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Spectral Efficiency of Joint Multiple Cell-Site Processors for Randomly Spread DS-CDMA Systems
Randomly spread direct-sequence code-division multiple-access (DS-CDMA) systems have also been considered in many information-theoretic analyses, in view of their widespread practical use. Important results for multiuser receivers in this setting were obtained in [5] and [6], while focusing on a single isolated cell. The asymptotic setup is considered in which both the number of users and the processing gain go to infinity, while their ratio goes to some finite constant (referred to as the “cell load”). This asymptotic regime enables the use of results from the theory of random matrices [7] to obtain various deterministic limiting information-theoretic performance measures of interest. The results are extended to the Wyner [1] linear cell-array multicell model with single-cell-site processing in [8], for nonfading channels, and in [9] for flat-fading channels, demonstrating the impact of undecodable out-of-cell interference in this setting. The capacity under outage constraint for “strongest-users-only” receivers is derived in [10], for a setting in which the users employ equal rates and transmit powers, and the receiver decodes the transmissions of the largest subset of intracell users that can be reliably decoded (see also [11] for the corresponding analysis in a single-cell system, and also [12], [13] for related analyses). In this correspondence, one step further is taken and the impact of employing joint multiple-cell-site processing on the performance of randomly spread DS-CDMA systems is explored. Adhering again to Wyner’s infinite linear cell-array model, it is assumed that the infinite array is divided into clusters of M cells. The receiver jointly processes the signals received at each of the cell sites within the cluster, assuming that the signatures and codebooks of all intracluster users are available to the receiver. As for out-of-cluster users whose signals are received at the cluster’s cell sites, their codebooks are assumed unknown to the receiver. The receiver is only aware of their signatures, using this information for interference mitigation. Note that according to Wyner’s linear model only the two neighboring cells outside the cluster’s edges affect the receiver. Full channel state information is assumed available to the receiver (only) with respect to both intra- and out-of-cluster transmissions. The particular case of M = 1 boils down to single-cell-site processing [8], [9]. Binary spreading sequences are assumed and a variant of the flatfading channel model analyzed in [6] and [9] is considered, focusing on Rayleigh fading. Unlike the common flat-fading model, it is assumed that a long chip-level interleaver is incorporated in each user’s transmitter. The idea is to interleave, prior to transmission, chips corresponding to the spreading sequences of different channel symbols, in a way that, effectively, at the output of the matching de-interleaver at the receiver different chips experience independent fades, corresponding to a homogenous fading model [5] (commonly a symbol-level interleaver is assumed, and chips within the same spreading sequence experience the same channel fade). This model is of real theoretical interest, being analytically tractable, as otherwise (assuming symbol-level interleaving) the performance of joint multicell processing in the setting in question is still an open problem. Furthermore, the chip-interleaved approach might be considered as an operative tool to enhance the system’s diversity factor, and should hence provide an upper bound to performance with symbol-level interleaving. Two types of joint multi-cell receivers are considered. I) The optimum joint processor that achieves the mutual information between the channel input due to intracluster users, and the
Oren Somekh, Member, IEEE, Benjamin M. Zaidel, Member, IEEE, and Shlomo Shamai (Shitz), Fellow, IEEE
Abstract—A chip-interleaved randomly spread direct-sequence code-division multiple-access (DS-CDMA) scheme is considered, employed in two variants of Wyner’s infinite linear cell-array model with flat fading. Focusing on the asymptotic setup in which both the number of users per cell and the processing gain go to infinity, while their ratio (the “cell load”) goes to some finite constant, the spectral efficiencies of the optimum and linear minimum mean-squared error (MMSE) joint multicell receivers are investigated. A dramatic performance enhancement as compared to singlecell-site processing is demonstrated. The asymptotic behavior of the two receivers in extreme signal-to-noise ratio (SNR) regimes and in a high cellload regime are analyzed as well. The impact of chip interleaving versus symbol interleaving is also investigated. Chip-level interleaving is found beneficial in several cases of interests, and is conjectured to be beneficial in general. Index Terms—Direct-sequence code-division multiple-access (DSCDMA), fading, multiuser detection, Shannon theory, spectral efficiency, Wyner’s model.
I. INTRODUCTION One of the most dramatic developments of the past two decades in communications technology has been the huge evolvement of cellular communications systems. Cellular systems now offer ubiquitous wireless access to a wide variety of multimedia services, for both outdoor and indoor applications, with a constantly increasing penetration and a growing demand for higher and higher user data rates. This evolution has led to an abundance of scientific researches in the quest for an efficient utilization of the available bandwidth, thus increasing the capacity of the systems in concern. Of particular interest in recent years is the attractive analytically tractable model for a multicell system suggested by Wyner in [1]. Accordingly, the system’s cells are ordered in either an infinite linear array, or in the familiar two-dimensional hexagonal pattern (also infinite). Only adjacent cell interference is assumed present, and characterized by a single parameter, a scaling factor 2 (0; 1]. Considering nonfading channels and a “wideband” transmission scheme, where all bandwidth is available for coding (as opposed to random spreading), the throughputs obtained with optimum and linear minimum mean-squared error (MMSE) joint processing of the received signals from all cell sites were derived in [1] (see also [2] for an earlier relevant work). These results were extended to flat-fading channels in [3], where it is observed that fading may increase the throughput under certain conditions. Single-cell-site processing in this multicell framework was considered in [4]. Manuscript received August 1, 2004; revised September 22, 2006. This work was supported by the REMON consortium. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Chicago, IL, June/July 2004. The authors are with the Department of Electrical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel (e-mail: orens@ tx.technion.ac.il;
[email protected];
[email protected]). Communicated by A. Lapidaoth, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2007.899561
0018-9448/$25.00 © 2007 IEEE
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Fig. 1. Wyner’s infinite linear cell-array model.
