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MIMO-OFDM systems with multi-user interference - Signals and ...

where s m is the data vector of size T that is intendedto be transmitted to **user** m, **and** its covariance Φ k m isdefined by Φ k m = E{s k m(s k m) H }. The thermal noisevector at the i th receiver, n k i is of size N r, whose spatialcovariance matrix is diagonal **with** equal power (σi k)2per antenna . The N r × T matrix Wi k **and** N t × T matrixVm k represent the receive **and** transmit weight vectors,respectively. Each column of these matrices correspondsto one of the(streams. Total ) transmit power at thek th subcarrier is tr VmH k Φkm Vmk . Assuming that thetransmitted streams from different **user**s are independentof each other **and** also independent of noise samples, thetotal covariance matrix at receiver i is given byX k i =W k iHHkii V k i Φ k i V k iHHkiiHWki + W k iHQki W k i ,whereQ k i = ∑ H k miVmΦ k k mVmH k Hk Hmi + σk2 i INr , (2)m≠iis the covariance of the **interference** **and** noise at the k thsubcarrier of the i th receiver.3. MAXIMUM THROUGHPUT POLICYIf the transmit signal is assumed to be Gaussian or thenumber of **user**s is large enough (using central limit theoremin this case), we can assume a Gaussian distributionfor the **interference**. Furthermore, we follow twoscenarios. In the first one, we assume a fix transmitpower policy per subcarrier where every **user** adapts itsdata rate **with** the total transmit power across its antennasheld constant. In the second scheme, the transmitpower is fixed per **user**. For notation simplicity, we willdrop the index k. The mutual information at subcarrierk can be expressed as:∣ ( ) ∣ ∣∣Wi HI i = log 2 H ii V i Φ i V H i H H ∣∣ii + Q i W i− log 2∣ ∣∣Wi H Q i W i∣ ∣∣ (3)( We would like)to maximize I i , **with** the constrainttr VmH k Φkm Vmk ≤ Pm. k Using the relation det(I +AB) =det(I + BA), we can change the mutual informationto( ) −1I i = log 2 |H H ii W i W H i Q i W i×W i H H ii V i Φ i V H i + I Nt |. (4)The MMSE receiver weight vectors are given byW i = µQ −1i H ii V i , (5)where µ is a real constant. With this choice, it is easy tosee that the mutual information is changed to (6), whichis the capacity obtained by optimum receiver processing.Therefore the chosen MMSE receiver weight vector isoptimum.[]I =log 2 det H H ii Q −1i H ii V i Φ i V H i + I Nt . (6)The eigenvalue decomposition of the matrix H H ii Q−1 i H iiis UΛU H , where U is an unitary matrix, **and** Λ is diagonal.Note that the number of eigen-modes of the systemis determined by the rank of this matrix. With thischoice, the mutual ∣ information becomes ∣ ∣∣UΛUI i = log H 2 V i Φ i V H ∣∣i + I Nt= log 2∣ ∣∣Λ 1/2 U H V i Φ i V H i UΛ 1/2 + I Nt∣ ∣∣ .(7)Using Hadamard inequality (|A| ≤ ∏ i a ii), the mutualinformation, I i is maximized if U H V i Φ i Vi H˜P U =i is diagonal **with** non-negative elements p i1 ,p i2 ,...p iNt .In this case[ ]I i =log 2 det ˜Pi Λ + I Nt = ∑ log 2 (1 + p ij λ j ) ,jwhere λ j ’s are the diagonal elements of Λ. Since Uis unitary, **and** tr(AB) =tr(BA), then tr(V i Φ i Vi H)=tr( ˜P i ).3.1. Constant Power per SubcarrierIf we assume that the total power of eigen-modes at eachsubcarrier is fixed, our problem becomes the maximizationof I i , **with** the constraint tr( ˜P i ) ≤ Pi k . The answeris the well-known water-filling solution [7] , which says(p ij = ν − 1 ) +, (8)λ jwhere ν is chosen to guarantee ∑ j p ij = Pi k, **and**(x) + = max(x, 0). This equation is used for nonzeroλ j ’s. If λ j =0, we take p ij =0.To find the optimum transmit weight vectors, we needto solve U H V i Φ i Vi HU = ˜P i or( )( ) HV i Φ 1/2i V i Φ 1/2i = U ˜Pi U H for V i .IfN t >T,[ ]1/2 P1then V i = U Φ −1/2i , where P 1 is an T × T0} {{ }Bdiagonal matrix **with** nonzero diagonal elements of ˜P ion its main diagonal. If N t = T , then V i = UBΦ −1/2˜P1/2ii ,**with** B = . In all cases, BB H = ˜P i . With thesechoices, we havex k i = Wik H(Hkii Vi k s k i + ∑ H k miVms k k m + n k i )m≠i} {{ }q= µ ∗ ΣΦ −1i s k i + µ ∗ Φ −1/2i B H U H H k iiHQ −1 q,

