Large System Analysis for Amplify & Forward SIMO Multiple Access Channel with Ill-conditioned Second Hop
aa r X i v : . [ c s . I T ] N ov Large System Analysis forAmplify & Forward SIMO Multiple AccessChannel with Ill-conditioned Second Hop
Symeon Chatzinotas
Abstract —Relaying has been extensively studied during thelast decades and has found numerous applications in wirelesscommunications. The simplest relaying method, namely amplifyand forward, has shown potential in MIMO multiple accesssystems, when Gaussian fading channels are assumed for bothhops. However, in some cases ill conditioned channels may appearon the second hop. For example, this impairment could affectcooperative BS systems with microwave link backhauling, whichinvolve strong line of sight channels with insufficient scattering.In this paper, we consider a large system analysis of suchas model focusing on both optimal joint decoding and jointMMSE filtering receivers. Analytical methods based on freeprobability are presented for calculating the ergodic throughput,the MMSE error and the average SINR. Furthermore, theperformance degradation of the system throughput is evaluatedconsidering second hop impairments such as ill-conditioning andrank deficiency, while high- and low-SNR limits are calculated forthe considered performance metrics. Finally, the cooperative BSsystem is compared to a conventional channel resource divisionstrategy and suitable operating points are proposed.
Index Terms —Amplify and Forward, Multiuser Detection, Ill-conditioned Channel, Rank-deficient Channel.
I. I
NTRODUCTION
The Dual Hop (DH) Amplify-and-Forward (AF) relay chan-nel has attracted a great deal of attention mainly due to itslow complexity and its manyfold benefits, such as coverageextension and decreased outage probability. Although the DHAF channel has been extensively studied in the literature[1]–[3], the effect of second hop condition number on itsperformance is not well quantified yet.Assuming Gaussian channel matrices in both hops, authorsin [1] approached the problem asymptotically using Silver-stein’s fixed-point equation and found closed-forms expres-sions for the Stieltjes transform. Under similar assumptions,a finite analysis was recently performed by [2]. On the otherhand, authors in [3] following a replica analysis tackled theproblem of Kronecker correlated Gaussian matrices.In addition, the MIMO MAC has been studied heavilyduring the last decades since it comprises a fundamentalchannel model for multiuser uplink cellular [4] and multibeamreturn link communications [5], [6]. The work in [7], [8]has combined AF relaying with a MAC and has performed afree-probabilistic analysis for channel capacity. Furthermore,the work in [9] has combined AF relaying with cooperative
Base Stations and has performed a replica analysis for channelcapacity and MMSE throughput.In our scenario, we study a DH AF SIMO MAC modellingcooperative BSs with microwave link backhauling and wefocus on the impact of ill-conditioned or rank-deficient MIMOchannel matrices in the second hop. The paradigm of BScooperation (also known as multicell joint decoding and net-work MIMO) was initially proposed almost three decades agoand its performance gain over conventional cellular systemswas demonstrated in two seminal papers [10], [11]. Themain assumption is the existence of a central processor (CP)which is interconnected to all the BSs through a backhaulof wideband, delayless and error-free links. In addition, thecentral processor is assumed to have perfect Channel StateInformation (CSI) about all the wireless links of the system.These assumptions enable the central processor to jointlydecode all the UTs of the system, rendering the conceptof intercell interference void. Since then, there has beenan ongoing research activity extending and modifying theinitial results for more practical propagation environments,transmission techniques and backhaul infrastructures in anattempt to better quantify the performance gain.More specifically, it was demonstrated in [12] that Rayleighfading promotes multiuser diversity which is beneficial forthe ergodic capacity performance. Subsequently, realistic path-loss models and user distribution were investigated in [13],[14] providing closed-form capacity expressions based onthe cell size, path loss exponent and user spatial p.d.f. Thebeneficial effect of MIMO links was established in [15], [16],where a linear scaling with the number of BS antennas wasproven. However, correlation between multiple antennas hasan adverse effect as shown in [4], especially when correlationaffects the BS-side.Regarding backhauling, the ideal assumptions of previousstudies can only be satisfied by fiber connectivity between allBSs and the central processor. However, in current backhaulinfrastructure microwave links are often used, especially inrural environments where the cable network is unavailable.Recent studies have tried to alleviate the perfect backhaulassumption by focusing in finite-rate errorless links to the CP[17], finite-rate errorless links between adjacent BSs [18] andfinite-sum-rate backhaul with imperfect CSI [19]. Contrary tothese approaches, this paper assumes microwave backhaulingfrom all BSs to the CP, operating over the same frequency. TheBSs amplify and forward the received signals to an antennaarray at the CP and thus the backhaul rate is limited by the
Fig. 1. Conceptional illustration of the system model. system geometry, the relaying power and the impairments ofthe second hop MIMO channel.In this direction, the main contributions of this paper are: • the derivation of the ergodic capacity and a lower boundon the average Minimum Mean Square Error (MMSE)for AF SIMO MAC with Ill-conditioned second hop • the derivation of high and low SNR limits for channelcapacity and MMSE performance • the evaluation of the condition number and normalizedrank of the second hop channel matrix on the systemperformance • the performance comparison to a conventional systemwhich employs resource division access to eliminatemultiuser interference.The remainder of this paper is structured as follows: SectionII introduces the system model, while section III describesthe free probability derivations and the main capacity andMMSE results. Section IV verifies the accuracy of the analysisby comparing with Monte Carlo simulations and evaluatesthe effect of various system parameters on the performance.Section V concludes the paper. A. Notation
Throughout the formulations of this paper, normal x , lower-case boldface x and upper-case boldface X font is used forscalars, vectors and matrices respectively. E [ · ] denotes theexpectation, ( · ) H denotes the conjugate transpose matrix, and ⊙ denotes the Hadamard product. The Frobenius norm ofa matrix or vector is denoted by k·k , the absolute value ofa scalar is denoted by |·| and the delta function is denotedby δ ( · ) . ( · ) + is equivalent to max(0 , · ) , {·} is the indicatorfunction and → denotes almost sure (a.s.) convergence.II. S YSTEM M ODEL
A. Input-output Model
Figure 1 is a conceptual illustration of the input-outputmodel, which included M users, K BSs and a CP equippedwith a K -antenna array. It can be seen that the BS-CP(Central Processor) microwave links (second hop) form an ill-conditioned SIMO MAC, whereas the user-BS-CP links canbe modelled as SIMO AF MAC. Gaussian input is consideredat the user-side, while neither users nor relays are aware of theChannel State Information (CSI). On the other hand, the CP is assumed to have perfect knowledge of system-wide CSI. Thedescribed channel model can be expressed as follows: y = H x + z y = H √ ν y + z ⇔ y = √ ν H H x + √ ν H z + z , (1)where the M × vector x denotes the user transmittedsymbol vector with individual Signal to Noise Ratio (SNR) µ ( E [ x x H ] = µ I ), y denotes the K × received symbolvector by the BSs and the K × vector z denotes AWGNat BS-side with E [ z ] = and E [ z z H ] = I . The receivedsignal y is amplified by ν and forwarded and as a result y denotes the K × received symbol vector by the CP and the K × vector z denotes AWGN at CP-side with E [ z ] = and E [ z z H ] = I . It should be noted that for the remainder ofthis document µ and ν will be referred to as First Hop Power(FHP) and Second Hop Power (SHP) respectively.The K × M channel matrix H and the K × K channelmatrix H represent the concatenated channel vectors for theuser-BS and BS-CP links respectively. The first hop Rayleighfading channel H ∼ CN ( , I ) can be modelled as a Gaus-sian matrix with independent identically distributed (i.i.d.)complex circularly symmetric (c.c.s.) elements. The BSs-CPchannel H under line of sight suffers from correlation dueto lack of scattering and thus it can be modelled as anill-conditioned deterministic channel with variable conditionnumber ζ = λ max ( H H H ) /λ min ( H H H ) or as a rank-deficient deterministic channel with variable normalized rank α = rank( H H H2 ) /K . The exact matrix models for H aredescribed in detail in sections III-B and III-D. B. Performance Metrics
The performance metrics considered in this work are thechannel capacity achieved by successive interference cancel-lation at the CP and the average Mimimum Mean Square Error(MMSE) achieved by joint MMSE filtering at the CP followedby single-user decoding. It should be noted that both of thesereceiver structures require multiuser processing at the CP. Onthe other hand, section II-C considers a conventional systemwhere Frequency or Time Division Multiple Access is usedin combination with single-user interference-free decoding atthe CP.The capacity per receive antenna of this channel model isgiven by [20]–[23]:
C = 1 K E h log det (cid:16) I + µν H H H H H H (cid:0) I + ν H H H (cid:1) − (cid:17)i (2) = 1 K E (cid:2) log det (cid:0) I + ν H H H + µν H H H H H H (cid:1)(cid:3) − K E (cid:2) log det (cid:0) I + ν H H H (cid:1)(cid:3) (3) ( a ) = 1 K E (cid:2) log det (cid:0) I + ν H H H (cid:0) I + µ H H H (cid:1)(cid:1)(cid:3) − K E (cid:2) log det (cid:0) I + ν H H H (cid:1)(cid:3) = C − C (4)where step ( a ) uses the property log det( I + AB ) =log det( I + BA ) . It can be observed that the positive term C corresponds to the mutual information due to relaying, whilethe negative term C represents the performance loss due tonoise amplification.The receiver complexity in order to achieve the channelcapacity is quite high since it involves successive interferencecancellation [24]. In this direction, we consider a less complexreceiver which involves multiuser MMSE filtering followedby single-user decoding. Since, this is a linear operation weassume that K = M . The performance of the MMSE receiveris dependent on the achieved MSE averaged over users andchannel realizations and is given by: mmse avg = E " M M X m =1 mmse m = E " M M X m =1 (cid:20)(cid:16) I + µν H H H H (cid:0) I + ν H H H (cid:1) − H H (cid:17) − (cid:21) m,m = E (cid:20) M tr (cid:26)(cid:16) I + µν H H H H (cid:0) I + ν H H H (cid:1) − H H (cid:17) − (cid:27)(cid:21) = E (cid:20) M tr n(cid:0) I + ν H (cid:0) I + µ H H H (cid:1) H H (cid:1) − (cid:0) I + ν H H H (cid:1)o(cid:21) . (5)The average SINR and the achieved throughput per receiveantenna using LMMSE is given by: SINR avg = E " M M X m =1 mmse − m − (6) C mmse = log (1 + SINR avg ) ≥ − log (mmse avg ) = − log (cid:18) M E (cid:20) tr (cid:26)(cid:16) I + ν H (cid:16) I + µ H H H (cid:17) H H (cid:17) − (cid:16) I + ν H H H (cid:17)(cid:27)(cid:21) (cid:19) . (7) Compared to existing literature, our work starts from eq. (4)since the original problem in eq. (3) yields quite involvedsolutions [1]–[3]. In addition, by decomposing the problem intwo components, deeper insights can be acquired. We followa free probabilistic analysis as in [4], [23], [25]–[27] to derivethe channel capacity, but we extend it for the described DHAF SIMO MAC including the noise amplification terms andill-conditioned second hop modelling. More importantly, weconsider the MMSE filtering receiver and we obtain a lowerbound on the average MMSE performance.To simplify the notations during the mathematical analysis,the following auxiliary variables are defined: M = I + µ H H H ˜M = I + ν H H H N = H H H ˜N = H H H K = H H H (cid:0) I + µ H H H (cid:1) = ˜NM˜K = H (cid:0) I + µ H H H (cid:1) H H β = MK where β ≥ is the ratio of horizontal to vertical dimensionsof matrix H (users/BS). C. Conventional System
In a conventional cellular system, the available resources(frequency or time) would have to be split in K pieces inorder to avoid multiuser interference from neighboring BSs.This entails that only K out of M users could be servedsimultaneously, namely one user per cell ( β = 1 ). Moreover,this is the usual approach employed by current standards inorder to avoid co-channel interference . On the plus side,each user or BS relay could concentrate its power on asmaller portion of the resource using Kµ and Kν respectively.Assuming a single user per cell ( K = M ), the conventionalchannel model for a single user-BS-CP link can be written as: y = h x + z y = h √ Kνy + z ⇔ y = √ Kνh h x + √ Kνh z + z (8)with x Gaussian input with E [ x ] = Kµ and z , z AWGNwith E [ z ] = E [ z ] = 1 . In this case, the per-antenna capacityat the CP would be: C co = E [log (1 + SNR)] = E (cid:20) log (cid:18) K νh µh Kνh (cid:19)(cid:21) , (9)where h and h are the channel coefficients of the firstand second hop respectively. The first and second hop aremodelled as Rayleigh fading and AWGN channels respectivelyand thus we can assume that h ∼ CN (0 , and h = 1 . Theperformance of the conventional and proposed transmissionschemes are compared in section IV-D.III. P ERFORMANCE A NALYSIS
In order to calculate the system performance analytically, weresort to asymptotic analysis which entails that the dimensionsof the channel matrices grow to infinity assuming propernormalizations. It has already been shown in many occasionsthat asymptotic analysis yields results which are also validfor finite dimensions [22], [28], [29]. In other words, theexpressions of interest converge quickly to a deterministicvalue as the number of channel matrix dimensions increases.In this direction, the components of eq. (4) can be writtenasymptotically as: C = 1 K lim K,M →∞ E (cid:2) log det (cid:0) I + ν H H H (cid:0) I + µ H H H (cid:1)(cid:1)(cid:3) = lim K,M →∞ E " K K X i =1 log (1 + νλ i ( K )) → Z ∞ log (1 + νx ) f ∞ K ( x ) d x, (10) In reality, higher frequency reuse can be used in order to exploit spatialseparation of cells. However, frequency reuse cannot be exploited in theconsidered system without creating multiuser interference in the CP throughthe AF relaying. C = 1 K lim K,N →∞ E (cid:2) log det (cid:0) I + ν H H H (cid:1)(cid:3) = lim K,N →∞ E " K K X i =1 log (cid:16) νλ i (cid:16) ˜N (cid:17)(cid:17) → Z ∞ log (1 + νx ) f ∞ ˜N ( x ) d x, (11)where λ i ( X ) is the i th ordered eigenvalue of matrix X and f ∞ X is the asymptotic eigenvalue probability density function(a.e.p.d.f.) of X . It should be noted that while the channeldimensions K, M grow to infinity, the matrix dimension ratio β is kept constant.Using a similar approach, the average MMSE when β = 1 can be expressed as: mmse avg = lim K,M →∞ E (cid:20) M tr (cid:26)(cid:16) I + ν ˜K (cid:17) − ˜M (cid:27)(cid:21) (12) ( a ) ≥ lim K,M →∞ E M M X m =1 λ M − m +1 (cid:16) ˜M (cid:17) νλ m (cid:16) ˜K (cid:17) → Z F − ˜M (1 − x )1 + νF − ˜K ( x ) d x (13)where step ( a ) follows from property tr { AB } ≥ P Mm =1 λ m ( A ) λ M − m +1 ( B ) in [30] and F − X denotes the in-verse function of the asymptotic eigenvalue cumulative densityfunction (a.e.c.d.f.). The last step follows from the fact that theordered eigenvalues can be obtained by uniformly samplingthe inverse c.d.f. in the asymptotic regime [5].To calculate the expression of eq. (10),(11),(13), it sufficesto derive the asymptotic densities of K , ˜N , ˜K , ˜M , whichcan be achieved through the principles of free probabilitytheory [31]–[34] as described in sections III-A and III-B.Free probability (FP) has been proposed by Voiculescu [31]and has found numerous applications in the field of wirelesscommunications. More specifically, FP has been applied forcapacity derivations of variance profiled [13], correlated [4]Rayleigh channels, as well as Rayleigh product channels [25].Furthermore, it has been used for studying cooperative relays[35], interference channels [23] and interference alignmentscenarios [26]. The advantage of FP methodology comparedto other techniques, such as Stieltjes method, replica analysisand deterministic equivalents, is that the derived formulas usu-ally require just a polynomial solution instead of fixed-pointequations. However, the condition for these simple solutions isthat the original aepdfs can be expressed in polynomial form[36]. For completeness, some preliminaries of Random MatrixTheory have been included in appendix A in order to facilitatethe comprehension of derivations in sections III-A, III-B andIII-D. A. Fading First Hop
The first hop from users to BSs can be modelled as aRayleigh fading channel, namely H ∼ CN ( , I ) . Definition III.1.
Considering a Gaussian K × M channelmatrix H ∼ CN ( , I ) , the a.e.p.d.f. of K H H H convergesalmost surely (a.s.) to the non-random limiting eigenvaluedistribution of the Marˇcenko-Pastur law [37], whose densityfunctions are given by f ∞ K H H H ( x ) → f MP ( x, β ) f MP ( x, β ) = (1 − β ) + δ ( x ) + q ( x − a ) + ( b − x ) + πx (14) where a = (1 − √ β ) , b = (1 + √ β ) and η -transform, Σ -transform and Shannon transform are given by [28] η MP ( x, β ) = 1 − φ ( x, β )4 x (15) φ ( x, β ) = r x (cid:16) p β (cid:17) + 1 − r x (cid:16) − p β (cid:17) + 1 ! Σ MP ( x, β ) = 1 β + x (16) V MP ( x, β ) = β log (cid:18) x − φ ( x, β ) (cid:19) + log (cid:18) xβ − φ ( x, β ) (cid:19) − x φ ( x, β ) . (17) Lemma III.1.
The cumulative density function of theMarˇcenko-Pastur law for β = 1 is given by: F MP ( x ) = p − x ( x −
4) + 2 arcsin ( − x/
2) + π π . (18) Proof: The c.d.f. follow from eq. (14) after integration for β = 1 . Lemma III.2.
The a.e.p.d.f. of M converges almost surely(a.s.) to: f ∞ M ( x, β, ¯ µ ) → q(cid:0) x − − ¯ µ + 2¯ µ √ β − ¯ µβ (cid:1) (cid:0) ¯ µ + 2¯ µ √ β + ¯ µβ − x + 1 (cid:1) µπ ( x − , (19) where ¯ µ = Kµ.
Proof: The a.e.p.d.f. can be calculated considering thetransformation z ( x ) = (1 + Kµ x ) , where z and x representthe eigenvalues of M and K H H H respectively: f ∞ M ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) z ′ ( z − ( x )) (cid:12)(cid:12)(cid:12)(cid:12) · f ∞ K H H H ( z − ( x )) = 1¯ µ f MP (cid:18) x − µ (cid:19) . (20) Theorem III.1.
The inverse η -transform of M is given by (21) . This analysis can be straightforwardly extended for cases where variablereceived power is considered for each BS due to variable transmit powersor propagation paths across users. In this case, the channel can be modeledas a variance-profiled Gaussian matrix and it can be tackled using a scalingapproximation as described in [4], [13]. η − M ( x ) = − x ¯ µ − β ¯ µ + ¯ µ − p x ¯ µ + 2 x ¯ µ β − x ¯ µ − x ¯ µ + β ¯ µ − β ¯ µ + 2 β ¯ µ + ¯ µ + 2 ¯ µ + 12 x ¯ µ . (21) Proof: See Appendix B.
Theorem III.2.
The a.e.c.d.f. of M for β = 1 is given by: F M ( x ) = p ( x −
1) (4¯ µ − x + 1) − (cid:16) µ − x +12¯ µ (cid:17) µ + π ¯ µ π ¯ µ . (22) Proof: The c.d.f. follows from eq. (19) after integration for β = 1 . Theorem III.3.
The inverse η -transform of K is given by: η − K ( x ) = Σ ˜N ( x − η − M ( x ) (23) Proof: Given the asymptotic freeness between determinis-tic matrix with bounded eigenvalues ˜N and unitarily invariantmatrix M , the Σ -transform of K is given by multiplicative freeconvolution: Σ K ( x ) = Σ ˜N ( x )Σ M ( x ) ( a ) ⇐⇒ (cid:18) − x + 1 x (cid:19) η − K ( x + 1) = Σ ˜N ( x ) (cid:18) − x + 1 x (cid:19) η − M ( x + 1) where step ( a ) combines Definition A.3 and eq. (16) . Thevariable substitution y = x + 1 yields eq. (23) .B. Ill-conditioned Second Hop Matrix H is modelled as a deterministic matrix withpower normalization tr( H H H ) = K . Due to the lack ofscattering in line-of-sight environments, this matrix may beill-conditioned. The simplest model would be to assume auniform distribution of eigenvalues with support [ ζ − , ζ ] andcondition number ζ . The a.e.p.d.f. and transforms for uniformeigenvalue distribution with variable condition number aregiven by: f ∞ ˜N ( x ) = ζζ − (cid:8) ζ − . . . ζ (cid:9) (24) η ˜N ( x ) = ζ (ln ( ζ ) − ln ( ζ + x ) + ln (1 + xζ ))( ζ − x (25) S ˜N ( x ) = ζ (ln ( ζ ) − ln ( xζ −
1) + ln ( x − ζ )) ζ − . (26)However, this model results in exponential expressions forthe R - and Σ -transforms which yields complex closed formexpressions. To construct an analytically tractable problem, weconsider the tilted semicircular law distribution which can ac-commodate a variable condition number and more importantlyits Σ -transform is given by a first degree polynomial [38]. Theorem III.4.
In the asymptotic regime preserving the powernormalization, the tilted semicircular law converges to thefollowing distribution: f ∞ ˜N = 2 ζπ ( ζ − x s ( ζx − + (cid:18) − xζ (cid:19) + (27) with support [ ζ − , ζ ] . In this case, the transforms of the tiltedsemicircular law are given by: η ˜N ( x ) = 1 + 2 ζx + ζ − p ζ ( x + ζ + ζx + ζ x )( ζ − (28) S ˜N ( x ) = − x + 2 ζ − ζ x + 2 p ζ ( − x + ζx + ζ − ζ x ) x ( ζ − (29) R ˜N ( x ) = 2 x ζ − p ζ ( ζ + 2 ζx − x − ζ x )( ζ − (30) Σ ˜N ( x ) = 1 − ( ζ − ζ x (31) Proof: The closed-form expressions for the transforms arederived by integrating over the aepdf (27) using the definitionsin app. B.
Theorem III.5.
The capacity term C is given in closed formusing the Shannon transform: C = V ˜N ( ν ) (33) and in the low SNR regime: lim ν → C = 4 ζ − (cid:0) ζ + 4 ζ + 1 (cid:1) log (4 ζ ) + 4 (cid:0) ζ + 1 (cid:1) ln ( ζ + 1)2 ( ζ − . (34) Proof: The first equation can be derived using eq. (11) and def. A.1. As a result, lim ν → C = V ˜N (0) . Theorem III.6.
The Stieltjes transform of K is given by thesolution of the cubic polynomial in (35) .Proof: The first step is to substitute eq. (21) and (30) into (23) . Using prop. A.1 and applying suitable change ofvariables: xη − K ( − x S K ( x )) + 1 = 0 . (36) The final form of the polynomial is derived through algebraiccalculations.
Remark III.1.
For M = K , the eigenvalues of K and ˜K areidentical. Thus, the a.e.p.d.f. of ˜K is given by eq. (35) andLemma A.1 for β = 1 . Lemma III.3.
The quantity C is given by eq. (10) , where f ∞ K is given by lem. A.1 and eq. (35) . Remark III.2.
The average MMSE mmse avg is given by eq. (13) where F − ˜M ( x ) can be calculated using Theorem III.2 and F − ˜K ( x ) using integration and inversion over the a.e.p.d.f. inRemark III.1. V ˜N ( x )= √ ζ √ x + ζ √ xζ +1+ ( ζ ) log ( xζ + ζ √ ζ √ x + ζ √ xζ +1 ) − ζ log ( ζ + x + xζ √ ζ √ x + ζ √ xζ +1 ) ( ζ − − xζ + ζ − ζ log(2) − ζ log( ζ )+ ( ζ ) log( ζ +1)( ζ − (32) (cid:16) ( ζ − ¯ µ x + 4 ζ ( ζ − ¯ µ x (cid:17) S K ( x ) + (cid:16) ( ζ − (cid:0) ζ + 3 + 2 ζ (cid:1) ¯ µ x − (cid:16) ζ ( ζ − β − ζ (cid:0) − ζ (cid:1)(cid:17) µ − ζ ( ζ − ¯ µ x − (4 ¯ µζ ) (cid:17) S K ( x ) + (cid:16) ( ζ + 1) (cid:0) ζ + 3 − ζ (cid:1) ¯ µ x − ζ + 1) ζ (cid:0) ( β −
1) ¯ µ + ¯ µ (cid:1)(cid:17) S K ( x )( ζ + 1) ¯ µ (35) C. Ill-conditioned Extreme µ/ν
Limits1) High- µ , low- ν limit: In this regime, we consider thecase of high FHP and low SHP for ill-conditioned secondhop. Assuming µ → ∞ , ν → with constant νµ , the quantity C can be written as: lim µ →∞ ν → C = lim µ →∞ ν → C = 1 K lim K,M, µ →∞ ν → E (cid:2) log det (cid:0) I + ν H H H (cid:0) I + µ H H H (cid:1)(cid:1)(cid:3) = 1 K lim K,M →∞ E h log det (cid:16) I + νµ ˜NN (cid:17)i (37)For ill-conditioned matices, the expression in (37) is identicalto the one derived in [38] for a single hop channel with oneside correlation modeled according to the tilted semicircularlaw. As a result, it can be seen that investing on FHP andminimizing SHP results in a spatially correlated channel thatdoes not suffer from noise amplification. In this case, highcondition number entails high correlation for the equivalentchannel. The free-probabilistic analysis of (37) yields a closed-form expression for S ˜NN , aepdf and capacity [38, eq.(13-16)].In the same direction, the average MMSE in this regime canbe simplified into: lim µ →∞ ν → mmse avg = lim K,M,N →∞ E (cid:20) M tr (cid:26)(cid:16) I + νµ ˜NN (cid:17) − (cid:27)(cid:21) = η ˜NN ( νµ ) . (38)Subsequently, using prop. A.1 and S ˜NN from [38], the fol-lowing closed-form expression can be derived for β ≤ : η ˜NN ( x ) = (cid:18) − ζ − xζ − β xζ +2 q x +2 β x + x +2 β x + β x − βx ( xζ − ( ζ − ) (cid:19)(cid:18) ( ζ − − xζ (4 xζ )2 (cid:19) . (39)
2) High- ν limit: In this regime, we consider a capac-ity bound on symmetric systems for high SHP. UsingMinkowski’s inequality and β = γ = 1 , the capacity can be lower bounded as: C ≥ log (cid:18) ν exp (cid:18) K E (cid:2) log det (cid:0) H H H (cid:0) I + µ H H H (cid:1)(cid:1)(cid:3)(cid:19)(cid:19) − log (cid:18) ν exp (cid:18) K E (cid:2) log det (cid:0) H H H (cid:1)(cid:3)(cid:19)(cid:19) = log (cid:18) ν exp (cid:18) K E h log det (cid:16) ˜N (cid:17)i + 1 K E [log det ( M )] (cid:19)(cid:19) − log (cid:18) ν exp (cid:18) K E h log det (cid:16) ˜N (cid:17)i(cid:19)(cid:19) In the high- ν limit, using lim ν →∞ log(1 + νx ) = log( νx )lim ν →∞ C ≥ K E [log det ( M )] = V N ( µ, β ) → V MP (¯ µ, β ) (40)where V MP is given by def. III.1. As a result, it is shown thatfor high SHP the performance becomes independent of thecharacteristics of the second hop and it is entirely governedby the first hop. More specifically, this tight bound correspondsto the capacity of the first hop, which acts like bottleneck inthis regime.Similarly, it can be seen that this result also applies for theaverage MMSE: lim ν →∞ mmse avg = E (cid:20) M tr n(cid:0) I + µ H H H (cid:1) − o(cid:21) = η N ( µ, β ) → η MP (¯ µ, β ) . (41) D. Rank-deficient Second Hop
Rank deficiency is an extreme form of ill-conditioningwhere zero eigenvalues appear. In some problems, rank de-ficient matrices are used to approximate ill-conditioned onesby substituting infidecimal eigenvalues with zero. In order tomodel rank deficiency, we consider a K × K channel matrix H with tr { H H H } = K and rank { H H H } = αK where α ∈ (0 , is the normalized rank, namely the matrix ranknormalized by the matrix dimension. By varying α from zeroto unity, we can recover unit- to full-rank matrices. Theorem III.7.
In the asymptotic regime, the capacity con-verges to C → α V MP (cid:18) α ¯ µνν + α , βα (cid:19) . (42) x f ∞ K Fig. 2. A.e.p.d.f. plots of matrix K . Parameters: β = 1 , ν = µ = 10 dB .The solid analytic curves follow tightly the simulation-generated bars. The a.e.p.d.f. of matrix K follows a scaled version of the MPlaw: f ∞ K = α ¯ µ f MP (cid:18) αx − α ¯ µ , βα (cid:19) . (43) Proof: See Appendix C.
Remark III.3.
For rank-deficient second-hop, the MMSEperformance degrades rapidly since the equivalent receivedimensions are fewer that the number of users. As a result,the MMSE receiver could only be used if the channel rank islarger that the number of served users α ≥ β . IV. N
UMERICAL R ESULTS
In order to verify the accuracy of the derived closed-formexpressions and gain some insights on the system performanceof the considered model, a number of numerical results arepresented in this section.
A. A.e.p.d.f. Results
The accuracy of the derived closed-form expressions forthe a.e.p.d.f. of matrices K , M is depicted in Figures 2 and3 for ill-conditioned second hop. The solid line in subfigure2 is drawn using Theorem III.6 in combination with lem.A.1 , in subfigure 3 using lem. (III.2). The histograms denotethe p.d.f. of matrices K , M calculated numerically based onMonte Carlo simulations for K = 10 . It can be seen that thereis a perfect agreement between the two sets of results whichverifies our analytic results. B. Capacity Results
Figures 4 and 5 depict the effect of condition number ζ and normalized rank α on the per-antenna channel capacity C of the DH AF SIMO MAC and the per-antenna channelcapacity C which corresponds to an ill conditioned or rank-deficient single hop SIMO MAC respectively. The analytic The far left bar in subfig. 3 corresponds to the zero-eigevalues Dirac delta δ ( x ) which occurs due to rank deficiency (e.g. see eq. (14)). x f ∞ K Fig. 3. A.e.p.d.f. plots of matrix M . Parameters: β = 1 , ν = µ = 10 dB .The solid analytic curves follow tightly the simulation-generated bars. −0 −10 −20 −30 −40 Normalized rank α (dB) Condition Number ζ (dB) C a p a c i t y C ( n a t / s / H z ) Ill-Conditioned AF DHIll-Conditioned Single HopRank-deficient AF DHRank-deficient Single Hop
Fig. 4. Per-antenna capacity scaling vs. condition number ζ and normalizedrank α in dBs. Parameters: µ = ν = β = 1 . Condition number ζ Second Hop Power ν C ( n a t / s / H z ) Fig. 5. Per-antenna capacity scaling vs. condition number ζ and secondhop power ν in dBs. Parameters: µ = 10 dB, β = 1 . Second Hop Power ν Condition number ζ mm s e a v g Fig. 6. Average MMSE scaling vs. condition number ζ and second hoppower ν in dBs. Parameters: µ = 10 dB, β = 1 . For high amplification, firsthop performance acts as bottleneck. −10 −5 0 5 10 15 20 25 3000.010.020.030.040.050.060.070.080.09 Second Hop Power ν mm s e a v g ζ = 0 dB ζ = 10 dB ζ = 20 dB Fig. 7. Average MMSE performance (solid line) and proposed lower bound(dashed line) vs second hop power ν . Parameters: µ = 10 dB solid curves are plotted using a) eq. (10) and eq. (11) forthe ill-conditioned DH AF, b) eq. (42) for the rank deficientDH AF, c) eq. (32) for the ill-conditioned single hop and d)eq. (51) for the rank deficient single hop. It can be seen thatthe performance degrades much more steeply with normalizedrank than condition number in all cases. Especially for the DHAF, it can be observed that rank deficiency is detrimental andquickly drives capacity to zero due to rank loss. On the otherhand, the degradation with condition number is much smoothersince the second hop channel matrix H is still full-rank.In addition, the per-channel capacity C is plotted versus thesecond hop power ν and condition number ζ . As it can beseen, it is possible to recover part of the lost performance dueto ill-conditioning by increasing the amplification level ν . C. MMSE Results
Figure 6 depicts the effect of condition number ζ andsecond hop power ν on the average MMSE. As expected, theaverage MMSE increases with ζ but decreases with ν . It Condition number ζ mm s e a v g ν = −
10 dB ν = 0 dB ν = 10 dB Fig. 8. Average MMSE performance (solid line) and proposed lower bound(dashed line) vs condition number ζ . Parameters: µ = 10 dB Condition number ζ (dB) C ( n a t / s / H z ) Channel CapacityMMSE ThroughputConventional Throughput
Fig. 9. Throughput comparison between proposed and conventional systemvs. condition number ζ in dBs. Parameters: µ = ν = 10 dB, β = 1 . Theproposed system is preferable for condition numbers up to dBs. can be seen that performance can be improved using strongeramplification but for high ν there is a saturation thresholdwhich is governed by the first hop performance as describedin sec. III-C2. Figures 7 and 8 depicts the accuracy of theproposed lower bound. The solid plots were calculated throughMonte Carlo simulations of eq. (5), whereas the dashed plotsrepresent our lower bound which was calculated using RemarkIII.2. It can be seen that the proposed bound is tight for lowvalues of ζ , but it progressively diverges as ν and ζ growlarge. D. Comparison
In this section, the performance of the proposed system iscompared to the conventional system (as described in sectionII-C) by fixing the user and BS power at dBs. As it canbe seen in Fig. 9, while the condition number increases, theperformance of the proposed system degrades and even fallsbelow conventional performance for extremely ill-conditioned BS-CP channels. There are two crossing points in and dBs for the MMSE throughput and channel capacityrespectively. However, a two-fold performance gain can stillbe harnessed for condition numbers up to dBs for MMSEreceiver and up to dBs for optimal receiver.V. C
ONCLUSION
In this paper, we have investigated the performance ofBS cooperation scenario with microwave backhauling to aCP, where multiple users and BSs share the same channelresources. The user signals are forwarded by the BSs to anantenna array connected to a CP which is responsible formultiuser joint processing. This system has been modelledas a DH AF SIMO MAC with a ill-conditioned or rank-deficient second hop due to lack of scattering in line-of-sightenvironments. Its performance in terms of channel capacityand MMSE performance has been analysed through a large-system free-probabilistic analysis. It can be concluded thatperformance degrades much more gracefully with conditionnumber than with loss of rank. As a result, a performancegain can be achieved compared to conventional resourcepartitioning even for highly ill-conditioned second hop. Fur-thermore, performance degradation due to ill conditioning canbe compensated through stronger amplification at BS-side untilit reaches the first hop performance in the high amplificationlimit. A
PPENDIX AR ANDOM M ATRIX T HEORY P RELIMINARIES
Let f X ( x ) be the eigenvalue probability distribution func-tion of a matrix X . Definition A.1.
The Shannon transform of a positive semidef-inite matrix X is defined as V X ( γ ) = Z ∞ log (1 + γx ) f X ( x ) dx. (44) Definition A.2.
The η -transform of a positive semidefinitematrix X is defined as η X ( γ ) = Z ∞
11 + γx f X ( x ) dx. (45) Definition A.3.
The Σ -transform of a positive semidefinitematrix X is defined as Σ X ( x ) = − x + 1 x η − X ( x + 1) . (46) Property A.1.
The Stieltjes-transform of a positive semidefi-nite matrix X can be derived by its η -transform using S X ( x ) = − η X ( − /x ) x . (47) Lemma A.1.
The a.e.p.d.f. of X is obtained by determiningthe imaginary part of the Stieltjes transform S for realarguments f ∞ X ( x ) = lim y → + π I {S X ( x + j y ) } . (48) A PPENDIX BP ROOF OF T HEOREM
III.1Starting from eq. (20) and following def. A.2: η M ( ψ ) = Z + ∞−∞
11 + ψx f ∞ M ( x ) dx = 1 γ Z + ∞−∞
11 + ψx f ˜N (cid:18) x − γ (cid:19) dx ( a ) = γ iπ I | ζ | =1 ( ζ − ζ ((1 + β ) ζ + √ β ( ζ + 1))( ζ (1 + ψ (1 + γ + γβ )) + √ βψγ ( ζ + 1)) dζ. (49)Step ( a ) requires the variable substitutions x = wγ + 1 , dx = γdw , followed by w = 1 + β + 2 √ β cos ω , dw =2 √ β ( − sin ω ) dω and finally ζ = e iω , dζ = iζdω [39]. Subse-quently, a Cauchy integration is performed by calculating thepoles ζ i and residues ρ i of eq. (49): ζ = 0 ,ζ , = − (1 + β ) ± (1 − β )2 √ β ,ζ , = − − ψ γ − ψ β γ − ± (cid:16) ψ p ψ + 2 ψ γ + 2 ψ β γ + ψ + 2 ψ γ + 2 ψ β γ + ψ γ − β ψ γ + ψ β γ (cid:17) √ βψ γ Using the residues which are located within the unit disk, theCauchy integration yields: η M ( ψ ) = − β ρ + ρ + ρ ) Inversion yields eq. (21).A
PPENDIX CP ROOF OF T HEOREM
III.7The components of eq. (4) can be written as: C = 1 K lim K,M →∞ E (cid:2) log det (cid:0) I K + ν H H H (cid:0) I K + µ H H H (cid:1)(cid:1)(cid:3) = 1 K lim K,M →∞ E h log det (cid:16) I K + να D (cid:0) I K + µ H H H (cid:1)(cid:17)i = 1 K lim K,M →∞ E h log det (cid:16) I αK + να (cid:0) I αK + µ ¯H ¯H H (cid:1)(cid:17)i = α log (cid:16) να (cid:17) + 1 K lim K,M →∞ E (cid:20) log det (cid:18) I αK + aKµνν + α ¯H ¯H H αK (cid:19)(cid:21) → α log (cid:16) να (cid:17) + α V MP (cid:18) α ¯ µνν + α , βα (cid:19) , (50) C = 1 K lim K →∞ E (cid:2) log det (cid:0) I K + ν H H H (cid:1)(cid:3) = 1 K lim K →∞ E h log det (cid:16) I K + να D (cid:17)i = 1 K lim K →∞ E h log det (cid:16) I αK (cid:16) να (cid:17)(cid:17)i → α log (cid:16) να (cid:17) , (51)where D is a K × K zero matrix with αK ones across itsdiagonal and ¯H is a αK × M submatrix of H . Substraction yields the capacity expression. The aepdf follows from theequivalent matrix K : K = 1 α (cid:18) I αK + aKµ ¯H ¯H H aK (cid:19) . (52)R EFERENCES[1] V. Morgenshtern and H. Bolcskei, “Crystallization in large wirelessnetworks,”
IEEE Trans. Inf. Theory , vol. 53, no. 10, pp. 3319 –3349,oct. 2007.[2] S. Jin, M. McKay, C. Zhong, and K.-K. Wong, “Ergodic capacityanalysis of amplify-and-forward MIMO dual-hop systems,”
IEEE Trans.Inf. Theory , vol. 56, no. 5, pp. 2204 –2224, may 2010.[3] J. Wagner, B. Rankov, and A. Wittneben, “Large n analysis of amplify-and-forward MIMO relay channels with correlated rayleigh fading,”
IEEE Trans. Inf. Theory , vol. 54, no. 12, pp. 5735 – 5746, Dec 2008.[4] S. Chatzinotas, M. Imran, and R. Hoshyar, “On the multicell processingcapacity of the cellular MIMO uplink channel in correlated Rayleighfading environment,”
IEEE Trans. Wireless Commun. , vol. 8, no. 7, pp.3704–3715, July 2009.[5] N. Letzepis and A. Grant, “Capacity of the multiple spot beam satellitechannel with rician fading,”
IEEE Trans. Inf. Theory , vol. 54, no. 11,pp. 5210 –5222, nov. 2008.[6] D. Christopoulos, J. Arnau, S. Chatzinotas, C. Mosquera, and B. Otter-sten, “MMSE performance analysis of generalized multibeam satellitechannels,”
Communications Letters, IEEE , no. 99, pp. 1–4, 2013.[7] S.Chatzinotas and B.Ottersten, “Capacity analysis of dual-hop amplify-and-forward MIMO multiple-access channels,” in
International Confer-ence on Wireless Communications and Signal Processing (WCSP 2012) ,Hunagshan, China, Oct 2012.[8] S. Chatzinotas, “MMSE filtering performance of dual-hop amplify-and-forward multiple-access channels,”
Wireless Communications Letters,IEEE , vol. 2, no. 1, pp. 122–125, 2013.[9] C.-K. Wen and K.-K. Wong, “On the sum-rate of uplink MIMOcellular systems with amplify-and-forward relaying and collaborativebase stations,”
Selected Areas in Communications, IEEE Journal on ,vol. 28, no. 9, pp. 1409 –1424, december 2010.[10] S. V. Hanly and P. A. Whiting, “Information-theoretic capacity of multi-receiver networks,”
Telecommun. Syst. , vol. 1, pp. 1–42, 1993.[11] A. Wyner, “Shannon-theoretic approach to a Gaussian cellular multiple-access channel,”
IEEE Trans. Inf. Theory , vol. 40, no. 6, pp. 1713–1727,Nov 1994.[12] O. Somekh and S. Shamai, “Shannon-theoretic approach to a Gaussiancellular multiple-access channel with fading,”
IEEE Trans. Inf. Theory ,vol. 46, no. 4, pp. 1401–1425, Jul 2000.[13] S. Chatzinotas, M. Imran, and C. Tzaras, “On the capacity of variabledensity cellular systems under multicell decoding,”
IEEE Commun. Lett. ,vol. 12, no. 7, pp. 496 – 498, Jul 2008.[14] S. Chatzinotas, M. A. Imran, and C. Tzaras, “Optimal informationtheoretic capacity of the planar cellular uplink channel,” in
IEEE 9thWorkshop on Signal Processing Advances in Wireless Communications(SPAWC’08) , Pernambuco, Brazil, Jul 2008, pp. 196–200.[15] D. Aktas, M. Bacha, J. Evans, and S. Hanly, “Scaling results on thesum capacity of cellular networks with MIMO links,”
IEEE Trans. Inf.Theory , vol. 52, no. 7, pp. 3264–3274, July 2006.[16] S. Chatzinotas, M. A. Imran, and C. Tzaras, “Uplink capacity of MIMOcellular systems with multicell processing,” in
IEEE International Sym-posium on Wireless Communication Systems (ISWCS’08) , Reykjavik,Iceland, Oct 2008, pp. 453–457.[17] A. Sanderovich, O. Somekh, H. Poor, and S. Shamai, “Uplink macrodiversity of limited backhaul cellular network,”
Information Theory,IEEE Transactions on , vol. 55, no. 8, pp. 3457–3478, 2009.[18] O. Simeone, O. Somekh, H. Poor, and S. Shamai, “Local base stationcooperation via finite-capacity links for the uplink of linear cellularnetworks,”
Information Theory, IEEE Transactions on , vol. 55, no. 1,pp. 190–204, 2009.[19] P. Marsch and G. Fettweis, “Uplink comp under a constrained backhauland imperfect channel knowledge,”
Wireless Communications, IEEETransactions on , vol. 10, no. 6, pp. 1730–1742, 2011.[20] G. J. Foschini and M. J. Gans, “On limits of wireless communicationsin a fading environment when using multiple antennas,”
Wirel. Pers.Commun. , vol. 6, no. 3, pp. 311–335, 1998.[21] R. Blum, “MIMO capacity with interference,”
IEEE J. Select. AreasCommun. , vol. 21, no. 5, pp. 793–801, June 2003. [22] A. Lozano and A. Tulino, “Capacity of multiple-transmit multiple-receive antenna architectures,”
IEEE Trans. Inf. Theory , vol. 48, no. 12,pp. 3117 – 3128, dec 2002.[23] S. Chatzinotas and B. Ottersten, “Free probability basedcapacity calculation of multiantenna Gaussian fading channelswith cochannel interference,”
Physical Communication , Leipzig, Germany, Jun 2009.[25] R. M¨uller, “A random matrix model of communication via antennaarrays,”
IEEE Trans. Inf. Theory , vol. 48, no. 9, pp. 2495–2506, Sep2002.[26] S. Chatzinotas and B. Ottersten, “Interference mitigation techniquesfor clustered multicell joint decoding systems,”
EURASIP Journal onWireless Communications and Networking, Multicell Cooperation forNext Generation Communication Systems Series , vol. 132, 2011.[27] S. Sharma, S. Chatzinotas, and B. Ottersten, “Eigenvalue based sensingand SNR estimation for cognitive radio in presence of noise correlation,”
Vehicular Technology, IEEE Transactions on , no. 99, pp. 1–1, 2013.[28] A. M. Tulino and S. Verd´u, “Random matrix theory and wirelesscommunications,”
Commun. Inf. Theory , vol. 1, no. 1, pp. 1–182, 2004.[29] C. Martin and B. Ottersten, “Asymptotic eigenvalue distributions andcapacity for MIMO channels under correlated fading,”
IEEE Trans.Wireless Commun. , vol. 3, no. 4, pp. 1350–1359, Jul 2004.[30] J. Lasserre, “A trace inequality for matrix product,”
Automatic Control,IEEE Transactions on , vol. 40, no. 8, pp. 1500 –1501, aug 1995.[31] D. Voiculescu, “Asymptotically commuting finite rank unitary operatorswithout commuting approximants,”
Acta Sci. Math. , vol. 45, pp. 429–431, 1983.[32] F. Hiai and D. Petz, “Asymptotic freeness almost everywhere for randommatrices,”
Acta Sci. Math. (Szeged) , vol. 66, pp. 801–826, 2000.[33] ——, “The semicircle law, free random variables and entropy,”
Mathe-matical Surveys and Monographs , vol. 77, 2000.[34] Z. D. Bai, “Methodologies in spectral analysis of large dimensionalrandom matrices, a review,”
Statistica Sinica , vol. 9, pp. 611–677, 1999.[35] Z. H. Husheng Li and H. Poor, “Asymptotic analysis of large cooperativerelay networks using random matrix theory,”
EURASIP Journal onAdvances in Signal Processing , 2008, article ID 235867.[36] N. Letzepis and A. Grant, “Shannon transform of certain matrixproducts,” in
Information Theory, 2007. ISIT 2007. IEEE InternationalSymposium on , june 2007, pp. 1646 –1650.[37] V. Marˇcenko and L. Pastur, “Distributions of eigenvalues of some setsof random matrices,”
Math. USSR-Sb. , vol. 1, pp. 507–536, 1967.[38] X. Mestre, J. Fonollosa, and A. Pages-Zamora, “Capacity of MIMOchannels: Asymptotic evaluation under correlated fading,”
IEEE J.Select. Areas Commun. , vol. 21, no. 5, pp. 829–838, Jun 2003.[39] Z. Bai and J. Silverstein,