ON BASE STATION COOPERATION SCHEMES FOR UPLINK NETWORK MIMO UNDER A CONSTRAINED BACKHAUL

Transcription

1 ON SE STTION COOPERTION SCHEMES FOR UPLINK NETWORK MIMO UNDER CONSTRINED CKHUL Patrick Marsch and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Dresden, Germany STRCT n increasing demand for higher spectral efficiencies in mobile communications will require next generation cellular systems to employ a dense reuse of spectrum in combination with smart interference mitigation or cancellation schemes. Recent publications have revealed the large potential spectral efficiency and fairness gains achievable with multi-cell cooperative schemes, where multiple base stations jointly receive or transmit signals connected to multiple terminals, often referred to as network MIMO. One main issue, however, is the large extent of backhaul capacity required between cooperating base stations. In this paper, we focus on the cellular uplink and investigate the information theoretical limits of joint detection under a constrained backhaul. We propose a framework that incorporates different concepts from information theory and allows us to observe the benefit of different kinds of information exchange between the base stations. Monte Carlo simulation results suggest that a next generation cellular system should ideally adapt the base station cooperation scheme according to the current channel realization. I. INTRODUCTION Multi-cell cooperative detection or transmission in cellular systems has been proposed by e.g. [14], revealing strong network capacity and fairness improvements. ssuming infinite cooperation between base stations, the capacity limits of joint detection in the uplink have been explored in the context of multiple access channels by e.g. [4], and more realistic bounds for practical OFDM systems - assuming infinite backhaul within large clusters of cooperating base stations - observed in [9]. To make multi-cell cooperative signal processing economically attractive for next generation cellular systems, it appears necessary to strongly reduce the extent of backhaul traffic generated between cooperating base stations. We have recently investigated techniques that achieve this to a certain extent by selecting only subsets of users for joint signal processing [7], possibly in connection with smart scheduling techniques [6]. In this paper, we focus on a toy scenario in the uplink and investigate information theoretical bounds of backhaulconstrained cooperative detection. The problem is similar to the Gaussian CEO problem [13], where multiple agents (in our case the base stations) make correlated observations of the transmitted symbols and report these through constrained links to a central processing unit. This is comparable to the concept of compress-and-forward [2], well-known in the context of relaying, where coding techniques such as Slepian-Wolf or Wyner-Ziv [15] are usually employed. Recently, the authors in [10] have combined compress-and-forward techniques based on [13] with decode-and-forward techniques [2] and investigated backhaul-constrained cooperative detection within a circular Wyner model in [11, 12]. Here, superposition coding is employed, such that each base station decodes a portion of its own terminal s transmission, and relays only the uncertainty about the remaining signal to a central unit. s opposed to [11, 12], we consider arbitrary channel realizations, and allow a direct cooperation between base stations (i.e. there is no central unit). s in [8], we consider that either quantized receive signals are exchanged between base stations to enable joint detection - similar to the concept in [10, 11] -, or decoded signals are exchanged, such that the involved base stations can pre-subtract the interference from certain terminals before detecting their own terminals. Our framework also incorporates the concept of locally decoded messages as in [11], as well as common messages decoded by both base stations, known to improve the rate region in non-cooperative interference channels [1, 5], and also concepts of frequency division multiplex (FDM). The paper is organized as follows. In section II., we describe our system model and basics of cooperative detection schemes. In sections III. and IV. we derive achievable rate regions for different cooperation schemes and state the concept of performance regions that also incorporate the backhaul required to achieve certain rates. The paper is concluded with simulation results in section V. and conclusions in section VI.. II. SYSTEM MODEL In this paper, we consider an uplink transmission from two terminals a and b with one transmit antenna each to two base stations and with an arbitrary number of receive antennas N bs each, as depicted in figure 1. We assume that the transmission takes place through a frequency-flat channel, for example a single sub-carrier of an OFDM system, described through [ ] h H = a h b = [h a h b ], (1) h a h b where h a, for instance, describes the channel coefficients between base station and terminal a, and h a,h b,h a,h b C [ 1]. We assume that both base stations (Ss) have perfect knowledge of H, and that all four involved entities are perfectly synchronized in time and frequency, such that the transmission is free of inter-symbol and inter-carrier interference. s in [8], we consider two basic concepts of cooperation: 1. Relaying certainty (decode and forward). base station decodes a terminal s signal and forwards the decoded

