Degrees of Freedom for the MIMO Interference Channel
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1 ISIT 2006, Seattle, USA, July 9 4, 2006 Degrees of Freedom for the MIMO Interference Channel Syed A Jafar Electrical Engineering and Computer Science University of California Irvine, California, syed@eceuciedu Maralle J Fakhereddin Department of Electrical Engineering California Institute of Technology, Pasadena, CA maralle@systemscaltechedu Astract We explore the availale degrees of freedom DoF for the two user MIMO interference channel, and find a general inner ound and a genie aided outer ound that give us the exact # of DoF in many cases We also study a share-and-transmit scheme and show how the gains of transmitter cooperation are entirely offset y the cost of enaling that cooperation so that the availale DoF are not increased I INTRODUCTION Multiple input multiple output MIMO systems have assumed great importance in recent times ecause of their remarkaly higher capacity compared to single input single output systems It is well known [] [3] that capacity of a point to point PTP MIMO system with M inputs and N outputs increases linearly as minm,n at high SNR For power and andwidth limited wireless systems, this opens up another dimension - space that can e exploited in a similar way as time and frequency Similar to time division and frequency division multiplexing, MIMO systems present the possiility of multiplexing signals in space For example, using singular value decomposition SVD of a MIMO channel, one can generate parallel channels in space similar to those created y dividing time or frequency into orthogonal slots The availaility of spatial DoF depends upon two factors: cooperation within inputs/outputs, and channel knowledge Previous work has shown that in the asence of channel knowledge, spatial DoF are lost [4], [5] Multiuser systems, with constrained cooperation etween inputs/outputs distriuted among multiple users, are especially challenging since, unlike PTP case, joint processing is not possile at inputs/outputs The availale spatial DoF are affected y the inaility to jointly process the signals at the distriuted inputs and outputs [6] investigated DoF as a function of distriuted and partial side information for multiple access MAC and roadcast BC channels In this paper, we quantify the loss in availale DoF under the distriuted processing constraints imposed y the two user interference channel It was recently shown in [7] that cooperation etween single antenna transmitters does not provide additional multiplexing gain in an interference channel In this paper, we explore the enefits of transmitter cooperation when the nodes have multiple antennas We estalish a general inneround and a genie ased outeround on the # of DoF for MIMO interference channel For many cases of practical interest, these ounds are shown to e tight and we have the exact # of DoF We also consider a simple cooperative scheme to understand why transmitter cooperation does not increase DoF Through this simple scheme, we are ale to show how the enefits of cooperation are completely offset y the cost of enaling it II DEGREES OF FREEDOM MEASURE In order to isolate the impact of distriuted processing from channel uncertainty, we assume that channel state is fixed and perfectly known at all transmitters and receivers Also, we assume that the channel matrices are sampled from a rich scattering environment Therefore we can ignore the measure zero event that some channel matrices are rank deficient It is well known that the capacity of a scalar additive white Gaussian noise AWGN channel scales as logsnr at high SNR On the other hand, for a single user MIMO channel with M inputs and N outputs, the capacity growth rate can e shown to e minm,n logsnr at high SNR This motivates the natural definition of spatial DoF as: C Σ ρ η lim ρ logρ, where C Σ ρ is the sum capacity just capacity in case of PTP channels at SNR ρ Inotherwords,DoFη represent the maximum multiplexing gain [3] of the generalized MIMO system For PTP case, M,N DoF are easily seen to correspond to the parallel channels that can e isolated using SVD, involving joint processing at the M inputs and N outputs, ie ηptp = minm,n 2 A The Multiple Access Channel The MAC channel is an example of a MIMO system where cooperation is allowed only etween the channel outputs Let the MAC consist of N outputs controlled y the same receiver and 2 users, each controlling M and M 2 inputs for a total of M = M + M 2 inputs For the MAC, the availale DoF are the same as with perfect cooperation etween all users ηmac =ηptp =minm + M 2,N 3 While the capacity region of the MIMO MAC is well known and the spatial multiplexing gain has also een explored in previous work, we include the following constructive proof to introduce zero forcing ZF notation which will e useful in the derivation of our main result for the interference channel /06/$ IEEE 452
