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Outage Probability for Optimum Combining of Arbitrarily Faded Signals in the Presence of Correlated Rayleigh Interferers Q. T. Zhang, Senior Member, IEEE, and X. W. Cui
Abstract—Exact outage-probability analysis for optimum combining of arbitrarily faded signals in the presence of correlated Rayleigh-faded interferers is not available in the literature. In this paper, we show that the conditional probability density of the reciprocal of the instantaneous signal-to-interference ratio (SIR), given the signal vector, can be represented as the higher order derivative of a simple exponential function in signal power whereby generic formulas for the outage probability and probability density function related to SIR can be determined. The new formulas take simple closed form in terms of the characteristic function of the signal vector. They are, therefore, widely applicable, leading to various results for correlated Rayleigh-, Rician-, and Nakagami-faded signals. Numerical examples are also presented for illustration. Index Terms—Correlated Nakagami fading, correlated Rayleigh fading, correlated Rician fading, optimum combining, outage probability.
I. INTRODUCTION
O
UTAGE probability is an important performance measure in the design of wireless communication systems to operate in a fading environment with cochannel interference. As an indication of a minimum requirement on the grade of service, outage probability represents the probability of unsatisfactory reception over the intended coverage area. To combat multipath fading, diversity reception is a commonly used and powerful means for enhancement of the system performance. Various techniques can be used for diversity combining. Among them, optimum combining has become a recent focus of many researchers due to its capability to exploit the channel-state information to achieve the maximum signal-to-interference ratio (SIR) at every instant of time. The outage probability for optimum combining was first analyzed in [10] under the assumption that both signal and interfering vectors follow the zero-mean complex Gaussian distribution with a common covariance matrix, so that the statistic can be used for the evaluation of outage Hotelling denote the -by-1 performance. To be precise, let and signal and th interfering vectors and let denote their Manuscript received September 8, 2003; revised February 4, 2004 and March 22, 2004. This work was supported by the Research Grants Council, Hong Kong Special Administrative Region, China under Project CityU 1248/02E. Q.T. Zhang is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected]). X. W. Cui is with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China, on leave at the City University of Hong Kong (e-mail:
[email protected]). Digital Object Identifier 10.1109/TVT.2004.830953
the identity matrix. The common covariance matrix and channel model treated in [10] can be more accurately described and , where notation by means that vector follows the -dimensional and covariance complex Gaussian distribution with mean allows the signal to matrix . The inclusion of constant have different power from the interference. We call this channel combination the correlated Rayleigh/correlated Rayleigh model for ease of illustration. Here and hereafter, signal and all interfering vectors are assumed, for mathematical tractability, to be mutually independent. A simpler scenario is addressed in and , [11], which assumes implying that all channel gains for both signal and interferers are subject to independent and identically distributed (i.i.d.) Rayleigh fading and, thus, the term the “i.i.d. Rayleigh/i.i.d. Rayleigh model.” Chayawan et al. [5] extend the methodology of Shah et al. to the i.i.d. Rician/i.i.d. Rayleigh model, viz. and with denoting the line-ofsight (LoS) vector. It should be emphasized that the term i.i.d. is used here to mean that the entries of a random vector are i.i.d. variables. The i.i.d. assumption allows for invoking a theorem in multivariate analysis [12] to assert that the SIR follows the distribution, whereby the outage performance noncentral can be determined. The concept of optimum combining can be extended to a wireless multiple-input–multiple-output (MIMO) system to exploit diversity at both the transmitter and receiver. Its optimal weighting coefficients are derived in [9]. Kang and Alouini [6] analyze the outage performance of such an optimal MIMO combiner by assuming that the desired station transmits its signal by using an antenna array, whereas each interferer employs a single transmit antenna. The channel assumption is the correlated Rayleigh/correlated Rayleigh model, one that is similar to that used in [10], such that a lemma of Khartri [8] can be employed as a basis for analysis. In [7], Kang and Alouini further extend their results to the optimal MIMO combiner operating on i.i.d. Rician/i.i.d. Rayleigh channels, for which the solution can be obtained by invoking another result of Khartri [8]. So far, nearly all research work on the outage performance analysis for optimum combining imposes a common restriction on fading channels, requiring that both signal and interfering vectors are complex Gaussian distributed and have an identical covariance structure. A more restricted assumption is required when the signal suffers from Rician fading; namely, all channel gains for both signal and interferers must be i.i.d. The assumptions of this kind, however, may not be valid in practice. In fact,
