Multiple access channel
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| Multiple access channel | |
|---|---|
| Name | Multiple access channel |
| Field | Information theory |
| Introduced | 1960s |
| Related | Multiple-input multiple-output; Broadcast channel; Relay channel; Network information theory |
Multiple access channel A multiple access channel is a foundational model in information theory and electrical engineering describing how several transmitters send information to a single receiver over a shared medium. It connects concepts from Claude Shannon's A Mathematical Theory of Communication, Robert G. Gallager's coding theory, the IEEE community, and modern wireless communication standards such as LTE and 5G NR. The model underpins research at institutions like Bell Labs, MIT, Stanford University, and Caltech and informs protocols in projects including IEEE 802.11, 3GPP, and ETSI.
Introduction
The multiple access channel (MAC) formalizes interactions among multiple senders and a single receiver using probabilistic channel maps introduced in early treatments by Abraham Wyner and Thomas M. Cover and later extended by Aaron D. Wyner and I. Csiszár in network contexts. It can be discrete, continuous, or fading, with canonical examples studied in the literature from Shannon to contemporary papers at ISIT and in journals like IEEE Transactions on Information Theory and Proceedings of the IEEE. Research on the MAC intersects with work on the broadcast channel, interference channel, and relay channel and benefits from mathematical tools developed in probability theory, convex optimization, and measure theory.
Channel Model and Definitions
The MAC model specifies input alphabets for each encoder, an output alphabet for the decoder, and a family of conditional probability measures P(y|x1,x2,...,xk) as in formulations by Shannon and generalized by Cover and Thomas and El Gamal. Key definitions include messages, encoders, decoders, blocklength, codebooks, average error probability, and memoryless channel assumptions used in proofs by Gallager and in textbooks from Cambridge University Press and Prentice Hall. Formal tools such as typicality, jointly typical sets, and mutual information play roles traced to works by David Slepian, Jacques Wolf, and I. Csiszár.
Capacity Region and Achievable Rates
The capacity region of the MAC is the closure of all achievable rate tuples characterized in seminal results by Ahlswede and Liao and succinctly presented in Cover and Thomas. For the two-user discrete memoryless MAC, the region is given by constraints involving mutual informations I(X1;Y|X2), I(X2;Y|X1), and I(X1,X2;Y), with converse proofs leveraging techniques from Fano's inequality and entropy bounds used by Wyner and Ziv. Extensions to Gaussian MACs connect to results by Tse and Viswanath and dovetail with capacity formulas employed in Shannon's continuous channel theory and practical power allocation schemes studied at Bell Labs and in 3GPP white papers.
Coding Schemes and Techniques
Achievability proofs for the MAC exploit random coding, joint typicality decoding, successive decoding, and superposition coding as developed by Cover, El Gamal, and Slepian and Wolf. Practical coding methods include turbo codes and LDPC codes whose analysis draws on work by Guanas, David MacKay, and Claude Berrou, as well as polar codes introduced by Erdal Arıkan and applied to multiuser settings in research from Ecole Polytechnique and University of Toronto. Rate-splitting, successive interference cancellation, and joint decoding strategies are informed by studies at Nokia Bell Labs, Qualcomm, and in standards bodies like 3GPP and IEEE 802.11.
Special Cases and Variants
Special cases include the two-user MAC, Gaussian MAC, fading MAC, and asymmetric MAC treated in analyses by Tse, Goldsmith, Larsson, and Gesbert. Variants encompass the compound MAC, cognitive MAC studied in work by Devroye and Jovicic, the multiple access relay channel explored by Kramer and Gastpar, and the cooperative MAC linked to results from Wyner and Ozarow. Connections to the interference channel, X-channel, and multiple-input multiple-output (MIMO) systems show interplay with advances from Telatar, Foschini, and Paulraj.
Applications and Practical Considerations
MAC theory informs spectrum sharing, uplink design, random access protocols, and resource allocation in systems engineered by Qualcomm, Huawei, Ericsson, and Nokia; it underlies standards like LTE, 5G NR, and technologies such as Wi-Fi and Bluetooth. Practical considerations include channel state information, power control, synchronization, and latency constraints addressed in work by Goldsmith, Tse, Viswanath, and algorithmic implementations in testbeds at MIT, Stanford University, and ETH Zurich. Advanced topics apply MAC results to network coding, distributed storage systems as in research at Microsoft Research and IBM Research, and to multiuser data compression studied by Slepian and Wolf.