M
signals received at the cluster’s cell sites, while accounting for the structure of the interference of out-of-cluster users. II) The linear MMSE receiver, that knows the signatures of all interfering users (both intracluster and out-of-cluster users), and mitigates their interference by means of a linear MMSE filter. The filter’s outputs are then followed by single-user decoders. The two receivers are analyzed and compared in terms of the cluster-averaged per-cell spectral efficiency (i.e., as opposed to individual rates, the achievable total number of bits per second per hertz (bits/s/Hz) that can transmitted arbitrarily reliably), taken as the figure of merit for system performance. The key tool for the analysis is the observation that system model in concern is completely equivalent to a certain class of multiple-input multiple-output (MIMO) channel models. It is well known that degrees of freedom can be equivalently provided by the processing gain in a spread-spectrum system, or by the number of antenna elements in a system with an antenna array. Hence, results obtained for MIMO channels can be directly applied to the problem of concern. With that in mind, recently obtained results by Tulino et al. in [14] are employed (see also [15] for a generalized full paper version). In [14], the spectral efficiency per receive antenna of a rather general class of single-user MIMO channels is derived, while assuming isotropic input and asymptotic conditions in terms of the number of both receive and transmit antennas (the results are argued to represent well systems with a finite and practical number of antennas). Tulino et al. also derive the achievable rate with a linear MMSE receiver, and identify an elegant relation between this rate and the maximum achievable rate, in an analogous manner to the relation shown in [6]. Appropriately interpreting the different quantities of [14], expressions for the per-cell spectral efficiencies of both the optimum and linear MMSE receiver are derived for equal transmit powers. The results are compared to previously obtained single-cell-site processing results [8], and the performance enhancement due to multiple-cell-site processing is demonstrated. The extreme signal-to-noise ratio (SNR) regimes are investigated as well. For the sake of comparison, and toward a more complete treatment of the multiple-cell-site processing problem, a modified multicell model is also considered in which the system cells form a circular array, obtained by simply assuming that the first and the M th cluster-cells are adjacent to each other. The circular structure of the resulting equivalent MIMO channel interpretation leads to particularly simple expressions for the spectral efficiency of both the optimum and linear MMSE receivers. To be more specific, the expressions are identical to those obtained for a single-cell system, as considered in [5], but where the average received power (SNR) of each user equals the sum of the average received power at its local cell site plus the average received power at the two adjacent cell sites (according to the Wyner model). Furthermore, as the analysis to follow shows, the average per-cell spectral efficiencies in the linear system model approach, as the cluster size M grows, the corresponding spectral efficiencies obtained in the circular array model. Hence, the circular array is a useful model for providing insight into the effect of joint multiple-cell-site processing, when the
cluster of cells to be jointly processed is large enough (numerical results show that the approach to the circular array limit is rather fast). The last part of this correspondence is devoted to the investigation of the impact of chip-level interleaving on system performance, while mainly confining the discussion to the optimum joint multiple-cell-site receiver. The focus is first on the circular-array model described above, for which the superiority of chip-level interleaving over symbol-level interleaving in the high cell load regime (the optimum choice in terms of spectral efficiency for the optimum receiver in the circular-array setup) is established. Next, following [16], the low-SNR regime is investigated for the cases in which chip-level and symbol-level interleavers are employed, and also for the corresponding nonfading setup. The superiority of the optimum spectral efficiency with chip-level interleaving in flat-fading channels is established again in this regime. Conditions in which flat-fading combined with symbol-level interleaving becomes beneficial in the low-SNR regime, as compared to nonfading channels, are also discussed. In order to get additional insight into the effect of chip-level interleaving, a single-cell microdiversity multipleantenna setup, as considered in [17] and [6], is also investigated. The analysis shows that the spectral efficiency attained with chip-level interleaving always upper bounds the one attained with symbol-level interleaving in extreme SNR and in the high cell-load regimes. It is therefore conjectured that the superiority of chip-level interleaving holds in general. The structure of this correspondence is as follows. Section II presents the system model. Next, the infinite linear and the circular cell-array models are considered in Section III, where expressions for the average per-cell spectral efficiencies of the optimum and linear MMSE receivers are presented. Section IV includes numerical results and compares the two multicell models. Section V investigates the impact of chip-level interleaving on system performance. Finally, Section VI ends this correspondence with a summary and some concluding remarks. II. PRELIMINARIES AND SYSTEM MODEL Denoting by K the number of users per cell, and by N the length of the spreading sequences (the “processing gain”), the focus is on the asymptotic setup in which K , N ! 1, while K=N ! < 1 (the “cell load”). Adhering to Wyner’s linear cell-array model, as depicted in Fig. 1, while assuming full chip, symbol and codeword synchronization, the baseband representation of the complex N -dimensional received signal vector at the mth cell site, at some arbitrary time index, is given by
y
m
= SS 01 H m
01xm01 + S m H m;mxm + SS m+1 H m;m+1xm+1 + nm
m;m
(2-1)
where stands for the Hadamard product, defined for arbitrary matrices
A and B as
[A B ]
i;j
[A ] [B ] : i;j
i;j
(2-2)
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In the above notation, bold lower case denotes vectors and bold upper case denotes matrices. The K -dimensional vector x m consists of the symbols transmitted by the K users operating in the mth cell. The users are assumed to employ Gaussian codebooks (which conforms with the capacity-achieving statistics), and it is also assumed that the users cannot cooperate their transmissions in any way. Full channel state information is assumed to be available to the multiple cell-site receiver, however, the users are assumed to be unaware of the instantaneous channel realizations. Therefore, the symbols transmitted by each of the users are assumed to be zero-mean independent and identically distributed (i.i.d.) (across time and users) circularly symmetric (proper) complex Gaussian random variables [18], with variance P , that designates the equal transmit power of all users. The entries of the N 2 K matrices fH m;n g are the channel chip-level fades affecting the signals transmitted by users in the nth cell, as observed by the mth cell-site receiver. With the underlying chip-level interleaver assumption, the entries of the fH m;n g matrices are taken as i.i.d. zero-mean circularly symmetric complex Gaussian random variables, with unit variance (corresponding to Rayleigh fading). The matrices are also assumed to be statistically independent for different values of m and n. The N 2 K matrices fS m g denote the signature matrices, with the columns of the matrix S m being the spreading sequences of the users operating in the mth cell. It is assumed that the users employ random binary spreading sequences, and hence, the entries of the signature matrices to be i.i.d. random variables taking the p p are assumed values f01= N; 1= N g with equal probability. Independence of the spreading sequences of different users is also assumed. The underlying assumption of full channel state information at the receiver implies that the channel fade matrices fH m;n g and the signature matrices fS m g are instantaneously known at the receiver. Finally, the N -dimensional vector n m denotes the additive white Gaussian noise (AWGN) at the receiver of the mth cell site, i.e., nm is a zero-mean circularly symmetric complex Gaussian random vector with E fnm nm y g = I N , where (1)y denotes the conjugate-transpose operation, and I N denotes the N 2 N identity matrix. The noise spectral level is thus normalized, without loss of generality, to unity, and P denotes, in fact, the transmit SNR of each of the users. The noise vectors at the receivers of different cell sites are assumed to be i.i.d. It is noted that although the spectral efficiency expressions are given in Sections III and V as functions of the transmit SNR (for simplicity of notation), the tradeoffs between power and bandwidth are evaluated by expressing the spectral efficiency as a function of the normalized transmit energy per bit, through the relation P = 1 NE C (P ), where C (P ) denotes the spectral efficiency. As already mentioned in the Introduction, the key tool used in the analysis to follow is a recent result by Tulino et al. obtained for MIMO channels with isotropic input in [14]. It is emphasized that although the generalized analysis found in [15] considers also the optimum input power distribution, we adhere in the following to the setting and notation of [14], as it is more appropriate for representing the uplink of practical cellular systems, assuming symmetry between users and noncooperative transmissions.