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the desired user differs from its cochannel interferers in the distance to the base station, in propagation conditions, and, thus, in the channel models. Generally speaking, the fading channels for the desired user differ from their counterparts for the interferers in the sense that they may follow different distributions and possess different covariance structures. The difficulty in handling such a general situation lies in the lack of directly useful results in the mathematical literature. In this paper, we tackle the general issue of outage performance analysis of optimum combining in the presence of correlated Rayleigh-faded interferers. The signal vector is allowed to follow an arbitrary fading distribution with a covariance structure different from its interference counterpart. The problem is formulated in Section II and the general expression for outage probability is derived in Section III. The general formula is then applied to signals subject to Rayleigh, Rician, and Nakagami fading, leading to particular results, as shown in Sections IV–VI. Illustrating results are presented in Section VII, followed by concluding remarks in Section VIII. II. FORMULATION Consider a wireless communication environment in which the base station employs an array of antennas for the reception of cochannel interferers of equal a desired signal corrupted by power. The received signal at the antenna array includes the desired signal, interference signals, and noise, given by (1) where and denote the channel complex gain vectors for desired user and for the th interferer, respectively. is the zero-mean additive white Gaussian noise vector with distribu. and represent the desired and th tion interfering complex transmit symbols using -ary phase-shift , . Throughout keying (PSK) so that this paper, we assume that and have been normalized so that their entries, on average, have unit power. The influence of the and , so that and path loss has been absorbed into represent the average received signal and interference power, respectively. A linear combining vector is used in the receiver to produce output (2) where, and afterward, we use superscripts and to denote the transposition and conjugate transposition, respectively. It is well known that the conditional signal-to-interference-plusnoise ratio (SINR) can be maximized to give (3) . The optimal combiner is where easily obtainable by using the principle of the Rayleigh quotient, . given by
III. METHODOLOGY AND GENERIC FORMULAS For mathematical tractability, we assume that the number of interferers is no less than the number of receive antennas, i.e., . Throughout this paper, it is assumed that the system performance is interference limited and thermal noise is negligible, so that we can rewrite the SIR as
(4) with
denoting (5)
We further assume that the interfering vectors are subject to Rayleigh fading and that each follows the zero-mean complex Gaussian distribution with covariance matrix ; namely, . For a system operating in a multiuser environment, outage probability is one of its important performance indices for the quality of service. Outage occurs whenever the SIR is less than a preset power protection ratio, say . Thus, the outage prob. To determine ability can be written as the outage probability, we need to find the probability density function (pdf) of . However, directly determining the pdf of is very difficult and, hence, we instead consider the reciprocal of the SIR , which is defined by (6) Hence, by denoting (7) we can reformulate the outage probability as (8) The pdf of is relatively easy to handle. Following the independence assumption on the interfering vectors, we can assert that is a complex Wishart distributed having the pdf [1], [2] (9) which can be simply denoted by CW . We can easily extend the result for real data by Muirhead [2, p. 95] to its complex counterpart to assert that given signal vector , also follows the complex Wishart distribution, as shown by CW . Note that the Wishart distribution CW reduces to a gamma distribution whose pdf can be expressed explicitly. Specifically, we can write (10)
ZHANG AND CUI: OUTAGE PROBABILITY FOR OPTIMUM COMBINING OF ARBITRARILY FADED SIGNALS
which involves both exponential and polynomial factors in , making it difficult to proceed further. We therefore rewrite it in the form of differentiation, producing (11)
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formance evaluation. However, we do not pursue this issue here due to space limitations. Equation (16) reveals that when operating in a Rayleigh/Rayleigh environment with equal covariance matrix, the pdf of is independent of the common covariance into the general formula matrix. We insert (14) to obtain the outage probability
where and
(12)
