Sum and Difference of Two Squared Correlated ... - Semantic Scholar
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
1367
Sum and Difference of Two Squared Correlated Nakagami Variates in Connection With the McKay Distribution Henrik Holm, Member, IEEE, and Mohamed-Slim Alouini, Senior Member, IEEE
Abstract—General formulas for the probability density function of the sum and the difference of two correlated, not necessarily identically distributed, squared Nakagami variates (or equivalently, gamma variates) are derived. These expressions are shown to be in the form of the McKay “Bessel function” distributions. In addition, formulas for the moments of these distributions, in terms of the Gauss hypergeometric function, are provided. An application of these new results relevant to the calculation of outage probability in the presence of self-interference is discussed. Index Terms—Correlated fading, gamma distribution, McKay distribution, Nakagami fading, outage probability.
I. INTRODUCTION
T
HE wide versatility, computational tractability, and experimental consistency [1] of the Nakagami- distribution [2] has made it popular as a fading model in the performance analysis of wireless systems (see, for example, [3]–[9]). One of the important features of the distribution is that the power of a signal perturbed by Nakagami fading is gamma distributed. Thus, in performance evaluation involving Nakagami fading, one can often rely on established results (in the statistics literature) about the gamma distribution. An important special case of the Nakagami distribution is the Rayleigh distribution, which arises in the situation of multipath transmission with no direct component, i.e., when all of the received power stems from scattered components. The corresponding distribution for the signal power is the exponential distribution. Another often-employed distribution when modeling wireless channels is the Rice distribution, which considers the multipath signal as a sum of a direct component and a reflected component. When the magnitude of the direct component is zero, the Rice also reduces to the Rayleigh distribution. Although the Nakagami- and the Rice
Paper approved by V. A. Aalo, the Editor for Diversity and Fading Channel Theory of the IEEE Communications Society. Manuscript received November 5, 2001; revised April 9, 2003 and January 27, 2004. This work was supported in part by the Research Council of Norway under Grant 11939/431, and in part by the National Science Foundation under Grants CCR-9983462 and ECS9979443. This paper was presented in part at the 35th Annual Conference on Information Sciences and Systems (CISS’01), Baltimore, MD, March 2001. H. Holm is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA, on a visit from the Department of Electronics and Telecommunications, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (e-mail: [email protected]). M.-S. Alouini is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.833019
distributions are not equal, except in the common Rayleigh special case, their probability density functions (PDFs) can be close parameter and when a proper mapping is used between the the Rice factor [2, eqs. (55) and (56)]. However, an advantage of the Nakagami- over the Rice model is the relatively simple expression for its PDF and joint PDF for the signal power. Indeed, while the gamma PDF contains no complicated special functions, the signal power in the Rician case is governed by a noncentral chi-square distribution with two degrees of freedom, which contains a modified Bessel function. On the other hand, the bivariate distribution of two squared correlated Nakagamivariates (or equivalently, two correlated gamma variates) has a relatively simple closed-form expression, while (to the best of our knowledge) no simple formula is yet known for the joint PDF of two correlated noncentral chi-square variates. Many of the performance-analysis problems in wireless communication systems over Nakagami- channels require determination of the statistics of functions of the squared envelope of Nakagami- faded signals. In his classical 1960 paper [2], relying on a series of papers [10]–[12] appearing in Japanese journals, Nakagami cites expressions for the distribution of the sum, the ratio, and the product of squares of two correlated Nakagami- variates (or equivalently, sum, ratio, and product of two correlated gamma variates). However, no expression for the PDF of the difference between two gamma variates is provided. This PDF can not be derived from the PDF of a sum, and it has a slightly different form, since the sum of gamma variates (always positive) naturally is positive, while the difference can clearly be negative. Apparently, even in the specialized statistics literature, only the difference between two uncorrelated gamma variates has been investigated [13]–[17]. One of the contributions of this paper is to show that starting from the bivariate Gamma distribution, we are able to derive a closed-form expression for the PDF, moments, and in certain particular cases, the cumulative distribution function (CDF) of the difference between two correlated, not necessarily identically distributed, gamma-distributed variates. Much effort has been made in finding general statistical distributions, or systems of distributions, for the purpose of fitting sample data to a distribution. The Pearson system [14], [17] is, for example, one of the most general systems of distributions. As an addition to the Pearson system, McKay [13] provides what he calls “a Bessel function distribution,” which is a distribution with particularly simple expressions for the moment-generating function (MGF) and for the cumulants. This distribution splits naturally in two forms, depending on the value of one of the
0090-6778/04$20.00 © 2004 IEEE
1368
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
parameters, where one of them describes a variate that can be only positive, while the other describes a variate that can take on all values. The two forms of this distribution will together be referred to as “the McKay distributions.” Since the sum and difference of two uncorrelated gamma-distributed variates are governed by these two forms, respectively [17], they will be of particular interest in the context of this paper. In particular, another contribution of this paper is to show that the sum and difference of correlated gamma variates are also governed by McKay’s distributions. The final contribution of this paper is an application of the mathematical results to the problem of self-interference, namely, derivation of the CDF of outage probabilities in a self-interference scenario. The paper is organized as follows. Section II defines the two McKay distributions. Expressions for the moments of these distributions are stated as theorems, as are the PDFs of the sum and difference between two uncorrelated gamma variates. Lastly, an important special case, relevant to Rayleigh fading, is investigated. Section III introduces the bivariate gamma distribution and derives the distributions of the sum and difference of correlated gamma variates. The application of this new result, relevant to the calculation of outage probability in the presence of self-interference, is presented in Section IV. Finally, a conclusion summarizing the main results is given in Section V.
is not defined when the argument is equal to zero. Howwhich is valid ever, with the aid of an approximation of , we provide an expression for the type for small and for II McKay distribution for . , and and are as in Definition 2 Proposition 1: When (3) Proof: Use the formula from [19, eq. (9.6.9)] to find the limit of (2). when We have not been able to find an expression for , but plotting of (2) indicates that the PDF approaches a Dirac “ ” distribution when approaches from above. A. Moments McKay provides expressions for the MGF and for the cumulants of his distribution functions, [13, eqs. (10) and (19)], respectively. In addition, it is also possible to find general closed-form expressions for the moments of the McKay distributions, as we show now. Theorem 1 (Moments of McKay’s Type I Distribution): Let be distributed according to the type I McKay distribution (Definition 1). Then the moments of can be expressed as
II. MCKAY’S BESSEL FUNCTION DISTRIBUTION In this section, we define McKay’s distributions and provide expressions for the moments of these distributions. Next, we remind the reader about the basic definition of the gamma distribution and state some previously known results regarding the sum and difference between independent gamma variates. Lastly, an important special case is considered. McKay defines his distribution in terms of one function with different forms due to different values of one of the parameters, namely, the parameter. However, we find it more convenient to treat the two forms as separate distributions; called and , respectively. follows the type Definition 1 (Type I McKay Distribution): , , and I McKay distribution with parameters 1 when the PDF of is given by
(4) is Gauss’ hypergeometric function where [18, Sec. 9.1]. Proof: See Appendix I. As indicated, the previous theorem applies solely to McKay’s . For , distribution of the first kind, i.e., when the type II McKay distribution is the appropriate distribution function, and then the moments must be expressed as in the following theorem. Theorem 2 (Moments of McKay’s Type II Distribution): Let follow the type II McKay distribution (Definition 2). Then the moments of can be expressed as
(1) where is the gamma function, is the modified Bessel function of the first kind and of order , and is the Heaviside unit step function [18]. follows the Definition 2 (Type II McKay Distribution): , , type II McKay distribution with parameters and when the PDF of is given by (5) (2) where is the modified Bessel function of the second kind and of order .
