Optimum Multiuser Noncoherent DPSK Detection in Generalized ...
Optimum Multiuser Noncoherent DPSK Detection in Generalized Diversity Rayleigh Fading Channels Matthias Brehler and Mahesh K. Varanasi compared to the fast fading channel. The mobility of the users in a scattering environment induces fading, which severely affects the bit error rate of both, coherent and noncoherent communications. To combat fading, diversity is a commonly used technique. While optimum (maximal ratio) combining is well-known for coherent communications and has also been specified in the context of optimum multiuser detection [1, 2], the optimum noncoherent combination of several diversity branches for DPSK modulation has been considered only recently [3] for the single-user channel. The resulting optimum single-user user receiver was shown to obtain substantial gains over the standard equal-gain combining strategy [4]. It was also applied to an effective single-user channel obtained after projecting out (or decorrelating) interfering users, where it was seen to be superior to previous approaches that are based on decorrelating and equal-gain combining [5–7]. [8] proposes several sub-optimum receivers (including the LMMSE receiver) which however are out-performed by the detector of [3]. In all comparisons, the lack of a fundamental performance bound, to which the various approaches can be compared to, is apparent. This paper presents the (jointly) optimum multiuser DPSK detector in generalized diversity Rayleigh fading (GDRF) and rigorously analyzes it, providing a benchmark for the sub-optimum approaches. Furthermore, the fundamental limitations of DPSK modulation for multiuser communications are revealed through the exact calculation of the asymptotic efficiency and near-far resistance [2]. Optimum multiuser DPSK modulation is also considered in [9] and [10] but in contrast to our presentation these references assume that the envelope of the fading coefficients is known (i.e. it has to be estimated) and the uncertainty is only in the phase. More specifically, [9] considers an asynchronous multiuser channel and reduces it to a single-user channel by the standard Gaussian approximation; [10] considers the synchronous problem and derives the jointly optimum detector and various sub-optimum approaches for the case of known envelopes but contains no results on performance analysis. Under the white Gaussian noise background regime, [11,12] derive the optimum (bi-) linear multiuser detectors for the synchronous and asynchronous channel, respectively, and [13, 14] introduce multistage receivers for these settings.
Abstract— The jointly optimum multiuser noncoherent detector for DPSK modulation over the generalized diversity Rayleigh fading (GDRF) channel is derived and analyzed. The GDRF channel includes time/frequency/receiver antenna diversity and allows fading correlations between the various diversity branches of each user. Noncoherent detection here refers to the case where the receiver has neither knowledge of the instantaneous phases nor of the envelopes of the users’ channels. Upper and lower bounds on the bit error probability of the optimum detector are derived for a given user. For fast fading, when the fading coefficients vary from one symbol interval to the next (but are still essentially constant over one symbol interval), the detector asymptotically (for high signal-to-noise ratios) reaches an error floor, which is bounded from below and above for different fast fading scenarios. For slow fading, when the channel is constant for at least two consecutive symbol intervals, the upper bound is shown to converge asymptotically to the lower bound. Thus the asymptotic efficiency of optimum multiuser DPSK detection can be determined and is found to be positive. In contrast to coherent detection however, it is smaller than unity in general. Since the asymptotic efficiency is independent of the interfering users’ signal strengths, the optimum detector is near-far-resistant. While optimum multiuser detection is exponentially complex in the number of users, its performance provides the benchmark for sub-optimal detectors. In particular, it is seen that the previously suggested post-decorrelative detectors can be far from satisfactory. Index Terms—DPSK modulation, diversity communications, error analysis, fading channels, maximal ratio combining, multiuser detection, noncoherent detection, optimum detection.
D
I. I NTRODUCTION
IFFERENTIAL phase shift keying (DPSK) is an especially attractive modulation scheme for multiuser communications, because in many applications the users are mobile and thus their channels keep changing. DPSK avoids the expense of elaborate channel estimation and tracking algorithms necessary for coherent demodulation, because one does not have to estimate the channel parameters: the information is encoded in the phase difference between two successive symbol transmissions. In this work, neither the instantaneous phase nor the envelope of the users’ channels are assumed to be known at the receiver. Due to the differential encoding, it is usually required for DPSK that the fading coefficients stay constant over two successive symbol intervals. Channels where such an assumption is valid will be referred to as slowly fading channels. However, fast fading channels are also considered where we define fast fading to mean that the fading parameters are allowed to vary from one symbol interval to the next (but not within a symbol interval). The slow fading assumption holds for lower vehicular speeds and/or higher data rates and/or higher carrier frequencies when
II. T HE G ENERALIZED D IVERSITY R AYLEIGH FADING (GDRF) C HANNEL
Submitted, July 2000; revised, December 2001 and October 2002. This work was supported in part by NSF Grant ANIR-9725778 and by US Army Research Office Grant DADD19-99-1-0291. The results of this paper were presented in part at the Conference on Information Sciences and Systems at Princeton University, Princeton, NJ, March 1998 and the Communications Theory Mini Conference, IEEE GLOBECOM, San Francisco, CA, Nov. 2000. The authors are with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309-0425 (e-mail: [email protected], [email protected]).
In this section the discrete-time model for the multiuser generalized diversity Rayleigh fading channel (GDRF) is developed. It allows for inter-diversity fading correlation, interdiversity signal correlation, and inter-user signal correlations and applies to variety of diversity situations [3]. It handles S receive antennas and Lw waveform diversity channels (mul1
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tipath, frequency diversity, or others). The total diversity order is L = SLw . Our presentation of the K -user GDRF channel draws heavily on the single-user channel presented in [3]. Extending the single-user discrete-time model [3, eq. (6)] to multiple users leads to p (n) = 1=2 (n) (n) + (n); (1)
R D c
q
where = Nw10 is the average signal to noise ratio (SNR) of the user of interest (which we assume without loss of generality to be the first user), is the KLKL dimensional generalized signal correlation matrix defined as
R
Z
R = T U (t)N 1UT (t) dt T 1 (t);
(2)
N
T (t); : : : ; T (t)T and the noise with (t) = 2 K correlation matrix (see [3, Section II] for more details, each k (t) corresponds to (t) defined therein), is the KL KL diagonal relative energy matrix = w1 1 diag f[w1 ; w2 ; : : : ; wK ]g L with wk the k th user’s average energy and the Kronecker (or tensor) product [15], (n) is the KL KL diagonal symbol matrix (n) = diag f[d1 (n); : : : ; dnK (n)]]g
L , with dk (n) = dk (n 1)bk (n) j2 l oM 1 and bk (n) 2 F = e M denotes the k th user’s MPSK l=0 information symbol, (n) is the KL-length channel fading information (CFI) vector (n) = [ T 1 (n); T2 (n); : : : ; TK (n)]T , which is CN ( ; (n; n)) distributed with
U
U
U
U U
U
I
D c
D
I
c 0
c
c
c
(n; m) = E c(n)cy (m) ; and
(3)
is the additive white (in time) Gaussian noise, which is CN ( ; ) distributed. The k th user’s generalized signal correlation matrix kk is the k th diagonal L L block of . Each user’s signature waveforms uk;l (t) are normalized such that their energy in the time T is unity. Hence, the diagonal elements of the matrix interval R T b = T (t) (t) dt are all unity. Similar to kk , we define the k th diagonal L L block of b as b kk .1 The k th diagonal L L submatrix of (n; m), kk (n; m), is the matrix correlation function of the k th user’s fading parameters. We assume that the fading paths of different users are statistically independent of each other, i.e., (n; m) will be blockdiagonal with kk (n; m) as diagonal elements. Wide sense stationarity of the fading process allows us to write (n; n) = (0) and (n; m) = y (m; n) = (m n). We define the slowly fading channel to have constant fading coefficients for the duration of two successive symbol intervals. Hence, for slow fading we have (1) = (0). In the fast fading channel,
R
0R
R
R
U U
R R
R
1 Note that in the case of only a single receiver antenna or uncorrelated noise b denotes the correlation matrix and b are equal. In [3] R in the antennas b kk of this paper. of the diversity waveforms only and is hence not equal to R b kk ), yet b kk is a block-extended version of R b of [3] (precisely IS R Here, R all normalizations are the same.
R
R
the fading coefficients can vary from one symbol interval to the next (but are still assumed to be essentially constant within one symbol interval). To ensure that wk is the total average received energy of user k , the fading coefficients (n) are normalized. The average received power due to the isolated transmission of the k th user in a noiseless channel should be equal to wk . This amounts to
c
tr kk (0) Rb kk = 1;
(4)
which is equivalent with [3, Eq. (9)], if the different definitions of b are considered. As we mask out the ISI in cases it arises (cf. [3]) we will be able to perform symbol-by-symbol decisions. Hence, we detect the data symbols in the zeroth time interval bk (0) = dk (0)dk ( 1) and only need to consider the KL-dimensional vectors of observations for the zeroth and the previous symbol, (0) and ( 1). To simplify notation in the sections to come we stack these vectors and obtain 2KL dimensional vec= T ( 1); T (0) T . As ( 1) and tor of observations (0) are zero-mean, complex Gaussian random variables (because (n) and (n) are) so is . We denote their correlation matrix with q . q depends in general on the M 2K possible combinations of the differentially encoded symbols ( 1) and (0). Since we assume that the fading paths of different users are uncorrelated, q will depend on the information symbols (0) = (0) ( 1) only: The product (n; m) (n) commutes ((n; m) 2 f 1; 0g2) because (n; m) is block-diagonal with blocks of size L L and (n) and are diagonal matrices with the same element repeated L times on their diagonals. As we will only consider data symbols in the zeroth symbol interval, we drop the time index in (0) and introduce the integer subscript i (1 i M K ) indicating the dependence of the K data symbols fbk gk=1 on the underlying hypothesis Hi . Hence, Hi denotes that i is the particular realization of the data symbols with bik the realization of user k . These definitions and the assumption of uncorrelated fading for different users let us write the correlation matrix of given the specific data symbols i (corresponding to the hypothesis Hi ) were encoded by the transmitter as
R
q
q
q
q
c
D B
q
q
q
K K
D D
q
D
K
D
D
B
B
q
B
KqjHi = =
R ( 1; 0)Bi R
(5)
R ( 1; 1)R + R :
RBi (0; 1) R
R (0; 0)R + R
III. D ETECTORS FOR DPSK M ODULATION IN C HANNEL
THE
GDRF
In this section we briefly visit known suboptimum detectors for noncoherent multiuser DPSK detection in GDRF channels and introduce the optimum, maximum likelihood detector. Previous approaches are mostly of the post-decorrelative type, i.e., the multiuser channel is simplified to an effective single-user channel by decorrelating the users. Although it is well-known that decorrelation is a sub-optimum procedure (cf. [1]), the notion of optimum multiuser DPSK detection in GDRF channels has only been introduced in a conference version of this paper [16]. Moreover, it is only recently that even the single-user optimum detection problem was solved ( [3]). Hence, we revisit
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
the derivation of the optimum multiuser detector of [16] in more detail in Section III-B; it assumes Rayleigh fading and knowledge of the users’ SNRs as well as of the fading statistics (but not of the realizations of the channel fading information vector). A. Post-Decorrelative Approaches For the single-user, independently and identically fading diversity channel ( (0) = ) the ad-hoc solution to DPSK reception is to combine the decision statistics of all branches by weighting them equally [4]. (It was shown in [3] that this adhoc solution corresponds to the GLRT detector presented in that paper). In [5, 6], and [7] the idea of equal-gain combining is applied to the multiuser decorrelated statistics2
I
z(n) = R
1 q(n):
(6)
The decision for each user is solely based on its corresponding decorrelated statistics, thereby neglecting the noise correlation. Hence, the decorrelating equal-gain combiner (D-EGC) for the first user can be formulated as
D EGC : ^i1 = = arg i 2f1min Refbi1 zy1 ( 1)z1 (0)g; 1 ; 2; :::; M g
(7)
where bi1 2 F is the first user’s data symbol in the zeroth symbol interval. The same decision rule h may be written in the effeci T tive single-user statistics ^ 1 = 1 ( 1)T ^ T ; 1 (0)T ^ T = ^ T ( 1); ^T (0)
q1
q
q1
D EGC : ^i1 = y ^ = arg i 2f1min q ; 2; :::; M g 1 1
z
0 ^ bi1 R
R z
R
bi1 ^ 2
R 0
2
R
(8)
q^1 ;
where ^ 1 is the upper-left L L block of of the decision rule is convenient for the performance analysis: Since all decision rules presented in this work can be expressed in such a quadratic form, the general, unified error probability analysis of [17] can be applied to each one of them. Note that the D-EGC neither accounts for inter-diversity branch correlation, nor fading paths with different signal strengths, nor for the noise correlation through the inter-diversity and inter-user signal correlation. Rather than applying the decorrelating operation and equalgain combining the decorrelated decision statistics, the singleuser detection problem is systematically approached in [3]. A rigorous problem formulation therein leads to the singleuser generalized likelihood ratio test (SU-GLRT) and minimum probability of error (SU-MEP) detector. Both of them by far out-perform the D-EGC. Having obtained the single-user MEP and GLRT detectors, it is straightforward to apply them to the effective single-user channel in a multiuser setting. For future reference we will refer to them in the multiuser context as the decorrelating single-user GLRT (D-SU-GLRT) and MEP (DSU-MEP) detectors.