We next need to average (11) over signal vector . Note that (11) depends on only through the exponential function and the expected value of simply equals the characteristic function (CHF) of with transfer variable . It follows that (13)
(17)
B. Unequal Covariance Structure When values of
where . Having obtained the pdf of , we are able to obtain the outage probability
, let . The CHF of
denote the eigennow becomes (18)
which, when inserted into (13), allows us to reexpress the pdf of as (19) (14) This is a very simple formula for optimum combining of arbitrarily faded signals in the presence of correlated Rayleighfaded interferers. The crucial step to make this simple expression available is representing the pdf of as the derivative of an exponential function, as shown in (11). The outage probability in (14) depends on the faded signal vector only through the CHF of and, hence, it is applicable to various faded signals and allows their branch correlation to be arbitrary, as long as their characteristic functions exist. IV. RAYLEIGH-FADED SIGNALS For a general Rayleigh-faded signal, we can assume that . The CHF of is, thus, given by
The right-hand side contains a higher order derivative of the product of rational functions. We will use the following reare sult for its simplification. Suppose that differentiable functions of . Then, we have (20)
where and
denotes the th derivative of with respect to is an -tuple such that with signifying the set of nonnegative integers. The proof of (20) is given in Appendix I. We apply (20) to (19) to obtain (21)
(15) There are two cases; namely, and . We will analyze them separately, since their treatment is different.
whereas, by inserting (18) into (14) and using (20) to simplify, it turns out that
A. Equal Covariance Structure . We simplify the CHF of as In this case, , which, when inserted into (13), produces the pdf of (22)
(16) It characterizes the reciprocal of the SIR at the optimum combiner’s output. The pdf so obtained can be used for the error-per-
This is the formula for the correlated Rayleigh/correlated Rayleigh model with unequal correlation structure. In the calculation of outage probability, we need to find out all possible -tuples, . For programming, one can certainly get this done by using do-loops. The drawback is its computational complexity and, furthermore, one needs to modify
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the program whenever is changed. We therefore propose a to meet the constraint that more efficient approach. Finding is equivalent to partitioning a set of balls into groups (a group can be empty). To this end, we put the balls on a line and partition them with partitioning positions and we have boards. Hence, there are in total ways for partitioning. If we label the positions for a ball and a partitioning board with 0 and 1, respectively, we can use the binary expression of an integer to fulfil the task, as outlined as follows.
Directly determining the higher order derivatives of ficult. Therefore, we consider its logarithm
(27) instead and find that (28) A similar technique was used in [20], but in different context. We are now in a position to use Faa di Bruno’s formula for differentiation of a composite function [19], [20] to obtain the th derivative of . It turns out that
For n = 0 : 2k+L01 0 1 convert n into a binary expression (vector), denoted by v . if sum(v) = L 0 1 then, this is one partition we need. We count the number of zeros (i.e., the run of zeros) before each and after the last partitioning board (i.e., each one) to obtain the values of t1 ; . . . ; tL . else if sum(v ) 6= L 0 1 continue end end
is dif-
(29) where
is an
-tuple defined by
. , the -tuple has one more conCompared to straint, i.e., . Thus, the technique for finding can be built on the top of that for . One can first generate the corresponding , which is then checked with the extra condition. Combining (29), (26), and (25) allows us to express the pdf of explicitly as
(30) V. RICIAN-FADED SIGNALS For a general Rician-faded signal, we assume that , for which the CHF of can be expressed explicitly as
We insert (24) into the general formula (14) and use (26) and (28) to simplify, yielding
(23) Just as in the case of Rayleigh fading, there are two cases for Rician fading, which need different treatment. A. Equal Covariance Structure In the first case, the signal and interfering vectors have an identical covariance structure. Namely, . Since the entries of and have unit average power, the scaling factor must meet the following that . This condition is useful in setting a SIR in numerical calculation. By denoting , we can rewrite (23) as (24) which, when inserted into (16), gives us
(31) Unlike the case for the Rayleigh/Rayleigh model with equal covariance matrix, the pdf of and outage probability for the Rician-faded signal corrupted by Rayleigh-faded interferers are influenced by the common covariance matrix through parameter . B. Unequal Covariance Structure
(25)
We next turn to the case with unequal correlation matrix for which and, hence, the CHF of can be rewritten as
Before proceeding further, let us denote (26)
(32)
ZHANG AND CUI: OUTAGE PROBABILITY FOR OPTIMUM COMBINING OF ARBITRARILY FADED SIGNALS
Fig. 1. Outage probability versus SIR: Rayleigh-faded signals with
R =R .
