Two-Unicast Wireless Networks: Characterizing the Degrees of Freedom

Shomorony, Ilan; Avestimehr, A. Salman

doi:10.1109/tit.2012.2214024

“…Consider, for example the setting M = N that is previously solved by Cadambe and Jafar in [7]. Cadambe and Jafar create this infinite chain of alignments using the asymptotic alignment scheme for M = N = 1 and implicitly create an infinite alignment chain in the non-asymptotic solution for, e.g., M = N = 2, as the chain closes upon itself to form a loop, i.e., V 1(1) = V 1 (2) . The chain closes upon itself mainly because in this setting (as well as all cases where M = N > 1) the optimal signal vectors are eigenvectors of the cumulative channel encountered in traversing the alignment chain starting from any transmitter and continuing until we return to the same transmitter.…”

Section: Thementioning

confidence: 99%

“…where the last inequality is obtained because knowingȲ n 1 we can decode W 1 , and thus we obtain S n 1a 2 which is a noisy version of X n 3a 2 . Then we use the fact that dropping the condition terms cannot decrease the differential entropy.…”

Section: Casementioning

confidence: 99%

See 1 more Smart Citation

Subspace alignment chains and the degrees of freedom of the three-user MIMO interference channel

Wang

¹

,

Gou

²

,

Jafar

³

2012

2012 IEEE International Symposium on Information Theory Proceedings

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We show that the 3-user M T × M R MIMO interference channel where each transmitter is equipped with M T antennas and each receiver is equipped with M R antennas has d(M, N ) = min M 2−1/κ , N 2+1/κ degrees of freedom (DoF) normalized by time, frequency, and space dimensions, whereWhile the DoF outer bound of d(M, N ) is established for every M T , M R value, the achievability of d(M, N ) DoF is established in general subject to a normalization with respect to spatial-extensions, i.e., the scaling of the number of antennas at all nodes. Specifically, we show that qd(M, N ) DoF are achievable for the 3-user qM T × qM R MIMO interference channel, for some positive integer q which may be seen as a spatial-extension factor. q is the scaling factor needed to make the value qd(M, N ) an integer. Given spatial-extensions, the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency. In the absence of spatial extensions, it is shown through examples how essentially the same interference alignment scheme may be applied over time-extensions over either constant or time-varying channels. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the DoF outer bound, as well as in the construction of the optimal interference alignment schemes.The DoF value d(M, N ) is a piecewise linear function of M, N , with either M or N being the bottleneck within each linear segment while the other value contains some redundancy, i.e., it can be reduced without reducing the DoF. The corner points of these piecewise linear segments correspond to two sets, A = {1/2, 2/3, 3/4, · · · } and B = {1/3, 3/5, 5/7, · · · }. The set A contains all those values of M/N and only those values of M/N for which there is redundancy in both M and N , i.e., either can be reduced without reducing DoF. The set B contains all those values of M/N and only those values of M/N for which there is no redundancy in either M or N , i.e., neither can be reduced without reducing DoF. Because A and B represent settings with maximum and minimum redundancy, essentially they are the basis for the DoF outer bounds and inner bounds, respectively.Our Our results show that M/N ∈ A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers. This may also be seen as a consequence of the maximum redundancy in the M/N ∈ A settings.

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“…Since we have already finished the proof for (M, M + 1) = (2,3) and (4,5) previously, now let us assume it works for the (M − 2, M − 1) case. That is to say, by providing genie signals G M −2 m , m ∈ {1, · · · , M − 2} to Receiver 1, we obtain a total of M − 2 sum rate inequalities, each obtained by averaging over user indices.…”

Section: A1 Casesmentioning

confidence: 99%

Subspace Alignment Chains and the Degrees of Freedom of the Three-User MIMO Interference Channel