R
3
B. The Minimum Error Probability Detector The minimum error probability (MEP) detector minimizes the probability of error in the joint detection of all K users. Once the transmission model is specified, it results from a standard maximum likelihood derivation. We assume that the statistics of the fading parameters, i.e., the covariance matrices (n; m), and the users’ individual signal strengths, i.e., the wk ( and , respectively) are known at the receiver. Consequently the observations given the data symbols i are CN ( ; qjHi ) distributed and since all hypotheses are assumed equally likely the MEP detector is simply
q
B
0K
MEP : ^i = arg i2f1; 2min qy Kqj1Hi q +ln KqjHi ; ; :::; M K g
M
(9)
M
where we define for any matrix j j = det( ). In contrast to the single-user case, the MEP decision rule for the multiuser GDRF channel is not independent of the determinant of qjHi . It is convenient to state the decision rule not in the decibut in the the decorrelated statistics , where sion statistics T = T ( 1) T (0) . The covariance matrix of the new decision statistics becomes
K
z
z
q
z z KzjHi = B( i 1;(0;1) 1)+ R
z
1
( 1; 0) Bi 1
(0; 0) + R
:
(10) The MEP decision rule can then equivalently be stated in the decorrelated decision statistics as
z
MEP : ^i = arg i2f1; 2min zy Kzj1Hi z +ln KzjHi : ; :::; M K g
(11)
C. Generalized Likelihood Ratio Test (GLRT) The MEP requires statistical knowledge of the fading corre-
1 . The last form lations and the users’ average energies. The estimation of these
2 In a single-user channel with inter-diversity signal correlations those papers also propose decorrelating the user’s diversity signals and equal-gain combining.
quantities can be circumvented through a generalized likelihood ratio test (GLRT) based approach. The multiuser GLRT was introduced for the noncoherent detection of DPSK in [16] and for general nonlinear multipulse modulation (NMM) in [18]. It maximizes the likelihood of the observations first over the unknown fading parameters ( 1) = (0) (assuming slow fading) and subsequently over the information symbols i . For DPSK and our notation the result is
c
c
B
G : ^i = arg i2f1; 2min ; :::; M K g 1 qy BIi R + Bi RBi I Bi q:
(12)
The GLRT receiver is exponentially complex in the number of users, as is the MEP. Examples in [16] demonstrate that it can suffer a performance loss of 2 to 3 dB relative to the MEP. Since in this paper we are concerned with analysis of optimum DPSK detection in Rayleigh fading, we do not consider the GLRT. Note however, that it may be analyzed using similar methods applied here to the MEP. Moreover, since the GLRT does not incorporate any assumptions about the fading distribution, it may be expected to perform well under a variety of fading conditions.
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IV. P ERFORMANCE A NALYSIS In this section the performance of the detectors presented in Section III is examined. To this end, lower and upper bounds on the bit error rate (BER) of the multiuser MEP detector are derived in Section IV-A. The analysis also applies to the decorrelating-type detectors with the simplification that the bounds coincide. For the multiuser MEP, the asymptotic (high SNR) convergence of the upper bound to the lower bound in slowly fading channels is proved in Section IV-B using the unified general analysis of [17]. The convergence of the bounds enables us to obtain exact formulas for the asymptotic efficiency and the near-far resistance of the detectors. For fast fading channels, the error floor is bounded from below and above. For simplicity, we restrict attention to binary modulation for both slow and fast fading. Without loss of generality, all performance measures are obtained for the first user. A. Bounds on Bit Error Rate The upper bound on the error probability of user one is derived by invoking a union bound and the lower bound by only considering the single error events, i.e., events for which there occurs an error for user one only. To derive these bounds we first consider the so-called pairwise error probability Pr fÆj < Æi g, the probability that the decision statistic Æj corresponding to hypothesis j is smaller than the decision statistic Æi of the true hypotheses Hi . This probability can be expressed as the probability that a quadratic form in the decision variables is smaller than some constant. The cumulative distribution function of such a quadratic form is for example derived in [17]. All introduced detectors can be expressed in the general form
q
: ^i = arg i2f1;min xy Fix + ci 2; :::; 2K g = arg i2f1;min Æ; 2; :::; 2K g i
F
x
z F
K xFx
xFx
Pe
= 2
K
2 X K
K
i=1
PejHi
X
Pr fÆj < Æi g :
i=1 8j2f1; :::; 2K g s:t: bi1 6=bj1
(13)
where bik denotes the k th user’s data symbol corresponding to hypothesis Hi . A lower bound on the first user’s error probability is simply obtained by averaging over all 2K pairwise error probabilities whose corresponding data symbols differ only for the first user, so that
Pe
21K
2 X K
Pr fÆj < Æi g :
i=1 j s:t: bj1 6=bi1 bjk =bik ; 2kK
(14)
B. Asymptotic Error Rate Analysis In [17] a unified asymptotic analysis of quadratic receives in Rayleigh fading channels is presented. In the multiuser context, such an analysis was applied for the first time to a problem similar to the one at hand arising in the error probability analysis of noncoherent multiuser detection for nonlinear modulation [19]. In the case considered here, we will find likewise that the bounds on the error probability converge asymptotically for high SNR in the case of slow fading. However, the results of [17] greatly simplify this task when compared to [19], because we merely have to find the asymptotic eigenvalues of certain matrices. For fast fading we bound the error floor from below and above; numerical examples show that these bounds are tight. To this end, we will first state some definitions and then reveal the results of the asymptotic analysis for the MEP detector in several sections. First the behavior of the constant cij is examined in Section IV-B.1: for slow fading it approaches a constant for increasing SNR, and it becomes zero for fast fading. In Section IV-B.2, the non-zero asymptotic eigenvalues of ij are found as the SNR approaches infinity. For slow fading eij L of the asymptotic eigenvalues are positive and linear in , and the eigenvalue minus unity is repeated eij L times (where eij is the number of errors in an error event). For fast fading eij L eigenvalues approach a positive constant and eij L eigenvalues a negative constant. The results of these sections combined with [17, Proposition 2] allow us to obtain the asymptotic bit error rate (BER) formulas for slow fading and bounds on the error floor for fast fading, respectively, in Section IV-B.3. Finally, we specify the asymptotic efficiency and and near-far resistance of the multiuser MEP detector in Section IV-B.4 for slow fading. For the remaining analysis we will just consider hypotheses Hi , Hj whose corresponding vectors of data symbols have the following structure
C
where Æi is implicitly defined. For the MEP detector the de = , i = zj1Hi , and ci = ln zjHi . cision statistics For the other approaches similar assignments can be made. In the remainder of this section we express everything in terms of i , ci , and the covariance matrix xjHi of the decision statistics . Since Pr fÆj < Æi g is the probability that the decision statistic y j + cj is smaller than y i + ci given Hi is the true hypothesis, we can apply the results of [17, Proposition 1] and obtain the pairwise error probabilities through residues. With the latter we union bound PejHi , the error probability of user one conditioned on hypothesis Hi . Computing PejHi precisely would require the evaluation of the probability of the union of all 2K 1 possible events Æj < Æi with j and i such that the first user’s bit in hypotheses Hi and Hj differs. This probability is upper bounded by the sum of the probabilities of those error events so that the first user’s error probability can be bounded by
x
2
2 X K
K
K
bTi = 1Teij +1Teij+ +1Toij+ 1Toij ; bTj = +1Teij 1Teij+ +1Toij+ 1Toij ; (15) where 1n is an n length vector containing n ones and eij + + eij + oij + + oij = K and eij + + eij = eij , the number of errors incurred when bi is mistaken for bj . As indicated in (15), we assume that all errors of an error event (Hi
! Hj ) occur
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
within the first eij users, and furthermore, that the first eij elements of ij are minus unity and the following eij + elements are plus unity where of course eij = eij + eij + . Additionally, we also order the K eij remaining users for which no error occurs in a way that the first oij + users send +1 and the next oij users send 1. The ordering can be performed without loss of generality, be~ i ! H~ j ) with arbitrary ordering we cause for any error event (H can reorder the users according to the above scheme, rearrange the users’ signal correlation matrix , and the correlation matrix of the decision statistics zjHi and hence calculate the error probability of the corresponding error event (Hi ! Hj ). Note that the block-extended matrices of data symbols i and j hence have a structure corresponding to (15) and we the matrices define for any matrix
e
K
R
B
B
M
MBi = 12 (M + Bi MBi) ;
(16)
MBj = 12 (M + Bj MBj ) ;
whose properties are examined in the appendix. With the help of the above ordering we define a 4 4 block where the diagopartitioning of any KL KL matrix nal elements have the following dimensions: size( 11 ) = eij L eij L, size( 22 ) = eij + L eij + L, size( 33 ) = oij + L oij + L, and size( 44 ) = oij L oij L. All other entries have appropriate dimensions. Every matrix subscript in the following sections on the asymptotic analysis refers to the submatrix according to this partitioning, unless stated otherwise. Furthermore, a subscript on [ ]ij ((i; j ) 2 f1; 2; 3; 4g2 ) refers to the corresponding submatrix of the expression in brackets according to this partitioning. Depending on the error event, one or more dimensions of a block in this partitioning might be zero. For consistency, we define the determinant of an empty matrix (dimension zero for a row or a column or both) to be unity and the product of empty matrices to be an empty matrix again. However, note that we defined in Section II the matrices kl and kl (n) as the L L submatrices of and (n), respecaccording to the partively. Consequently, a submatrix of titioning introduced here is written as, for example, 11 and is not to be confused with 11 , which is always the upper-left L L submatrix of . For (n) the submatrix according to the new partiotioning is written with square brackets, for example, as [ (n)]11 in contrast to 11 (0), the correlation matrix of the first user’s L diversity branches. It will also be convenient in what follows to define a 2 2 block-matrix partitioning of any 2KL 2KL matrix as
M
M
M M
M
X
X
R
R
R
R
R
K = KKulll KKurlr
;
(17)
X
C
C
C
C
C
C
C
C
K
C
K
C
D
B.1 Asymptotic Analysis of the Determinant Ratio Proposition 1 (Limit of the logarithm of the determinant ratio) The limit of the logarithm of the determinant ratio cij is independent of the interfering users’ signal strengths. For slow fading it can be stated as
c^ij
zjH i lim cij = lim ln
!1 !1 zjHj 11 14 22 41 44 32 ln 11 13 22 31 33 42
=
Q Q Q Q
=
Q R
1. where = For fast fading c^ij
K K
Q Q Q Q
Q Q Q Q
Q23 Q33 ; Q24 Q44
= lim c =0: !1 ij
Proof: Note the property
K
where all blocks have dimension KL KL (u denotes upper, l lower or left as appropriate, and r right). For simplicity, a combination of both partitionings will be written as, for example, ul23 or [ ]ur14 . For the analysis to come it will be crucial that the upper left block of ij is equal to its lower right block ([ ij ]ul = [ ij ]lr )
K
and that its upper right block is equal to its lower left one ([ ij ]ur = [ ij ]ll ). Hence we can exploit the easy-to-prove fact that the eigenvalues of such a matrix are equal to the eigenvalues of [ ij ]ul + [ ij ]ur and [ ij ]ul [ ij ]ur . For ij to have such a structure, it is sufficient that zjHi and zjHj have the same structure (and hence their inverses). The latter is true for slow fading, because (n; m) equals (0) for all four values of (n; m) 2 f 1; 0g2 (recall also that the product (n; m) (n) commutes). For fast fading, we assume, besides wide sense stationarity of the fading processes, (1) = (1)y to obtain the same upper-right and lower-left block, as desired. This assumption implies (1) = ( 1), i.e., the cross-correlation functions of the fading processes are all even at one time lag. More specifically, we choose (1) = (0), where (0 < < 1) is a single fade rate for all users. This choice implies of course that the relative movement between the users and a base station is the same for all users, which is clearly unlikely in practical applications. However, the basic system behavior is captured by this choice and controlled by one simple parameter. Furthermore, the analysis is readily available for cases of a specific fade rate for each user or even a specific fade rate for each user’s diversity signal (the latter choice requires a diagonal (0), though). The fade rates can be connected to the normalized Doppler bandwidth BD T by, for example, = J0 (BD T ) for fading with power spectrum density given by Jakes’s model [20]. Note that the choice = 1 corresponds to the slowly fading case.
R
5
A B = jA + BjjA Bj; B A
(18)
which follows from the fact that the eigenvalues of the first ma. Furthermore, we trix are the eigenvalues of + and can write
A B
KzjHi = where
(0) 0 0 (0)
A B
I + 1Q Bi1 ; Bi I+ Q
Q = (R(0) ) 1 = 1 1 (0)Q:
(19) (20)
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Since the determinant of a product of matrices is equal to the product of the determinants and zjHj can be written in a similar manner, we obtain
K
cij
= ln
K K
I + 1Q Bi1 Bi I + Q = ln jMij ; = ln 1 jMj j I + Q Bj1 Bj I+ Q
zjHi zjH j
which allows to reduce the problem to finding the determinants of the matrices i and j . Consider fast fading. We find the limit of the determinant of i by applying equation (18) as
M
M
M
I + 1 Q + Bi I + 1 Q lim jMi j = lim !1 = jI + Bi j jI Bi j :
!1
Mj j is lim jMj j = jI + Bj j jI
!1
B
i
Similarly, the limit of j
B
jj :
B
= ln(1) = 0
Q Q14 Qii4 Q44
3 Q11 Q1ii 5; Q = 4 Qii1 Qiiii Q41 Q4ii where the blocks Q11 and Q44 have the dimensions according the partiotioning introduced in Section IV-B and Qiiii has got the dimension (eij + + oij + )L (eij + + oij + )L. The remaining blocks have the appropriate dimensions. The determinant of Mi
can then be written as
jM j
i
i :
I + 1Q + B I + 1Q B
The first determinant can be approximated3 by
I + 1 Q + Bi = 1 Q11
1 Q1ii
1 Q14 = 1 Qii1 1 Qiiii + 2I 1 Qii4 1 Q41
1 Q4ii
1 Q44 11 Q14 j2I j j2I j for 1; 1 Q Q41 Q44 22 33
3 All approximations in this paper are in the sense that if x( ) is the quantity
dependent on and e x( ) its approximation, then lim
1 Qiiii j2I11 j j2I44 j
Q22 Q23 j2I11j j2I44j : Q32 Q33 Similarly one obtains an approximation for Mj . As all the factors of 1 and 2 cancel out one obtains c^ij for slow fading as stated in the theorem but with blocks of Q. But as all blocks involved have block-sizes that are integer multiples of L and (0) is block-diagonal with block-sizes L L, the blocks of ((0) ) 1 cancel out and we can state c^ij in blocks of 1 =
Q=R
1
.