G=
III. MULTICELL SETUPS: ANALYSIS A. Infinite Linear Array Setup This section focuses on the infinite linear array setup, as described in the Introduction. The notation (1)IA shall be used to denote quantities related to this model. Unless stated otherwise, all the derivations in this section are valid for M 2. Without any loss of generality, the expressions to follow relate to the cluster of cells numbered f1; 2; . . . ; M g. The signal vector received by the joint multiple-cell-site receiver for the cluster in concern is given by (see (2-1)) IA M +1 M yM 1 [MN 21] = G [MN 2(M +2)K ] x 0 [(M +2)K 21] + n 1 [MN 21] ;
where
yM 1
0
. ...
..
.
0
=
T
y T1 ; y T2 ; . . . ; y TM
+1 xM = xT0 ; xT1 ; . . . ; xTM ; xTM +1 0
and
nM 1 The MN
=
nT1 ; nT2 ; . . . ; nTM
T
T
:
2 (M + 2)K channel transfer matrix GIA equals SS H ; 0 0
GIA =
1 0
0 0
G
0
(3-2)
SS M +1 H M;M +1
where G is the MN 2 MK matrix given by (3-3), shown at the bottom of the page. The spectral efficiency of the optimum receiver in this setting is given by IA CM
=
1
M
lim
N;K !1 !
1
N
M E I xM 1 ; y1
(3-4)
where the expectation is over the spreading signatures and channel fades, and one has to bear in mind that the cell sites at the edges of the M -cells cluster receive signals from users of adjacent clusters (see (3-1)). According to the underlying assumptions of the current setting, these other-cluster transmissions cannot be decoded by the receiver, which is only aware of the structure of their received signals. Using Kolmogorov’s identity and the underlying assumptions of Wyner’s model (interference emerges from adjacent cells only), the mutual information in (3-4) can be rewritten for M 2 as
S 1 H 1;1 SS 2 H 1;2 0 SS 1 H 2;1 S 2 H 2;2 SS 3 H 2;3 0 SS 2 H 3;2 S 3 H 3;3 ..
(3-1)
M I xM 1 ; y1
0 I x ; xM ; yM jxM = I xM ; y M 0 I (x ; y jx ; x ) 0 I (xM ; y M jxM 0 ; xM ) = I xM ; y M 0 2I (x ; y jx ; x ) =I
+1 xM ; yM 1 0 0
+1
0
0
1
+1
0
...
0
SS 4 H 3;4 ..
.
+1
+1
1
1
1
1
2
1
0
1
1
1
2
(3-5)
0 0
...
SS M 01 H M;M 01 S M H M;M
:
(3-3)
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where the last equality follows from the symmetry between the two cells at the cluster’s edges. Substituting (3-5) into (3-4), the spectral efficiency of the optimum receiver is given by
IA = C 0 2 C ; CM M M I
where the M 2 (M + 2) matrix P M (representing the discrete-index asymptotic power profile [19]) is given by
(3-6)
PM =
where
.. .
0
CM M1 N;Klim!1 N1 E I xM0 +1 ; y M1
! 1 1 GIA IA y = N;K lim !1 N E log det I + PG G M !
(3-7)
lim
N;K !1 !
0M (m) =
1 E fI (xx ; y jx ; x )g 0 1 1 2 N
1 2 GI I y = N;K lim !1 N E log det I + PG G !
:
(3-8)
The N 2 K sponding to other-cluster users of cell 0, whose signals are received at cell-site 1, given by
S 0 H 1;0 :
(3-9)
Examining (3-6), it is observed that CM can be interpreted as the average per-cell spectral efficiency in the case in which the receiver also tries to decode the transmissions of users in the two cluster-adjacent cells (assuming that their codebooks are now also known at the receiver). That is, the setup corresponds to an extended cluster of (M +2) cells, consisting of the original cluster of M cells and the two clusteradjacent cells (cells 0 and (M + 1)), while only the signals received at the M cells sites of the original cluster are being processed by the receiver. The quantity CI may be interpreted as the spectral efficiency of an optimum receiver in a single isolated cell setup with homogenous Rayleigh flat fading [5], [6], and with a scaled SNR of 2 P . Recall now that with the underlying assumption of binary spreading sequences, and independent circularly symmetric Gaussian channel fades, the entries of the channel transfer matrix GIA are marginally Gaussian. Furthermore, they are independent and hence jointly Gaussian (and uncorrelated). With this observation, one can establish an analogy between the multicell DS-CDMA setup discussed here, and the single-user MIMO channel of [14], and invoke [14, Proposition 1] to derive a set of equations whose solution gives the spectral efficiency of (3-6). Starting with the first term in (3-6), this reads
CM = M1
M +1
log 1 + 0M (m)(1 + 2 2 ) P
m=0 M
.