Wang¹,

Gou

²

,

Jafar

³

2014

IEEE Trans. Inform. Theory

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We show that the 3-user M T × M R MIMO interference channel where each transmitter is equipped with M T antennas and each receiver is equipped with M R antennas has d(M, N ) = min M 2−1/κ , N 2+1/κ degrees of freedom (DoF) normalized by time, frequency, and space dimensions, whereWhile the DoF outer bound of d(M, N ) is established for every M T , M R value, the achievability of d(M, N ) DoF is established in general subject to a normalization with respect to spatial-extensions, i.e., the scaling of the number of antennas at all nodes. Specifically, we show that qd(M, N ) DoF are achievable for the 3-user qM T × qM R MIMO interference channel, for some positive integer q which may be seen as a spatial-extension factor. q is the scaling factor needed to make the value qd(M, N ) an integer. Given spatial-extensions, the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency. In the absence of spatial extensions, it is shown through examples how essentially the same interference alignment scheme may be applied over time-extensions over either constant or time-varying channels. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the DoF outer bound, as well as in the construction of the optimal interference alignment schemes.The DoF value d(M, N ) is a piecewise linear function of M, N , with either M or N being the bottleneck within each linear segment while the other value contains some redundancy, i.e., it can be reduced without reducing the DoF. The corner points of these piecewise linear segments correspond to two sets, A = {1/2, 2/3, 3/4, · · · } and B = {1/3, 3/5, 5/7, · · · }. The set A contains all those values of M/N and only those values of M/N for which there is redundancy in both M and N , i.e., either can be reduced without reducing DoF. The set B contains all those values of M/N and only those values of M/N for which there is no redundancy in either M or N , i.e., neither can be reduced without reducing DoF. Because A and B represent settings with maximum and minimum redundancy, essentially they are the basis for the DoF outer bounds and inner bounds, respectively.Our Our results show that M/N ∈ A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers. This may also be seen as a consequence of the maximum redundancy in the M/N ∈ A settings.

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“…Recently, the degrees of freedom (DoF) region is completely characterized for layered two unicast Gaussian networks without feedback [8]. The result resembles that in the linear deterministic case [7] with a change of performance measure from rate to DoF.…”

Section: Introductionmentioning

confidence: 98%

On two unicast wireless networks with destination-to-source feedback

Wang

¹

2012

2012 IEEE International Symposium on Information Theory Proceedings

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Abstract-In this paper we study the role of feedback in layered two unicast wireless networks with arbitrary number of nodes and connectivity. The feedback model allows destinations to feedback their received signals to their respective sources. In the case of linear deterministic networks, we fully characterize the capacity region when the two individual minimum cut values are equal to 1 and show that feedback only helps increase capacity whenever the capacity region without feedback has (1, 1/2) or (1/2, 1) as its corner point but not both. Therefore, feedback helps balance the resource utilization of the two users, similar to the role of feedback in the two-user interference channel [1]. I. INTRODUCTIONIn multi-user wireless networks, feedback helps balance the resource utilization among different users [1] by treating part of the interference received at destinations as useful side information. Through the feedback from destinations to sources, the side information can be exploited efficiently, and the system resource utilization is balanced across different users. In the context of the two-user interference channel, in the approximate characterization of the feedback capacity region, only the sum rate upper bound and the two individual rate bounds are active. In contrast, when feedback is not available [2], additionally bounds on 2R 1 + R 2 and R 1 + 2R 2 will be active, which implies that resource utilization is not balanced in certain regimes. The resource utilization interpretation [1] is best visualized using the linear deterministic model [3].Channel output feedback in interference channels with constant channel gains has been well studied [4] [1] [5] [6]. With a good understanding on the role of feedback in single-hop multi-user networks, one of the natural follow-up questions is whether or not it extends to networks with arbitrary number of nodes and connectivity. In this paper we make a step towards answering this question by studying a class of layered two unicast linear deterministic networks [3] where the channel strengths are either unity or zero and the individual minimum cut values are equal to 1. This class of networks was studied in the previous work [7] when feedback is not available. The non-feedback capacity region is completely characterized in [7], and similar to the two-user interference chanel [2], there are networks where in the characterization of the capacity region, bounds on 2R 1 +R 2 or R 1 +2R 2 are active. Moreover, the class of two unicast linear deterministic networks can be partitioned into five different categories according to their capacity regions: {T, T 12 , T 21 , P, S}, as shown in Fig. 1.Our main result is that, when feedback is available from the destinations to their respective sources, it helps increase the

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Two-Unicast Wireless Networks: Characterizing the Degrees of Freedom

Cited by 70 publications

References 37 publications

Subspace alignment chains and the degrees of freedom of the three-user MIMO interference channel

Subspace alignment chains and the degrees of freedom of the three-user MIMO interference channel

Subspace Alignment Chains and the Degrees of Freedom of the Three-User MIMO Interference Channel

On two unicast wireless networks with destination-to-source feedback

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