System Performance of Interference Alignment under TDD Mode with Limited Backhaul Capacity

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1 Syste Perforance of Interference Alignent under TDD Mode with Liited Bachaul Capacity Matha Deghel, Mohaad Assaad and Mérouane Debbah Departent of Telecounications SUPELEC, Gif-sur-Yvette, France Matheatical and Algorithic Sciences Lab, HUAWEI, France E-ail: {atha.deghel, ohaad.assaad, Abstract This paper considers a MIMO interference syste where interference alignent (IA) technique is adopted to anage the proble of interference. We consider a tie division duplex (TDD) syste where each transitter estiates its channel state inforation (CSI) by probing the receivers. In addition, the transitters share their local CSI estiate between each other using a bachaul lins of liited capacity. A quantization over the bachaul is therefore required to reduce the aount of inforation to exchange. We study in this paper the ipact of this quantization on the syste perforance and deterine the optial nuber of transitter-receiver pairs that axiizes the syste throughput. I. INTRODUCTION Interference is one of the ajor drawbacs in wireless counication systes due to the large nuber of users counicating on the sae channel. This proble has otivated the researchers to investigate transitting schees that can itigate interference. Interference alignent (IA) was introduced in [] as one of the ost efficient interference anageent technique. It is based on the concept of designing precoding schee that confine the interfering signals observed at each receiver into a low diensional subspace, providing a larger subspace to decode the desired signal. In [], IA has been shown to achieve axiu ultiplexing gain in MIMO channels. One disadvantage of IA is that it requires global channel state inforation (CSI) at each of the transitters, which is difficult to obtain in practical systes. Therefore, IA under liited feedbac was studied and several quantization techniques were proposed, in order to aid the transitters to acquire (probe) CSI nowledge fro receivers and then to share it between each other. For instance, in [] a copression schee for the cloud radio access networs is proposed. In [], the Grassannian Manifold quantization technique was adopted to reduce the inforation exchange. Another schee proposed in [] is used in sending the channel conditions fro users to transitters. An iportant factor to consider, which is related to the CSI acquisition process, is the CSI probing (acquisition) cost. We consider a TDD ode where receivers (users) send training sequences in the uplin so that the transitter can estiate their channels. Since this schee uses orthogonal sequences, their lengths are proportional to the nuber of active users in the syste. In other words, after acquiring the CSI of L users, the rate is ultiplied by Lθ, where θ is the fraction (of the slot) that taes the CSI acquisition of one user []. Fro the above, it can be seen that choosing the nuber of pairs L is iportant to investigate. If L is high, it eans that we have ore users to probe and then less tie for data transission. This proble was studied in [7] by optiizing the transission rate which is function of the overhead caused by the probing process for an analog feedbac strategy. The context here is different fro the aforeentioned wor. We assue prefect local CSI at the transitters, but each of which needs to send its local nowledge to all other transitters over finite capacity bachaul lins. Then, the interference alignent solution is coputed independently at each transitting node. The reainder of this paper is organized as follows. In Section II, the syste odel is described. Section III analyzes the perforance of the syste when the IA technique is used and when a quantization schee is perfored over the bachaul lins of finite capacity. Therein, we derive the transission rate of the syste, then we define and solve an optiization proble which sees to axiize this rate with respect to the nuber of pairs and bits. Finally, we give nuerical results in Section IV and conclude in Section V. Notation: Boldface uppercase sybols (i.e., A) represent atrices whereas lowercases (i.e., a) are used for vectors.; (.) denotes the conjugate transpose of a vector or atrix; I is used for square identity atrix;. denotes the absolute value;. represents the nor of second degree; CN (a, A) is a coplex Gaussian rando vector with ean a and covariance atrix A. II. SYSTEM MODEL In this section, we present the MIMO interference networ under consideration, where we apply the IA technique over finite capacity bachaul lins. We review the concept of IA and we propose a CSI sharing schee to reduce the aount of inforation exchange required to achieve IA. A. MIMO Interference Channel Model Consider the L-user MIMO interference networ illustrated in Fig. in which each transitter (T X ) is equipped with N t antennas and each receiver (R X ) has N r antennas. Transitter has d ( in(n t, N r )) data streas to send to its intended receiver (user).