..
.
0 ...
..
.
0
..
. 2
..
.
1
(3-11)
.. . 2
m+1 1 [P M ]l;m ; [P ] P (1 + 2 2 ) l=m01 1 + lk+1 =l01 1+0 (k)(1+2 ) P m = 0; 1; . . . ; M + 1; (3-12)
where “out-of-range” indices should be ignored. Finally
matrix GI in (3-8) is the channel transfer matrix corre-
GI
..
0 0
with rows enumerated as m = f1; . . . ; M g, and columns enumerated as k = f0; . . . ; (M + 1)g. The values of the discrete-index function +1 are given by the unique solutions to the following set f0M (m)gmM=0 of equations:
and
CI
2 1 2 0 . . . 0 2 1 2 0 . . .
m+1 [P M ]m;k P + 1 log 1 + 2 M m=1 k=m01 1 + 0M (k)(1 + 2 ) P M +1 0 M1 0M (m)7M (m) log e (3-10) m=0
7M (m) =
(1 + 2 2 ) P ; 1 + 0M (m)(1 + 2 2 ) P
m = 0; 1; . . . ; M + 1:
(3-13) It is noted that the quantities 0M (m) and 7M (m), have actually practical interpretations. Following [14], 0M (m)(1 + 2 2 ) P represents the asymptotic signal-to-interference-plus-noise ratio (SINR) at the output of the linear MMSE receiver, as a function of the cluster-cell index. Also, the mean-squared error at the output of this receiver equals 7M (m)= (1 + 2 2 ) P . In order to complete the derivation of the spectral efficiency of the optimum receiver, it remains to derive an expression for CI of (3-8). However, with the single-cell homogeneous fading model interpretation discussed above, denoting the equivalent average transmit power I 2 P , CI is explicitly given by [5], [6] in this model as Pav
CI = log 1 + PavI 0 41 F PavI ; I ; 0 1 F P I ; + log 1 + Pav av 4
e F P I ; 0 4log av P I
(3-14)
av
where
F (x; z)
p
p
2
x 1+ z 2+10 x 10 z 2+1
:
(3-15)
Turning to the linear MMSE receiver, then under the assumption of Gaussian codebooks, its average per-cell spectral efficiency equals
IA CM
1 M lim K 1 E I x ; y M m;k 1 M m=1 N;K !1 k=1 N !
:
(3-16)
But as stated earlier, the quantity 0M (m) 1 + 2 2 P is recognized as the SINR at the output of the joint multicell linear MMSE receiver for users of the mth intracluster cell [14]. Hence, the average per-cell spectral efficiency of the linear MMSE receiver is given by
M IA = 1 IA (m)(1 + 2 2 ) P : CM log 1 + 0M M m=1
(3-17)
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Fig. 2. S
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as a function of for (a) M > 2, (b) M = 2, and (c) M = 1.
Note that although the cluster receiver processes the signals of users from M + 2 cells (M intracluster cells, and the two cluster-adjacent cells), the average in (3-17) is over intracluster cells only, as codebooks of out-of-cluster users are assumed to be unknown at the receiver. The reader is referred to [19, Sec. 5] for an elaboration on the multicell DS-CDMA–single-user MIMO channel analogy, and for the complete derivation of the above spectral efficiency results. The following propositions summarize some basic properties of the two receivers in the infinite linear array setup.
Proof: See [19, Appendix B.2.1] and [19, Appendix B.3] for derivations of the low- and high-SNR spectral efficiency slopes of the optimum receiver, respectively. As can be observed, (3-19) is the product of the low-SNR slope of the spectral efficiency of the optimum receiver in the single-cell nonfading setup [6] (i.e., 2 =(1 + )), and a term that goes to unity as the cluster size M grows. Note also that (3-19) monotonically increases with the cell load , thus establishing the optimality of increasing without bound for the optimum receiver in the low-SNR regime. The limiting slope as ! 1 is given by
E that enProposition III.1: The minimum transmit and receive N able reliable communications for both the optimum and linear MMSE receivers equal
Ebr IA = log 2; e N0 min
Ebt IA = N0 min
loge 2
1 + 2 2 1
0M :
S IA !1 0 lim
(3-18)
1
E
Considering N , the joint multiple-cell-site processing schemes min 1 as comare observed to produce an energy gain of 1 + 2 2 1 0 M pared to single cell-site processing. Proposition III.2: The low- and high-SNR spectral efficiency slopes of the optimum receiver are given by =
2
1 + 1 + 4 2
1
0
1 + 2 1 (1+ )
2
M
1
0M 1
+ 4 4
2
1
0
M
(3+ ) 2(1+ )
(3-19)
MM+2 1 = 1 0 M2 ; MM+2 < 1 1 : 10 M; These results are valid for arbitrary values of K , N and M ;
2
S0IA
=
2
1 + 2 1 + 4 2
1
2
1
2 2
1
2
2
(3-21)
Proposition III.3: The low-SNR spectral efficiency slope of the linear MMSE receiver is given by (3-22) at the bottom of the page. Proof: See [19, Appendix B.2.2].
and
S IA
0M : (1 + 2 ) 0 M
2 1 + 2
The high-SNR spectral efficiency slope of the optimum receiver is depicted as a function of the cell load in Fig. 2. Examining the high-SNR slope it is observed that taking ! 1 is no longer optimum. In fact, the optimum value of in terms of the high-SNR slope approaches MM+2 , indicating that some amount of spreading is optimum in this regime. It is interesting to note that when M = 1 [9], taking ! 1 turns the receiver interference limited, that is, its E ! 1. This behavior is also spectral efficiency goes to a limit as N observed for M = 2 (joint two cell-site processing). In contrast, for M 3, the receiver is no longer interference limited when ! 1, although as mentioned earlier, this is a strictly suboptimum choice for the cell load in the high-SNR regime.
Proof: See [19, Appendix B.1].
S0IA
=
(3-20)
Note that as was the case for the optimum receiver, the low-SNR slope of the linear MMSE receiver also coincides, as the cluster size M grows, with the corresponding slope in the single-cell nonfading setup,
2.