0
1The third parameter to the first McKay distribution can also be c < 1. Then, the distribution function is defined for negative, instead of positive, .
Proof: See Appendix II. The hypergeometric function is commonly available in numerical software, so the moments can be calculated easily. Expressions for the moments can also be found by using multiple derivatives of the MGF. In addition, it is always possible to find an expression for the th moment in terms of the first
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
cumulants [20, Ch. 3], and vice versa. As an example, using [13, eq. (19)] gives the following expression for the mean and the variance of the first McKay distribution:
(6) . Hence, the second moment can be found as Before concluding this section, we mention that in the special , the formulas case that arises when is restricted to (4) and (5) for types I and II, respectively, become particularly simple, as shown in Appendix III.
1369
Equation (9) can be solved after inserting expressions for the PDFs, (7). Comparison with (1) after insertion of parameters from (8) concludes the proof. This result is also provided without proof in the classical textbook by Johnson et al. [17, Sec. 12-4.4], but we could not trace where the result was originally derived and presented. Theorem 4 (Difference Between Two Independent Gamma , be mutually independent random variVariates): Let and , reables, distributed as is distributed according to the spectively. Then second of McKay’s forms with parameters
B. Sum and Difference of Two Independent Gamma Variates Definition 3 (Gamma Distribution): follows a gamma distribution with shape parameter and scale parameter when the PDF of is given by (7) to denote that We use the shorthand notation follows a gamma distribution with shape parameter and scale parameter . As mentioned before, the gamma distribution is related to the Nakagami distribution. Namely, when a signal is Nakagami distributed with Nakagami fading parameter and average fading power , then the power (i.e., the square) of the signal is Gamma and scale parameter distributed with shape parameter . Theorem 3 (Sum of Two Independent Gamma Variates): Let , be mutually independent random variables, distributed and , respectively. Then as is distributed according to the first of McKay’s forms with parameters
(8) and , so the restrictions that applies to the Thus, McKay distribution I are fulfilled. Proof: The PDF of the sum of two independent variates is [21, Sec. 6-2] (9) By combining the integration formulas [18, eqs. 3.388-1 and 3.383-2] to yield the following formula:
(10)
(11) where we see that the requirements and are met, and are positive. since Proof: The PDF of the difference between two indepencan be derived from [21, Sec. 6-2], dent variates and it is expressed as follows: (12) Two integration formulas [18, Eqs. 3.383-3 and 3.383-8] can be combined, and together they produce the following rule:
(13) After inserting the expressions for the individual PDFs, (7) and (12) can be solved using (13). Insertion of parameters from (11) and subsequent comparison with (2) concludes the proof. As is the case for Theorem 3, this result is provided without proof in the classical textbook by Johnson et al. [17, Sec. 12-4.4], and again, we were not able to trace where the result was originally derived and presented. C. Special Case of In the important special case of Rayleigh fading, the previous PDF and moment expressions can be used by simply setting . As can be seen from (8) and (11), this means . that the parameter in the McKay distributions is It turns out that this restriction leads to particularly simple expressions for both the PDFs and the moments, involving only elementary functions (i.e., no special functions like the hypergeometric function or modified Bessel functions.) It is also possible to find simple closed-form expressions for the CDFs. ): When , the Corollary 1 (McKay Type I, type I McKay distribution reduces to the following expression: (14)
1370
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
Proof: Insert
in (1), use the identity from [19, eq. (10.2.13)], and collect terms. Equation (14) is recognized as the PDF of the sum of two correlated exponential variates with the same scale factor . This and and can be seen by setting comparing with [22, eq. (31)] or [23, Sec. 10.6]. (Note that the definition of in these references is different from the case we are studying here.). Like Definition 1, Definition 2 can be simplified when . The following corollary is a counterpart to Corollary 1. ): When , the Corollary 2 (McKay Type II, type II McKay distribution reduces to the following expression:
, scale parameters common shape parameter , , respectively, and correlation coefficient if their joint PDF is given by
(19)
(15) (equivalent to ) in (2), use the Proof: Insert from [19, eq. (10.2.17)], identity and collect terms. The hypergeometric functions appearing in (4) and (5) are , as is displayed in the also possible to simplify when following corollary. ): Let . Then Corollary 3 (Moments When the expression for the moments of both McKay distributions can be written as follows:
, to We use the shorthand notation denote that and follow a bivariate gamma distribution. , the joint PDF reduces to the product of two uniWhen variate gamma PDFs. A. Sum of Two Correlated Gamma Variates Theorem 5 (Sum of Two Correlated Gamma Variates): Let , . The sum follows the type I McKay distribution with parameters
(16) Proof: See Appendix III. Because of the simple PDFs when , it is possible to find closed-form expressions for the CDFs of both McKay distributions. , and Corollary 4 (CDF of McKay Type I): When , the CDF of the McKay type when restricting, as before, I distribution can be written as follows:
(17) Proof: Integrate (14) from to and collect terms. Corollary 5 (CDF of McKay Type II): When , and , the CDF of the McKay type II distriwhen restricting bution can be written as follows: for for
.
(20)
where, since , the restrictions and are met. Proof: See Appendix IV. Thus, Nakagami and Nishio’s result [2], [10] is, in fact, on the McKay form. Hence, McKay’s first form, which is known to be valid for the sum of independent gamma-distributed variates, is, in fact, also valid for the sum of correlated ones. Corollary 6 (Sum of Two Correlated Exponential Vari, . The sum ates): Let has a PDF as follows:
(18)
, integrate (15) from to and Proof: For collect terms. For , find by integrating (15) to and collecting terms. from (21) III. SUM AND DIFFERENCE OF CORRELATED GAMMA VARIATES When the gamma-distributed variates are correlated, the bivariate gamma distribution, which takes into account the correlation, must be employed. and Definition 4 (Bivariate Gamma Distribution): are governed by a bivariate gamma distribution with
Proof: Insert values for and from (20) in (14), and collect terms. Of course, the previous corollary also applies to the uncorrelated case by inserting corresponding values for and (from in (21). (8)) in (14), or by inserting
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
1371
B. Difference Between Two Correlated Gamma Variates Theorem 6 (Difference Between Two Correlated Gamma , . The difference Variates): Let follows the PDF given by
for (22a) When
, there exists an expression for the special case (22b)
Proof: See Appendix V. Corollary 7 (Relation to McKay’s Distribution Function): The difference between two correlated gamma variates is distributed according to the type II McKay distribution with parameters
(23) and are met, as long as where the conditions . Proof: Can be readily seen by inserting the parameters , , and from (23) in (2). Corollary 8 (Difference Between Two Correlated Exponential Variates): This corollary is a counterpart to Corollary 6. , . The difference Let has a PDF as follows:
Fig. 1. Comparison between the analytical PDF and the PDF obtained via Monte Carlo simulation, for = 2, = 20, = 10, and = 0:5.
A normalized histogram of this difference was plotted and is , , , shown in Fig. 1 for the special case of . As can be seen from this figure, the analytical and and the simulation results are in perfect agreement. IV. APPLICATION TO OUTAGE PROBABILITY WITH SELF-INTERFERENCE A. System Model When transmitting in a multipath fading environment, one will receive contributions that appear as delayed, in average downscaled, and correlated versions of the desired signal. If these contributions are not handled properly, they will degrade the quality of the decoded signal because of the self-interference, as pointed out in [27] for multicarrier systems. A measure for the effect of interference and of noise is the signal-to-interference-and-noise ratio (SINR). We consider a single-carrier system in the presence of one self-interfering signal and define the SINR as follows. be the instantaneous power of Definition 5 (SINR): Let the desired signal, the instantaneous power of the reflected signal, and the additive noise variance. We define the SINR as (25)
(24) Proof: Insert values for and from (23) in (15), and collect terms. This implies, of course, that an expression for the CDF of the difference between two exponentially distributed variates can be expressed similarly to (18), with values for and from (23) inserted.