2 ISIT 2006, Seattle, USA, July 9 4, 2006 ZF, which is normally a suoptimal strategy, is sufficient in this case as well as in MIMO BC channel to utilize all DoF Converse: The converse is straightforward ecause, for the same # of inputs and outputs, ηmac ηptp = minm + M 2,N In other words, the lack of cooperation at the inputs can not increase DoF Achievaility: The N received signal Y at the MAC receiver 2 Y = H k X k + N = V H V X + Z, 4 k= where N is the N AWGN vector, H k is the N M k channel matrix for user k, andx k is the M k transmitted vector for user k V H = V H is the M + M 2 N matrix otained y vertically stacking the matrices H and H 2 Similarly, V X = V X is the M +M 2 matrix otained y vertically stacking X and X 2 Transforming the output vector Y new = V H V H VH Y using generalized Moore-Penrose inverse and ignoring the zero gain channels result in the minm,n parallel channels Y new i =V X i+n new i, i minm,n, 5 where N new i N0,λ i are Gaussian noise terms and λ i is the i th diagonal term of V H V H The noise terms may e correlated across different channels ut the correlations are inconsequential since each channel is encoded and decoded separately Dividing power equally among the minm,n channels, we can achieve minm,n ρ ηmac lim log + ρ logρ minm,n = lim ρ minm,n i= i= [minm,n logρ+ logρ log λ 2 i minm,n λ 2 i ]=minm,n Note that the channel gains or the exact power allocation does not affect the DoF as long as the SNR on each channel is proportional to ρ Comining the converse and the achievaility, we have estalished that ηmac=minm + M 2,N B The Broadcast Channel The BC channel is an example of a MIMO system where cooperation is allowed only etween the channel inputs Let the BC consist of M inputs controlled y the same transmitter and 2 users, each controlling N and N 2 outputs for a total of N = N + N 2 outputs In a similar fashion as the MAC, it is possile to show that y ZF at the BC transmitter, minm,n parallel channels can e created, so that the total DoF are the same as with perfect cooperation etween all the users ηbc =ηmac =ηptp =minm,n 7 M M2 Fig T T2 H H 2 Z Z 2 R R2 N N2 M,N,M 2,N 2 Interference channel III INTERFERENCE CHANNEL Consider an M,N, M 2,N 2 interference channel with two transmitters T and T 2, and two receivers R and R 2, where T has a message for R only and T 2 has a message for R 2 only T and T 2 have M and M 2 antennas respectively R and R 2 have N and N 2 antennas respectively We denote the channels for link with N xm channel gain matrix H, for link 2 y N 2 xm 2 matrix H 2, for the channel etween T and R 2 y N 2 xm channel matrix Z 2, and etween T 2 and R y N xm 2 matrix Z Figure shows an illustration of this interference channel We assume that we arrange the links so that link always has the most # of antennas either at its transmitter or receiver, ie maxm,n maxm 2,N 2 A Inneround on the Availale Degrees of Freedom For the M,N, M 2,N 2 interference channel we prove the following inneround on the availale DoF ηint minm,n + minm 2 N,N 2 + M >N + minm 2,N 2 M + M <N, 8 where is the indicator function and x + =max0,x While we conjecture that this ound is tight for any M,N,M 2,N 2, we can prove a converse only with some additional assumptions on the # of antennas A general achievaility proof is outlined next Sketch of Achievaility Proof: According to our model, either M N,M 2,N 2 or N M,M 2,N 2 First,we consider the case when M N,M 2,N 2 Step : From SVD, Z 2 = UΛV H, where U and V are N 2 xn 2 and M xm unitary matrices respectively and Λ is the diagonal matrix of singular values of Z 2 By applying SVD to Z 2, we decompose the channel into minm,n 2 parallel channels Therefore, there are M N 2 effective inputs at T that are not connected to R 2, and do not cause any interference to R 2 Step 2: Similarly, applying SVD to Z creates minm 2,N parallel connections There are M 2 N + effective inputs at T 2 that are not connected to R, and therefore do not cause any interference with R 453