0
1 + 2
2
M
1+ (1+2 )
1
0M 1
+ 4 4
2
1
0
M
3+4 (1+2 )
;
M
2:
(3-22)
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i.e., 2 =(1+2 ) [6]. The slope of (3-22) monotonically increases with to a limiting slope of 1 1 + 2 2 1 0 M lim S0IA = 2 2 !1 (1 + 2 2 ) 0 M (1 + 4 2 ) 2
(3-23)
which again establishes the optimality of taking ! 1 for the linear MMSE receiver in the low-SNR regime, as shown for the optimum receiver. Proposition III.4: For ! 1, the spectral efficiency of the opIA , is given by the solution of the following imtimum receiver, CM plicit equation:
IA CM
M
(1 + 2 2 ) 1 1 + 2 2 1 0 M
= log 1 +
0 M2 log
Ebr C IA N0 M
2
1+
1 + 2 2
2:
10 M 1
Ebr C IA N0 M
; (3-24)
Proof: See [19, Appendix B.4.2]. As can be observed, as the cluster size M grows, (3-24) coincides with the spectral efficiency in the nonfading single-cell setup [5], for E values. which taking ! 1 is optimum for all N
Proposition III.5: For ! MMSE receiver is given by 2 1 IA = 1 + 2 1 0 M CM (1 + 2 2 )
log e 0
M
2:
Note that the last term in (3-25) equals the spectral efficiency of the linear MMSE receiver in the single-cell nonfading setup (for ! 1), and the two results coincide as M ! 1. Also, the MMSE receiver E becomes interference limited for ! 1, and as N ! 1 the spectral efficiency approaches the limit of
;
!1
1 1 + 2 2 1 0 M (1 + 2 2 )
MK l=1
log e:
(3-26)
In this subsection, we consider a slightly modified system model. It is now assumed that the M system cells are arranged in a circle, so that the first cell and the M th cell are adjacent to one another (with M 3). As shall be evident in the sequel, this modified structure has an inherent (circular) symmetry that leads to simple analytical results. Furthermore, the circular model becomes particularly useful, as it will be shown in the following that as M ! 1, the spectral efficiency results for the infinite linear array setup coincide with those of the circular array setup. The notation (1)C shall be used to denote quantities related to the latter setup.
C = log Copt
SS 1 H M;1
2
C 1 + Pav
=
1
.
0
..
. ...
MN
MN l=1
[G C ]l;j
E
2
:
(3-28)
0 41 F PavC ;
C 0 1F + log 1 + Pav 4 C = log Cms
C; Pav
C 1 + Pav
e F P C ; 0 4log av P C av
0 41 F PavC ;
(3-29)
(3-30)
respectively, where F (1; 1) is defined by (3-15). It is noted that the results are independent of the number of cells M (M 3). It is also worth mentioning that the same results may be derived using the multicell DS-CDMA–single-user MIMO channel analogy established in Section III-A. Applying [14, Proposition 1] to the circular array setup yields a similar set of equations as in (3-12), the solutions of which are independent of m, leading again to the results of (3-29) and (3-30) (see [19, Sec. 6] for more details). E required for reliable commuIn view of the above, the minimum N nication in the circular setup is given by [6]
Ebr C = log 2; e N0 min
Ebt C = N0 min
loge 2 : (1 + 2 2 )
(3-31)
Taking ! 1, and expressing the spectral efficiency of the linear E yields MMSE receiver in terms of the received N
r 01 C = log e 0 Eb Cms : N0
(3-32)
For the optimum receiver it can be shown that the spectral efficiency for ! 1 is given by
S 1 H 1;1 SS 2 H 1;2 0 SS 1 H 2;1 S 2 H 2;2 SS 3 H 2;3 0 SS 2 H 3;2 S 3 H 3;3 ..
[G C ]i;l
E
Hence, according to [14, Proposition 3], the spectral efficiency results of the circular array setup are identical to the results derived for the nonfading (equivalently homogenous fading) single-cell setup [5], but C = with an increased average transmit power per user satisfying Pav 2 (1 + 2 )P . Joint multiple cell-site processing is thus observed to completely eliminate the effect of other-cell interference while fully exploiting the total received power from each user, as received by the antennas of three cell-sites according to Wyner’s model. Using the above equivalence the average per-cell spectral efficiencies of the optimum and the linear MMSE receivers can be explicitly expressed as [5]
B. Circular Array Setup
GC =
MK
and
Ebr 01 ; N0
(3-25)
=
1
1, the spectral efficiency of the linear
Proof: See [19, Appendix B.4.1].
IA CM
The overall received signal, as seen by the joint multiple-cell-site receiver, can be described by (3-1), while replacing the channel transfer matrix G IA by the following MN 2 MK matrix given in (3-27) at the bottom of the page, and the input vector by x M 1 . Careful examination of the power profile matrix [14], [19] associated with GC , reveals that it is block-circulant, and therefore asymptotically doubly regular [14] since for every row i and column j
C = log Copt
... ...
SS 4 H 3;4 ..
.
r C Eb 1 + Copt
N0
:
(3-33)
SS M H 1;M 0
...
SS M 01 H M;M 01 S M H M;M
(3-27)
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007
Fig. 3. Spectral efficiency of the optimum receiver versus transmit
in the infinite linear array model, for = 1=2 and optimum choice of .
This result coincides with the spectral efficiency of the single-user AWGN channel. Furthermore, it coincides with the corresponding result obtained in [3] for the setup in which no random spreading is employed, demonstrating thus the optimality of taking ! 1 for the optimum receiver. To obtain the low- and high-SNR spectral efficiency slopes of the optimum receiver, one can directly apply again the single-cell results of [6] yielding
S0C
=
and
C S1
2
(3-34)
1+
; 1 > 1:
=
1;
(3-35)
The optimality of increasing without bound is clearly evident from the fact that both slopes monotonically increase with . The low-SNR slope for the linear MMSE receiver is given by [6]
S0C
=
2
1 + 2
;
(3-36)
demonstrating the optimal choice of ! 1 for the linear MMSE receiver at the low-SNR regime. The high-SNR slope of the linear MMSE receiver is given by [6]
C S1
=
; < 1 1=2; = 1 0; > 1:
(3-37)
The following theorem defines the relation between the spectral efficiencies of the optimum receiver in the two multicell setups considered in this correspondence. Theorem III.6:
IA lim C M !1 M
=
C : Copt
2631
(3-38)
Proof: See the Appendix for an outline of the proof, and [19, Appendix A] for full details.