B. Outage Probability If the SINR falls below a predetermined protection ratio an outage is declared. Denoting the outage probability by we have
, ,
C. Numerical Validation Monte Carlo simulation is well suited for checking the results derived in the previous section. Correlated gamma-distributed random variables can be generated as described in [24]–[26]. We used the algorithm described in [25, App. A] to generate 10 000 and computed the difference. pairs
(26) In a Nakagami fading environment with Nakagami parameter , will be gamma distributed, where is the short-term average of the desired faded signal power. The self-interfering signal is also gamma distributed, i.e.,
1372
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
Fig. 2. Outage probability as a function of (S =(S + N ) ) with = 0 dB, N = 40 dB, and S = 10 dB, for m = 2 and for = 0:1; 0:3; 0:5; 0:7; 0:9.
0
.2 The desired and interfering signals , are correlated with correlation coefficient so that the pair . Multiplication of a gamma variate with a constant results in a new gamma variate with mean scaled by the multiplication factor, so and are correlated with correlation coefficient , and . thus, the pair As such, the random variable can be viewed as the difference between two correlated gamma variates, and it follows the distribution given in (22). Thus, the outage probability can be expressed as the CDF of evaluated at
(27) In the general case of Nakagami fading, we are not able to find a closed-form solution for this integral, but because of the smooth tails, numerical integration over the PDF, as given in (27), can be easily performed with standard mathematical software such as Matlab or Mathematica. However, the CDF can be found in closed form for the following two special cases. 1) Rayleigh Fading Case: For the special case of Rayleigh fading, (i.e, ), there exists a closed-form expression for the CDF of (and hence, the outage probability), namely (18) with the parameters and defined using (23), as follows:
(28) Note that used.
is positive, so the second line of (18) should be
2We assume here that the desired and interfering signals have identical shape parameter m. Considering different shape parameters for the desired and interfering signal would fall outside the scope of this paper, and as such, it is a subject for further research.
Fig. 3. Outage probability as a function of (S =(S + N ) ) with = N = 40 dB, and S = 10 dB, for m = 1; 2; 4; 8; and for = 0:9.
0 dB,
0
2) Interference-Limited Case: In the case of interferencelimited transmission, the noise is negligible and the noise variance is approximated by zero. In this case, a solution to the integral in (27) can be found in terms of Gaussian hypergeometric functions by means of [18, eq. (6.621-3)]. After some manipulations, and with the help of the transformations [19, eqs. (15.3.7), (15.3.3), and (6.6.8)], it can be shown that the outage probability can be put in the following compact closed form: (29) where
(with given in (28)), and is the normalized incomplete Beta function defined in [19, eq. (6.6.2)] and tabulated by Pearson in [28]. In the case that the correlation , . Once substituted in (29), this leads to a result which is in agreement with [3, eq. (6a)] and [4, eq. (13)] for the outage probability of cellular systems when the number of cochannel interferers in the latter equations is set to one. In the case of integer, it can be shown that (29) reduces to the following finite sum:
(30) C. Numerical Examples The outage probability depends on a number of factors: ; ; ; ; ; and . When is more than approximately three orders of magnitude larger than , it turns out that depends on the fraction more than on its individual components. However, when the magnitude of decreases in comparison with , the effect of on the outage probability diminishes. We will focus in our numerical examples on the former case. The outage probability was calculated from the formula in (27), by using numerical integration for various values of the parameters; plots of as a function of are shown in Figs. 2 and 3, and as a function of in Figs. 4 and 5. In all these plots, we have set dB, dB,
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
1373
APPENDIX I MOMENTS OF MCKAY’S TYPE I DISTRIBUTION In this appendix, we give a proof of Theorem 1. Let be distributed according to the PDF stated in Definition 1. We find the moments of as given by
(31)
Fig. 4. Outage probability as a function of for m = 2 and for with = 0 dB, N = 40 dB, and S = 10 dB.
0
where we have assigned . The following formula can be deduced from [18, eq. (6.621-1)], by using [19, eq. (9.6.3)], where is the Bessel function of the first kind and order
(32) valid for
,
, and
. Let
(33) (noting that , so that the requirements for (32) are fulfilled), and use the “doubling formula” ([18, eq. (8.335-1)]) to arrive at the following expression: Fig. 5. Outage probability as a function of for m = 1; 2; 4; 8, and for (S =(S + N ) ) = 10 dB with = 0 dB, N = 40 dB, and S = 10 dB.
0
(34) dB, and has then been varied to create values for the fraction in the desired range. Note that dB, decreases as increases. for Also, for dB, , regardless of the value of the other parameters.
where is Gauss’ hypergeometric function [18, Sec. 9.1]. This concludes the proof of Theorem 1.
V. CONCLUSION
be In this appendix, we give a proof of Theorem 2. Let distributed according to the PDF stated in Definition 2. We find the moments of as given by
This paper offered general formulas for the PDF of the sum and the difference of two correlated, not necessarily identically distributed, squared Nakagami variates and established connections between these PDFs and the McKay “Bessel function” distributions. In addition, formulas for the moments of these distributions in terms of the Gauss hypergeometric function were derived. As an illustration of the mathematical formalism, an application of these new results, relevant to the calculation of outage probability in the presence of self-interference, was presented, and some numerical examples were provided and discussed.
APPENDIX II MOMENTS OF MCKAY’S TYPE II DISTRIBUTION
(35)
1374
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
where is as defined in Appendix I. The integral can be solved with in terms of Gauss’ hypergeometric function the aid of [18, eq. (6.621-3)], cited here for convenience
): In (5), let . Lemma 2 (Moments, Type II, (by the seNext, employ that function, [18, eq. (9.100)]) and ries representation of the that (which follows from [18, eq. (9.121-1)]). Then the expression for the moments reduces to the following formula:
(36) and . Employ the same where it is required that parameters , , and as in (33), and , noting that , so the requirements are fulfilled. After some algebraic manipulations, we arrive at the desired expression
(39)
(37)
This shows that the expressions for the moments of the two McKay distributions are, in fact, identical for the special case of .
which ends the proof of Theorem 2.
APPENDIX IV SUM OF TWO CORRELATED GAMMA VARIATES
APPENDIX III MOMENTS FOR MCKAY DISTRIBUTIONS (SPECIAL CASE) , In the special case that arises when is restricted to the formula for the moments becomes particularly simple. In the following, this will be shown with the aid of two lemmas. First, the expression for the moments of a type I McKay distribution is investigated. ): In (4), let Lemma 1 (Moments, Type I, . By using the formula (deduced from [19, eq. (15.1.10)]), we get the following expression:
Let , . Nakagami and Nishio [10, eq. (43)] and [2, eq. (142)] provide the PDF of the square root of the sum of the squares of two Nakagami-distributed variates. This is equivalent to the square root of the sum of two gamma. The expression is as distributed variates, i.e., follows:
(40) where (41) By using a standard transformation of variables, it can be shown that follows the distribution described by
(38) (42) Note that so that the function can be replaced by the factorial. Next, we turn to the expression for the type II McKay distribution.
By inserting , , and from (20) into (1) and comparing the result with (42), we see that Nakagami’s result is indeed on the form of McKay’s distribution.
