversed, so their capacity regions are identical.
TTrue
FFalse
Answer: False
The BC and MAC are duals in a specific mathematical sense — the Gaussian BC and MAC capacity regions are related by a duality transformation (the sum-power constraint maps between them). But they are NOT 'the same channel reversed.' The MAC has independent senders with no coordination; the BC has a single sender with full control. The MAC uses SIC at the receiver; the BC uses superposition coding at the transmitter. The achievability techniques, converse proofs, and information-theoretic challenges are different. The MAC capacity region is completely known for general channels; the general BC is not.
Question 3 Short Answer
Explain superposition coding and why it outperforms time-division (TDMA) on the degraded broadcast channel.
Think about your answer, then reveal below.
Model answer: In superposition coding, the sender transmits both messages simultaneously: X = X_1 + X_2, where X_2 (for the weaker user) has higher power alpha*P and X_1 (for the stronger user) has power (1-alpha)*P. The weak user decodes X_2 while treating X_1 as noise. The strong user first decodes X_2 (because they have a better channel), subtracts it, then decodes X_1. Both users receive data simultaneously without dividing time or bandwidth. TDMA gives user 1 a fraction t of the time at full power and user 2 the remaining (1-t), so each user's rate is scaled by their time fraction. Superposition coding beats TDMA because the strong user can 'see through' the weak user's message, extracting their own message from the residual — this concurrent transmission utilizes the channel more efficiently than orthogonal sharing.
The intuition is that the channel quality difference between users is an asset, not a liability. The strong user's ability to decode the weak user's message first creates a natural layering that TDMA cannot exploit. This is the information-theoretic basis for NOMA (non-orthogonal multiple access) in 5G downlink.