# Interference channel

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In [information theory](/source/information_theory), the '''interference channel''' is the basic model used to analyze the effect of [interference](/source/Interference_(communication)) in communication channels. The model consists of two pairs of users communicating through a shared channel. The problem of interference between two mobile users in close proximity or [crosstalk](/source/crosstalk) between two parallel [landlines](/source/landline) are two examples where this model is applicable.

Unlike in the [point-to-point channel](/source/Point-to-point_(telecommunications)), where the amount of information that can be sent through the channel is limited by the noise that distorts the transmitted signal, in the interference channel the presence of the signal from the other user may also impair the communication. However, since the transmitted signals are not purely random (otherwise they would not be decodable), the receivers may be able to reduce the effect of the interference by partially or totally decoding the undesired signal.

== Discrete memoryless interference channel ==
The mathematical model for this channel is the following:

center|800px|Interference channel model

where, for <math>i\in\{1,2\}</math>:
* <math>W_i</math> is the message to be transmitted by user <math>i</math>;
* <math>X_i</math> is the channel input symbol (<math>X_i^n</math> is a sequence of <math>n</math> symbols) of user <math>i</math>;
* <math>Y_i</math> is the channel output symbol (<math>Y_i^n</math> is a sequence of <math>n</math> symbols) of user <math>i</math>;
* <math>\hat{W}_i</math> is the estimate of the transmitted message by user <math>i</math>; and
* <math>p(y_1,y_2|x_1,x_2)</math> is the noisy memoryless channel, which is modeled by a [conditional probability distribution](/source/conditional_probability_distribution).

The capacity of this channel model is not known in general; only for special cases of <math>p(y_1,y_2|x_1,x_2)</math> the capacity has been calculated, e.g., in the case of strong interference or deterministic channels.<ref name="ElGamalKim">{{cite book |last1=El Gamal |first1=Abbas |author1-link=Abbas El Gamal |last2=Kim |first2=Young-Han |title=Network Information Theory |publisher=Cambridge University Press |year=2011 |isbn=978-1-107-00873-1}}</ref><ref>{{cite journal |last=Carleial |first=A. B. |title=Interference channels |journal=IEEE Transactions on Information Theory |volume=24 |issue=1 |pages=60–70 |date=January 1978 |doi=10.1109/TIT.1978.1055812}}</ref>

A widely used inner bound is the Han–Kobayashi achievable region, introduced by [Te Sun Han](/source/Te_Sun_Han) and Kingo Kobayashi in 1981.<ref name="HanKobayashi1981">{{cite journal |last1=Han |first1=Te Sun |last2=Kobayashi |first2=Kingo |title=A new achievable rate region for the interference channel |journal=IEEE Transactions on Information Theory |volume=27 |issue=1 |pages=49–60 |date=January 1981 |doi=10.1109/TIT.1981.1056307}}</ref> In this scheme, each sender splits its message into a private part and a common part, so that receivers can decode part of the interfering transmission while treating the remaining interference as noise.<ref name="ElGamalKim" /> For the Gaussian interference channel, a later refinement of the Han–Kobayashi approach was shown to achieve the capacity region to within one bit.<ref>{{cite journal |last1=Etkin |first1=Raul H. |last2=Tse |first2=David N. C. |last3=Wang |first3=Hua |title=Gaussian Interference Channel Capacity to Within One Bit |journal=IEEE Transactions on Information Theory |date=December 2008 |volume=54 |issue=12 |pages=5534–5562 |doi=10.1109/TIT.2008.2006447}}</ref>

==References==
{{reflist}}
<!-- * [A. El Gamal](/source/Abbas_El_Gamal), Y.-H. Kim, ''Network Information Theory'', [Cambridge University Press](/source/Cambridge_University_Press), 2011. {{ISBN|978-1-107-00873-1}} -->
===Further references===
{{refbegin}}
* [R. Ahlswede](/source/Rudolf_Ahlswede), “The Capacity Region of a Channel with Two Senders and Two Receivers,” [The Annals of Probability](/source/The_Annals_of_Probability), vol. 2, No. 5, pp. 805–814, Oct. 1974.
{{refend}}

===Extensions===
{{refbegin}}
* O. Sahin and E. Erkip, “Achievable Rates for the Gaussian Interference Relay Channel,” in [IEEE Global Telecommunications Conference, 2007](/source/Global_Communications_Conference), Nov. 2007, pp. 1627–1631.
* I. Marić, R. Dabora, and A. J. Goldsmith, “Relaying in the Presence of Interference: Achievable Rates, Interference Forwarding, and Outer Bounds,” [IEEE Trans. Inf. Theory](/source/IEEE_Trans._Inf._Theory), vol. 58, no. 7, pp. 4342–4354, Jul. 2012.
* G. Bassi, P. Piantanida, and S. Yang, “Capacity Bounds for a Class of Interference Relay Channels,” [IEEE Trans. Inf. Theory](/source/IEEE_Trans._Inf._Theory), vol. 61, no. 7, pp. 3698–3721, Jul. 2015.
{{refend}}

Category:Information theory
Category:Telecommunication theory

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