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
APPENDIX V DIFFERENCE BETWEEN TWO CORRELATED GAMMA VARIATES
1375
ries expansion, changing the order of integration and summation, and then combining terms, (43) can be rewritten as
In this appendix, we give a direct proof of Theorem 6.3 Let , and let their difference . Similarly to the proof of Theorem 4, the PDF of the difference can be derived between two correlated variates from [21, Sec. 6-2]. The PDF can be calculated as follows:
(47)
(43)
To solve the integral in (47), we rely on the integration formula employed in the proof of Theorem 4, i.e., (13). Since the integral in (47) is in the form of (13) and , , and remembering that , the conditions of (13) are met. In particular, inserting (13) in (47) and collecting similar terms leads to
where we have used . Equation (43) must be treated separately for the cases and . For , we have
(48) (44) which can be viewed as a Laplace transform, and to which a closed-form solution can be found in [18, eq. (17.13-110)] as
The sum can be solved by the aid of the multiplication theorem for Bessel functions. Lemma 3 (Multiplication Theorem for Bessel Functions): The multiplication theorem for modified Bessel functions states [19, eq. (9.6.51)]
(49) where for the modified Bessel function of the second kind, represents . Thus, , and we have the following formula: (45) which can be further simplified by combining terms to (46) The previous expression is valid for . For , we rely on the series expansion for the Bessel [18, eq. function (8.445)] to solve the integral in (43). In particular, using the se3By direct proof, we mean that we are providing in this appendix a proof that starts from the bivariate gamma distribution. Alternatively, this theorem can be proved by first using a linear transformation of random variables to convert two correlated gamma variates into independent ones, and then by using the result of Theorem 4.
(50) Since the sum in (48) is on the form of (50) with , , and , we are able to solve the sum in (48) in closed form.4 After using (50) in (48) and collecting terms, we arrive at the final desired result, given by
can be shown that (4 =( + ) ) < 1 except when = 1 and = . The latter case is the trivial one when X = X ) 1 = 0 and is, thus, not particularly interesting. The condition in (50), j 0 1j < 1, is
4It
therefore met so that Lemma 3 can safely be used to simplify (48).
1376
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
(51) which proves Theorem 6. Note that (46), the special case for , can also be found by using the formula the limit when devised in Proposition 1. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers and the editor for their time and insight. Their suggestions considerably improved the quality of the paper. REFERENCES [1] A. U. H. Sheikh, M. Handforth, and M. Abdi, “Indoor mobile radio channel at 946 MHz: Measurements and modeling,” in Proc. 43rd IEEE Vehicular Technology Conf., Secaucus, NJ, May 1993, pp. 73–76. [2] M. Nakagami, “The m-distribution—A general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, Ed. New York: Pergamon, June 1960, pp. 3–36. [3] A. A. Abu-Dayya and N. C. Beaulieu, “Outage probabilities of cellular mobile radio systems with multiple Nakagami interferers,” IEEE Trans. Veh. Technol., vol. 40, pp. 757–768, Nov. 1991. [4] Y.-D. Yao and A. U. H. Sheikh, “Investigations into cochannel interference in microcellular mobile radio systems,” IEEE Trans. Veh. Technol., vol. 41, pp. 114–123, May 1992. [5] T. Eng and L. B. Milstein, “Coherent DS-CDMA performance in Nakagami multipath fading,” IEEE Trans. Commun., vol. 43, pp. 1134–1143, Feb.–Apr. 1995. [6] V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. Commun., vol. 43, pp. 2360–2369, Aug. 1995. [7] M.-S. Alouini and M. K. Simon, “Multichannel reception of digital signals over correlated Nakagami fading channels,” in Proc. 36th Allerton Conf. Communication, Control, Computing, Allerton Park, IL, Sept. 1998, pp. 146–155. [8] P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun., vol. 47, pp. 44–52, Jan. 1999. [9] M. Z. Win and J. H. Winters, “Exact error probability expressions for MRC in correlated Nakagami channels with unequal fading parameters and branch powers,” in Proc. IEEE Global Communications Conf., Dec. 1999, pp. 2331–2335. [10] M. Nakagami and M. Nishio, “The distribution of the sum of squares of two correlated m-variables” (in Japanese), J. Inst. Elect. Commun. Engrs. Japan, vol. 38, no. 10, pp. 782–787, 1955. ¯ “The distribution of the ratio of the squares of two correlated [11] M. Ota, m-variables,” Inform. Theory Res. Committee Japan Rep., 1956. ¯ “The distribution of the product of two cor[12] M. Nakagami and M. Ota, related m-variables,” Radio Wave Prop. Res. Committee Japan Rep., 1957. [13] A. T. McKay, “A Bessel function distribution,” Biometrika, vol. XXIV, no. 1/2, pp. 39–44, May 1932. [14] K. Pearson, S. A. Stouffer, and F. N. David, “Further applications in statistics of the T (x) Bessel function,” Biometrika, vol. XXIV, no. 3/4, pp. 293–350, Nov. 1932. [15] R. G. Laha, “On some properties of the Bessel function distributions,” Bull. Calcutta Math. Soc., vol. 46, pp. 59–72, 1954. [16] M. D. Springer, The Algebra of Random Variables. New York: Wiley, 1979. [17] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. New York: Wiley, 1994, vol. 1. [18] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic, 2000.
[19] Handbook of Mathematical Functions, M. Abramowitz and I. E. Stegun, Eds., U.S. Dept. Commerce, 1972. [20] M. G. Kendall, The Advanced Theory of Statistics, 3rd ed. London, U.K.: Griffin, 1947, vol. 1. [21] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [22] L. Fang, G. Bi, and A. C. Kot, “New method of performance analysis for diversity reception with correlated Rayleigh-fading signals,” IEEE Trans. Veh. Technol., vol. 49, pp. 1807–1812, Sept. 2000. [23] W. C. Y. Lee, Mobile Communications Engineering, 1st ed. New York: McGraw-Hill, 1982. [24] C. Tellambura and A. D. S. Jayalath, “Generation of bivariate Rayleigh and Nakagami-m fading envelopes,” IEEE Commun. Lett., vol. 4, pp. 170–172, May 2000. [25] Y.-C. Ko, “Analysis techniques for the performance evaluation of wireless communication systems and estimation of wireless channels,” Ph.D. dissertation, Univ. Minnesota, Minneapolis, MN, Oct. 2001. [26] Q. T. Zhang, “A decomposition technique for efficient generation of correlated Nakagami fading channels,” IEEE J. Select. Areas Commun., vol. 18, pp. 2385–2392, Nov. 2000. [27] A. Ligeti, “Outage probability in the presence of correlated lognormal useful and interfering components,” IEEE Commun. Lett., vol. 4, pp. 15–17, Jan. 2000. [28] K. Pearson, Table of Incomplete Beta Functions. London, U.K.: Cambridge Univ. Press, 1932.
Henrik Holm (S’96–M’03) was born in Sarpsborg, Norway, on January 25, 1972. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Department of Telecommunications, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, in 1997 and 2002, respectively. Since May 2002, he has been a Postdoctoral Associate at the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, with funding from the NTNU. His current research interests are in the areas of channel estimation and modulation techniques for wireless communications.