B.2 Asymptotic Eigenvalue Analysis
2(K
eij )L of the eigenvalues of the matrix
2eij L eigenvalues of Cij approach arbitrarily closely 1 with multiplicity eij L,
the real and positive eij L eigenvalues of the matrix 1 1
2 11 [ (0)]11 11 13 33 31 , the real and positive eij+ L eigenvalues of the matrix 1 1
2 22 [ (0)]22 22 24 44 42 , 1. where = For fast fading with fade rate the non-zero eigenvalues of ij approach arbitrarily closely the real and positive value 2(1 ) 1 with multiplicity eij L, the real and negative value 2(1 ) 1 with multiplicity eij L . Proof: That all eigenvalues are real follows from Sylvester’s law of inertia [21, Theorem 4.5.8], because ij is the product of two Hermitian matrices and hence has real eigenvalues. We make use of the special block-structure of the involved matrices and define the matrices
Q R
Q Q
Q Q Q Q Q Q
C
for fast fading. Now consider slow fading. For this derivation, we introduce a 3 3 block-partitioning of the matrix
i =
For slow fading and high values of the SNR, the remaining
B
2
I + 1 Q Bi
Proposition 2:
M
c^ij
Cij = KzjHi Kzj1Hj I are zero.
Note that according to our fast fading assumptions, is unequal plus or minus unity. With these limits, it is not hard to see that the determinants of i and j are equal (as i and j differ in the sign of some elements, but are added and subtracted for the calculation of both determinants), and hence their ratio is one, and we obtain
M
which follows from applying a determinantal identity to a 2 2 partitioned matrix and from [17, Appendix C] (take out 1 ). Using corresponding techniques one also finds that
!1 x=ex = 1.
C
Y = (0); X = Y + R 1 = Y + Q: K
With these definitions, the correlation matrices zjHj and zjHj can be more compactly expressed. Applying the matrix inversion lemma for the inverse of a 2 block-partitioned matrix (cf. [21]) to zjHj yields
K
K
h
i
h
i
Kzj1Hj ul = Kzj1Hj lr = = X 1 + 2 X 1 YBj X 2 YBj X 1 YBj 1 YBj X h i h i Kzj1Hj ur = Kzj1Hj ll = = X 1YBj X 2 YBj X 1 YBj ; where a formulae for the inverse of a small-rank adjustment of a matrix [21, Section 0.7.4], also known as Woodbury’s identity, was used. Consequently, after some manipulations, including
1
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
h
7
i
Q
Q
1
Q Q Q Q
1 1 positive as is a Schur 24 44 42 Bj 22 = 22 complement of the positive-definite matrix ) and Leij h i as goes to infinity. By performing the same derivation for 1 [ ij ]ul = zjHi zjHj = [ ij ]ul ur one finds the remaining non-zero eigenvalues of ij ul 1 for slow fading, which are given in the proposition. 1 j = ( j i) 2 ; j+ j j out of the inverse term Consider fast fading. Taking in (21) allows us to make an approximation for fast fading Somewhat more easily, one obtains the upper-right element as the application of Woodbury’s identity, the upper-left element of ij becomes
C C
K K I B B B BX X
C
YB X YB
[Cij ]ur = (Bj Bi )Y X Since [Cij ]ul = [Cij ]lr and [Cij ]ur = [Cij ]ll , this fully specifies Cij . Furthermore, as Bj Bi has (K eij )L zero rows and columns, Cij has 2(K eij )L zero rows and hence the
claimed number of eigenvalues equal to zero. For the remaining eigenvalues, it is sufficient to consider [ ij ]ul [ ij ]ur denoted as [ ij ]ulur . We will only calculate [ ij ]ul+ur in detail, because similar arguments apply for [ ij ]ul ur . So we can write
C
C
[Cij ]ul+ur = (Bj Bi ) Bj + (Bj X Y) X
C
1 2 YBj X 1 YBj :
Y / and hence = (Y + Q) 1 Y 1 Y 1 QY
1
1 for 1;
where once again the formulae for the inverse of a small-rank adjustment was applied. Inserting this approximation and the definition for into (21) yields
X [Cij ]ul+ur (Bj Bi ) Bj + (Bj Y + Bj Q Y) Y + Q
Y + 2Bj QBj
Q
Bj + 12 (Bj I)YQB1j + Bj 21 QQB1j 02 0 0 03 h0 i B6 Y22 QB1j 22 0 777 6 0 = (Bj Bi ) B B6 @4 0 0 0 05 0 0 31 2 1 ILe 0 2 ij 1 C 6 I Le 2 ij 0 7 7C ; + 64 0 5A 0
(Bj Bi )
where indicates non-zero matrix elements that are of no interest for the eigenvalues (since they do not depend on ) and we 1 make use of the properties of Bj and Bj given in the ap/ we can apply [17, Appendix C]: the eigenpendix. As values of theh last matrix are arbitrarily close toi the eigenvali h 1 = 2 22 [ (0)] 1 ues of 2 22 Bj 22 Bj (which are
Q
Y
Y Q
22
C
B.3 Asymptotic Bit Error Rate and Bounds on the Error Floor
MEP PAs
=
QQ
Q
22
2 X K
1
L j2 11 (0)j 2K
i=1 8j2f1; :::; 2K g s:t: eij =1
8 eijX L 1 > 2eij L 1 > c^ij > e > > eij L >
> > > > > :
aij ;
Q13 Q331Q31 1 Q22 Q24Q441Q42
1 : where aij Q11
For the remaining analysis, we have to distinguish between fast and slow fading. Consider slow fading, i.e., = 1, which allows us to simplify (using Bj )
[Cij ]ul+ur
The last matrix obviously has eij L non-zero eigenvalues: the eigenvalue 2(1+) 1 with multiplicity eij L, and the eigenvalue 2(1 ) 1 with multiplicity eij + L. As in the slowly fading case, the remaining eigenvalues are obtained by applying the corresponding techniques to [ ij ]ul ur .
(21)
2
Bj + Bj + Bj QY 1 I I + QY 1 2 I + 2 Bj QBj Y 1 1 (Bj Bi ) Bj + (Bj I) I 2 I 1 :
[Cij ]ul+ur (Bj Bi )
Proposition 3: For slow fading ( (0) = (1)), the upper bound on the BER of the multiuser MEP detector obtained in Section IV-A converges asymptotically to the lower bound as SNR ! 1. The asymptotic expression for the BER is independent of the interfering users’ signal strengths and can be stated as
Now consider that
X
C
Y
1 2 YBj X 1 YBj :
C C
I
k
( c^ij )k k!
k=0
eX ij L
2eij L 1 eij L
k=0
1
k
k c^
for c^ij
1
=
0;
for c^ij > 0;
ij
k!
and c^ij is given through Proposition 1. For fast fading ( (1) = (0), 0 < < 1) we bound the MEP from below and above as first user’s error floor Pfloor
MEP Pfloor MEP Pfloor
1
4
2
K X K
e
e=1 eL X
k=1
L X L 2L
1 1
k=1
2eL 1 eL
L
1 1
4 k
2
1
1
k
2 1+
k
;
eL
2 1+
k
;
where we simplified eij to e because only the number of errors matters and not their position.4
jj
4 There is a typo in [3]: should be equal to instead of the given expression. With this correction, the results can be seen to coincide for the single-user case.
8
ACCEPTED BY THE IEEE TRANSACTIONS ON INFORMATION THEORY
Proof: Consider slow fading first. In Proposition 2 we found the 2eij L non-zero eigenvalues of ij for high values of the SNR; eij L of these eigenvalues are 1 and eij L positive and linear in the SNR. As in Proposition 1, we found the constant cij for large SNR (c^ij ), we can apply the results of [17, Proposition 2] and approximate the pairwise error probabilities Pr fÆj < Æi gMEP for high values of the SNR aij Pr fÆj < Æi gMEP eij L
j2 11 [ (0)]11 22 [ (0)]22 j ;
C
where aij is defined in the proposition. As c^ij is independent of the SNR (and hence aij ), the probability of a single error event (eij = 1) dominates the union bound on the error probability. Hence, one can neglect all terms with eij > 1 in the calculation of the union bound and is left with the same expression for the upper bound as for the lower bound. The asymptotic expression of the error probability comes arbitrarily close to the true error probability, i.e, PeMEP lim MEP = 1:
!1 PAs For eij = 1 we have 11 = L (or 22 = L ) and [ (0)]11 = 11 (0) (or [ (0)]22 = 11 (0)), which is exploited to simplify the asymptotic expression. Consider fast fading. For fast fading, one inserts the eigenvalues found in Proposition 2 into the definitions of the bounds. Hence, the characteristic function has two poles of multiplicity eij L. The poles (corresponding to eigenvalues of ij ) merely depend on the number of errors and not on which users 1 possible error are detected erroneously. There are K e 1 events for e errors within K users, when an error for the first user must occur. This allows us to replace the sum over j by a sum over the number of errors e and the binomial coefficient. Example: Two-User Asymptotic Error Probability Let the two-user signal correlation matrix be defined as
I
I
C
R = RR1121 RR1222
R
:
R
jR11 j R11 R12R221R21 ;
(23)
which is greater or equal than zero. So we write the asymptotic error probability succinctly as
=
L X 1 2L 1 c^k : k! L
L j211 (0)R11 j 2 k=0
(24)
Generally, when the two users signals are correlated (and hence 1 21 < j j), the asymptotic two-user error 11 12 22 11 probability is larger than the single user asymptotic error rate: k = 0 in the summation yields the single user asymptotic error rate. As the terms for k > 0 are all positive, the two-user asymptotic error rate is larger than the single user’s rate. Consequently, the asymptotic efficiency of DPSK in GDRF channels will not be unity as it is in the case of coherent BPSK modulation [1, Proposition 2].
R
R R R
R
MEP
=
B K @2
2L 1
L
1
X
L
C aij A
i=1 8j2f1; :::; 2K g s:t: eij =1
and coincides with the near-far resistance. Proof: As we obtained an expression for the multiuser asymptotic BER for the MEP in the previous section (which easily specializes to the single-user channel), the asymptotic efficiency is straightforwardly calculated as given in the proposition. Furthermore, the multiuser asymptotic BER does not depend on the energies of the interfering users and hence the asymptotic efficiency coincides with the near-far-resistance. Example: Two-User Asymptotic Efficiency For two users the above expressions simplify to 1 X k ! L1 L 2 L 1 1 2 L k c^ MEP = 1 + 2U ; L 2 k=1 L k! (25) where we specified c^ in (23) in the example of the previous section, and we took out of the sum over k the term with k = 0 to demonstrate that the asymptotic efficiency is smaller than one. Note however that it is independent of the interfering users’ energies and always larger than zero (hence the detector in near-far resistant). V. N UMERICAL R ESULTS
c^ = ln
MEP PAs ;2U
Asymptotic efficiency is a performance measure that captures the performance degradation of a specific user due to interfering users in the limit of SNR ! 1 ( [1, 2]). Since for fast fading the BER reaches an error floor, the definition of asymptotic efficiency as given in [1, 2] cannot be applied to this case. Hence, we only give the asymptotic efficiency for slow fading and discuss a two user example. Proposition 4: The asymptotic efficiency of the multiuser MEP detector is 0 1 1
(22)
The four hypotheses one has to average over yield only two different values for aij and c^ij . Exploiting that cij = cji and writing the inverse in terms of blocks of , allows us to state the asymptotic probability with the single constant
Q
B.4 Asymptotic Efficiency and Near-Far Resistance
In this section we present numerical examples to illustrate our analytical findings. In Section V-A we consider slowly fading channels and in Section V-B fast fading channels. In both cases six users are considered who employ length-31 Gold-sequences. The signal correlation matrix is calculated as in [1] using the ISI mask described therein. For the decorrelating detectors the correlation matrix is the inverse of the upper left LL submatrix of the inverse of the multiuser correlation matrix . There are four paths for each user (hence is 24 dimensional). In the majority of our examples we will choose the four paths to fade independently but with unequal power distribution. The average relative power of each fading path is chosen according to the simplified GSM ‘urban’ test profile. For simplicity we choose the kk (0) equal for all users. Normalizing kk (0) properly leads to kk (0) = diagf0:22; 0:42; 0:26; 0:10g for all users. For the examples considering inter-diversity branch correlations, we calculate each kk from a Toeplitz matrix with Æ jk lj as its (k; l)th element. Normalization according to (4) leads to different kk in general, although the same value of Æ is chosen for each user.