2 ha backhaul link a h a h b h b Figure 1: Uplink transmission considered in this paper. data to other base stations which then pre-subtract this known interference before decoding other terminals. We call this distributed interference subtraction (DIS). 2. Relaying uncertainty (compress and forward). base station quantizes and forwards received signals to a partnering base station, where a joint decoding of terminals is performed. This corresponds to a concept often referred to as a distributed antenna system (DS). We consider superposition coding, i.e. we assume that terminal a invests its transmit power into superimposed messages,v a,w a,j a, and terminal b transmits messages U b,v b,w b,j b, where the corresponding transmit powers are denoted as p Ua,p Ub,p Va,p Vb,p Wa,p Wb,p Ja,p Jb. Each message consists of N symbols consecutively transmitted over the channel, hence e.g. = {s [1],s [2],,s [N] }. The base stations perform the following decoding and cooperation steps: Messages,V a and U b,v b are decoded conventionally by Ss and, respectively (hence without cooperation), a concept also employed in [11]. Messages W a and W b are also decoded conventionally, but by both base stations and, corresponding to the concept of common messages in [1, 5] Now, one phase of information exchange takes place over the backhaul. The Ss exchange the already decoded messages V a and V b (corresponding to the DIS concept stated before), and compress and forward the remaining undecoded signals (corresponding to the DS concept) Finally, messages J a,j b are jointly decoded, benefiting from the previous information exchange over the backhaul In the sequel, we will use the following set notation: S all S a S b S S S b = {,U b,v a,v b,w a,w b,j a,j b } : all messages = {,V a,w a,j a } : messages transmitted by a = {U b,v b,w b,j b } : messages transmitted by b = {,V a,w a,w b } : messages conv. decoded by = {U b,v b,w a,w b } : messages conv. decoded by = {J a,j b } : messages jointly decoded by and The transmission of each symbol can be stated as px s [n] y [n] X = H a px s [n] X b + n, (2) where y C [2N bs 1] are the signals received by the Ss. ll transmitted symbols are assumed to be mutually uncorrelated circularly symmetric Gaussian scalars with X S all : E{s [n] X } = 0 and E{(s[n] X } = 1, and n C[2N bs 1] denotes the thermal noise plus interference from outside the modelled system and received by the Ss, which we assume to be uncorrelated and circularly symmetric Gaussian with a diagonal covariance matrix [ E{nn H } = Φ nn = X )H s [n] Φ nn 0 [ N bs ] 0 [ N bs ] Φ nn ], (3) where Φ nn,φ nn C [N bs N bs ] are the noise covariance matrices connected to base stations and, respectively. III. CHIEVLE RTES We now want to derive the achievable rates for the transmissions of the two terminals as a function of a given power allocation p = [p Ua,p Ub,p Va,p Vb,p Wa,p Wb,p Ja,p Jb ]. First, we use the notation from [5] to state the achievable rate region of all conventionally decoded messages X S S as the set of all rate points R conv (p) = {(R Ua,R Ub,R Va,R Vb,R Wa,R Wb )} that fulfill X S S : R X 0 and S S, S = S \ S : R X I(Y ; S S )[p] (4) S S, S = S \ S : R X I(Y ; S S )[p] (5) where for K {,} the transinformation I(Y K ; S S )[p] is given in equation (6). This notation incorporates the concept of joint decoding [5], i.e. the decoding performance at one base station is independent of any concrete decoding order employed by the other S. For the joint decoding of messages J a,j b (after an information exchange has taken place over the backhaul), we consider four different decoding strategies: 1. oth messages are jointly decoded by S. We can then state the achievable rate region of messages J a,j b as all R jt,1 (p,q )={(R Ja,R Jb )} fulfilling R Ja,R Jb 0 and S S, S = S \ S : R X I(Y ; S S )[p,,q,1,1], (8) where q R + 0 denotes the number of bits employed for the quantization of each received symbol at base station, before forwarding the signals to S. The transinformation I(Y ; S S )[p,q,q,θ,µ] is given in equation (7) and explained later.