Mohamed-Slim Alouini (S’94–M’99–SM’03) was born in Tunis, Tunisia. He received the “Diplôme d’Ingénieur” degree from the Ecole Nationale Supérieure des Télécommunications (TELECOM Paris), Paris, France, and the “Diplôme d’Etudes Approfondies” (D.E.A.) degree in electronics from the University of Pierre and Marie Curie (Paris VI), Paris, France, both in 1993. He received the M.S.E.E. degree from the Georgia Institute of Technology (Georgia Tech), Atlanta, in 1995, and the Ph.D. degree in electrical engineering from the California Institute of Technology (Caltech), Pasadena, in 1998. He also received the Habilitation degree from the University of Paris VI, Paris, France, in 2003. He joined the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, in September 1998, where he is now an Associate Professor and where his current research interests include statistical characterization and modeling of fading channels, performance analysis of diversity combining techniques and MIMO systems, capacity and outage analysis of multiuser wireless systems subject to interference and/or jamming, and design and performance evaluation of multiresolution, hierarchical, and adaptive modulation schemes. He has published several papers on the above subjects, and he is coauthor of Digital Communication over Fading Channels (New York: Wiley, 2000). Dr. Alouini is a recipient of a National Semiconductor Graduate Fellowship Award, the Charles Wilts Prize for outstanding independent research, corecipient of the 1999 Prize Paper Award of the IEEE Vehicular Technology Conference (VTC’99-Fall), Amsterdam, The Netherlands, and a 1999 CAREER Award from the National Science Foundation. He has also received a McKnight Land-Grant Professorship from the Board of Regents, University of Minnesota in 2001, the Best Instructor Award by the Institute of Technology Student Board, University of Minnesota, in 2001 and 2002, and the Taylor Career Development Award from the Institute of Technology of the University of Minnesota in 2003. He is an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, and for the Wiley Journal on Wireless Systems and Mobile Computing.
1367
Sum and Difference of Two Squared Correlated Nakagami Variates in Connection With the McKay Distribution Henrik Holm, Member, IEEE, and Mohamed-Slim Alouini, Senior Member, IEEE
Abstract—General formulas for the probability density function of the sum and the difference of two correlated, not necessarily identically distributed, squared Nakagami variates (or equivalently, gamma variates) are derived. These expressions are shown to be in the form of the McKay “Bessel function” distributions. In addition, formulas for the moments of these distributions, in terms of the Gauss hypergeometric function, are provided. An application of these new results relevant to the calculation of outage probability in the presence of self-interference is discussed. Index Terms—Correlated fading, gamma distribution, McKay distribution, Nakagami fading, outage probability.
I. INTRODUCTION
T
HE wide versatility, computational tractability, and experimental consistency [1] of the Nakagami- distribution [2] has made it popular as a fading model in the performance analysis of wireless systems (see, for example, [3]–[9]). One of the important features of the distribution is that the power of a signal perturbed by Nakagami fading is gamma distributed. Thus, in performance evaluation involving Nakagami fading, one can often rely on established results (in the statistics literature) about the gamma distribution. An important special case of the Nakagami distribution is the Rayleigh distribution, which arises in the situation of multipath transmission with no direct component, i.e., when all of the received power stems from scattered components. The corresponding distribution for the signal power is the exponential distribution. Another often-employed distribution when modeling wireless channels is the Rice distribution, which considers the multipath signal as a sum of a direct component and a reflected component. When the magnitude of the direct component is zero, the Rice also reduces to the Rayleigh distribution. Although the Nakagami- and the Rice
Paper approved by V. A. Aalo, the Editor for Diversity and Fading Channel Theory of the IEEE Communications Society. Manuscript received November 5, 2001; revised April 9, 2003 and January 27, 2004. This work was supported in part by the Research Council of Norway under Grant 11939/431, and in part by the National Science Foundation under Grants CCR-9983462 and ECS9979443. This paper was presented in part at the 35th Annual Conference on Information Sciences and Systems (CISS’01), Baltimore, MD, March 2001. H. Holm is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA, on a visit from the Department of Electronics and Telecommunications, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (e-mail: [email protected]). M.-S. Alouini is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.833019
distributions are not equal, except in the common Rayleigh special case, their probability density functions (PDFs) can be close parameter and when a proper mapping is used between the the Rice factor [2, eqs. (55) and (56)]. However, an advantage of the Nakagami- over the Rice model is the relatively simple expression for its PDF and joint PDF for the signal power. Indeed, while the gamma PDF contains no complicated special functions, the signal power in the Rician case is governed by a noncentral chi-square distribution with two degrees of freedom, which contains a modified Bessel function. On the other hand, the bivariate distribution of two squared correlated Nakagamivariates (or equivalently, two correlated gamma variates) has a relatively simple closed-form expression, while (to the best of our knowledge) no simple formula is yet known for the joint PDF of two correlated noncentral chi-square variates. Many of the performance-analysis problems in wireless communication systems over Nakagami- channels require determination of the statistics of functions of the squared envelope of Nakagami- faded signals. In his classical 1960 paper [2], relying on a series of papers [10]–[12] appearing in Japanese journals, Nakagami cites expressions for the distribution of the sum, the ratio, and the product of squares of two correlated Nakagami- variates (or equivalently, sum, ratio, and product of two correlated gamma variates). However, no expression for the PDF of the difference between two gamma variates is provided. This PDF can not be derived from the PDF of a sum, and it has a slightly different form, since the sum of gamma variates (always positive) naturally is positive, while the difference can clearly be negative. Apparently, even in the specialized statistics literature, only the difference between two uncorrelated gamma variates has been investigated [13]–[17]. One of the contributions of this paper is to show that starting from the bivariate Gamma distribution, we are able to derive a closed-form expression for the PDF, moments, and in certain particular cases, the cumulative distribution function (CDF) of the difference between two correlated, not necessarily identically distributed, gamma-distributed variates. Much effort has been made in finding general statistical distributions, or systems of distributions, for the purpose of fitting sample data to a distribution. The Pearson system [14], [17] is, for example, one of the most general systems of distributions. As an addition to the Pearson system, McKay [13] provides what he calls “a Bessel function distribution,” which is a distribution with particularly simple expressions for the moment-generating function (MGF) and for the cumulants. This distribution splits naturally in two forms, depending on the value of one of the
0090-6778/04$20.00 © 2004 IEEE
1368
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
parameters, where one of them describes a variate that can be only positive, while the other describes a variate that can take on all values. The two forms of this distribution will together be referred to as “the McKay distributions.” Since the sum and difference of two uncorrelated gamma-distributed variates are governed by these two forms, respectively [17], they will be of particular interest in the context of this paper. In particular, another contribution of this paper is to show that the sum and difference of correlated gamma variates are also governed by McKay’s distributions. The final contribution of this paper is an application of the mathematical results to the problem of self-interference, namely, derivation of the CDF of outage probabilities in a self-interference scenario. The paper is organized as follows. Section II defines the two McKay distributions. Expressions for the moments of these distributions are stated as theorems, as are the PDFs of the sum and difference between two uncorrelated gamma variates. Lastly, an important special case, relevant to Rayleigh fading, is investigated. Section III introduces the bivariate gamma distribution and derives the distributions of the sum and difference of correlated gamma variates. The application of this new result, relevant to the calculation of outage probability in the presence of self-interference, is presented in Section IV. Finally, a conclusion summarizing the main results is given in Section V.
is not defined when the argument is equal to zero. Howwhich is valid ever, with the aid of an approximation of , we provide an expression for the type for small and for II McKay distribution for . , and and are as in Definition 2 Proposition 1: When (3) Proof: Use the formula from [19, eq. (9.6.9)] to find the limit of (2). when We have not been able to find an expression for , but plotting of (2) indicates that the PDF approaches a Dirac “ ” distribution when approaches from above. A. Moments McKay provides expressions for the MGF and for the cumulants of his distribution functions, [13, eqs. (10) and (19)], respectively. In addition, it is also possible to find general closed-form expressions for the moments of the McKay distributions, as we show now. Theorem 1 (Moments of McKay’s Type I Distribution): Let be distributed according to the type I McKay distribution (Definition 1). Then the moments of can be expressed as
II. MCKAY’S BESSEL FUNCTION DISTRIBUTION In this section, we define McKay’s distributions and provide expressions for the moments of these distributions. Next, we remind the reader about the basic definition of the gamma distribution and state some previously known results regarding the sum and difference between independent gamma variates. Lastly, an important special case is considered. McKay defines his distribution in terms of one function with different forms due to different values of one of the parameters, namely, the parameter. However, we find it more convenient to treat the two forms as separate distributions; called and , respectively. follows the type Definition 1 (Type I McKay Distribution): , , and I McKay distribution with parameters 1 when the PDF of is given by
(4) is Gauss’ hypergeometric function where [18, Sec. 9.1]. Proof: See Appendix I. As indicated, the previous theorem applies solely to McKay’s . For , distribution of the first kind, i.e., when the type II McKay distribution is the appropriate distribution function, and then the moments must be expressed as in the following theorem. Theorem 2 (Moments of McKay’s Type II Distribution): Let follow the type II McKay distribution (Definition 2). Then the moments of can be expressed as
(1) where is the gamma function, is the modified Bessel function of the first kind and of order , and is the Heaviside unit step function [18]. follows the Definition 2 (Type II McKay Distribution): , , type II McKay distribution with parameters and when the PDF of is given by (5) (2) where is the modified Bessel function of the second kind and of order .