E that enables reliIt is noted that the coincidence of the minimum N able communications, and of the low- and high-SNR slopes in the two setups, as M ! 1, can also be observed from the explicit expressions of these quantities as derived in Section III-A. IV. NUMERICAL RESULTS In this section, we bring some numerical results that demonstrate the performance enhancement of joint multiple-cell-site processing, and the interrelations between the two multicell setups considered in this correspondence. Figs. 3 and 4 show the average per-cell spectral efficiencies in the infinite linear array setup of the optimum receiver (3-6), and the linear MMSE receiver (3-17), respectively. The spectral effiE for the optimum ciencies are plotted as a function of the transmit N E choice of (which is in general a function of N ). The results were evaluated for = 12 , which corresponds to the case in which the total average intercell interference power equals one-half of the total average intracell received power, and can be considered as a “practical” level of intercell interference. Note that the low-SNR slopes of E to take the receivers imply that it is optimum for low values of N E ! 1. However, beyond some critical value of N , the optimum choice for decreases from infinity and takes on finite values, approaching eventually, for the optimum receiver, the value of M=(M + 2) in the high-SNR regime. The region in which the optimum choice of decreases from infinity explains the “knee effect” observed in the spectral efficiency curves for both receivers. The optimum values of E for the linear MMSE receiver are depicted in Fig. 5 as a function of N . For the sake of comparison, we included in Figs. 3 and 4 the corresponding spectral efficiencies for the case of M = 1 [8], and for the circular array setup. Comparing the results, the dramatic effect of employing joint multiple cell-site processing on system performance is clearly evident. The approach of the spectral efficiency in the infinite linear cell-array setup to the spectral efficiency obtained in the circular array setup, as the cluster size M gets large, is also clearly observed for both receivers. In order to emphasize the impact of joint multiple-cell-processing even further, Fig. 6 shows the average per-cell spectral efficiency of the joint multiple-cell-site linear MMSE receiver for M = 2 and M = 3, together with the spectral efficiencies obtained with single-
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Fig. 4. Spectral efficiency of the linear MMSE receiver versus transmit
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007
in the infinite linear array model, for = 1=2 and optimum choice of .
Fig. 5. The optimum cell load for the linear MMSE receiver as function of the transmit
cell-site processing (M = 1) of both the optimum and linear MMSE receivers [9]. The spectral efficiencies are plotted in Fig. 6 as a funcE for the optimum choice of , and for = 1=2. tion of the transmit N As can be observed, already with a joint two-cell-site processor, the linear MMSE receiver outperforms the optimum single-cell-site proE regime, below some threshold E , cessing receiver in the low N N E and in the high N region, beyond a threshold. Furthermore, with no more than joint three-cell-site processing, the linear MMSE receiver outperforms the optimum single-cell-site processing receiver for all E . This result is of particular practical interest in view values of N of the fundamental receiver complexity difference between the two settings. The complexity of the optimum single-cell-site processing receiver grows exponentially with the number of users per cell, while that of the joint three-cell linear MMSE receiver grows only linearly with the number of users per cell.
, for = 1=2.
V. THE IMPACT OF CHIP-LEVEL INTERLEAVING As discussed in the Introduction to this correspondence, incorporating a chip-level interleaver results in a homogenous fading process [5], in which each spreading chip of each user experiences independent fades. This comes in contrast to the conventional use of a symbol-level interleaver, resulting in a fading process in which the whole spreading sequence (N chips) of each user experiences the same fade, and the fades are independent from symbol to symbol. Hence, the terms homogeneous fading, and nonhomogeneous fading, shall be used interchangeably in the following to refer to the chip-level interleaved and symbol-level interleaved setups, respectively. Although most of this correspondence is dedicated to the investigation of multicell setups employing chip-level interleaving, for which analytical results can be obtained following [14], it is of great interest to compare the results
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007
2633
Fig. 6. Comparison of joint multiple-cell-site linear MMSE processing and optimum single-cell-site processing, for the optimum choice of
G
CS
=
The overall received signal while employing symbol-level interleaving can be described in the same manner as for the chip-interleaved setup of Section III-B, while replacing the channel transfer matrix GC of (3-27) by (5-1), shown at the bottom of the page. The notation CS (1) shall be used henceforth to denote quantities related to the symbol-level interleaved setting. S m in (5-1) denotes the binary N 2 K signature matrix of the K users of the mth cell, and H n;m denotes the K 2 K diagonal matrix of channel fades affecting the signals of the mth cell users, when received at the nth cell-site antenna. It is assumed that all channel fades are zero-mean, unit-variance i.i.d. random variables, and perfectly known to the receiver. Note, that when a chip-level interleaver is employed, the resulting fading process is mathematically described by the Hadamard multiplication of the signature matrix by the fading matrix. In contrast, the use of a
SS 4H 3 4
..
..
;
;
;
SS 1 H
M;1
.
0
.
...
SS H 1
...
;
;
..
= 1=2.