where Σ=B H ΛB is a diagonal matrix, for all 3 abovementionedcases. This result shows that at the receiver,**and** at each subcarrier, the **multi**ple streams are orthogonalto each other. The receiver covariance matrix forN t = T isE[x k i x k Hi ]=µ2 ˜P2 i Λ 2 Φ −1i + µ 2 ˜Pi ΛΦ −1i . (9)So the SNR for the j th stream is p ij λ j . Similar resultscan be found for other cases.3.2. Constant power per **user**In reality the total transmit power for each base stationis limited. As a practical limitation, here we assume thattotal power allocated to each **user** is fixed. the overallmutual information between the i th receiver, **and** its correspondingtransmitter, can be found from I i = ∑ Kk=1 Ik i .If ˜P k i = Uk Hi Vki Φ k i Vk Hi Uki for each k is diagonal,by Hadamard inequality, each term of the above sum ismaximized regardless of any power constraint. Sinceeach term is positive, the sum will also be maximized.Therefore, we want to maximize∑KTI i = log 2 (1 + p ir λ ir ) , (10)r=1constrained to ∑ KTr=1 p ir = P i , where P i is the powerallocated to **user** i. The index r is used to represent allstreams **and** all subcarriers. For a given r, the index ofthe stream is found from j =(r mod T )+1, **and** theindex of subcarrier from k = ⌊(r/T)⌋. Again, The answeris the water-filling solution presented in (8). TheSNRs are again p ir λ ir at corresponding subcarriers **and**streams.3.3. Iterative WaterfillingNote that each transmitter optimizes its transmit spectrumindependently of the other transmitters but knowingthe **interference** covariance matrix at its receiver. Sinceeach transmitter is repeating the same process, the **interference**at the receiver is going to be changed **and** theabove steps should be repeated until the transmit **and****interference** covariance matrices converge. Therefore,the iterative water-filling [4] is used to find the powersat each eigen-mode at each subcarrier of each transmitter.Note that the authors in [4] have used the iterativewater-filling for a single-cell single-carrier system,while we are proposing this scheme for **multi**-cell **multi**carrier**multi**ple antenna **systems**.For the first scheme, in which the constant powerper subcarrier is considered, the iterative water-filling atsubcarrier k is proposed as follows, starting from the receivern =1:Algorithm I [Constant power per subcarrier]1. Find the **interference** from Eq. (2).CDF10.90.80.70.60.50.40.30.20.1CDF of rates for fixed power per **user**Multi CellSingle CellSingle User00 5 10 15 20 25 30 35 40RatesFig. 1: Achievable rate CDF, for fixed power per carrier2. Find the eigenvalue decomposition ofH k nnHQk −1n Hknn = U k nΛ k nU k nH.3. Find the power of each eigen-mode **with** nonzeroeigenvalue from (8) For λ j =0, take p ij =0.Create a diagonal matrix ˜P k n, whose diagonal elementsare p ij .4. Find the transmit weight vectors Vn k = U k nBΦ k n−1/2,1/2when B is ˜P n for N t = T . For the other casesB is found correspondingly.5. Set n = n +1**and** continue from step 1 untilconvergence.For the second scheme, where the constant powerper **user** is considered, the iterative water-filling is similarto Algorithm I, where step 3 is described as follows:Step 3. for Algorithm II: For r = 1...KT, findthe power of each eigen-mode **with** nonzero eigenvaluefrom (8), For λ ir =0, take p ir =0. For each subcarrier,create a diagonal matrix ˜P k n, whose diagonal elementsare corresponding p ir ’s.For each iterative water-filling scheme, the choiceof receive weight vectors at a receiver has no effect onthe amount of **interference** at other **user**s. Therefore, thereceive weight vectors are evaluated using Eq. (5), afterthe iterative algorithms are converged.We have shown that the above iteration will convergeto a fixed point starting from any initial transmit **and** receiverprocessing matrices.4. SIMULATIONWe simulate a wireless network consisting of 100 **and**125 co-channel base stations placed in a hexagonal pattern.We assume one or two mobiles r**and**omly distributedin a cell according to a uniform distribution. We usean **OFDM** system **with** 32 subchannels for transmission.The communication channel is assumed to follow theCOST207 Typical Urban 6-ray channel model. Each

10.9CDF of rates for different number of streams for fixed power per **user**123410.9CDF of rates for different number of streams for fixed power per **user**12340.80.80.70.70.60.60.50.50.40.40.30.30.20.20.10.100 20 40 60 80 100 120Fig. 2: Achievable rate CDF, for fixed power per **user****OFDM** symbol is assumed to be 1µs long which correspondsto a b**and**width of 1MHz. The white Gaussianthermal noise power at each receiver is calculatedbased on a noise figure of 3dB **and** 1MHz b**and**width.We consider 4 transmit antennas **and** 4 receive antennas.Independent **multi**ple data streams are transmitted fromeach base stations to each mobile in its correspondingcell. Two different cases are considered.Algorithms 2 have been simulated for **multi**-cell system.Fig. 1 shows the CDF of the achievable rate for125 cells **with** of reuse 3, one mobile per cell, **and** twotransmitted streams. In this figure, we have shown thecase where the in-cell **interference** plus noise is used asan approximation for noise covariance to calculate thetransmit weights. It can be seen that when the **multi**cell**interference** is taken into account, the overall rate ishigher. The same CDF for different number of streamsper subcarrier is shown in Figs. 2 **and** 3 for 100 cells**with** one **and** two mobiles per cell, respectively. In thefirst case, as we increase the number of streams, moreb**and**width can be assigned per **user** **and** the achievablerate per **user** is increased. However, as we increase thenumber of co-channel mobiles per cell, more processingis needed to combat the effect of in-cell **interference**s,**and** therefore the performance of two streams is betterthan that of four streams.5. CONCLUSIONWe have proposed iterative water-filling solutions for**multi**-**user** **multi**-cell wireless **systems** where **multi**pleantennas are deployed at both transmitters **and** receivers.The proposed algorithm assigns **multi**ple independentsubstreams for each **user** to increase data rate for each**user** **and** is performed as a distributed scheme. That is,each transmitter has only the knowledge of **interference**at its own receiver **and** the channel response in that link.We have shown that the algorithm converges to a fixedpoint solution where the data rate for each **user** is lo-00 10 20 30 40 50 60 70 80 90Fig. 3: Achievable rate CDF, for fixed power per **user**cally optimized. We have evaluated the performance ofthe proposed scheme through numerical analysis.6. REFERENCES[1] G. J. Foschini, “Layered Space-Time Architecturefor Wireless Communication in a Fading EnvironmentWhen using Multi-element Antennas,” BellLabs Tech Journal, pp. 41–59, Fall 1996.[2] F. Farrokhi, A. Lozano, G. Foschini, **and** R. A.Valenzuela, “Spectral Efficiency of FDMA/TDMAWireless Systems **with** Transmit **and** Receive AntennaArrays,” IEEE Trans. on Wireless Comm.,vol. 1, Oct 2002.[3] F. Farrokhi, A. Lozano, G. Foschini, **and** R. A.Valenzuela, “Very high capacity **multi**ple-acess systemusing **multi**ple transmit **and** recieve antennas,”Bell Labs Technical Memo, Dec 1999.[4] W. Yu, W. Rhee, S. Boyd, **and** J. M. Cioffi, “IterativeWater-Filling for Gaussian Vector Multiple AccessChannels,” IEEE Intrn. Symp. on Inf. Theory, 2001.[5] W. Su, Z. Safar, M. Olfat, **and** K. J. R. Liu, “ObtainingFull Diversity Space Frequency Codes fromSpace-Time Codes via Mapping,” IEEE Trans. onSignal Proc., special issue on Signal Processing for**MIMO** **systems**, Dec 2002, Nov 2003.[6] M. Olfat, F. R. Farrokhi, **and** K. J. R. Liu, “PowerAllocation for **OFDM** using Adaptive Beamformingover Wireless Networks,” submitted to IEEE Trans.on Comm. (under revision), May 2003.[7] T. M. Cover **and** J. A. Thomas, Elements of InformationTheory. John Wiley, 1991.