0
1The third parameter to the first McKay distribution can also be c < 1. Then, the distribution function is defined for negative, instead of positive, .
Proof: See Appendix II. The hypergeometric function is commonly available in numerical software, so the moments can be calculated easily. Expressions for the moments can also be found by using multiple derivatives of the MGF. In addition, it is always possible to find an expression for the th moment in terms of the first
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
cumulants [20, Ch. 3], and vice versa. As an example, using [13, eq. (19)] gives the following expression for the mean and the variance of the first McKay distribution:
(6) . Hence, the second moment can be found as Before concluding this section, we mention that in the special , the formulas case that arises when is restricted to (4) and (5) for types I and II, respectively, become particularly simple, as shown in Appendix III.
1369
Equation (9) can be solved after inserting expressions for the PDFs, (7). Comparison with (1) after insertion of parameters from (8) concludes the proof. This result is also provided without proof in the classical textbook by Johnson et al. [17, Sec. 12-4.4], but we could not trace where the result was originally derived and presented. Theorem 4 (Difference Between Two Independent Gamma , be mutually independent random variVariates): Let and , reables, distributed as is distributed according to the spectively. Then second of McKay’s forms with parameters
B. Sum and Difference of Two Independent Gamma Variates Definition 3 (Gamma Distribution): follows a gamma distribution with shape parameter and scale parameter when the PDF of is given by (7) to denote that We use the shorthand notation follows a gamma distribution with shape parameter and scale parameter . As mentioned before, the gamma distribution is related to the Nakagami distribution. Namely, when a signal is Nakagami distributed with Nakagami fading parameter and average fading power , then the power (i.e., the square) of the signal is Gamma and scale parameter distributed with shape parameter . Theorem 3 (Sum of Two Independent Gamma Variates): Let , be mutually independent random variables, distributed and , respectively. Then as is distributed according to the first of McKay’s forms with parameters
(8) and , so the restrictions that applies to the Thus, McKay distribution I are fulfilled. Proof: The PDF of the sum of two independent variates is [21, Sec. 6-2] (9) By combining the integration formulas [18, eqs. 3.388-1 and 3.383-2] to yield the following formula:
(10)
(11) where we see that the requirements and are met, and are positive. since Proof: The PDF of the difference between two indepencan be derived from [21, Sec. 6-2], dent variates and it is expressed as follows: (12) Two integration formulas [18, Eqs. 3.383-3 and 3.383-8] can be combined, and together they produce the following rule:
(13) After inserting the expressions for the individual PDFs, (7) and (12) can be solved using (13). Insertion of parameters from (11) and subsequent comparison with (2) concludes the proof. As is the case for Theorem 3, this result is provided without proof in the classical textbook by Johnson et al. [17, Sec. 12-4.4], and again, we were not able to trace where the result was originally derived and presented. C. Special Case of In the important special case of Rayleigh fading, the previous PDF and moment expressions can be used by simply setting . As can be seen from (8) and (11), this means . that the parameter in the McKay distributions is It turns out that this restriction leads to particularly simple expressions for both the PDFs and the moments, involving only elementary functions (i.e., no special functions like the hypergeometric function or modified Bessel functions.) It is also possible to find simple closed-form expressions for the CDFs. ): When , the Corollary 1 (McKay Type I, type I McKay distribution reduces to the following expression: (14)
1370
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
Proof: Insert
in (1), use the identity from [19, eq. (10.2.13)], and collect terms. Equation (14) is recognized as the PDF of the sum of two correlated exponential variates with the same scale factor . This and and can be seen by setting comparing with [22, eq. (31)] or [23, Sec. 10.6]. (Note that the definition of in these references is different from the case we are studying here.). Like Definition 1, Definition 2 can be simplified when . The following corollary is a counterpart to Corollary 1. ): When , the Corollary 2 (McKay Type II, type II McKay distribution reduces to the following expression:
, scale parameters common shape parameter , , respectively, and correlation coefficient if their joint PDF is given by
(19)
(15) (equivalent to ) in (2), use the Proof: Insert from [19, eq. (10.2.17)], identity and collect terms. The hypergeometric functions appearing in (4) and (5) are , as is displayed in the also possible to simplify when following corollary. ): Let . Then Corollary 3 (Moments When the expression for the moments of both McKay distributions can be written as follows:
, to We use the shorthand notation denote that and follow a bivariate gamma distribution. , the joint PDF reduces to the product of two uniWhen variate gamma PDFs. A. Sum of Two Correlated Gamma Variates Theorem 5 (Sum of Two Correlated Gamma Variates): Let , . The sum follows the type I McKay distribution with parameters
(16) Proof: See Appendix III. Because of the simple PDFs when , it is possible to find closed-form expressions for the CDFs of both McKay distributions. , and Corollary 4 (CDF of McKay Type I): When , the CDF of the McKay type when restricting, as before, I distribution can be written as follows:
(17) Proof: Integrate (14) from to and collect terms. Corollary 5 (CDF of McKay Type II): When , and , the CDF of the McKay type II distriwhen restricting bution can be written as follows: for for
.
(20)
where, since , the restrictions and are met. Proof: See Appendix IV. Thus, Nakagami and Nishio’s result [2], [10] is, in fact, on the McKay form. Hence, McKay’s first form, which is known to be valid for the sum of independent gamma-distributed variates, is, in fact, also valid for the sum of correlated ones. Corollary 6 (Sum of Two Correlated Exponential Vari, . The sum ates): Let has a PDF as follows:
(18)
, integrate (15) from to and Proof: For collect terms. For , find by integrating (15) to and collecting terms. from (21) III. SUM AND DIFFERENCE OF CORRELATED GAMMA VARIATES When the gamma-distributed variates are correlated, the bivariate gamma distribution, which takes into account the correlation, must be employed. and Definition 4 (Bivariate Gamma Distribution): are governed by a bivariate gamma distribution with
Proof: Insert values for and from (20) in (14), and collect terms. Of course, the previous corollary also applies to the uncorrelated case by inserting corresponding values for and (from in (21). (8)) in (14), or by inserting
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
1371
B. Difference Between Two Correlated Gamma Variates Theorem 6 (Difference Between Two Correlated Gamma , . The difference Variates): Let follows the PDF given by
for (22a) When
, there exists an expression for the special case (22b)
Proof: See Appendix V. Corollary 7 (Relation to McKay’s Distribution Function): The difference between two correlated gamma variates is distributed according to the type II McKay distribution with parameters
(23) and are met, as long as where the conditions . Proof: Can be readily seen by inserting the parameters , , and from (23) in (2). Corollary 8 (Difference Between Two Correlated Exponential Variates): This corollary is a counterpart to Corollary 6. , . The difference Let has a PDF as follows:
Fig. 1. Comparison between the analytical PDF and the PDF obtained via Monte Carlo simulation, for = 2, = 20, = 10, and = 0:5.