Fig. 2. Outage probability versus SIR: Rayleigh-faded signals with
R =6 R .
where
. Eigendecompose such that is the matrix of eigenvectors of and is the diagonal matrix of its eigenvalues. By , the th entry of by and , we can simplify (32), ending up with
, where denoting
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(33) Proceeding in a way similar to the derivation of (30), we obtain the pdf of as
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Fig. 3.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 4, JULY 2004
Variation of outage probability with various signal correlation. Rician-faded signals with
Fig. 4. Variation of outage probability with Rician factor. Rician-faded signals with
R
R
of the same structure as
of the same structure as
R.
R.
(34) By substituting (33) into (14), it follows that
(35) which is the formula for the correlated Rician/correlated Rayleigh model with unequal covariance matrices.
ZHANG AND CUI: OUTAGE PROBABILITY FOR OPTIMUM COMBINING OF ARBITRARILY FADED SIGNALS
Fig. 5. Variation of outage probability with various combinations of signal/interference correlation. Rician-faded signals with
VI. NAKAGAMI-FADED SIGNALS For Nakagami-faded signals, we only consider the case of independent Rayleigh-faded interference, i.e., . We assume that the desired signal is subject to Nakagami fading with power covariance matrix and fading parameter . is the covariance of the power vector , where and is the th entry of . The CHF of is then given by (36) where the entry of is related to the th entry of by . Let denote the eigenvalues of . With these notations, the CHF of can be rewritten as (37) Inserting it into (13) and (14) and simplifying it results in
(38) and
(39)
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R
structure different from
R.
is the gamma function. This is the respectively, where outage-probability formula for the correlated Nakagami/uncorrelated Rayleigh model. VII. NUMERICAL RESULTS To examine our theoretical results, let us consider the orderoptimal combiner. Suppose the correlation between antennas and is dictated by the exponential model [21], [22] (40) where denotes the absolute value of correlation coefficient between two adjacent antennas and stands for the antenna separation normalized by the wavelength. The -bycovariance matrix is simply denoted by . In this section, the power-protection ratio is set to be 12 dB, which is roughly the same as that used in the global system for mobile communications (GSM) system. In all the numerical examples, we assume that the interfering vectors are subject to Rayleigh fading. Let us first consider the case where both desired signal and interfering vectors suffer from Rayleigh fading with equal antenna branch correlation; namely, , whose th entry is given by . Fig. 1 shows two sets of curves for and , respectively. Each set consists of three curves representing the situations with 4, 6, and 8 interferers, respectively. As expected, for a given , the outage performance improves with increased diversity order . For a given diversity order, on the other hand, increasing the number of interferers will degrade the outage performance. Let us consider again the case of Rayleigh fading in which, however, the signal and interference have different branch correlations. To see the way the covariance matrices affect the outage probability, we set and . The results for three settings, , and
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Fig. 6. Outage probability versus SIR for Nakagami-faded signals.
, are shown in Fig. 2, where . It is observed that the outage performance improves with decreased signal correlation, but degrades as the interference correlation decreases. Less correlation in the signal vector implies more information that is available to the optimum combiner and, thus, the better performance. A reverse explanation, however, is applied to the interference. We next turn to the case for Rician-faded signal. We assume that the signal vector follows the -dimensional Gaussian distribution with mean and covariance matrix . Here, the Rician factor is defined as the power ratio of the LoS component to the multipath component. Vector with denoting the all-one column vector. Variation of the outage probability with for is shown in Fig. 3, where and . It is observed again that the smaller the signal correlation, the better the outage performance. We are interested in the influence of the Rician factor on the outage probability and the results are plotted in Fig. 4. As expected, the larger the , the better the outage performance. We further study the case where signal and interference have different covariance structures, ending up with results shown in Fig. 5, where and . We can make similar observations to that for a Rayleigh-faded signal. Finally, we study the case with a Nakagami-faded signal with a particular focus on the influence of signal correlation on the outage performance. The results are graphed in Fig. 6 with and fading parameter . Note that, for Nakagami fading, the covariance matrix is given for the signal power. Again, we find that the less signal correlation leads to a better performance.
in the presence of correlated Rayleigh-faded interferers. We first determined the conditional pdf of the reciprocal of the instantaneous SIR, assuming that the signal gain vector is given, ending up with the complex Wishart distribution with a dimension of one. We then represented this conditional pdf in terms of the higher order derivative of an exponential function, which enables us to obtain a generic but simple expression for the outage probability. Finally, we applied the generic formula to various faded signals, leading to various results for Rayleigh-, Rician-, and Nakagami-faded signals. The new formulas are illustrated by diverse numerical examples.
APPENDIX PROOF OF (20) We use the method of induction. Let us first examine the case of the product of two functions; namely, . According to the Leibnitz’s rule, the th derivative of the product of two functions is given by [18]
(41) which is indeed in the form of (20). In the second step, we assume that the rule given in (20) is true for the product of functions, viz
VIII. CONCLUSION In this paper, we derived the outage performance of the optimum combiner for the detection of an arbitrarily faded signal
(42)
ZHANG AND CUI: OUTAGE PROBABILITY FOR OPTIMUM COMBINING OF ARBITRARILY FADED SIGNALS
and need to check the case of the product of straightforward to show that
functions. It is
(43)
(44) (45) where (43) follows from (41) and (44) follows from (42). This last line is exactly what we need and thus completes the proof. REFERENCES [1] A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Math. Stat., vol. 35, pp. 475–501, 1964. [2] R. J. Muirhead, Aspects of Multivariate Statistical Analysis. New York: Wiley, 1982. [3] M. Nakagami, “The -distribution: A general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, Ed. New York: Pergamon, 1960. [4] W. Braun and U. Dersch, “Physical mobile radio channel model,” IEEE Trans. Veh. Technol., vol. 40, pp. 472–482, May 1991. [5] C. Chayawan and A. Aalo, “On the outage probability of optium combining and maximum ratio combining scherness in an interference-limited Rece fading channel,” IEEE Trans. Commun., vol. 50, pp. 532–535, Apr. 2002. [6] M. Kang and M.-S. Alouini, “Performance analysis of MIMO systems with cochannel interference over Rayleigh fading channels,” in Proc. IEEE Int. Conf. Communications (ICC’02), vol. 1, May 2002, pp. 391–395. [7] , “Performance analysis MIMO MRC systems over Rician channels,” in Proc. 56th IEEE Int. Vehicular Technology Conf. (VTC’02), vol. 2, Sept. 2002, pp. 869–873. [8] C. G. Khartri, “Distribution of the largest or smallest characteristic root under null hypothesis concerning complex multivariate normal populations,” Ann. Math. Stat., vol. 35, pp. 1807–1810, 1964. [9] K.-K. Wong, R. S. K. Cheng, K. B. Letaief, and R. D. Rurch, “Adaptive antennas at mobile and base stations in an OFDM/TDMA system,” IEEE Trans. Commun., vol. 49, pp. 195–206, Jan. 2001. [10] A. Shah and A. M. Haimovich, “Performance analysis of optimum combining in wireless communications with Rayleigh fading and cochannel interference,” IEEE Trans. Commun., vol. 46, pp. 473–479, Apr. 1998. [11] , “Performance analysis of maximal ratio combining and comparison with optimum combining for mobile radio communications with cochannel interference,” IEEE Trans. Veh. Technol., vol. 49, pp. 1454–1463, July 2000. [12] R. J. Muirhead, Aspects of Multivariate Statistical Aspects. New York: Wiley, 1982.
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[13] Q. T. Zhang, “Exact analysis of postdetection combining for DPSK and NFSK systems over arbitrarily correlated Nakagami channels,” IEEE Trans. Commun., vol. 46, pp. 1459–1467, Nov. 1998. , “Maximal ratio combining over Nakagami fading channels with [14] arbitrary branch covariance matrix,” IEEE Trans. Veh. Technol., vol. 48, pp. 1141–1150, July 1999. [15] M. Z. Win, G. Chrisikos, and J. H. Winters, “MRC performance for M-ary modulation in arbitrarily correlated Nakagami fading channels,” IEEE Commun. Lett., vol. 4, pp. 301–303, Oct. 2000. [16] M. Z. Win and J. H. Winters, “On maximal ratio combining in correlated Nakagami with unequal fading parameters and SNR’s among branches: An analytical framework,” in Proc. IEEE Wireless Communications Networking Conf., vol. 3, Sept. 1999, pp. 1058–1064. , “Exact error probability expressions for MRC correlated Nak[17] agami channels with unequal fading parameters and branches powers,” in Proc. IEEE Global Telecomm. Conf., Symp. Communication Theory, vol. 1, Dec. 1999, pp. 2331–2335. [18] W. H. Beyer, CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC, 1987. [19] C.-J. de la Vallee Poussin, Cours D’Analyze Infinitesimale, 12th ed. Paris, France: Gauthier-Villars, Libraire Univ. Louvain, 1959, vol. 1. [20] M. Z. Win and R. K. Mallik, “Error analysis of noncoherent -ary FSK with postdetection EGC over correlated Nakagami and Rician channels,” IEEE Trans. Commun., vol. 50, pp. 378–383, Mar. 2002. [21] M. Chiani, M. Z. Win, and A. Zanella, “On the capacity of spatially correlated MIMO Rayleigh-fading channels,” IEEE Trans. Inform. Theory, vol. 49, pp. 2363–2371, Oct. 2003. [22] S. L. Loyka, “Channel capacity of MIMO architecture using the exponential correlation matrix,” IEEE Commun. Lett., vol. 5, pp. 369–371, Sept. 2001.
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Q. T. Zhang (S’84–M’85–SM’95) received the B.Eng. degree from Tsinghua University, Beijing, China, and the M.Eng. degree from South China University of Technology, Guangzhou, China, both in wireless communications, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada. He held a research position and was Adjunct Assistant Professor at McMaster University after his graduation. In January 1992, he joined the Satellite and Communication Systems Division, Spar Aerospace Ltd., Montreal, PQ, Canada, as a Senior Member of Technical Staff. At Spar Aerospace, he participated in the development and manufacturing of the Radar Satellite (Radarsat). He was subsequently engaged in the development of the advanced satellite communication systems for the next generation. He joined Ryerson Polytechnic University, Toronto, ON, Canada, in 1993 and became a Full Professor in 1999. In 1999, he took one-year sabbatical leave at the National University of Singapore and is now with the City University of Hong Kong, Kowloon, Hong Kong. His research interests include transmission and reception over fading channels with current focus on wireless multiple-input–mulitpleoutput (MIMO) and ultrawide-band (UWB) systems. Dr. Zhang is an Associate Editor for the IEEE COMMUNICATION LETTERS.
X. W. Cui received the B.S. degree in 1999 from Tsinghua University, Beijing, China, where is currently working toward the Ph.D. degree. His research interests include information theory and statistical signal processing for wireless communication.