A. Symbol-Interleaved System Model
S 1H 1 1 SS 2H 1 2 0 SS 1H 2 1 S 2H 2 2 SS 3H 2 3 0 SS 2H 3 2 S 3H 3 3 ;
and
of the cluster size M , as evident from the numerical results of Section IV. It is noted, however, that the same analysis can be repeated for the infinite-array setup in a rather straightforward manner. Finally, a multiple-antenna model is investigated, which also demonstrates that homogenous fading can be beneficial in terms of the optimum spectral efficiency, as compared to nonhomogeneous fading. The investigation considers several scenarios including the low- and high-SNR regimes, and the case in which the cell load is high. The following leads us to conjecture that homogeneous fading is beneficial in terms of the optimum spectral efficiency for all SNRs and cell load values.
to the ones obtained in the more practical multicell setups employing symbol-level interleavers. When single-cell-site processing is employed, it is already well known from [5], [6], [8], and [9], that while homogenous fading has no effect on the spectral efficiency as compared to nonfading channels, flat fading decreases the spectral efficiency of symbol-level interleaved systems (assuming no channel state information at the transmitter). Unfortunately, the derivation of the spectral efficiency with nonhomogeneous fading in the multicell system setups considered here poses some considerable analytical difficulties, and is still an open problem. Therefore, in order to compare the two fading models, we focus on the optimum joint multiple-cell-site receiver and do the following. First, an upper bound is used to demonstrate the superiority of the optimum spectral efficiency obtained in the homogenous fading multicell model, in the high cell load region ( ! 1). Next, the low-SNR regime is considered for which the same behavior is also observed. Furthermore, we show that in the low-SNR regime fading may turn out beneficial, in terms of the optimum spectral efficiency, as compared to the corresponding setup in the absence of fading. This conclusion comes in contrast to the case of single-cell-site processing. The above comparison is confined to the circular array setup of Section III-B (with the necessary adaptations to the nonhomogeneous and nonfaded setups), as it leads to more tractable results on one hand while capturing the impact of joint multiple-cell-site processing on the other. In addition, this model also provides a good approximation and an upper bound to the spectral efficiencies obtained in the more “realistic” infinite linear-array setup of Section III-A, for rather moderate values
;
M
...
;
.
SS 01 H M
M;M
0
;M
...
01 S M H M;M
:
(5-1)
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007
symbol-level interleaver yields a fading process that is described using regular matrix multiplication of the signature matrix by the diagonal fading matrix. B. High Cell Load Upper Bound For every channel realization and any N , M , and K , the conditional channel input–output mutual information of the symbol-level interleaved circular array setup (see (3-1) and (5-1)) satisfies the following relations: CSM M I xM 1 jS 1 ; H 1 ;y
M y CSM 1 jS 1 ; H
=h
0 h y CSMjxM ; S M ; H = h y CSMjS M ; H 0 h nM 1
1
M
(a)
m=1
(b)
=
1
1
1
1
M h yCS2 jS 1M ; H yCS
x ;y
= MI
3 1
= M log det
2
0 h (n )
S 1H 2;1 ( S
2
N;Klim!1 N1 E !
= log e
e
I N + PG~ G~ y
(5-2)
log det
+ 2e + log
(5-3)
E1 1
1 P
+ ( 0 1) log e (5-4)
1
1 30 e P
E1
1
P
0 2P e 2
E1
=
1
2 P
= 1: (5-5)
2 (K 0 1)(1 + 2 4 ) 1+ 1+ (1 + )(1 + 2 2 )2
01 01
2 2(K 0 1) 2 (2 + 2 ) 10 K+ (K + )(1 + 2 2 )2
(5-8)
where K is the kurtosis of an individual fading coefficient.1 In the particular case of = 0, the above expression boils down, as expected, to the low-SNR slope of the optimum receiver in the single isolated cell setting 2 =(K + ) [6]. The detailed derivation of the above result is given in [19, Appendix C]. 2) Nonfaded Circular Array Setup: Consider the symbol-level interleaved circular array setup defined in Section V-A in the absence of fading. In this setting, one should take H n;m = I K in (5-1), where I K is a K 2 K identity matrix, and the notation (1)CNF shall be used to denote related quantities. Using similar arguments to those of SecE tion V-C1, the minimum required N for reliable communication is given by
Ebt CNF N0 min
=
loge 2 ; (1 + 2 2 )
(5-9)
while the low-SNR spectral efficiency slope of the optimum receiver is given by
S0CNF
2
e dt (t > 0) is the exponential integral funcwhere E1 (x) t x tion, and is the unique solution to the following implicit equation
+ P
S0CS
=
I N + PG~G~ y
(5-7)
where m2 is the second power moment of an individual fading coefficient (see [19, Sec. 8.3.1] for more details). Turning to the low-SNR spectral efficiency slope of the optimum receiver, and applying [16, Theorem 13], it follows that
S 2H 2;2 SS 3H 2;3 ) :
1 P
E1
loge 2 Ebt CS = N0 min m2 (1 + 2 2 )
1
where the expectation is taken over the spreading sequences and the relevant channel fades. Careful inspection of (5-3) reveals that the right-hand side (RHS) of the inequality equals the spectral efficiency achieved by an optimum receiver employed in a single isolated cell accommodating 3K users. The 3K users, operating in a flat-fading environment, use unequal transmit powers, with K users transmitting at power P , while the remaining 2K users are transmitting at power 2 P . This setup, which is treated in [9] may also be interpreted as the extended cluster portion of the optimum spectral efficiency in the infinite linear array setup analyzed in Section III-A, for the particular case of M = 1 (see (3-7)). This spectral efficiency is given for the particular case of Rayleigh fading by
C3FK
1) Symbol-Level Interleaved Circular Array Setup: Consider the symbol-level interleaved circular array setup defined in Section V-A. Applying [16, Theorem 8], it can be shown that the minimum required E N for reliable communication is
jS ; H
Hence, the per-cell spectral efficiency of the optimum joint multiplecell-site receiver in this setup is upper-bounded by CS CM
(5-6)
which is identical to (3-33), designating the limiting spectral efficiency of the optimum receiver in the chip-interleaved setup as ! 1. Thus, we have established that in the high cell-load region, the optimum spectral efficiency with chip-level interleaving coincides with the upper bound to the corresponding spectral efficiency in the symbolinterleaved setting.