A normalized histogram of this difference was plotted and is , , , shown in Fig. 1 for the special case of . As can be seen from this figure, the analytical and and the simulation results are in perfect agreement. IV. APPLICATION TO OUTAGE PROBABILITY WITH SELF-INTERFERENCE A. System Model When transmitting in a multipath fading environment, one will receive contributions that appear as delayed, in average downscaled, and correlated versions of the desired signal. If these contributions are not handled properly, they will degrade the quality of the decoded signal because of the self-interference, as pointed out in [27] for multicarrier systems. A measure for the effect of interference and of noise is the signal-to-interference-and-noise ratio (SINR). We consider a single-carrier system in the presence of one self-interfering signal and define the SINR as follows. be the instantaneous power of Definition 5 (SINR): Let the desired signal, the instantaneous power of the reflected signal, and the additive noise variance. We define the SINR as (25)
(24) Proof: Insert values for and from (23) in (15), and collect terms. This implies, of course, that an expression for the CDF of the difference between two exponentially distributed variates can be expressed similarly to (18), with values for and from (23) inserted.
B. Outage Probability If the SINR falls below a predetermined protection ratio an outage is declared. Denoting the outage probability by we have
, ,
C. Numerical Validation Monte Carlo simulation is well suited for checking the results derived in the previous section. Correlated gamma-distributed random variables can be generated as described in [24]–[26]. We used the algorithm described in [25, App. A] to generate 10 000 and computed the difference. pairs
(26) In a Nakagami fading environment with Nakagami parameter , will be gamma distributed, where is the short-term average of the desired faded signal power. The self-interfering signal is also gamma distributed, i.e.,
1372
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
Fig. 2. Outage probability as a function of (S =(S + N ) ) with = 0 dB, N = 40 dB, and S = 10 dB, for m = 2 and for = 0:1; 0:3; 0:5; 0:7; 0:9.
0
.2 The desired and interfering signals , are correlated with correlation coefficient so that the pair . Multiplication of a gamma variate with a constant results in a new gamma variate with mean scaled by the multiplication factor, so and are correlated with correlation coefficient , and . thus, the pair As such, the random variable can be viewed as the difference between two correlated gamma variates, and it follows the distribution given in (22). Thus, the outage probability can be expressed as the CDF of evaluated at
(27) In the general case of Nakagami fading, we are not able to find a closed-form solution for this integral, but because of the smooth tails, numerical integration over the PDF, as given in (27), can be easily performed with standard mathematical software such as Matlab or Mathematica. However, the CDF can be found in closed form for the following two special cases. 1) Rayleigh Fading Case: For the special case of Rayleigh fading, (i.e, ), there exists a closed-form expression for the CDF of (and hence, the outage probability), namely (18) with the parameters and defined using (23), as follows:
(28) Note that used.
is positive, so the second line of (18) should be
2We assume here that the desired and interfering signals have identical shape parameter m. Considering different shape parameters for the desired and interfering signal would fall outside the scope of this paper, and as such, it is a subject for further research.
Fig. 3. Outage probability as a function of (S =(S + N ) ) with = N = 40 dB, and S = 10 dB, for m = 1; 2; 4; 8; and for = 0:9.
0 dB,
0
2) Interference-Limited Case: In the case of interferencelimited transmission, the noise is negligible and the noise variance is approximated by zero. In this case, a solution to the integral in (27) can be found in terms of Gaussian hypergeometric functions by means of [18, eq. (6.621-3)]. After some manipulations, and with the help of the transformations [19, eqs. (15.3.7), (15.3.3), and (6.6.8)], it can be shown that the outage probability can be put in the following compact closed form: (29) where
(with given in (28)), and is the normalized incomplete Beta function defined in [19, eq. (6.6.2)] and tabulated by Pearson in [28]. In the case that the correlation , . Once substituted in (29), this leads to a result which is in agreement with [3, eq. (6a)] and [4, eq. (13)] for the outage probability of cellular systems when the number of cochannel interferers in the latter equations is set to one. In the case of integer, it can be shown that (29) reduces to the following finite sum:
(30) C. Numerical Examples The outage probability depends on a number of factors: ; ; ; ; ; and . When is more than approximately three orders of magnitude larger than , it turns out that depends on the fraction more than on its individual components. However, when the magnitude of decreases in comparison with , the effect of on the outage probability diminishes. We will focus in our numerical examples on the former case. The outage probability was calculated from the formula in (27), by using numerical integration for various values of the parameters; plots of as a function of are shown in Figs. 2 and 3, and as a function of in Figs. 4 and 5. In all these plots, we have set dB, dB,
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
1373
APPENDIX I MOMENTS OF MCKAY’S TYPE I DISTRIBUTION In this appendix, we give a proof of Theorem 1. Let be distributed according to the PDF stated in Definition 1. We find the moments of as given by
(31)
Fig. 4. Outage probability as a function of for m = 2 and for with = 0 dB, N = 40 dB, and S = 10 dB.
0
where we have assigned . The following formula can be deduced from [18, eq. (6.621-1)], by using [19, eq. (9.6.3)], where is the Bessel function of the first kind and order
(32) valid for
,
, and
. Let
(33) (noting that , so that the requirements for (32) are fulfilled), and use the “doubling formula” ([18, eq. (8.335-1)]) to arrive at the following expression: Fig. 5. Outage probability as a function of for m = 1; 2; 4; 8, and for (S =(S + N ) ) = 10 dB with = 0 dB, N = 40 dB, and S = 10 dB.
0
(34) dB, and has then been varied to create values for the fraction in the desired range. Note that dB, decreases as increases. for Also, for dB, , regardless of the value of the other parameters.
where is Gauss’ hypergeometric function [18, Sec. 9.1]. This concludes the proof of Theorem 1.
V. CONCLUSION
be In this appendix, we give a proof of Theorem 2. Let distributed according to the PDF stated in Definition 2. We find the moments of as given by
This paper offered general formulas for the PDF of the sum and the difference of two correlated, not necessarily identically distributed, squared Nakagami variates and established connections between these PDFs and the McKay “Bessel function” distributions. In addition, formulas for the moments of these distributions in terms of the Gauss hypergeometric function were derived. As an illustration of the mathematical formalism, an application of these new results, relevant to the calculation of outage probability in the presence of self-interference, was presented, and some numerical examples were provided and discussed.
APPENDIX II MOMENTS OF MCKAY’S TYPE II DISTRIBUTION
(35)
1374
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
where is as defined in Appendix I. The integral can be solved with in terms of Gauss’ hypergeometric function the aid of [18, eq. (6.621-3)], cited here for convenience
): In (5), let . Lemma 2 (Moments, Type II, (by the seNext, employ that function, [18, eq. (9.100)]) and ries representation of the that (which follows from [18, eq. (9.121-1)]). Then the expression for the moments reduces to the following formula:
(36) and . Employ the same where it is required that parameters , , and as in (33), and , noting that , so the requirements are fulfilled. After some algebraic manipulations, we arrive at the desired expression
(39)
(37)
This shows that the expressions for the moments of the two McKay distributions are, in fact, identical for the special case of .
which ends the proof of Theorem 2.