2 Transitter - Transitter Transitter + Figure : L-User MIMO interference networ. Receiver - Receiver Receiver + Under this syste odel, the received signal at user can be expressed as follows: L γi P d i y v j i xj i + z, () d i H i j in which H i C Nr Nt is the channel atrix between T X i and R X with independent and identically distributed zero ean and unit variance Gaussian eleents, γ i represents the path loss of channel H i, P is the total power at each transitter equally allocated aong its streas, x j i denotes the jth data strea fro transitter i, v j i C Nt is the corresponding precoding vector of unit nor and z is the additive white Gaussian noise distributed according to CN (, σ I Nr ). We denote by α i the fraction γ ip d i, for all, i in {,..., L}. We assue a TDD transission where tie is slotted and where a transitter can acquire the channel state inforation of a receiver by probing, which consues a fraction θ of a slot. Probing ore users taes ore tie, and hence leaves a saller fraction of resources for actual data transission. As alluded earlier, if L users (i.e. it eans we have L active transitter-receiver pairs) are probed, then the actual rate becoes ( Lθ)R, where R is the transission rate without considering the probing cost. B. Application of Interference Alignent Technique For the sae of siplicity, we restrict ourselves to a perstrea zero-forcing receiver. Specifically, Receiver uses the cobiner vector u CNr of unit nor to detect the -th strea fro transitter, such as ˆx (u ) y inter-strea interference (ISI) {}}{ desired signal { }}{ α (u ) H v x d + α (u ) H v j xj inter-user interference (IUI) j j {}}{ noise L d i {}}{ + αi (u ) H i v j i xj i + (u ) z. () i j As can be seen in (), two sources of interferences (ISI and IUI) affect the detection at each receiver. To anage this proble, we use the IA technique which consists on designing the set of precoder and cobiner vectors such that []: (u ) H i v j i, (, ) (i, j). () We adit that each receiver obtains a perfect nowledge of the cobiner vector designed at its paired transitter. In the ideal case where we have perfect global CSI at all the transitters and for an achievable degree of freedo vector d [d,..., d L ], ILI and ISI can be canceled copletely at the receivers since the conditions for perfect alignent are satisfied. The CSI sharing echanis over the liited bachaul is detailed in the following. C. CSIT Sharing Over Finite Capacity Bachaul Lins As entioned before, global CSI is required in order to solve the interference alignent proble given in (). We assue that the transitters have a perfect nowledge of their local CSI, eaning that the i-th transitter estiates perfectly the channels H i, for,..., L. However, the local CSI (excluding the direct lins) of other transitters are obtained via bachaul lins of liited capacity. Transitter - Transitter Transitter + Figure : CSIT sharing over bachaul lins. We focus on the scenario shown in Fig. where each T X receives all the required CSI and independently designs the IA vectors []. But, since the bachaul lins that connect transitters to each other are of liited capacity, a codeboobased quantization is adopted to reduce the huge aount of inforation exchange. In detail, let h i represents the vectorization of the channel atrix H i and suppose that all the transitters share a predeterined codeboo CB of size B. Transitter i selects the index n o of the optial codeword in CB according to: n o arg ax n B h iĥn, where h i h i h i and B is the nuber of bits used to quantize H i. After quantizing all the atrices of its local CSI, transitter i sends the corresponding optial indexes to all other transitters which share the sae codeboo, allowing these transitters to reconstruct the quantized local nowledge of transitter i. Using the odel in [], we define the quantization error as e i ĥ i h i. The cuulative distribution h i function (CDF) of e i is then given by: Pr(e i ε) B ε Q for ε B Q, where Q Nt N r. III. SYSTEM PERFORMANCE ANALYSIS In this section, we analyze the perforance for IA in a MIMO interference networ with bachaul lins of liited

3 capacity. We first derive the total transission rate under the considered syste. Then, we provide an optiization proble with its corresponding solution for axiizing this rate. A. Transission Rate Under Finite Capacity Bachaul Lins As shown in the previous section, the IA vectors are designed based on the available CSI nowledge which is obtained after the transitting nodes share their perfect local nowledge between each other. It results that the IA technique is capable to copletely cancel the ISI since local CSI is perfectly nown, but not the ILI due to the quantization process which leads to iperfect global CSI at the transitting nodes. Under such conditions, the signal to interference plus noise ratio (SINR) for strea at receiver is be given by: ζ α (û ) H ˆv σ + L d i () α i (û ) H iˆv j i i j where ˆv and û are the precoding and cobining vectors, respectively, designed based on the available CSI described in the previous section. We denote by I the interference ter that appears in the denoinator of (). L d i I α i (û ) H iˆv j i L d i α i h i s,j i j i j L d i α i h i ( h i ) s,j, () i j where w i is a unit nor vector isotropically distributed in the null space of ĥi, s,j,i ˆv j i (û ) ( is the Kronecer product) and h i is the noralized vector of channel h i. Following the odel used in [], the channel direction h i can be written as follows: h i e i ĥ i + ei w i, where ĥi is the channel quantization vector of h i and w i is independent of e i, with w i (of unit nor) isotropically distributed in the null space of ĥ i. The product ( h i ) s,j,i can then be expressed as ( h i ) s,j,i e i (ĥ i ) s,j,i + e i (w i ) s,j,i (wi e i ) s,j,i. Therefore, I can be rewritten as: L d i I α i h i e i (w i ) s,j. () i,i j Transission Rate : Based on [8], we define the transission rate (throughput) achieved as the probability to get an SINR greater than a given threshold τ. In practice, this can be interpreted by the fact that if the SINR is lower than a certain value, then the transitted signal can not be decoded correctly. Thus, we can write the throughput that corresponds to strea of pair as:,i,i R Pr(ζ τ) (7) Proposition. The transission rate corresponding to strea at user can be given by R e σ τ α MGF I ( t), (8) where t τ α and MGF I is the oent generating function of the rando variable I. Proof. The proof is provided in Appendix A. The above proposition provides a general forula for the transission rate under the considered syste, which depends on the MGF expression of I. This latter expression is given in the following proposition. Proposition. The expression MGF I ( t) is given by MGF I ( t) L ( α itd i + ) Q F (b i, Q; a i + b i ; δ i + δ α i td i ), (9) where F is the hypergeoetric function, δ B Q, a i (Q+)di Q Q and b i (Q )a i. Proof. Refer to Appendix B for the proof. Let R denote the total transission rate of the syste which can be calculated by taing the su of all transission rates over all streas and pairs, such as: R L d R L d e σ τ α MGF I ( t). () The above result does not consider the cost of probing that, as entioned in Section II, will reduce the transission rate by a factor Lθ, where θ is the fraction of tie to probe one user. Under all the above considerations, the total transission rate can be given by the following proposition. Proposition. If we consider the probing cost, the total transission rate R p can be rewritten as: [ L d R p ( Lθ) e σ τ L α ( α iτd i + ) Q α B Q i ] F (b i, Q; a i + b i ; ) + α B Q α i τd i. () Proof. Proposition is proved by cobining the probing cost, the expression in () and the result of Proposition. B. Throughput Maxiization The total transission rate in () is a function of several paraeters. Aong these paraeters, we focus on the nuber of bits B and the nuber of pairs L. We analyze the syste perforance by axiizing the expression of the transission rate in (), function of L and B, under the constraint of finite (total) capacity C of bachaul lins. But, as it can be seen

4 in (), solving this proble for the general case is of high coplexity. Therefore, before proceeding in the analysis, we ae the following assuptions: (i) all the transitters have the sae nuber of streas d and (ii) all the direct lins and all the cross lins have equal path loss γ and γ, respectively. Under these assuptions, we can rewrite () as [ L d R p ( Lθ) e σ τ L α ( α τd + ) Q α B Q F (b, Q; a + b; ( α τd α B Q + α B Q α τd i ) ] + ) Q F (b, Q; a + b; ( Lθ)Lde σ τ α + α B Q α τd ) L. () where α γp d and α γp d. Now, we can define the optiization proble as follows: axiize B,L R p (B, L) () subject to L(L ) B C, () where L(L ) B is the total nuber of bits exchanged on the bachaul lins of liited capacity C. This expression is obtained fro the fact that we have L transitters, each of which shares L channels to L other transitters. Rear. To ensure the feasibility of the interference alignent proble, one additional condition (given in [9]) to consider is that N t + N r d(l + ), which puts a liitation on the axiu nuber of pairs. We propose the following algorith to solve the axiization proble defined by () and (). Optiization algorith Fix the capacity C of bachaul lins. for B to N B do for L to N L do if condition () is not statisfied then put R p (B, L). else copute the transission rate R p (B, L). end if end for end for Choose B and L which correspond to the ax value of R p. Note that the value of N L should be chosen based on the condition given in the rear before. IV. NUMERICAL RESULTS In this section we present the nuerical results. We assue a hoogeneous syste where the nuber of antennas N t N r, d, τ.8 and γ. We tae N B Transission Rate R p (b/s/hz).... L B Figure : The transission rate R p for different cobinations of nuber of bits and nuber of pairs. the factor θ.. Nuber of pairs L 7 θ. θ. θ. 8 Capacity (Mb/s) Figure : Optial nuber of pairs L for different values of the capacity. The factor θ {.,.,.}. which is the axiu nuber of bits that we can use for the quantization process, and N L which satisfies the condition given in the previous section. In addition, we use log ( P σ ) to represent the SNR in db. Fig. plots the transission rate of the syste in () for different cobinations of the nuber of bits B and the nuber of pairs L, when θ., γ. and SNR db. As can be seen fro this figure, for a fixed B, the transission rate R p is very sensitive to the variation of the nuber of pairs L. However, for a fixed L, the function R p is less sensitive to the variation of the nuber of bits B. Now, we consider the sae path loss γ as before and we use the algorith given in Section III to obtain the optial values of B and L that axiize the rate R p for different values of the total capacity C and the fraction θ. Fig. displays the variation of the optial nuber of pairs L as a function of the capacity C, for different values of θ. The ore we increase the capacity, the ore we relax the constraint in (9), then the possibility that the optial value of L increases is higher. Moreover, if the fraction θ is low (i.e. low probing cost), the optial nuber of pairs can reach higher values which will raise the syste rate. For instance, for θ., the nuber of pairs L reaches 7 pairs at its

5 7 Nuber of Bits B Optial Nuber of Pairs θ. θ. θ. 8 Capacity (Mb/s) Figure : Optial nuber of pairs B for different values of the capacity. The factor θ {.,.,.} Path Loss Coefficient γ Figure 7: The variation of the optial nuber of transitter-receiver pairs in function of the path loss coefficient γ. 7 Maxiu Transission Rate (b/s/hz) θ., C θ., C θ., C θ., C θ., C θ., C SNR (db) Maxiu Transission Rate (b/s/hz)....8 Path Loss Coefficient γ Figure : The variation of the axiu transission rate in function of the SNR. We consider different values of the factor θ {.,.,.} and C {, } Mb/s. axiu instead of for θ.. Fig. shows the variation of the optial nuber of bits B as a function of the capacity C. For the sae θ, B increases until it reaches the axial value. If we fix the value of C, we can see that B taes larger values for higher θ. This can be explained by the fact that the transission of the syste is ore sensitive to L than to B. Fig. plots the variation of the axiu transission rate as function of the SNR for different values of θ and the capacity C. For large values of θ, L is low which explains the decrease of the rate. For the sae θ, if we increase C we get larger argins for L and B, and then better rates. Fig. 7 and 8 represent the variation of the optial nuber of pairs and the axiu transission rate, respectively, in function of the path loss coefficient γ. While increasing γ, the optial nuber of pairs and the axiu rate decrease, because we pass fro a low interfering scenario to a highly interfering one. V. CONCLUSION In this paper, we consider a TDD syste with L-user interference channel under the IA technique. The TDD strategy Figure 8: The variation of the axiu transission rate in function of the path loss coefficient γ. constrains the axial nuber of pairs we can choose. Given the requireent of IA to share CSI between transitters over bachaul lins of finite capacity, we use a quantization schee to reduce the aount of inforation to exchange. The throughput of the syste is derived and it depends on the nuber of quantization bits and the nuber L of pairs. We have investigated the effect of each of these paraeters on the variation of the throughput of the syste. We have also proposed an algorith to axiize the transission rate function of L and B, for a given capacity C. It is shown that this rate is ore sensitive to the variation of L and less to B, and that this sensitivity depends on the fraction θ of the slot reserved for probing. REFERENCES [] V. Cadabe and S. Jafar, Interference alignent and degrees of freedo of the K-user interference channel, IEEE Trans. Inf. Theory, vol., no. 8, pp., Aug 8. [] T. Gou and S. Jafar, Degrees of freedo of the K user M N MIMO interference channel, IEEE Trans. Inf. Theory, vol., no., pp. 7, Dec. [] S. Par, O. Sieone, O. Sahin, and S. Shaai, Perforance evaluation of ultiterinal bachaul copression for cloud radio access networs, CoRR, vol. abs/.9,. [Online]. Available:

6 [] M. Rezaee, M. Guillaud, and F. Lindqvist, CSIT sharing over finite capacity bachaul for spatial interference alignent, CoRR, vol. abs/.8,. [] X. Chen and C. Yuen, Perforance analysis and optiization for interference alignent over MIMO interference channels with liited feedbac, IEEE Trans. Signal Process., vol., no. 7, pp , April. [] P. Chaporar and A. Proutiere, Optial joint probing and transission strategy for axiizing throughput in wireless systes, IEEE J. Select. Areas Coun., vol., no. 8, pp., Oct. 8. [Online]. Available: [7] O. E. Ayach, A. E. Lozano, and R. W. Heath, On the overhead of interference alignent: Training, feedbac, and cooperation, CoRR, vol. abs/.,. [Online]. Available: [8] K. Huang and V. Lau, Stability and delay of zero-forcing SDMA with liited feedbac, IEEE Trans. Inf. Theory, vol. 8, no., pp. 99, Oct. [9] C. Yetis, T. Gou, S. Jafar, and A. Kayran, On feasibility of interference alignent in MIMO interference networs, IEEE Trans. Signal Process., vol. 8, no. 9, pp , Sept. [] O. Ayach and R. Heath, Interference alignent with analog channel state feedbac, IEEE Trans. Wireless Coun., vol., no., pp., February. [] B. Jhannesson and N. Giri, On approxiations involving the beta distribution, Counications in Statistics - Siulation and Coputation, vol., no., pp. 89, 99. [] T. Yoo, N. Jindal, and A. Goldsith, Multi-antenna downlin channels with liited feedbac and user selection, IEEE J. Select. Areas Coun., vol., no. 7, pp. 78 9, Septeber 7. [] S. Nadarajah and S. Kotz*, On the product and ratio of gaa and beta rando variables, Allgeeines Statistisches Archiv, vol. 89, no., pp. 9,. [] H. Batean, Tables of Integral Transfors. New Yor: McGraw-Hill Boo Copany, 9. APPENDIX A. Transission Rate Calculation. Using (7), the transission rate that corresponds to strea of pair is given by R Pr(ζ τ). In the expression of ζ given in (), we denote g as g α (û ) H ˆv. The rando variable (û ) H ˆv has an exponential distribution with paraeter (see []), then g has an exponential distribution with paraeter α. Thus, the SINR expression ζ can be represented as ζ g I. The transission rate in (7) can be +σ re-expressed as: R Pr(ζ τ) g Pr( I + τ) Pr(g σ τi + τσ ) CCDF g (τi + τσ )f(i )di, () x where CCDF g (x) e α is the copleentary cuulative distribution function of rando variable g, and f(i ) is the probability density function of I. Thus, we get R CCDF g (τi + τσ )f(i )di e ( τi +τσ ) α f(i )di e σ τ α MGF I ( τ ). () α B. Derivation of the Moent generating function of I. Fro (), I L α i h i d i e i (w i ) s,j. i j Since w i and s,j,i are independent and identically distributed (i.i.d.) isotropic vectors in the null space of ĥi, (w i ) s,j,i is i.i.d. β(, Q ) distributed for all i, where Q N t N r. d i (w i ) s,j is the su of i.i.d. Beta variables, which j,i can be approxiated to another Beta distribution []. Thus, we have di (w i ) s,j d i β(a i, b i ), in which a i j,i (Q+)d i Q Q and b i (Q )a i. According to the theory adopted in [], e i h i has Γ(Q, B B Q ) as distribution, where Q is the inverse scale paraeter. Let δ B Q. It follows that I L ρ i X i Y i,,i i where ρ i α i d i, X i Γ(Q, δ) and Y i β(a i, b i ). Note that Q and δ are the shape and rate paraeters, respectively. X i Y i is the product of a Gaa and Beta rando variables. Then, the pdf of Z i X i Y i (for z i > ) is given by [] f Zi (z i ) δq Γ(b i) e δzi Ψ(b i, +Q a i ; δz i ), where Γ(Q)B(a i,b i) zq i Ψ is the Kuer function defined in []. Now, we calculate the MGF of Z i at t: MGF Zi ( t) κ + + e tzi f Zi (z i )dz i z Q i e tzi δzi Ψ(b i, + Q a i ; δz i )dz i (i) κ Γ(Q)Γ(a i) δ Q Γ(a i + b i ) (ii) ( t δ + ) Q ( ) Q t δ + F (b i, Q; a i + b i ; + δ ) t F (b i, Q; a i + b i ; + δ ), (7) t where κ δq Γ(b i) Γ(Q)B(a. i,b i) The equality (i) is obtained using the relation fro []. The equality (ii) holds since the Beta function B(a, b) Γ(a)Γ(b) Γ(a+b). L We can write I ρ i Z i, which is the su of,i weighed rando variables Z i with ρ i as weights. The MGF of I at t is then given by: MGF I ( t) L MGF Zi ( tρ i ) i L ( α id i t + ) Q F (b i, K; a i + b i ; δ i The desired result follows fro (8). + δ ). (8) α i d it