3 K {,} : I(Y K ; S S )[p] = ld I + j {a,b} j S h K j p X (h K j ) H } {{ } Covariance of signals to be decoded j {a,b} j\{s S } h K j p X (h K j ) H + Φ K nn } {{ } Covariance of interference and noise 1 (6) I(Y ; S S )[p,q,q,θ,µ] = Scaling Covariance of signals to be decoded Scaling {}}{{ 1 µ }}{{}}{ Ξ(q,q ) 1 2 h j p X h H j Ξ(q,q ) 1 2 j {a,b} j S ld I+ Ξ(q,q ) 1 [ ] 2 h j p X h H h j + b θp Ub (h b )H 0 [ N bs] }{{} 0 [ Scaling j {a,b} \{S S N bs ] h a (1 θ)p Ua (h a ) H +Φ }{{} nn Ξ(q,q ) 1 2 +Φ qq (q,q ) }{{}}{{} } }{{} 3. }{{} Scaling 4. Quant. noise 2. Interference affecting only one S 1. Interference affecting both Ss (7) 2. Similarly, both messages can be jointly decoded by S, yielding an achievable rate region R jt,2 (p,q ) = {(R Ja,R Jb )} fulfilling R Ja,R Jb 0 and S S, S = S \ S : R X I(Y ; S S )[p,q,,0,1], (9) where q R + 0 denotes the number of quantization bits employed at base station. 3. oth messages are decoded individually by and, respectively, but exploiting the interference pre-subtraction, array and diversity gain due to the previous exchange of information. This yields an achievable rate region R jt,3 (p,q,q )={(R Ja,R Jb )} s.t. R Ja,R Jb 0 and R Ja I(Y ;J a )[p,,q,1,1] (10) R Jb I(Y ;J b )[p,q,,0,1] (11) 4. Frequency division multiplex (FDM) is used, i.e. terminal a concentrates its transmit power for message J a into only a portion 0 µ 1 of the available bandwidth, and terminal b transmits J b over the remaining, orthogonal bandwidth portion 1 µ. There is hence no interference between the two messages, and we can state R jt,4 (p,q,q,µ)={(r Ja,R Jb )} s.t. R Ja,R Jb 0 and 0 µ 1 : µr Ja + (1 µ)r Jb µ I(Y ;J a J b )[p,,q,1,µ] + (1 µ) I(Y ;J b J a )[p,q,,0,1 µ] (12) The achievable sum rate region of terminals a and b - as a function of power allocation p, quantization resolution q,q, and the FDM parameter µ - can now be stated as R(p,q,q,µ) = {(R Ua + R Va + R Wa + R Ja,R Ub + R Vb + R Wb + R Jb ) : (R Ua,R Ub,R Va,R Vb,R Wa,R Wb ) R conv (R Ja,R Jb ) R jt,1 R jt,2 R jt,3 R jt,4 } (13) We now explain term I(Y ; S S )[p,q,q,θ,µ] in eq. (7) in detail. It is assumed here that the Ss have already decoded messages S S and exchanged information over the backhaul. In the denominator, we have four interference terms. First, we have to consider the messages J a, J b that have not been decoded yet and interfere both Ss. The second term describes interference that has a different impact on the two Ss. If e.g. S forwards quantized signals to, the signals originally received by are interfered by message U b, whereas the quantized signals provided by contain interference from message which, however, can eventually be removed, as the message is already known to. oolean θ realizes this behaviour. The last two interference terms denote thermal noise Φ nn and quantization noise Φ qq, respectively, where the latter is given through rate distortion theory [3] as Φ qq (q,q ) = [ 2 q Φ yy 0 [ N bs ] 0 [ N bs ] 2 q Φ yy ] (14) Here, Φ yy and Φ yy denote the covariance of the signals that are to be quantized at Ss and, respectively, given as Φ yy =h a p Ja (h a ) H +h b (p Ub +p Vb +p Jb )(h b ) H +Φ nn (15) Φ yy =h b p Jb (h b ) H +h a (p Ua +p Va +p Ja )(h a ) H +Φ nn (16) and q,q denote the number of quantization bits used per received symbol at Ss and, respectively. oth the desired

4 7 chievable sum rate if the common rate is maximized Sum rate [bits/channel use] DIS schemes beneficial in mid backhaul regime DS schemes beneficial in high backhaul regime 5.5 DIS schemes DS schemes DIS/DS combined 5 DIS with superpos. DS with superpos. DIS/DS with superpos. 4.5 FDM FDM beneficial in low ll schemes combined backhaul regime Cut set bound ackhaul [bits/channel use] Figure 2: Performance regions for an example channel (dashed line indicates points where common rate is maximized). signals to be decoded in the nominator of equation (7), as well as the interference and thermal noise are scaled down by matrix ) (1 2 q I 0 [ N Ξ(q,q ) = bs ] ) (1 2 q (17) I 0 [ N bs ] assuring that any signal power before quantization is equal to the signal power after quantization plus the quantization noise. Eqs. (14), (17) imply that quantization exploits the signal correlation at the N bs antennas of each S, but not the signal correlation between both Ss, as e.g. in Wyner-Ziv compression [15]. IV. CHIEVLE PERFORMNCE In [8], we have introduced the concept of performance regions that capture both achievable rates and the backhaul required to achieve certain rates. n achievable performance region is defined as the set of all rates and backhaul fulfilling P(P a max,p b max) = {(R a,r b,β) : (R a,r b ) R(p,q,q,µ) β=r Va +R Vb +q +q } (18) where denotes the convex hull - implying time-sharing - around all performance points based on p,q,q,µ fulfilling p X Pmax, a p X Pmax, b 0 µ 1, q,q 0 (19) a b and Pmax,P a max b R + 0 denote the maximum transmit powers of terminals a and b, respectively. V. SIMULTION RESULTS We provide simulation results for an example channel realization and MonteCarlo results for many channel realizations. We compare the following cooperation schemes that represent the complete parameter space from eq. (19), except that: DIS schemes. Here, the Ss only cooperate by exchanging decoded messages for interference pre-subtraction. Hence, parameters q,q are set to zero, and only decoding strategies 1-3 from section III. are employed. DS schemes. Here, the base stations only cooperate by exchanging quantized signals for joint decoding of messages. Hence, parameters p Va,p Vb are set to zero, and only decoding strategies 1-3 are employed. FDM schemes. Here, we assume that the terminals invest their complete transmit power into messages J a and J b and employ decoding strategy 4. We further distinguish whether superposition coding is used or not, as this appears rather complicated to use in practical systems. Hence, for simulation results where superposition coding is not explicitly stated, the transmit power of each terminal was always invested entirely into one of the possible messages.. Simulation Methodology It is difficult to calculate the performance region for a given channel and power constraints due to the large parameter space

5 Sum rate [bits/channel use] Sum rate maximized MonteCarlo results for N bs =1 and ρ=3d DIS 4.6 DS DIS/DS combined 4.4 DIS with superpos. DS with superpos. 4.2 Common rate DIS/DS with superpos. maximized DIS/DS with sp. + FDM ackhaul [bits/channel use] Figure 3: MonteCarlo simulation results. described by eq. (19) and the four joint decoding strategies described in section III., particularly as the rates in eqs. (6) and (7) are non-convex in the power parameters. We thus perform an initial brute-force search over the parameter space at a moderate resolution and determine the decoding strategies and power allocations supporting the convex performance region. For these points, we then perform more detailed local searches, determine the supporting points again, so that after few iterations we obtain results where the power allocation is optimized to a granularity of less than 0.5% of P a max,p b max, respectively.. n Example Channel Figure 2 shows the performance region of an example channel H = [ i, i; i, i] for different cooperation schemes, Pmax a =Pmax b =1 and Φ nn =0.1 I. We plot the achievable rates of terminals a and b on the x- and y-axis, respectively, and the backhaul β on the z-axis. The top row shows the performance region for DIS, DS and FDM without superposition coding. In the DIS case, we see two dominant performance points where either of the Ss forwards decoded data to the other, yielding partial interference cancellation, but no array or diversity gain. The lower plots in Fig. 2 reveal the superiority of certain cooperation schemes in different backhaul regimes. Whereas DIS concepts can be best for a low backhaul (enabling an efficient usage of the backhaul), DS schemes are clearly superior due to the obtained spatial multiplexing gain in high backhaul regimes. FDM concepts are at most beneficial in regimes of very low backhaul. C. Monte Carlo Simulations For figure 3, many channel realizations were drawn from an i.i.d Rayleigh distribution fulfilling E{(h a ) H h a } = E{(h b )H h b } = 1 and E{(h b )H h b } = E{(h a ) H h a } = 1/ρ, where ρ is a measure for the isolation of the two interfering cells. Here, the sum rate of both terminals is plotted against the required backhaul, if either the sum rate itself or the common rate is maximized. Contrary to the example channel shown before, we see that on average, DIS schemes are strongly inferior to DS schemes, but combining both concepts leads to a slightly improved average performance. In fact, the benefit of employing both DIS and DS will be larger in a practical system, as it is simple to implement DIS, but DS will perform much worse than the rate distortion bound observed here. s expected, FDM concepts show a slight benefit in regimes of very low backhaul. Interestingly, there is not much benefit of using superposition coding, so that an adaptive DIS / DS / FDM concept without superposition coding appears practical while yielding close to optimal performance. VI. CONCLUSIONS In this paper, we investigated different forms of base station cooperation in uplink network MIMO under a constrained backhaul. We incorporated various concepts from information theory into a common framework and observed that a system should ideally adapt the cooperation scheme to the channel, while superposition coding appears only marginally attractive. REFERENCES [1]. Carleial. Interference channels. IEEE Transactions on Information Theory, 24(1):60 70, Jan [2] T. Cover and.e. Gamal. 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