APPENDIX IV SUM OF TWO CORRELATED GAMMA VARIATES
APPENDIX III MOMENTS FOR MCKAY DISTRIBUTIONS (SPECIAL CASE) , In the special case that arises when is restricted to the formula for the moments becomes particularly simple. In the following, this will be shown with the aid of two lemmas. First, the expression for the moments of a type I McKay distribution is investigated. ): In (4), let Lemma 1 (Moments, Type I, . By using the formula (deduced from [19, eq. (15.1.10)]), we get the following expression:
Let , . Nakagami and Nishio [10, eq. (43)] and [2, eq. (142)] provide the PDF of the square root of the sum of the squares of two Nakagami-distributed variates. This is equivalent to the square root of the sum of two gamma. The expression is as distributed variates, i.e., follows:
(40) where (41) By using a standard transformation of variables, it can be shown that follows the distribution described by
(38) (42) Note that so that the function can be replaced by the factorial. Next, we turn to the expression for the type II McKay distribution.
By inserting , , and from (20) into (1) and comparing the result with (42), we see that Nakagami’s result is indeed on the form of McKay’s distribution.
HOLM AND ALOUINI: SUM AND DIFFERENCE OF TWO SQUARED CORRELATED NAKAGAMI VARIATES
APPENDIX V DIFFERENCE BETWEEN TWO CORRELATED GAMMA VARIATES
1375
ries expansion, changing the order of integration and summation, and then combining terms, (43) can be rewritten as
In this appendix, we give a direct proof of Theorem 6.3 Let , and let their difference . Similarly to the proof of Theorem 4, the PDF of the difference can be derived between two correlated variates from [21, Sec. 6-2]. The PDF can be calculated as follows:
(47)
(43)
To solve the integral in (47), we rely on the integration formula employed in the proof of Theorem 4, i.e., (13). Since the integral in (47) is in the form of (13) and , , and remembering that , the conditions of (13) are met. In particular, inserting (13) in (47) and collecting similar terms leads to
where we have used . Equation (43) must be treated separately for the cases and . For , we have
(48) (44) which can be viewed as a Laplace transform, and to which a closed-form solution can be found in [18, eq. (17.13-110)] as
The sum can be solved by the aid of the multiplication theorem for Bessel functions. Lemma 3 (Multiplication Theorem for Bessel Functions): The multiplication theorem for modified Bessel functions states [19, eq. (9.6.51)]
(49) where for the modified Bessel function of the second kind, represents . Thus, , and we have the following formula: (45) which can be further simplified by combining terms to (46) The previous expression is valid for . For , we rely on the series expansion for the Bessel [18, eq. function (8.445)] to solve the integral in (43). In particular, using the se3By direct proof, we mean that we are providing in this appendix a proof that starts from the bivariate gamma distribution. Alternatively, this theorem can be proved by first using a linear transformation of random variables to convert two correlated gamma variates into independent ones, and then by using the result of Theorem 4.
(50) Since the sum in (48) is on the form of (50) with , , and , we are able to solve the sum in (48) in closed form.4 After using (50) in (48) and collecting terms, we arrive at the final desired result, given by
can be shown that (4 =( + ) ) < 1 except when = 1 and = . The latter case is the trivial one when X = X ) 1 = 0 and is, thus, not particularly interesting. The condition in (50), j 0 1j < 1, is
4It
therefore met so that Lemma 3 can safely be used to simplify (48).
1376
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 8, AUGUST 2004
(51) which proves Theorem 6. Note that (46), the special case for , can also be found by using the formula the limit when devised in Proposition 1. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers and the editor for their time and insight. Their suggestions considerably improved the quality of the paper. REFERENCES [1] A. U. H. Sheikh, M. Handforth, and M. Abdi, “Indoor mobile radio channel at 946 MHz: Measurements and modeling,” in Proc. 43rd IEEE Vehicular Technology Conf., Secaucus, NJ, May 1993, pp. 73–76. [2] M. Nakagami, “The m-distribution—A general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, Ed. New York: Pergamon, June 1960, pp. 3–36. [3] A. A. Abu-Dayya and N. C. Beaulieu, “Outage probabilities of cellular mobile radio systems with multiple Nakagami interferers,” IEEE Trans. Veh. Technol., vol. 40, pp. 757–768, Nov. 1991. [4] Y.-D. Yao and A. U. H. Sheikh, “Investigations into cochannel interference in microcellular mobile radio systems,” IEEE Trans. Veh. Technol., vol. 41, pp. 114–123, May 1992. [5] T. Eng and L. B. Milstein, “Coherent DS-CDMA performance in Nakagami multipath fading,” IEEE Trans. Commun., vol. 43, pp. 1134–1143, Feb.–Apr. 1995. [6] V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. Commun., vol. 43, pp. 2360–2369, Aug. 1995. [7] M.-S. Alouini and M. K. Simon, “Multichannel reception of digital signals over correlated Nakagami fading channels,” in Proc. 36th Allerton Conf. Communication, Control, Computing, Allerton Park, IL, Sept. 1998, pp. 146–155. [8] P. Lombardo, G. Fedele, and M. M. Rao, “MRC performance for binary signals in Nakagami fading with general branch correlation,” IEEE Trans. Commun., vol. 47, pp. 44–52, Jan. 1999. [9] M. Z. Win and J. H. Winters, “Exact error probability expressions for MRC in correlated Nakagami channels with unequal fading parameters and branch powers,” in Proc. IEEE Global Communications Conf., Dec. 1999, pp. 2331–2335. [10] M. Nakagami and M. Nishio, “The distribution of the sum of squares of two correlated m-variables” (in Japanese), J. Inst. Elect. Commun. Engrs. Japan, vol. 38, no. 10, pp. 782–787, 1955. ¯ “The distribution of the ratio of the squares of two correlated [11] M. Ota, m-variables,” Inform. Theory Res. Committee Japan Rep., 1956. ¯ “The distribution of the product of two cor[12] M. Nakagami and M. Ota, related m-variables,” Radio Wave Prop. Res. Committee Japan Rep., 1957. [13] A. T. McKay, “A Bessel function distribution,” Biometrika, vol. XXIV, no. 1/2, pp. 39–44, May 1932. [14] K. Pearson, S. A. Stouffer, and F. N. David, “Further applications in statistics of the T (x) Bessel function,” Biometrika, vol. XXIV, no. 3/4, pp. 293–350, Nov. 1932. [15] R. G. Laha, “On some properties of the Bessel function distributions,” Bull. Calcutta Math. Soc., vol. 46, pp. 59–72, 1954. [16] M. D. Springer, The Algebra of Random Variables. New York: Wiley, 1979. [17] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. New York: Wiley, 1994, vol. 1. [18] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic, 2000.
[19] Handbook of Mathematical Functions, M. Abramowitz and I. E. Stegun, Eds., U.S. Dept. Commerce, 1972. [20] M. G. Kendall, The Advanced Theory of Statistics, 3rd ed. London, U.K.: Griffin, 1947, vol. 1. [21] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [22] L. Fang, G. Bi, and A. C. Kot, “New method of performance analysis for diversity reception with correlated Rayleigh-fading signals,” IEEE Trans. Veh. Technol., vol. 49, pp. 1807–1812, Sept. 2000. [23] W. C. Y. Lee, Mobile Communications Engineering, 1st ed. New York: McGraw-Hill, 1982. [24] C. Tellambura and A. D. S. Jayalath, “Generation of bivariate Rayleigh and Nakagami-m fading envelopes,” IEEE Commun. Lett., vol. 4, pp. 170–172, May 2000. [25] Y.-C. Ko, “Analysis techniques for the performance evaluation of wireless communication systems and estimation of wireless channels,” Ph.D. dissertation, Univ. Minnesota, Minneapolis, MN, Oct. 2001. [26] Q. T. Zhang, “A decomposition technique for efficient generation of correlated Nakagami fading channels,” IEEE J. Select. Areas Commun., vol. 18, pp. 2385–2392, Nov. 2000. [27] A. Ligeti, “Outage probability in the presence of correlated lognormal useful and interfering components,” IEEE Commun. