Chapter Summary

Chapter Summary: The Multiple Access Channel

Key Points

• 1.
  MAC Capacity Region

  The capacity region of the two-user DM-MAC is the convex hull of all rate pairs satisfying $R_{1} \leq I(X_1;Y|X_2)$, $R_{2} \leq I(X_2;Y|X_1)$, and $R_{1} + R_{2} \leq I(X_1,X_2;Y)$, over all product input distributions. For a fixed distribution, this is a pentagon in the rate plane.

• 2.
  Joint Typicality and SIC

  Joint typicality decoding achieves the full pentagon by searching over all codeword pairs. Successive cancellation decoding has lower complexity and achieves the corner points of the dominant face. The entire dominant face is obtained by time-sharing between the two SIC decoding orders.

• 3.
  Gaussian MAC

  For the Gaussian MAC ($Y = X_1 + X_2 + Z$), Gaussian inputs are optimal and no time-sharing is needed. The sum rate $\frac{1}{2}\log(1+(P_1+P_2)/N)$ strictly exceeds the TDMA sum rate, establishing the information-theoretic superiority of non-orthogonal multiple access (NOMA).

• 4.
  MIMO MAC

  The MIMO MAC capacity region is parameterized by the users' input covariance matrices and achieved by Gaussian inputs with SIC. When the channel vectors are orthogonal (as in massive MIMO), the MAC penalty vanishes and each user achieves its interference-free capacity.

• 5.
  $K$-User MAC and Polymatroidal Structure

  The $K$-user MAC capacity region requires $2^K - 1$ subset inequalities. The region is a polymatroid, characterized by the submodularity of the mutual information set function. SIC with a specific decoding order achieves one of $K!$ corner points.

• 6.
  Fading MAC and Multiuser Diversity

  With CSIR only, the fading MAC capacity is the ergodic average of the fixed-channel capacity. With CSIT, opportunistic scheduling (serving the best user) is sum-rate optimal. Multiuser diversity provides a throughput gain that scales as $\log\log K$, motivating channel-aware schedulers in 4G/5G systems.