R
R
R
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
A. Slow Fading Figure 1 compares all of the presented detectors for slow fading and equal energies of all users (wk =w1 = 1). Applying the single-user MEP detector to the decorrelated multiuser channel as proposed in [3] gives an improvement of 5dB over the adhoc decorrelating equal gain combining (D-EGC) detector of [5, 6], and [7]. However, the multiuser MEP can improve significantly on the decorrelating approaches: in this example it out-performs the decorrelating single-user MEP (D-SU-MEP) detector by 6dB, and hence the D-EGC detector by 11dB. The upper bounds we obtained on the BER of the MEP detector converges to the lower bound, approaching the asymptote, as we found in our asymptotic analysis. The 2dB gap between the single-user and multiuser channel for MEP detection illustrates that the asymptotic efficiency of DPSK in GDRF channels is not equal to unity (cf. Section IV-B.4). The 3dB gap between optimum DPSK and optimum coherent BPSK modulation in singleuser GDRF channels can be observed in Figure 1, because the asymptotic efficiency of coherent MEP detection is unity. In Figure 2 the bounds are displayed for two fixed values of the first user’s SNR over varying interfering users energyratios. The lower bound almost collapses for low SNRs with the SU-MEP. The looseness of the upper bound for weak interfering users can be traced back to the derivation of the asymptotic error rates, where we made use of the fact that for 1 (recall that contains the power ratios of all users relative to the first). When the power of the interfering users tends to zero, this will be true for increasing values of only. Since the multiuser MEP detector collapses to the single user MEP detector when the energy of the interfering users tends to zero, one might argue that the lower bound captures the true BER on the MEP and the upper bound is loose, because multiple error events in the union bound become more likely for weak interfering users. For strong interfering users the upper bound on the MEP detector becomes tighter, as the multiple error events in the union bound become less likely. The asymptote on the BER is not affected by the varying interfering users’ energies, as expected from the analytical results. So far we have only considered independent fading examples. Figure 3 shows the BERs achieved by the different detectors for a fixed SNR of 20dB as function of Æ , the correlation within the diversity branches. (The same definition for the line-styles applies as in the previous figures. At 20dB the upper bound has already converged, it is not visible in the figure). Setting the fading correlation parameter Æ to zero corresponds to independent fading and Æ equal to one corresponds to fully correlated fading. For increasing Æ the performance of all detectors degrades. For higher SNRs the same gaps to the optimum detectors show up as in the case of independent fading.
I
For fast fading we have to choose the correlation matrix of the fading coefficients at time lag one, (1). As we have restricted the analysis to (1) = (0) (cf. Section IV-B), we will only consider numerical examples with such a (1). Recall that the fade rate is connected to the normalized Doppler bandwidth BD T by = J0 (BD T ) for fading with power spectrum den-
sity given by Jakes’s model [20]. BD T has to be multiplied with the factor 2c rfbc (c is the speed of light, rb = 1=T the bit rate, and fc the carrier frequency) to obtain a corresponding velocity. Figure 4 depicts the BERs of the MEP and D-EGC detectors for fast fading with fade rates = 0:9 and = 0:975. Note the flooring of the BERs. Moreover, not surprisingly, all detectors perform worse for the faster fading case and the BER floor is considerably higher. The multiuser MEP and D-SU-MEP reach the same error floor as the MEP in the single-user channel for both fade rates. This shows that for very high SNRs, fast fading is the limiting factor on detection performance. Whereas the BER floor reached by the D-EGC might be regarded as acceptable in comparison to the D-SU-MEP or the MEP, for practically relevant ranges of the SNR the performance of the D-EGC is not acceptable: For a medium SNR of 15dB the D-EGC has a dismal BER of 0.2 making it essentially useless. The BER for the D-SU-MEP can improve on this BER by a factor of ten and the multiuser MEP by a factor of 100. VI. C ONCLUSIONS In this work we present a systematic approach to the problem of optimum multiuser noncoherent DPSK detection in generalized diversity Rayleigh fading (GDRF) channels. It extends the optimum single-user approach of [3] to the multiuser channel. With the exception of [3], previous approaches to DPSK detection in diversity Rayleigh fading channels have so far been guided by the ad-hoc post-decorrelative equal-gain combining rule. [3] introduced the optimum detection approach for the single-user DPSK GDRF channel. Until now, when multiuser DPSK detection in GDRF channels has been considered, the single-user decision rules were applied to the decorrelated signals, i.e., in an effective single-user channel, suffering from noise-enhancement through the decorrelating operation. On the other hand, the multiuser MEP detector obtained here is the optimum strategy for the joint detection of all users in Rayleigh fading. It is the noncoherent maximal ratio combiner and requires knowledge about the fading statistics and the users’ SNRs. A rigorous analysis of its error probability is undertaken and a lower and upper bound on its BER are derived. For slow fading, when the fading coefficients are essentially constant over two successive symbol intervals, the upper bound converges asymptotically to the lower bound. Hence, we can obtain an asymptotic BER and the asymptotic efficiency and show that the latter is in general smaller than unity. Moreover, the asymptotic efficiency turns out to coincide with the near-far resistance proving the near-far resistance of the MEP detector for slow fading. For fast fading an upper and a lower bound on the bit error floor are obtained. A PPENDIX P ROPERTIES
B. Fast Fading
9
OF A
DATA -S YMBOL -W EIGHTED S UM M ATRICES
OF
T WO
MBi , MBj defined MM MMB1j . One easily finds MBj and subsequently MB1j through the ap-
In this appendix we present results for 1 in (16), and the matrices Bi and
plication of a formula for the inverse of a 2 2 block-partitioned matrix (cf. [21]), where four blocks of the 44 block-partitioned
10
ACCEPTED BY THE IEEE TRANSACTIONS ON INFORMATION THEORY
M
matrix Bj make up one of the can then calculate
2 2 blocks.
With these we
=
M 21 M24M441M41 M11 M14M441M41
1
;
MMB1i 24 = 1 M14 M44 M41 M 1 M14 1 ; MMB1j = = M M M 24 21 11 11 h i h i 3 2 1 = 1 1 MM I i MMBj 12 h 0 i MMBj 14 7 Bi 31 6 h 7 6 1 1 = M31 M34 M441 M41 M11 M14 M441 M41 1 ; I MM 0 7 6 MMBj B j 6 21 h 23 h i i 7; 1 = 7 6 1 1 MM B 0 MM I MM 7 6 i 34 Bj 32 h Bj 34 5 4 h i i 1 1 = M34 M31 M111 M14 M44 M41 M111 M14 1 ; MMBj 41 0 MMBj 43 I MMB1i 42 = where = M42 M43 M331 M32 M22 M23 M331 M32 1 ; h i MMB1j 12 = MMB1i 43 = = M43 M42 M221 M23 M33 M32 M221 M23 1 : = M12 M14 M441 M42 M22 M24 M441 M42 1 ; h i MMB1j 14 = Throughout this section we require of course that all the inverted matrices actually exist. If M is a positive definite correlation 1 = M14 M12 M221 M24 M44 M42 M221 M24 ; matrix this is always the case, as we invert diagonal blocks and h i 1 Schur complements, which are both positive definite for a posiMMBj 21 = M . tive definite matrix = M21 M23 M331 M31 M11 M13 M331 M31 1 ; h i R EFERENCES MMB1j 23 = [1] M. K. Varanasi, “Parallel group detection for synchronous CDMA communication over frequency-selective Rayleigh fading channels,” IEEE = M23 M21 M111 M13 M33 M31 M111 M13 1 ; Trans. Inform. Theory, vol. 43, no. 1, pp. 116–128, Jan. 1996. h i [2] S. Verd´u, Multiuser Detection, Cambridge Univ. Press, New York, NY, 1998. MMB1j 32 = [3] M. K. Varanasi, “A systematic approach to the design and analysis of optimum DPSK receivers for generalized diversity communications = M32 M34 M441 M42 M22 M24 M441 M42 1 ; over Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 9, h i pp. 1365–1375, Sept. 1999. 1 MMBj 34 = [4] J. G. Proakis, Digital Communications, McGraw-Hill, New York, 3rd edition, 1995. Zvonar and D. Brady, “Linear multipath-decorrelating receivers for = M34 M32 M221 M24 M44 M42 M221 M24 1 ; [5] Z.CDMA frequency-selective fading channels,” IEEE Trans. Commun., h i vol. 44, no. 6, pp. 650–653, June 1996. MMB1j 41 = [6] Z. Zvonar, “Multiuser detection and diversity combining for wireless 1 CDMA systems,” in Wireless and Mobile Communications, J. M. Holtz1 1 = M41 M43 M33 M31 M11 M13 M33 M31 ; man and D. J. Goodman, Eds. Kluwer Academic Press, 1994. h i [7] H. C. Huang and S. C. Schwartz, “A comparative analysis of linear mulMMB1j 43 = tiuser detectors for fading multipath channels,” in Proc. IEEE Global Telecommun. Conf., Dec. 1994, pp. 11–15. 1 Duel-Hallen and S. Andrijic, “DPSK diversity combining for CDMA = M43 M41 M111 M13 M33 M31 M111 M13 : [8] A. frequency selective fading channels,” in Proc. Asilomar Conference on One can also show that
MMB1i = 2 6 6 4
1 MM 1 I MM B Bi 13 0 1 i 12 1 MMBi 21 I 0 MMBi 24 MMB1i 31 0 I MMB1i 34 0 MMB1i 42 MMB1i 43 I
[9]
3 7 7; 5
[10] [11] [12] [13]
where
MMB1i 12 = = M12 M13 M331 M32 M22 M23 M331 M32 MMB1i 13 = = M13 M12 M221 M23 M33 M32 M221 M23 MMB1i 21 =
1 1
[14]
; ;
[15] [16]
Signals, Systems and Computers, Pacific Grove, CA, Nov. 1997. A. Abrardo, “Noncoherent MLSE detection of M -DPSK for DS-CDMA wireless systems,” IEEE Trans. Veh. Technol., vol. 50, no. 4, pp. 900–909, July 2001. G. N. Karystinos and D. A. Pados, “On DPSK demodulation of DS/CDMA signals,” in Proc. IEEE Global Telecommun. Conf., Rio de Janeiro, Brazil, Dec. 1999, vol. 5, pp. 2487–2492. M. K. Varanasi and B. Aazhang, “Optimally near-far resistant multiuser detection in differentially coherent synchronous channels,” IEEE Trans. Inform. Theory, vol. 37, no. 4, pp. 1006–1018, July 1991. M. K. Varanasi, “Noncoherent detection in asynchronous multiuser channels,” IEEE Trans. Inform. Theory, vol. 39, no. 1, pp. 157–176, Jan. 1993. C. J. Hegarty and B. Vojcic, “Two-stage multiuser detection for noncoherent CDMA,” in Proc. Allerton Conf. on Comm. Control, and Comput., Monticello, IL, Oct. 1995, pp. 1063–1072. C. J. Hegarty and B. Vojcic, “Two-stage multiuser detection for asynchronous CDMA using DPSK signalling,” in Proc. Comm. Th. Mini Conf., London, UK, 1996, IEEE GLOBECOM, pp. 132–136. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1994. M. K. Varanasi and M. Brehler, “A systematic approach to noncoherent detection for DPSK modulation in multiuser correlated diversity Rayleigh fading channels,” in Proc. Conf. Inform. Sciences and Systems. Princeton University, Mar. 1998, pp. 236–241.
[17] M. Brehler and M. K. Varanasi, “Asymptotic error probability analysis of quadratic receivers in Rayleigh fading channels with applications to a unified analysis of coherent and noncoherent space–time receivers,” IEEE Trans. Inform. Theory, vol. 47, no. 5, pp. 2383–2399, Sept. 2001. [18] E. Visotsky and U. Madhow, “Noncoherent multiuser detection for CDMA systems with nonlinear modulation: A non-Bayesian approach,” IEEE Trans. Inform. Theory, vol. 47, no. 4, pp. 1352–1367, May 2001. [19] A. Russ and M. K. Varanasi, “Noncoherent multiuser detection for nonlinear modulation over the Rayleigh fading channel,” IEEE Trans. Inform. Theory, vol. 47, no. 1, pp. 295–307, Jan. 2001. [20] W. C. Jakes, Microwave Mobile Communications, An IEEE Press Classic Reissue, New York, 1993, Originally An American Telephone and Telegraph Company Publication, 1974. [21] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1993.
Matthias Brehler (S’00–M’03) started his studies in electrical engineering at the Technische Universit¨at M¨unchen in 1992. From 1995 to 1996, he attended the University of Colorado, Boulder, where he received the M.S. degree in electrical engineering. In 1998 he received the Diplom–Ingenieur degree from the Technische Universit¨at M¨unchen. After completing his military service at the Universit¨at der Bundeswehr M¨unchen (Federal Armed Forces University, Munich), he returned to Boulder in 1999, where he obtained his Ph.D. in electrical engineering in Fall 2002 and is currently working as a Research Associate. His research interests include multiuser, multi-carrier, and space-time communications.
Mahesh K. Varanasi (S’87–M’89–SM’95) received the B.E. degree in electronics and communication engineering from Osmania University, Hyderabad, India, in 1984, and the M.S. and Ph.D. degrees in electrical engineering from Rice University, Houston, TX, in 1987 and 1989, respectively. In 1989, he joined the faculty of the University of Colorado at Boulder in the Electrical and Computer Engineering Department where he is now a Professor. His teaching interests include communication theory, information theory, and signal processing. His research interests include multiuser detection, space-time communications, equalization, signal design, diversity communications over fading channels, and power- and bandwidth-efficient multiuser communications.
Fig. 1. BER comparison of the decorrelating equal-gain combiner, postdecorrelative single-user MEP, multiuser MEP, single-user channel MEP, and optimum multiuser detectors. There is a 11dB gap between the D-EGC and the multiuser MEP detector.
Fig. 2. The upper and lower bound on the BER of the first user as a function of the relative energies of the other users.
Fig. 3. The BER of the first user for the different detectors as a function of the correlation within the diversity branches. The upper bounds converge to the lower and are hence not visible.
Fig. 4. All users have the same power. For faster fading, the performance of all detectors degrades, as expected.
0
10
−1
10
−2
10
−3
BER of user 1
10
−4
10
−5
slow fading, wk/w1=1
10
D−EGC D−SU−MEP MEP asymptote MEP upper bnd MEP lower bnd SU channel MEP optimum coherent
−6
10
−7
10
−8
10
−9
10
0
5
10
15 20 SNR (dB) of user 1
25
30
0
10
−1
10
−2
_ γ = 10dB
BER of user 1
10
slow fading MEP asymptote MEP upper bnd MEP lower bnd SU channel MEP
−3
10
−4
10
−5
10
_ γ = 20dB
−6
10
−2
10
−1
10
0
10 wk/w1
1
10
2
10
0
10
slow corr. fading, 20dB, wk/w1=1 −1
10
−2
BER of user 1
10
−3
10
−4
10
−5
10
−6
10
−7
10
0
0.1
0.2
0.3 0.4 0.5 Correlation parameter δ
0.6
0.7
0.8
0
10
−1
10
−2
BER of user 1
10
−3
10
ρ=0.9
−4
10
fast fading, w /w =1 k
−5
10
−6
10
1
D−EGC D−SU−MEP MEP upper bnd MEP lower bnd SU channel MEP
ρ=0.975
−7
10
0
10
20 30 SNR (dB) of user 1
40
50
Abstract— The jointly optimum multiuser noncoherent detector for DPSK modulation over the generalized diversity Rayleigh fading (GDRF) channel is derived and analyzed. The GDRF channel includes time/frequency/receiver antenna diversity and allows fading correlations between the various diversity branches of each user. Noncoherent detection here refers to the case where the receiver has neither knowledge of the instantaneous phases nor of the envelopes of the users’ channels. Upper and lower bounds on the bit error probability of the optimum detector are derived for a given user. For fast fading, when the fading coefficients vary from one symbol interval to the next (but are still essentially constant over one symbol interval), the detector asymptotically (for high signal-to-noise ratios) reaches an error floor, which is bounded from below and above for different fast fading scenarios. For slow fading, when the channel is constant for at least two consecutive symbol intervals, the upper bound is shown to converge asymptotically to the lower bound. Thus the asymptotic efficiency of optimum multiuser DPSK detection can be determined and is found to be positive. In contrast to coherent detection however, it is smaller than unity in general. Since the asymptotic efficiency is independent of the interfering users’ signal strengths, the optimum detector is near-far-resistant. While optimum multiuser detection is exponentially complex in the number of users, its performance provides the benchmark for sub-optimal detectors. In particular, it is seen that the previously suggested post-decorrelative detectors can be far from satisfactory. Index Terms—DPSK modulation, diversity communications, error analysis, fading channels, maximal ratio combining, multiuser detection, noncoherent detection, optimum detection.
D
I. I NTRODUCTION
IFFERENTIAL phase shift keying (DPSK) is an especially attractive modulation scheme for multiuser communications, because in many applications the users are mobile and thus their channels keep changing. DPSK avoids the expense of elaborate channel estimation and tracking algorithms necessary for coherent demodulation, because one does not have to estimate the channel parameters: the information is encoded in the phase difference between two successive symbol transmissions. In this work, neither the instantaneous phase nor the envelope of the users’ channels are assumed to be known at the receiver. Due to the differential encoding, it is usually required for DPSK that the fading coefficients stay constant over two successive symbol intervals. Channels where such an assumption is valid will be referred to as slowly fading channels. However, fast fading channels are also considered where we define fast fading to mean that the fading parameters are allowed to vary from one symbol interval to the next (but not within a symbol interval). The slow fading assumption holds for lower vehicular speeds and/or higher data rates and/or higher carrier frequencies when
II. T HE G ENERALIZED D IVERSITY R AYLEIGH FADING (GDRF) C HANNEL
Submitted, July 2000; revised, December 2001 and October 2002. This work was supported in part by NSF Grant ANIR-9725778 and by US Army Research Office Grant DADD19-99-1-0291. The results of this paper were presented in part at the Conference on Information Sciences and Systems at Princeton University, Princeton, NJ, March 1998 and the Communications Theory Mini Conference, IEEE GLOBECOM, San Francisco, CA, Nov. 2000. The authors are with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309-0425 (e-mail: [email protected], [email protected]).
In this section the discrete-time model for the multiuser generalized diversity Rayleigh fading channel (GDRF) is developed. It allows for inter-diversity fading correlation, interdiversity signal correlation, and inter-user signal correlations and applies to variety of diversity situations [3]. It handles S receive antennas and Lw waveform diversity channels (mul1
2
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tipath, frequency diversity, or others). The total diversity order is L = SLw . Our presentation of the K -user GDRF channel draws heavily on the single-user channel presented in [3]. Extending the single-user discrete-time model [3, eq. (6)] to multiple users leads to p (n) = 1=2 (n) (n) + (n); (1)
R D c
q
where = Nw10 is the average signal to noise ratio (SNR) of the user of interest (which we assume without loss of generality to be the first user), is the KLKL dimensional generalized signal correlation matrix defined as
R
Z
R = T U (t)N 1UT (t) dt T 1 (t);
(2)
N
T (t); : : : ; T (t)T and the noise with (t) = 2 K correlation matrix (see [3, Section II] for more details, each k (t) corresponds to (t) defined therein), is the KL KL diagonal relative energy matrix = w1 1 diag f[w1 ; w2 ; : : : ; wK ]g L with wk the k th user’s average energy and the Kronecker (or tensor) product [15], (n) is the KL KL diagonal symbol matrix (n) = diag f[d1 (n); : : : ; dnK (n)]]g
L , with dk (n) = dk (n 1)bk (n) j2 l oM 1 and bk (n) 2 F = e M denotes the k th user’s MPSK l=0 information symbol, (n) is the KL-length channel fading information (CFI) vector (n) = [ T 1 (n); T2 (n); : : : ; TK (n)]T , which is CN ( ; (n; n)) distributed with
U
U
U
U U
U
I
D c
D
I
c 0
c
c
c
(n; m) = E c(n)cy (m) ; and
(3)
is the additive white (in time) Gaussian noise, which is CN ( ; ) distributed. The k th user’s generalized signal correlation matrix kk is the k th diagonal L L block of . Each user’s signature waveforms uk;l (t) are normalized such that their energy in the time T is unity. Hence, the diagonal elements of the matrix interval R T b = T (t) (t) dt are all unity. Similar to kk , we define the k th diagonal L L block of b as b kk .1 The k th diagonal L L submatrix of (n; m), kk (n; m), is the matrix correlation function of the k th user’s fading parameters. We assume that the fading paths of different users are statistically independent of each other, i.e., (n; m) will be blockdiagonal with kk (n; m) as diagonal elements. Wide sense stationarity of the fading process allows us to write (n; n) = (0) and (n; m) = y (m; n) = (m n). We define the slowly fading channel to have constant fading coefficients for the duration of two successive symbol intervals. Hence, for slow fading we have (1) = (0). In the fast fading channel,
R
0R
R
R
U U
R R
R
1 Note that in the case of only a single receiver antenna or uncorrelated noise b denotes the correlation matrix and b are equal. In [3] R in the antennas b kk of this paper. of the diversity waveforms only and is hence not equal to R b kk ), yet b kk is a block-extended version of R b of [3] (precisely IS R Here, R all normalizations are the same.
R
R
the fading coefficients can vary from one symbol interval to the next (but are still assumed to be essentially constant within one symbol interval). To ensure that wk is the total average received energy of user k , the fading coefficients (n) are normalized. The average received power due to the isolated transmission of the k th user in a noiseless channel should be equal to wk . This amounts to
c
tr kk (0) Rb kk = 1;
(4)
which is equivalent with [3, Eq. (9)], if the different definitions of b are considered. As we mask out the ISI in cases it arises (cf. [3]) we will be able to perform symbol-by-symbol decisions. Hence, we detect the data symbols in the zeroth time interval bk (0) = dk (0)dk ( 1) and only need to consider the KL-dimensional vectors of observations for the zeroth and the previous symbol, (0) and ( 1). To simplify notation in the sections to come we stack these vectors and obtain 2KL dimensional vec= T ( 1); T (0) T . As ( 1) and tor of observations (0) are zero-mean, complex Gaussian random variables (because (n) and (n) are) so is . We denote their correlation matrix with q . q depends in general on the M 2K possible combinations of the differentially encoded symbols ( 1) and (0). Since we assume that the fading paths of different users are uncorrelated, q will depend on the information symbols (0) = (0) ( 1) only: The product (n; m) (n) commutes ((n; m) 2 f 1; 0g2) because (n; m) is block-diagonal with blocks of size L L and (n) and are diagonal matrices with the same element repeated L times on their diagonals. As we will only consider data symbols in the zeroth symbol interval, we drop the time index in (0) and introduce the integer subscript i (1 i M K ) indicating the dependence of the K data symbols fbk gk=1 on the underlying hypothesis Hi . Hence, Hi denotes that i is the particular realization of the data symbols with bik the realization of user k . These definitions and the assumption of uncorrelated fading for different users let us write the correlation matrix of given the specific data symbols i (corresponding to the hypothesis Hi ) were encoded by the transmitter as
R
q
q
q
q
c
D B
q
q
q
K K
D D
q
D
K
D
D
B
B
q
B
KqjHi = =
R ( 1; 0)Bi R
(5)
R ( 1; 1)R + R :
RBi (0; 1) R
R (0; 0)R + R
III. D ETECTORS FOR DPSK M ODULATION IN C HANNEL
THE
GDRF
In this section we briefly visit known suboptimum detectors for noncoherent multiuser DPSK detection in GDRF channels and introduce the optimum, maximum likelihood detector. Previous approaches are mostly of the post-decorrelative type, i.e., the multiuser channel is simplified to an effective single-user channel by decorrelating the users. Although it is well-known that decorrelation is a sub-optimum procedure (cf. [1]), the notion of optimum multiuser DPSK detection in GDRF channels has only been introduced in a conference version of this paper [16]. Moreover, it is only recently that even the single-user optimum detection problem was solved ( [3]). Hence, we revisit
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
the derivation of the optimum multiuser detector of [16] in more detail in Section III-B; it assumes Rayleigh fading and knowledge of the users’ SNRs as well as of the fading statistics (but not of the realizations of the channel fading information vector). A. Post-Decorrelative Approaches For the single-user, independently and identically fading diversity channel ( (0) = ) the ad-hoc solution to DPSK reception is to combine the decision statistics of all branches by weighting them equally [4]. (It was shown in [3] that this adhoc solution corresponds to the GLRT detector presented in that paper). In [5, 6], and [7] the idea of equal-gain combining is applied to the multiuser decorrelated statistics2
I
z(n) = R
1 q(n):
(6)
The decision for each user is solely based on its corresponding decorrelated statistics, thereby neglecting the noise correlation. Hence, the decorrelating equal-gain combiner (D-EGC) for the first user can be formulated as
D EGC : ^i1 = = arg i 2f1min Refbi1 zy1 ( 1)z1 (0)g; 1 ; 2; :::; M g
(7)
where bi1 2 F is the first user’s data symbol in the zeroth symbol interval. The same decision rule h may be written in the effeci T tive single-user statistics ^ 1 = 1 ( 1)T ^ T ; 1 (0)T ^ T = ^ T ( 1); ^T (0)
q1
q
q1
D EGC : ^i1 = y ^ = arg i 2f1min q ; 2; :::; M g 1 1
z
0 ^ bi1 R
R z
R
bi1 ^ 2
R 0
2
R
(8)
q^1 ;
where ^ 1 is the upper-left L L block of of the decision rule is convenient for the performance analysis: Since all decision rules presented in this work can be expressed in such a quadratic form, the general, unified error probability analysis of [17] can be applied to each one of them. Note that the D-EGC neither accounts for inter-diversity branch correlation, nor fading paths with different signal strengths, nor for the noise correlation through the inter-diversity and inter-user signal correlation. Rather than applying the decorrelating operation and equalgain combining the decorrelated decision statistics, the singleuser detection problem is systematically approached in [3]. A rigorous problem formulation therein leads to the singleuser generalized likelihood ratio test (SU-GLRT) and minimum probability of error (SU-MEP) detector. Both of them by far out-perform the D-EGC. Having obtained the single-user MEP and GLRT detectors, it is straightforward to apply them to the effective single-user channel in a multiuser setting. For future reference we will refer to them in the multiuser context as the decorrelating single-user GLRT (D-SU-GLRT) and MEP (DSU-MEP) detectors.
R
3
B. The Minimum Error Probability Detector The minimum error probability (MEP) detector minimizes the probability of error in the joint detection of all K users. Once the transmission model is specified, it results from a standard maximum likelihood derivation. We assume that the statistics of the fading parameters, i.e., the covariance matrices (n; m), and the users’ individual signal strengths, i.e., the wk ( and , respectively) are known at the receiver. Consequently the observations given the data symbols i are CN ( ; qjHi ) distributed and since all hypotheses are assumed equally likely the MEP detector is simply
q
B
0K
MEP : ^i = arg i2f1; 2min qy Kqj1Hi q +ln KqjHi ; ; :::; M K g
M
(9)
M
where we define for any matrix j j = det( ). In contrast to the single-user case, the MEP decision rule for the multiuser GDRF channel is not independent of the determinant of qjHi . It is convenient to state the decision rule not in the decibut in the the decorrelated statistics , where sion statistics T = T ( 1) T (0) . The covariance matrix of the new decision statistics becomes
K
z
z
q
z z KzjHi = B( i 1;(0;1) 1)+ R
z
1
( 1; 0) Bi 1
(0; 0) + R
:
(10) The MEP decision rule can then equivalently be stated in the decorrelated decision statistics as
z
MEP : ^i = arg i2f1; 2min zy Kzj1Hi z +ln KzjHi : ; :::; M K g
(11)
C. Generalized Likelihood Ratio Test (GLRT) The MEP requires statistical knowledge of the fading corre-
1 . The last form lations and the users’ average energies. The estimation of these
2 In a single-user channel with inter-diversity signal correlations those papers also propose decorrelating the user’s diversity signals and equal-gain combining.
quantities can be circumvented through a generalized likelihood ratio test (GLRT) based approach. The multiuser GLRT was introduced for the noncoherent detection of DPSK in [16] and for general nonlinear multipulse modulation (NMM) in [18]. It maximizes the likelihood of the observations first over the unknown fading parameters ( 1) = (0) (assuming slow fading) and subsequently over the information symbols i . For DPSK and our notation the result is
c
c
B
G : ^i = arg i2f1; 2min ; :::; M K g 1 qy BIi R + Bi RBi I Bi q:
(12)
The GLRT receiver is exponentially complex in the number of users, as is the MEP. Examples in [16] demonstrate that it can suffer a performance loss of 2 to 3 dB relative to the MEP. Since in this paper we are concerned with analysis of optimum DPSK detection in Rayleigh fading, we do not consider the GLRT. Note however, that it may be analyzed using similar methods applied here to the MEP. Moreover, since the GLRT does not incorporate any assumptions about the fading distribution, it may be expected to perform well under a variety of fading conditions.
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IV. P ERFORMANCE A NALYSIS In this section the performance of the detectors presented in Section III is examined. To this end, lower and upper bounds on the bit error rate (BER) of the multiuser MEP detector are derived in Section IV-A. The analysis also applies to the decorrelating-type detectors with the simplification that the bounds coincide. For the multiuser MEP, the asymptotic (high SNR) convergence of the upper bound to the lower bound in slowly fading channels is proved in Section IV-B using the unified general analysis of [17]. The convergence of the bounds enables us to obtain exact formulas for the asymptotic efficiency and the near-far resistance of the detectors. For fast fading channels, the error floor is bounded from below and above. For simplicity, we restrict attention to binary modulation for both slow and fast fading. Without loss of generality, all performance measures are obtained for the first user. A. Bounds on Bit Error Rate The upper bound on the error probability of user one is derived by invoking a union bound and the lower bound by only considering the single error events, i.e., events for which there occurs an error for user one only. To derive these bounds we first consider the so-called pairwise error probability Pr fÆj < Æi g, the probability that the decision statistic Æj corresponding to hypothesis j is smaller than the decision statistic Æi of the true hypotheses Hi . This probability can be expressed as the probability that a quadratic form in the decision variables is smaller than some constant. The cumulative distribution function of such a quadratic form is for example derived in [17]. All introduced detectors can be expressed in the general form
q
: ^i = arg i2f1;min xy Fix + ci 2; :::; 2K g = arg i2f1;min Æ; 2; :::; 2K g i
F
x
z F
K xFx
xFx
Pe
= 2
K
2 X K
K
i=1
PejHi
X
Pr fÆj < Æi g :
i=1 8j2f1; :::; 2K g s:t: bi1 6=bj1
(13)
where bik denotes the k th user’s data symbol corresponding to hypothesis Hi . A lower bound on the first user’s error probability is simply obtained by averaging over all 2K pairwise error probabilities whose corresponding data symbols differ only for the first user, so that
Pe
21K
2 X K
Pr fÆj < Æi g :
i=1 j s:t: bj1 6=bi1 bjk =bik ; 2kK
(14)
B. Asymptotic Error Rate Analysis In [17] a unified asymptotic analysis of quadratic receives in Rayleigh fading channels is presented. In the multiuser context, such an analysis was applied for the first time to a problem similar to the one at hand arising in the error probability analysis of noncoherent multiuser detection for nonlinear modulation [19]. In the case considered here, we will find likewise that the bounds on the error probability converge asymptotically for high SNR in the case of slow fading. However, the results of [17] greatly simplify this task when compared to [19], because we merely have to find the asymptotic eigenvalues of certain matrices. For fast fading we bound the error floor from below and above; numerical examples show that these bounds are tight. To this end, we will first state some definitions and then reveal the results of the asymptotic analysis for the MEP detector in several sections. First the behavior of the constant cij is examined in Section IV-B.1: for slow fading it approaches a constant for increasing SNR, and it becomes zero for fast fading. In Section IV-B.2, the non-zero asymptotic eigenvalues of ij are found as the SNR approaches infinity. For slow fading eij L of the asymptotic eigenvalues are positive and linear in , and the eigenvalue minus unity is repeated eij L times (where eij is the number of errors in an error event). For fast fading eij L eigenvalues approach a positive constant and eij L eigenvalues a negative constant. The results of these sections combined with [17, Proposition 2] allow us to obtain the asymptotic bit error rate (BER) formulas for slow fading and bounds on the error floor for fast fading, respectively, in Section IV-B.3. Finally, we specify the asymptotic efficiency and and near-far resistance of the multiuser MEP detector in Section IV-B.4 for slow fading. For the remaining analysis we will just consider hypotheses Hi , Hj whose corresponding vectors of data symbols have the following structure
C
where Æi is implicitly defined. For the MEP detector the de = , i = zj1Hi , and ci = ln zjHi . cision statistics For the other approaches similar assignments can be made. In the remainder of this section we express everything in terms of i , ci , and the covariance matrix xjHi of the decision statistics . Since Pr fÆj < Æi g is the probability that the decision statistic y j + cj is smaller than y i + ci given Hi is the true hypothesis, we can apply the results of [17, Proposition 1] and obtain the pairwise error probabilities through residues. With the latter we union bound PejHi , the error probability of user one conditioned on hypothesis Hi . Computing PejHi precisely would require the evaluation of the probability of the union of all 2K 1 possible events Æj < Æi with j and i such that the first user’s bit in hypotheses Hi and Hj differs. This probability is upper bounded by the sum of the probabilities of those error events so that the first user’s error probability can be bounded by
x
2
2 X K
K
K
bTi = 1Teij +1Teij+ +1Toij+ 1Toij ; bTj = +1Teij 1Teij+ +1Toij+ 1Toij ; (15) where 1n is an n length vector containing n ones and eij + + eij + oij + + oij = K and eij + + eij = eij , the number of errors incurred when bi is mistaken for bj . As indicated in (15), we assume that all errors of an error event (Hi
! Hj ) occur
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
within the first eij users, and furthermore, that the first eij elements of ij are minus unity and the following eij + elements are plus unity where of course eij = eij + eij + . Additionally, we also order the K eij remaining users for which no error occurs in a way that the first oij + users send +1 and the next oij users send 1. The ordering can be performed without loss of generality, be~ i ! H~ j ) with arbitrary ordering we cause for any error event (H can reorder the users according to the above scheme, rearrange the users’ signal correlation matrix , and the correlation matrix of the decision statistics zjHi and hence calculate the error probability of the corresponding error event (Hi ! Hj ). Note that the block-extended matrices of data symbols i and j hence have a structure corresponding to (15) and we the matrices define for any matrix
e
K
R
B
B
M
MBi = 12 (M + Bi MBi) ;
(16)
MBj = 12 (M + Bj MBj ) ;
whose properties are examined in the appendix. With the help of the above ordering we define a 4 4 block where the diagopartitioning of any KL KL matrix nal elements have the following dimensions: size( 11 ) = eij L eij L, size( 22 ) = eij + L eij + L, size( 33 ) = oij + L oij + L, and size( 44 ) = oij L oij L. All other entries have appropriate dimensions. Every matrix subscript in the following sections on the asymptotic analysis refers to the submatrix according to this partitioning, unless stated otherwise. Furthermore, a subscript on [ ]ij ((i; j ) 2 f1; 2; 3; 4g2 ) refers to the corresponding submatrix of the expression in brackets according to this partitioning. Depending on the error event, one or more dimensions of a block in this partitioning might be zero. For consistency, we define the determinant of an empty matrix (dimension zero for a row or a column or both) to be unity and the product of empty matrices to be an empty matrix again. However, note that we defined in Section II the matrices kl and kl (n) as the L L submatrices of and (n), respecaccording to the partively. Consequently, a submatrix of titioning introduced here is written as, for example, 11 and is not to be confused with 11 , which is always the upper-left L L submatrix of . For (n) the submatrix according to the new partiotioning is written with square brackets, for example, as [ (n)]11 in contrast to 11 (0), the correlation matrix of the first user’s L diversity branches. It will also be convenient in what follows to define a 2 2 block-matrix partitioning of any 2KL 2KL matrix as
M
M
M M
M
X
X
R
R
R
R
R
K = KKulll KKurlr
;
(17)
X
C
C
C
C
C
C
C
C
K
C
K
C
D
B.1 Asymptotic Analysis of the Determinant Ratio Proposition 1 (Limit of the logarithm of the determinant ratio) The limit of the logarithm of the determinant ratio cij is independent of the interfering users’ signal strengths. For slow fading it can be stated as
c^ij
zjH i lim cij = lim ln
!1 !1 zjHj 11 14 22 41 44 32 ln 11 13 22 31 33 42
=
Q Q Q Q
=
Q R
1. where = For fast fading c^ij
K K
Q Q Q Q
Q Q Q Q
Q23 Q33 ; Q24 Q44
= lim c =0: !1 ij
Proof: Note the property
K
where all blocks have dimension KL KL (u denotes upper, l lower or left as appropriate, and r right). For simplicity, a combination of both partitionings will be written as, for example, ul23 or [ ]ur14 . For the analysis to come it will be crucial that the upper left block of ij is equal to its lower right block ([ ij ]ul = [ ij ]lr )
K
and that its upper right block is equal to its lower left one ([ ij ]ur = [ ij ]ll ). Hence we can exploit the easy-to-prove fact that the eigenvalues of such a matrix are equal to the eigenvalues of [ ij ]ul + [ ij ]ur and [ ij ]ul [ ij ]ur . For ij to have such a structure, it is sufficient that zjHi and zjHj have the same structure (and hence their inverses). The latter is true for slow fading, because (n; m) equals (0) for all four values of (n; m) 2 f 1; 0g2 (recall also that the product (n; m) (n) commutes). For fast fading, we assume, besides wide sense stationarity of the fading processes, (1) = (1)y to obtain the same upper-right and lower-left block, as desired. This assumption implies (1) = ( 1), i.e., the cross-correlation functions of the fading processes are all even at one time lag. More specifically, we choose (1) = (0), where (0 < < 1) is a single fade rate for all users. This choice implies of course that the relative movement between the users and a base station is the same for all users, which is clearly unlikely in practical applications. However, the basic system behavior is captured by this choice and controlled by one simple parameter. Furthermore, the analysis is readily available for cases of a specific fade rate for each user or even a specific fade rate for each user’s diversity signal (the latter choice requires a diagonal (0), though). The fade rates can be connected to the normalized Doppler bandwidth BD T by, for example, = J0 (BD T ) for fading with power spectrum density given by Jakes’s model [20]. Note that the choice = 1 corresponds to the slowly fading case.
R
5
A B = jA + BjjA Bj; B A
(18)
which follows from the fact that the eigenvalues of the first ma. Furthermore, we trix are the eigenvalues of + and can write
A B
KzjHi = where
(0) 0 0 (0)
A B
I + 1Q Bi1 ; Bi I+ Q
Q = (R(0) ) 1 = 1 1 (0)Q:
(19) (20)
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Since the determinant of a product of matrices is equal to the product of the determinants and zjHj can be written in a similar manner, we obtain
K
cij
= ln
K K
I + 1Q Bi1 Bi I + Q = ln jMij ; = ln 1 jMj j I + Q Bj1 Bj I+ Q
zjHi zjH j
which allows to reduce the problem to finding the determinants of the matrices i and j . Consider fast fading. We find the limit of the determinant of i by applying equation (18) as
M
M
M
I + 1 Q + Bi I + 1 Q lim jMi j = lim !1 = jI + Bi j jI Bi j :
!1
Mj j is lim jMj j = jI + Bj j jI
!1
B
i
Similarly, the limit of j
B
jj :
B
= ln(1) = 0
Q Q14 Qii4 Q44
3 Q11 Q1ii 5; Q = 4 Qii1 Qiiii Q41 Q4ii where the blocks Q11 and Q44 have the dimensions according the partiotioning introduced in Section IV-B and Qiiii has got the dimension (eij + + oij + )L (eij + + oij + )L. The remaining blocks have the appropriate dimensions. The determinant of Mi
can then be written as
jM j
i
i :
I + 1Q + B I + 1Q B
The first determinant can be approximated3 by
I + 1 Q + Bi = 1 Q11
1 Q1ii
1 Q14 = 1 Qii1 1 Qiiii + 2I 1 Qii4 1 Q41
1 Q4ii
1 Q44 11 Q14 j2I j j2I j for 1; 1 Q Q41 Q44 22 33
3 All approximations in this paper are in the sense that if x( ) is the quantity
dependent on and e x( ) its approximation, then lim
1 Qiiii j2I11 j j2I44 j
Q22 Q23 j2I11j j2I44j : Q32 Q33 Similarly one obtains an approximation for Mj . As all the factors of 1 and 2 cancel out one obtains c^ij for slow fading as stated in the theorem but with blocks of Q. But as all blocks involved have block-sizes that are integer multiples of L and (0) is block-diagonal with block-sizes L L, the blocks of ((0) ) 1 cancel out and we can state c^ij in blocks of 1 =
Q=R
1
.
B.2 Asymptotic Eigenvalue Analysis
2(K
eij )L of the eigenvalues of the matrix
2eij L eigenvalues of Cij approach arbitrarily closely 1 with multiplicity eij L,
the real and positive eij L eigenvalues of the matrix 1 1
2 11 [ (0)]11 11 13 33 31 , the real and positive eij+ L eigenvalues of the matrix 1 1
2 22 [ (0)]22 22 24 44 42 , 1. where = For fast fading with fade rate the non-zero eigenvalues of ij approach arbitrarily closely the real and positive value 2(1 ) 1 with multiplicity eij L, the real and negative value 2(1 ) 1 with multiplicity eij L . Proof: That all eigenvalues are real follows from Sylvester’s law of inertia [21, Theorem 4.5.8], because ij is the product of two Hermitian matrices and hence has real eigenvalues. We make use of the special block-structure of the involved matrices and define the matrices
Q R
Q Q
Q Q Q Q Q Q
C
for fast fading. Now consider slow fading. For this derivation, we introduce a 3 3 block-partitioning of the matrix
i =
For slow fading and high values of the SNR, the remaining
B
2
I + 1 Q Bi
Proposition 2:
M
c^ij
Cij = KzjHi Kzj1Hj I are zero.
Note that according to our fast fading assumptions, is unequal plus or minus unity. With these limits, it is not hard to see that the determinants of i and j are equal (as i and j differ in the sign of some elements, but are added and subtracted for the calculation of both determinants), and hence their ratio is one, and we obtain
M
which follows from applying a determinantal identity to a 2 2 partitioned matrix and from [17, Appendix C] (take out 1 ). Using corresponding techniques one also finds that
!1 x=ex = 1.
C
Y = (0); X = Y + R 1 = Y + Q: K
With these definitions, the correlation matrices zjHj and zjHj can be more compactly expressed. Applying the matrix inversion lemma for the inverse of a 2 block-partitioned matrix (cf. [21]) to zjHj yields
K
K
h
i
h
i
Kzj1Hj ul = Kzj1Hj lr = = X 1 + 2 X 1 YBj X 2 YBj X 1 YBj 1 YBj X h i h i Kzj1Hj ur = Kzj1Hj ll = = X 1YBj X 2 YBj X 1 YBj ; where a formulae for the inverse of a small-rank adjustment of a matrix [21, Section 0.7.4], also known as Woodbury’s identity, was used. Consequently, after some manipulations, including
1
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
h
7
i
Q
Q
1
Q Q Q Q
1 1 positive as is a Schur 24 44 42 Bj 22 = 22 complement of the positive-definite matrix ) and Leij h i as goes to infinity. By performing the same derivation for 1 [ ij ]ul = zjHi zjHj = [ ij ]ul ur one finds the remaining non-zero eigenvalues of ij ul 1 for slow fading, which are given in the proposition. 1 j = ( j i) 2 ; j+ j j out of the inverse term Consider fast fading. Taking in (21) allows us to make an approximation for fast fading Somewhat more easily, one obtains the upper-right element as the application of Woodbury’s identity, the upper-left element of ij becomes
C C
K K I B B B BX X
C
YB X YB
[Cij ]ur = (Bj Bi )Y X Since [Cij ]ul = [Cij ]lr and [Cij ]ur = [Cij ]ll , this fully specifies Cij . Furthermore, as Bj Bi has (K eij )L zero rows and columns, Cij has 2(K eij )L zero rows and hence the
claimed number of eigenvalues equal to zero. For the remaining eigenvalues, it is sufficient to consider [ ij ]ul [ ij ]ur denoted as [ ij ]ulur . We will only calculate [ ij ]ul+ur in detail, because similar arguments apply for [ ij ]ul ur . So we can write
C
C
[Cij ]ul+ur = (Bj Bi ) Bj + (Bj X Y) X
C
1 2 YBj X 1 YBj :
Y / and hence = (Y + Q) 1 Y 1 Y 1 QY
1
1 for 1;
where once again the formulae for the inverse of a small-rank adjustment was applied. Inserting this approximation and the definition for into (21) yields
X [Cij ]ul+ur (Bj Bi ) Bj + (Bj Y + Bj Q Y) Y + Q
Y + 2Bj QBj
Q
Bj + 12 (Bj I)YQB1j + Bj 21 QQB1j 02 0 0 03 h0 i B6 Y22 QB1j 22 0 777 6 0 = (Bj Bi ) B B6 @4 0 0 0 05 0 0 31 2 1 ILe 0 2 ij 1 C 6 I Le 2 ij 0 7 7C ; + 64 0 5A 0
(Bj Bi )
where indicates non-zero matrix elements that are of no interest for the eigenvalues (since they do not depend on ) and we 1 make use of the properties of Bj and Bj given in the ap/ we can apply [17, Appendix C]: the eigenpendix. As values of theh last matrix are arbitrarily close toi the eigenvali h 1 = 2 22 [ (0)] 1 ues of 2 22 Bj 22 Bj (which are
Q
Y
Y Q
22
C
B.3 Asymptotic Bit Error Rate and Bounds on the Error Floor
MEP PAs
=
Q
22
2 X K
1
L j2 11 (0)j 2K
i=1 8j2f1; :::; 2K g s:t: eij =1
8 eijX L 1 > 2eij L 1 > c^ij > e > > eij L >
> > > > > :
aij ;
Q13 Q331Q31 1 Q22 Q24Q441Q42
1 : where aij Q11
For the remaining analysis, we have to distinguish between fast and slow fading. Consider slow fading, i.e., = 1, which allows us to simplify (using Bj )
[Cij ]ul+ur
The last matrix obviously has eij L non-zero eigenvalues: the eigenvalue 2(1+) 1 with multiplicity eij L, and the eigenvalue 2(1 ) 1 with multiplicity eij + L. As in the slowly fading case, the remaining eigenvalues are obtained by applying the corresponding techniques to [ ij ]ul ur .
(21)
2
Bj + Bj + Bj QY 1 I I + QY 1 2 I + 2 Bj QBj Y 1 1 (Bj Bi ) Bj + (Bj I) I 2 I 1 :
[Cij ]ul+ur (Bj Bi )
Proposition 3: For slow fading ( (0) = (1)), the upper bound on the BER of the multiuser MEP detector obtained in Section IV-A converges asymptotically to the lower bound as SNR ! 1. The asymptotic expression for the BER is independent of the interfering users’ signal strengths and can be stated as
Now consider that
X
C
Y
1 2 YBj X 1 YBj :
C C
I
k
( c^ij )k k!
k=0
eX ij L
2eij L 1 eij L
k=0
1
k
k c^
for c^ij
1
=
0;
for c^ij > 0;
ij
k!
and c^ij is given through Proposition 1. For fast fading ( (1) = (0), 0 < < 1) we bound the MEP from below and above as first user’s error floor Pfloor
MEP Pfloor MEP Pfloor
1
4
2
K X K
e
e=1 eL X
k=1
L X L 2L
1 1
k=1
2eL 1 eL
L
1 1
4 k
2
1
1
k
2 1+
k
;
eL
2 1+
k
;
where we simplified eij to e because only the number of errors matters and not their position.4
jj
4 There is a typo in [3]: should be equal to instead of the given expression. With this correction, the results can be seen to coincide for the single-user case.
8
ACCEPTED BY THE IEEE TRANSACTIONS ON INFORMATION THEORY
Proof: Consider slow fading first. In Proposition 2 we found the 2eij L non-zero eigenvalues of ij for high values of the SNR; eij L of these eigenvalues are 1 and eij L positive and linear in the SNR. As in Proposition 1, we found the constant cij for large SNR (c^ij ), we can apply the results of [17, Proposition 2] and approximate the pairwise error probabilities Pr fÆj < Æi gMEP for high values of the SNR aij Pr fÆj < Æi gMEP eij L
j2 11 [ (0)]11 22 [ (0)]22 j ;
C
where aij is defined in the proposition. As c^ij is independent of the SNR (and hence aij ), the probability of a single error event (eij = 1) dominates the union bound on the error probability. Hence, one can neglect all terms with eij > 1 in the calculation of the union bound and is left with the same expression for the upper bound as for the lower bound. The asymptotic expression of the error probability comes arbitrarily close to the true error probability, i.e, PeMEP lim MEP = 1:
!1 PAs For eij = 1 we have 11 = L (or 22 = L ) and [ (0)]11 = 11 (0) (or [ (0)]22 = 11 (0)), which is exploited to simplify the asymptotic expression. Consider fast fading. For fast fading, one inserts the eigenvalues found in Proposition 2 into the definitions of the bounds. Hence, the characteristic function has two poles of multiplicity eij L. The poles (corresponding to eigenvalues of ij ) merely depend on the number of errors and not on which users 1 possible error are detected erroneously. There are K e 1 events for e errors within K users, when an error for the first user must occur. This allows us to replace the sum over j by a sum over the number of errors e and the binomial coefficient. Example: Two-User Asymptotic Error Probability Let the two-user signal correlation matrix be defined as
I
I
C
R = RR1121 RR1222
R
:
R
jR11 j R11 R12R221R21 ;
(23)
which is greater or equal than zero. So we write the asymptotic error probability succinctly as
=
L X 1 2L 1 c^k : k! L
L j211 (0)R11 j 2 k=0
(24)
Generally, when the two users signals are correlated (and hence 1 21 < j j), the asymptotic two-user error 11 12 22 11 probability is larger than the single user asymptotic error rate: k = 0 in the summation yields the single user asymptotic error rate. As the terms for k > 0 are all positive, the two-user asymptotic error rate is larger than the single user’s rate. Consequently, the asymptotic efficiency of DPSK in GDRF channels will not be unity as it is in the case of coherent BPSK modulation [1, Proposition 2].
R
R R R
R
MEP
=
B K @2
2L 1
L
1
X
L
C aij A
i=1 8j2f1; :::; 2K g s:t: eij =1
and coincides with the near-far resistance. Proof: As we obtained an expression for the multiuser asymptotic BER for the MEP in the previous section (which easily specializes to the single-user channel), the asymptotic efficiency is straightforwardly calculated as given in the proposition. Furthermore, the multiuser asymptotic BER does not depend on the energies of the interfering users and hence the asymptotic efficiency coincides with the near-far-resistance. Example: Two-User Asymptotic Efficiency For two users the above expressions simplify to 1 X k ! L1 L 2 L 1 1 2 L k c^ MEP = 1 + 2U ; L 2 k=1 L k! (25) where we specified c^ in (23) in the example of the previous section, and we took out of the sum over k the term with k = 0 to demonstrate that the asymptotic efficiency is smaller than one. Note however that it is independent of the interfering users’ energies and always larger than zero (hence the detector in near-far resistant). V. N UMERICAL R ESULTS
c^ = ln
MEP PAs ;2U
Asymptotic efficiency is a performance measure that captures the performance degradation of a specific user due to interfering users in the limit of SNR ! 1 ( [1, 2]). Since for fast fading the BER reaches an error floor, the definition of asymptotic efficiency as given in [1, 2] cannot be applied to this case. Hence, we only give the asymptotic efficiency for slow fading and discuss a two user example. Proposition 4: The asymptotic efficiency of the multiuser MEP detector is 0 1 1
(22)
The four hypotheses one has to average over yield only two different values for aij and c^ij . Exploiting that cij = cji and writing the inverse in terms of blocks of , allows us to state the asymptotic probability with the single constant
Q
B.4 Asymptotic Efficiency and Near-Far Resistance
In this section we present numerical examples to illustrate our analytical findings. In Section V-A we consider slowly fading channels and in Section V-B fast fading channels. In both cases six users are considered who employ length-31 Gold-sequences. The signal correlation matrix is calculated as in [1] using the ISI mask described therein. For the decorrelating detectors the correlation matrix is the inverse of the upper left LL submatrix of the inverse of the multiuser correlation matrix . There are four paths for each user (hence is 24 dimensional). In the majority of our examples we will choose the four paths to fade independently but with unequal power distribution. The average relative power of each fading path is chosen according to the simplified GSM ‘urban’ test profile. For simplicity we choose the kk (0) equal for all users. Normalizing kk (0) properly leads to kk (0) = diagf0:22; 0:42; 0:26; 0:10g for all users. For the examples considering inter-diversity branch correlations, we calculate each kk from a Toeplitz matrix with Æ jk lj as its (k; l)th element. Normalization according to (4) leads to different kk in general, although the same value of Æ is chosen for each user.
R
R
R
BREHLER AND VARANASI: OPTIMUM MULTIUSER NONCOHERENT DPSK DETECTION IN GDRF CHANNELS
A. Slow Fading Figure 1 compares all of the presented detectors for slow fading and equal energies of all users (wk =w1 = 1). Applying the single-user MEP detector to the decorrelated multiuser channel as proposed in [3] gives an improvement of 5dB over the adhoc decorrelating equal gain combining (D-EGC) detector of [5, 6], and [7]. However, the multiuser MEP can improve significantly on the decorrelating approaches: in this example it out-performs the decorrelating single-user MEP (D-SU-MEP) detector by 6dB, and hence the D-EGC detector by 11dB. The upper bounds we obtained on the BER of the MEP detector converges to the lower bound, approaching the asymptote, as we found in our asymptotic analysis. The 2dB gap between the single-user and multiuser channel for MEP detection illustrates that the asymptotic efficiency of DPSK in GDRF channels is not equal to unity (cf. Section IV-B.4). The 3dB gap between optimum DPSK and optimum coherent BPSK modulation in singleuser GDRF channels can be observed in Figure 1, because the asymptotic efficiency of coherent MEP detection is unity. In Figure 2 the bounds are displayed for two fixed values of the first user’s SNR over varying interfering users energyratios. The lower bound almost collapses for low SNRs with the SU-MEP. The looseness of the upper bound for weak interfering users can be traced back to the derivation of the asymptotic error rates, where we made use of the fact that for 1 (recall that contains the power ratios of all users relative to the first). When the power of the interfering users tends to zero, this will be true for increasing values of only. Since the multiuser MEP detector collapses to the single user MEP detector when the energy of the interfering users tends to zero, one might argue that the lower bound captures the true BER on the MEP and the upper bound is loose, because multiple error events in the union bound become more likely for weak interfering users. For strong interfering users the upper bound on the MEP detector becomes tighter, as the multiple error events in the union bound become less likely. The asymptote on the BER is not affected by the varying interfering users’ energies, as expected from the analytical results. So far we have only considered independent fading examples. Figure 3 shows the BERs achieved by the different detectors for a fixed SNR of 20dB as function of Æ , the correlation within the diversity branches. (The same definition for the line-styles applies as in the previous figures. At 20dB the upper bound has already converged, it is not visible in the figure). Setting the fading correlation parameter Æ to zero corresponds to independent fading and Æ equal to one corresponds to fully correlated fading. For increasing Æ the performance of all detectors degrades. For higher SNRs the same gaps to the optimum detectors show up as in the case of independent fading.
I
For fast fading we have to choose the correlation matrix of the fading coefficients at time lag one, (1). As we have restricted the analysis to (1) = (0) (cf. Section IV-B), we will only consider numerical examples with such a (1). Recall that the fade rate is connected to the normalized Doppler bandwidth BD T by = J0 (BD T ) for fading with power spectrum den-
sity given by Jakes’s model [20]. BD T has to be multiplied with the factor 2c rfbc (c is the speed of light, rb = 1=T the bit rate, and fc the carrier frequency) to obtain a corresponding velocity. Figure 4 depicts the BERs of the MEP and D-EGC detectors for fast fading with fade rates = 0:9 and = 0:975. Note the flooring of the BERs. Moreover, not surprisingly, all detectors perform worse for the faster fading case and the BER floor is considerably higher. The multiuser MEP and D-SU-MEP reach the same error floor as the MEP in the single-user channel for both fade rates. This shows that for very high SNRs, fast fading is the limiting factor on detection performance. Whereas the BER floor reached by the D-EGC might be regarded as acceptable in comparison to the D-SU-MEP or the MEP, for practically relevant ranges of the SNR the performance of the D-EGC is not acceptable: For a medium SNR of 15dB the D-EGC has a dismal BER of 0.2 making it essentially useless. The BER for the D-SU-MEP can improve on this BER by a factor of ten and the multiuser MEP by a factor of 100. VI. C ONCLUSIONS In this work we present a systematic approach to the problem of optimum multiuser noncoherent DPSK detection in generalized diversity Rayleigh fading (GDRF) channels. It extends the optimum single-user approach of [3] to the multiuser channel. With the exception of [3], previous approaches to DPSK detection in diversity Rayleigh fading channels have so far been guided by the ad-hoc post-decorrelative equal-gain combining rule. [3] introduced the optimum detection approach for the single-user DPSK GDRF channel. Until now, when multiuser DPSK detection in GDRF channels has been considered, the single-user decision rules were applied to the decorrelated signals, i.e., in an effective single-user channel, suffering from noise-enhancement through the decorrelating operation. On the other hand, the multiuser MEP detector obtained here is the optimum strategy for the joint detection of all users in Rayleigh fading. It is the noncoherent maximal ratio combiner and requires knowledge about the fading statistics and the users’ SNRs. A rigorous analysis of its error probability is undertaken and a lower and upper bound on its BER are derived. For slow fading, when the fading coefficients are essentially constant over two successive symbol intervals, the upper bound converges asymptotically to the lower bound. Hence, we can obtain an asymptotic BER and the asymptotic efficiency and show that the latter is in general smaller than unity. Moreover, the asymptotic efficiency turns out to coincide with the near-far resistance proving the near-far resistance of the MEP detector for slow fading. For fast fading an upper and a lower bound on the bit error floor are obtained. A PPENDIX P ROPERTIES
B. Fast Fading
9
OF A
DATA -S YMBOL -W EIGHTED S UM M ATRICES
OF
T WO
MBi , MBj defined MM MMB1j . One easily finds MBj and subsequently MB1j through the ap-
In this appendix we present results for 1 in (16), and the matrices Bi and
plication of a formula for the inverse of a 2 2 block-partitioned matrix (cf. [21]), where four blocks of the 44 block-partitioned
10
ACCEPTED BY THE IEEE TRANSACTIONS ON INFORMATION THEORY
M
matrix Bj make up one of the can then calculate
2 2 blocks.
With these we
=
M 21 M24M441M41 M11 M14M441M41
1
;
MMB1i 24 = 1 M14 M44 M41 M 1 M14 1 ; MMB1j = = M M M 24 21 11 11 h i h i 3 2 1 = 1 1 MM I i MMBj 12 h 0 i MMBj 14 7 Bi 31 6 h 7 6 1 1 = M31 M34 M441 M41 M11 M14 M441 M41 1 ; I MM 0 7 6 MMBj B j 6 21 h 23 h i i 7; 1 = 7 6 1 1 MM B 0 MM I MM 7 6 i 34 Bj 32 h Bj 34 5 4 h i i 1 1 = M34 M31 M111 M14 M44 M41 M111 M14 1 ; MMBj 41 0 MMBj 43 I MMB1i 42 = where = M42 M43 M331 M32 M22 M23 M331 M32 1 ; h i MMB1j 12 = MMB1i 43 = = M43 M42 M221 M23 M33 M32 M221 M23 1 : = M12 M14 M441 M42 M22 M24 M441 M42 1 ; h i MMB1j 14 = Throughout this section we require of course that all the inverted matrices actually exist. If M is a positive definite correlation 1 = M14 M12 M221 M24 M44 M42 M221 M24 ; matrix this is always the case, as we invert diagonal blocks and h i 1 Schur complements, which are both positive definite for a posiMMBj 21 = M . tive definite matrix = M21 M23 M331 M31 M11 M13 M331 M31 1 ; h i R EFERENCES MMB1j 23 = [1] M. K. Varanasi, “Parallel group detection for synchronous CDMA communication over frequency-selective Rayleigh fading channels,” IEEE = M23 M21 M111 M13 M33 M31 M111 M13 1 ; Trans. Inform. Theory, vol. 43, no. 1, pp. 116–128, Jan. 1996. h i [2] S. Verd´u, Multiuser Detection, Cambridge Univ. Press, New York, NY, 1998. MMB1j 32 = [3] M. K. Varanasi, “A systematic approach to the design and analysis of optimum DPSK receivers for generalized diversity communications = M32 M34 M441 M42 M22 M24 M441 M42 1 ; over Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 9, h i pp. 1365–1375, Sept. 1999. 1 MMBj 34 = [4] J. G. Proakis, Digital Communications, McGraw-Hill, New York, 3rd edition, 1995. Zvonar and D. Brady, “Linear multipath-decorrelating receivers for = M34 M32 M221 M24 M44 M42 M221 M24 1 ; [5] Z.CDMA frequency-selective fading channels,” IEEE Trans. Commun., h i vol. 44, no. 6, pp. 650–653, June 1996. MMB1j 41 = [6] Z. Zvonar, “Multiuser detection and diversity combining for wireless 1 CDMA systems,” in Wireless and Mobile Communications, J. M. Holtz1 1 = M41 M43 M33 M31 M11 M13 M33 M31 ; man and D. J. Goodman, Eds. Kluwer Academic Press, 1994. h i [7] H. C. Huang and S. C. Schwartz, “A comparative analysis of linear mulMMB1j 43 = tiuser detectors for fading multipath channels,” in Proc. IEEE Global Telecommun. Conf., Dec. 1994, pp. 11–15. 1 Duel-Hallen and S. Andrijic, “DPSK diversity combining for CDMA = M43 M41 M111 M13 M33 M31 M111 M13 : [8] A. frequency selective fading channels,” in Proc. Asilomar Conference on One can also show that
MMB1i = 2 6 6 4
1 MM 1 I MM B Bi 13 0 1 i 12 1 MMBi 21 I 0 MMBi 24 MMB1i 31 0 I MMB1i 34 0 MMB1i 42 MMB1i 43 I
[9]
3 7 7; 5
[10] [11] [12] [13]
where
MMB1i 12 = = M12 M13 M331 M32 M22 M23 M331 M32 MMB1i 13 = = M13 M12 M221 M23 M33 M32 M221 M23 MMB1i 21 =
1 1
[14]
; ;
[15] [16]
Signals, Systems and Computers, Pacific Grove, CA, Nov. 1997. A. Abrardo, “Noncoherent MLSE detection of M -DPSK for DS-CDMA wireless systems,” IEEE Trans. Veh. Technol., vol. 50, no. 4, pp. 900–909, July 2001. G. N. Karystinos and D. A. Pados, “On DPSK demodulation of DS/CDMA signals,” in Proc. IEEE Global Telecommun. Conf., Rio de Janeiro, Brazil, Dec. 1999, vol. 5, pp. 2487–2492. M. K. Varanasi and B. Aazhang, “Optimally near-far resistant multiuser detection in differentially coherent synchronous channels,” IEEE Trans. Inform. Theory, vol. 37, no. 4, pp. 1006–1018, July 1991. M. K. Varanasi, “Noncoherent detection in asynchronous multiuser channels,” IEEE Trans. Inform. Theory, vol. 39, no. 1, pp. 157–176, Jan. 1993. C. J. Hegarty and B. Vojcic, “Two-stage multiuser detection for noncoherent CDMA,” in Proc. Allerton Conf. on Comm. Control, and Comput., Monticello, IL, Oct. 1995, pp. 1063–1072. C. J. Hegarty and B. Vojcic, “Two-stage multiuser detection for asynchronous CDMA using DPSK signalling,” in Proc. Comm. Th. Mini Conf., London, UK, 1996, IEEE GLOBECOM, pp. 132–136. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1994. M. K. Varanasi and M. Brehler, “A systematic approach to noncoherent detection for DPSK modulation in multiuser correlated diversity Rayleigh fading channels,” in Proc. Conf. Inform. Sciences and Systems. Princeton University, Mar. 1998, pp. 236–241.
[17] M. Brehler and M. K. Varanasi, “Asymptotic error probability analysis of quadratic receivers in Rayleigh fading channels with applications to a unified analysis of coherent and noncoherent space–time receivers,” IEEE Trans. Inform. Theory, vol. 47, no. 5, pp. 2383–2399, Sept. 2001. [18] E. Visotsky and U. Madhow, “Noncoherent multiuser detection for CDMA systems with nonlinear modulation: A non-Bayesian approach,” IEEE Trans. Inform. Theory, vol. 47, no. 4, pp. 1352–1367, May 2001. [19] A. Russ and M. K. Varanasi, “Noncoherent multiuser detection for nonlinear modulation over the Rayleigh fading channel,” IEEE Trans. Inform. Theory, vol. 47, no. 1, pp. 295–307, Jan. 2001. [20] W. C. Jakes, Microwave Mobile Communications, An IEEE Press Classic Reissue, New York, 1993, Originally An American Telephone and Telegraph Company Publication, 1974. [21] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1993.
Matthias Brehler (S’00–M’03) started his studies in electrical engineering at the Technische Universit¨at M¨unchen in 1992. From 1995 to 1996, he attended the University of Colorado, Boulder, where he received the M.S. degree in electrical engineering. In 1998 he received the Diplom–Ingenieur degree from the Technische Universit¨at M¨unchen. After completing his military service at the Universit¨at der Bundeswehr M¨unchen (Federal Armed Forces University, Munich), he returned to Boulder in 1999, where he obtained his Ph.D. in electrical engineering in Fall 2002 and is currently working as a Research Associate. His research interests include multiuser, multi-carrier, and space-time communications.
Mahesh K. Varanasi (S’87–M’89–SM’95) received the B.E. degree in electronics and communication engineering from Osmania University, Hyderabad, India, in 1984, and the M.S. and Ph.D. degrees in electrical engineering from Rice University, Houston, TX, in 1987 and 1989, respectively. In 1989, he joined the faculty of the University of Colorado at Boulder in the Electrical and Computer Engineering Department where he is now a Professor. His teaching interests include communication theory, information theory, and signal processing. His research interests include multiuser detection, space-time communications, equalization, signal design, diversity communications over fading channels, and power- and bandwidth-efficient multiuser communications.
Fig. 1. BER comparison of the decorrelating equal-gain combiner, postdecorrelative single-user MEP, multiuser MEP, single-user channel MEP, and optimum multiuser detectors. There is a 11dB gap between the D-EGC and the multiuser MEP detector.
Fig. 2. The upper and lower bound on the BER of the first user as a function of the relative energies of the other users.
Fig. 3. The BER of the first user for the different detectors as a function of the correlation within the diversity branches. The upper bounds converge to the lower and are hence not visible.
Fig. 4. All users have the same power. For faster fading, the performance of all detectors degrades, as expected.
0
10
−1
10
−2
10
−3
BER of user 1
10
−4
10
−5
slow fading, wk/w1=1
10
D−EGC D−SU−MEP MEP asymptote MEP upper bnd MEP lower bnd SU channel MEP optimum coherent
−6
10
−7
10
−8
10
−9
10
0
5
10
15 20 SNR (dB) of user 1
25
30
0
10
−1
10
−2
_ γ = 10dB
BER of user 1
10
slow fading MEP asymptote MEP upper bnd MEP lower bnd SU channel MEP
−3
10
−4
10
−5
10
_ γ = 20dB
−6
10
−2
10
−1
10
0
10 wk/w1
1
10
2
10
0
10
slow corr. fading, 20dB, wk/w1=1 −1
10
−2
BER of user 1
10
−3
10
−4
10
−5
10
−6
10
−7
10
0
0.1
0.2
0.3 0.4 0.5 Correlation parameter δ
0.6
0.7
0.8
0
10
−1
10
−2
BER of user 1
10
−3
10
ρ=0.9
−4
10
fast fading, w /w =1 k
−5
10
−6
10
1
D−EGC D−SU−MEP MEP upper bnd MEP lower bnd SU channel MEP
ρ=0.975
−7
10
0
10
20 30 SNR (dB) of user 1
40
50