3 ISIT 2006, Seattle, USA, July 9 4, 2006 M N N M2 N Link Link 2 N minm N,N2 Fig 2 Achievaility proof for M,N,M 2,N 2 Interference channel when M M 2,N,N 2 Step 3: For link, all N effective outputs are used y R Step 4: T transmits to R using N effective inputs such that at most N + N 2 M + effective inputs that are active are also connected to R 2 Step 5: Link 2 uses only those effective inputs/outputs that are not connected to an active effective input/output of link Step 6: Link is left with N effective inputs and N effective outputs, ie the # of DoF for link = N Step 7: For link 2, T 2 is left with M 2 N + effective inputs while R 2 is left with minm N,N 2 effective outputs, ie the # of DoF for link 2 = minm 2 N,minM N,N 2 + = minm 2 N,N 2 + since M M 2 y assumption Hence proved For the case when N M,M 2,N 2, the same logic is followed Then, the total # of DoF is minm,n + minm 2,N 2 M + By adding the results from the two cases, we otain a general achievale proof of 8 An illustration of this proof is shown in figure 2 B Outerounds on the Availale Degrees of Freedom To start with, notice that a trivial outeround is otained from the PTP case, ie ηint minm + M 2,N + N 2 Indeed this outeround coincides with the inneround when either minm,m 2 N + N 2 or minn,n 2 M + M 2 In general, while the capacity region of the interference channel is not known even with single antennas at all nodes, various outerounds have een otained [8] [0] that have een useful in finding the capacity region in some special cases [], [2] Most of the existing outerounds are for single antenna systems For our purpose, we develop a genie ased outeround for MIMO interference channel where the # of antennas at either receiver is the # of transmit antennas at the interfering transmitter, ie either N M 2 or N 2 M Wefindthat, in many cases, this outeround is sufficiently tight to estalish the # of DoF Note that for this section, since we do not use the assumption that maxm,n maxm 2,N 2, the proof for the cases N M 2 or N 2 M is identical Theorem : For the M,N, M 2,N 2 interference channel with N M 2, the sum capacity is ounded aove y that of the corresponding M,M 2,N MAC channel with additive noise N N0, I N modified to N N0, K where K = I N Z Z Z Z + αz Z, α = min σmaxz 2, σmaxh 2 2 Proof: Let us define N a N 0, I N Z Z Z Z N N 0, Z Z Z Z αz Z N c N 0,αZ Z, as three N jointly Gaussian and mutually independent random vectors The positive semidefinite property of the respective covariance matrices is easily estalished from the definition of α Without loss of generality we assume N = N a + N + N c N = N a + N c Furthermore, ecause N and N 2 have the same marginal distriutions and the capacity of the interference channel does not depend on the correlation etween N and N 2,the capacity region is not affected if we assume N = N 2 Since a part of the proof is similar to the corresponding proof for the single antenna case, we will summarize the common steps, and emphasize only the part that is unique to MIMO interference channel Consider any achievale scheme for any rate point within the capacity region of the interference channel, so that R and R 2 can correctly decode their intended messages from their received signals with sufficiently high proaility Step : We replace the original additive noise N at R with N as defined in Theorem We argue that this does not make the capacity region smaller ecause the original noise statistics can easily e otained y locally generating and adding noise N at R Therefore, since R was originally capale of decoding its intended message with noise N,it is still capale of decoding its intended message with N Step 2: Suppose that a genie provides R 2 with side information containing the entire codeword X SinceX 2 is independent of X, R 2 simply sutracts out the interference from its received signal Thus, the channel Z 2 can e eliminated without making the capacity region smaller Step 3: By our assumption, R can decode its own message and therefore it can sutract X from its own received signal as well In this manner, after the interfering signals have een sutracted out we have Y = Z X 2 + N, Y 2 = H 2 X 2 + N 2 454
4 ISIT 2006, Seattle, USA, July 9 4, 2006 To complete the proof we need to show that if R 2 can decode X 2 then so can R This would imply that R can decode oth messages, hence giving us the MAC outer ound Step 4: Without loss of generality, let us perform SVD H 2 = U 2 Λ 2 V 2 on the channel etween T 2 and R 2 Thisisa lossless operation that leads to: Y 2new = X 2new + Λ 2 N 2, 9 where X 2new = V 2 X 2 To save space we allow some notation ause as we use generalized inverse and ignore the terms that correspond to zero diagonal channel gains in Λ 2 Note that these channels are useless for R 2 Also, we use the same symol for rotated versions of noise that are statistically equivalent Step 5: Next, we show that R can otain a stronger channel to X 2new so that if R 2 can decode it, so can R Tothis end, let R use ZF to otain: Y new = X 2new + V 2 Z Z Z N, = X 2new + αn 2 Now oth R and R 2 have a diagonal channel with input X 2new and uncorrelated additive white noise components on each diagonal channel Moreover, the strongest channel for R 2 has noise However the noise on any channel σmax 2 H2 for R is only α which is smaller Thus, we argue once again that R can locally generate noise and add it to its received signal to create a statistically equivalent noise signal as seen y R 2 Inotherwords,R has a less noisy channel to T 2 and therefore can decode any signal that R 2 can Since R can decode T s message y assumption, we have the MAC outeround The MAC outeround leads directly to the following outeround on the # of DoF Corollary : For the M,N, M 2,N 2 interference channel with N M 2, the # of DoF ηint minm + M 2,N Similarly, if N 2 M,thenηINT minm + M 2,N 2 The outeround and inneround are tight in many cases where we have the exact # of DoF Some examples are provided in the following tale M,N M 2,N 2 ηint,,, 2, 2 2 2, 2, 2, 2 2, 3, 2 2, 3 2 2, 3 2, 3 3 2, 3, 3 3 2, 2 3, 2 2 IV EFFECT OF TRANSMIT COOPERATION ON THE NUMBER OF DEGREES OF FREEDOM Comparing the interference channel and the BC channel otained y full cooperation etween the transmitters, it is clear that the availale DoF are severely limited y the lack of transmitter cooperation in the interference channel As an example, consider the interference channel with M,N = n, and M 2,N 2 =,n From the preceding section we know there is only one availale degree of freedom in this channel However, if full cooperation etween the transmitters is possile the resulting BC channel has M,N,N 2 = n+,,n The # of DoF is now n+ Therefore, transmitter cooperation would seem highly desirale Rather surprisingly, it has een shown recently [7] that for the,,, interference channel, allowing the transmitters to cooperate through a wireless link etween them even with full duplex operation, does not increase DoF For MIMO interference channels, as suggested y the example aove, the potential enefits of cooperation are even stronger and it is not known if transmitter cooperation can increase DoF The capacity results of [7] do not seem to allow direct extensions to MIMO interference channels To gain insights into the cost and enefits of cooperation in a MIMO interference channel, we consider a specific scheme where transmitters first share their information in a full duplex mode as a MIMO channel step and susequently transmit together as BC channel We will refer to this scheme as the share-and-transmit scheme A Degrees of Freedom with Share-and-Transmit Consider an M,N, M,N interference channel M N Also assume that each transmitter is sending information with rate R Note that while we make the preceding simplifying assumptions for simplicity of exposition, the following analysis and the main result extend directly to the general case of unequal # of antennas and unequal rates From 8, we know that the # of DoF for this interefernce channel with no transmitter cooperation is minm,n + minm,n M + = M +minm,n M + For the shareand-transmit scheme, we compute DoF as follows We first find the capacity of the sharing link C s and the capacity of transmission C t Then, we find the total capacity of the system C y evaluating the total amount of data transmitted divided y the total time it requires to transmit this data, ie 2R C = R C s + 2R 0 C t Dividing y logsnr where SNR is large, we otain the total #ofdofas C lim SNR log SNR = 2 DOFsharing + 2 DOFtransmit The # of DoF for the sharing link is that of MIMO PTP channel with M transmit and receive antennas = minm,m = M After transmitters share their information, they can fully cooperate as a 2M,N,N BC channel The # of DoF for this channel is min2m,2n =2minM,N Therefore, which gives the total # of DoF for the share-and-transmit scheme, ecomes = M Note that, 2M minm,n M+minM,N M +minm,n M + M 2 455
5 ISIT 2006, Seattle, USA, July 9 4, 2006 Rate Rate Distance= 4 Share & Transmit Transmit logtransmit Power Fig 4 Fig 3 Rate vs logtransmit Power with same distance Distance=5 Share & Transmit Transmit logtransmit Power Rate vs logtransmit Power with 5 distance for transmitting Therefore, we conclude that for this specific scheme transmitter cooperation in the high SNR regime does not provide any advantage to the # of DoF in the MIMO interference channel V SIMULATION RESULTS In this section, we verify the result discussed in the previous section, and discuss the effect of transmitter cooperation when the sharing links etween the transmitters are stronger than the transmission links For simplicity, we consider a 4,, 4, interfernce channel, and plot the rate versus the logarithm of the transmit power Note that we assume the noise to e 0- mean unit-variance Gaussian additive noise The share-and-transmit scheme is implemented as explained in section IV-A For the no cooperation scheme, T has a message for R only and dedicates its availale power to its link with R The same is true for T 2 and R 2 Note that since the transmit signal space is much larger than the receive signal space, T can decompose its channel with R as well as its channel with R 2 to create one non-interfering link to R and another to R 2 T 2 is ale to achieve this as well, and each receiver can then decode its message without interference In fig 3, we fix the distance etween each transmitter and receiver to e equal to that etween T and T 2 In this case, the transmitters allocate the same resources to their sharing link as to their transmission links Fig 3 indicates that the share-andtransmit scheme always has a lower rate for the same transmit power than the no cooperation scheme, which agrees with our result in section IV In fig 4, the distance etween each transmitter and receiver is 5 that etween T and T 2 Note that in this case, the sharing link is stronger than the transmission links since it does not suffer any path loss whereas the transmission links do Fig 4 shows that share-and-transmit scheme outperforms the no cooperation scheme As expected, when the sharing link is stronger, cooperation etween transmit nodes results in performance improvement over the no cooperation scheme Note that while our simulations are for the interference channel, similar results have een otained for the MAC in [3] VI CONCLUSIONS We investigate the availale DoF for MIMO interference channel The distriuted nature of the antennas significantly limits DoF For an interference channel with a total of N transmit antennas and a total of N receive antennas, the availale # of DoF can vary from N to ased on how the antennas are distriuted among the two transmitters and receivers Through an example of a share-and-transmit scheme, we show how the gains of transmitter cooperation are entirely offset y the cost of enaling that cooperation so that the availale DoF are not increased REFERENCES [] G J Foschini and M J Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Commun : Kluwer Academic Press, no 6, pp 3 335, 998 [2] E Telatar, Capacity of multi-antenna Gaussian channels, European Trans on Telecomm ETT, vol 0, pp , Novemer 999 [3] L Zheng and D N Tse, Packing spheres in the Grassmann manifold: A geometric approach to the non-coherent multi-antenna channel,, IEEE Trans Inform Theory, vol 48, pp , Fe 2002 [4] S Jafar, Isotropic fading vector roadcast channels: the scalar upperound and loss in degrees of freedom, To appear in the IEEE Trans Inform Theory See syed/ [5] A Lapidoth, On the high-snr capacity of non-coherent networks, Sumitted to IEEE Trans Inform Theory See [6] S Jafar, Degrees of freedom in distriuted MIMO communications, IEEE Communication Theory Workshop, 2005 [7] A Host-Madsen and Z Yang, Interference and cooperation in multisource wireless networks, in IEEE Communication Theory Workshop, June 2005 [8] A B Carliel, Outer ounds on the capacity of Gaussian interference channels, IEEE Trans Inform Theory, vol 29, pp , July 983 [9] G Kramer, Outer ounds on the capacity of Gaussian interference channels, IEEE Trans Inform Theory, vol 50, pp , Mar 2004 [0] S Vishwanath and S Jafar, On the capacity of vector Gaussian interference channels, in Proceedings of IEEE Information Theory Workshop, Oct 2004 [] R Ahlswede, The capacity region of a channel with two senders and two receivers, in Ann Pro, pp , Oct 974 [2] A B Carliel, A case where interference does not reduce capacity, IEEE Trans Inform Theory, vol 2, pp , Sep 975 [3] S Cui, A Goldsmith, and ABahai, Energy efficiency of MIMO and cooperative MIMO in sensor networks, IEEE Journal on Selected Areas in Communications, vol 22, August
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