SM
fH n;m g denotes all where h(1) denotes differential entropy, H channel fades, and S 1M denotes the set of all spreading signatures employed in all M cells. In (5-2), (a) follows from the fact that h(x; y ) h(x) + h(y) for any x, y arbitrary random variables (rvs), (b) follows from the symmetrical nature of the circular array setup, and the N 2 3K ~ is defined as matrix G G~
= log 1 + (1 + 2 2 )P + o( 01 )
C3FK
C. The Low-SNR Regime
0 h (nm )
h yCSm jS 1M ; H
Numerical analysis shows that for any finite value of the cell load , this upper bound surpasses the curve of the spectral efficiency of the optimum receiver in the chip-level interleaved circular array setup (3-29). However, when the cell load is high, it is easily verified that
=
2 2 2 (4 + 2 ) 1+ 1+ (1 + )(1 + 2 2 )2
01
:
(5-10)
As was the case in Section V-C1, for the particular of = 0 the above slope coincides with the corresponding slope of the single-cell setting 2 =(1 + ) [6]. More details on the derivation can be found in [19, Appendix D]. 3) Comparison: We are now ready to compare the symbol-level interleaved, chip-level interleaved, and nonfading circular array setups in the low-SNR regime. 1The kurtosis for a random variable a is defined as
K = E [ ja j
]=(E [
ja j
]) .
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007
Proposition V.1: The following statements hold with respect to the spectral efficiency of the optimum receiver in the low-SNR regime. I) In the presence of fading, a chip-level interleaver is beneficial over a symbol-level interleaver. II) Homogenous fading (chip-level interleaver) is beneficial over no fading. III) Rayleigh flat fading (symbol-level interleaver) is beneficial over no pfading in a certain range of the interference factor , for > 0 = ' : . Proof: Assuming that the second power moment of an individual , both fading setups require the fading coefficient satisfies m2 E to enable reliable communication (see (3-31) and (5-7) same N
( 33 1) 16 0 296
=1
min
E
for the chip-level interleaved and symbol-level interleaved N , remin spectively). Now, since K for any arbitrary random variable, the expression in brackets in the first equality in (5-8) is larger than , and the first statement is established in view of (3-34). The second statement is easily proved by noticing that the expression in brackets in (5-10) is larger than (for > ). To establishes the third statement, we search for the conditions in which the ratio of the low-SNR slopes in the symbol-level interleaved setup (5-8), and in the nonfaded setup (5-10), is greater than unity. Focusing on the particular case of Rayleigh fading K , this yields after some algebra the following quadratic inequality (in 2 ):
1
1
0
1 0 8 2 + 2(1 0 ) 4 < 0:
p
(5-11)
Investigation of this inequality yields that for < ( 33 0 1)=16 ' 0:296 there is no solution to (5-11), and for this range p of values fading is not beneficial. If the cell load lies in the range ( 33 0 1)=16 < 3=10, fading turns out to be beneficial for an interval of the intercell interference factor 2 [ l ; h ) where 8 0 64 2 + 8 0 8 ; l = (5-12a) 4(1 0 ) 8 + 64 2 + 8 0 8 : h = (5-12b) 4(1 0 ) Finally, for 3=10, fading is beneficial for 2 [ l ; 1], where l is given by (5-12a).
It is also worth mentioning that l reduces to zero as increases ). This coincides with the results (note that l is continuous at of [3], stating that when the number of users per cell goes to infinity, and all bandwidth is devoted to coding, fading is beneficial for all 2 ; .
=1
[0 1]
D. Multiple-Antenna Model We consider in the following a chip-interleaved single-cell multipleantenna setup based on the one presented and analyzed in [17] and [6]. A chip-level interleaver is employed such that the spreading chips of each user are affected by i.i.d. fading coefficients at each receive antenna (independence across chips and antennas is assumed). The notation 1 MA shall be used to denote quantities related to this setup. The received signal vector is given by
()
S [N 2K ] H 1[N 2K ]
x[K 21] + n[LN 21]
.. .
S [N 2K ] H L[N 2K ]
= 1 ... = 1
(5-13)
where H ` , ` ; ; L, is the matrix of i.i.d. zero-mean channel fading coefficients related to the `th receive antenna, satisfying 1 E j H ` n;k j2 L , 8 n N , k K.
( )
This setup can be analyzed by observing that due to the resulting homogenous fading model, the chip-interleaved multiple-antenna setup is equivalent to a nonfaded single-cell model with a single receive antenna, where the users employ random spreading sequences of length LN (with i.i.d. chips). In this case, the spectral efficiencies of the optimum and linear MMSE receivers are given by (3-29) and (3-30), respectively, while replacing the cell load by =L and Pav by P . In addition, since the spectral efficiency in bits/s/Hz should be normalized with respect to the actual length of the spreading sequences N , the expressions obtained for both receivers according to the single-antenna interpretation should be scaled by the number of antennas L (see [19] for the detailed expressions). Considering the extreme-SNR regimes, it can be similarly shown that
1
Ebt MA = Ebr MA = log 2 e N0 min N0 min
1
( = 2)
y MA [LN 21] =
2635
the low-SNR spectral efficiency slopes are
S0MA =
2 L ; S0MA = 2 L L + 2 L+
(5-14)
and the high-SNR slopes of the receivers are given by MA S1
; L;,
=
In addition, for reduce to
MA S1
= min( ; L):
(5-15)
1 it is easily verified that the spectral efficiencies
1 0 LMA log e + o( 01 ) 1 + o( 01) CLMA = (L 0 1) 1 0 LMA log e + L log MA CLMA = L
L
(5-16)
MA , the linear MMSE multiuser efficiency parameter, satisfies where L
LMA =
1 01 1 + L P + o( ):
(5-17)
To get insight into the impact of chip-level interleaving, the above results should be compared to the original multiple-antenna setup discussed in [17] and [6], where symbol-level interleaving is employed (resulting in a nonhomogeneous fading process). The corresponding spectral efficiencies can be found in [6, Sec. V], as well as the extreme-SNR and high cell-load characterization. This comparison yields the following observations. The minimum (transmit and receive) energy per bit required for reliable communication is identical in both setups. From [6, eq. (192)], [6, eq. (201)], and (5-14), while recalling that the kurtosis of an arbitrary random variable is greater than one, it is clear that the low-SNR slopes of the chip-level interleaved setup surpass the low-SNR slopes of the symbol-level interleaved multiple-antenna setup. Turning to the high-SNR regime and considering [6, eq. (199)], [6, eq. (202)], and (5-15), it is concluded that the high-SNR slopes of the linear MMSE receiver in both multiple-antenna setups coincide, while for the optimum receiver the difference between the high-SNR slopes in the two setups is
0; 1S1 = (L 0 1)=2; L01
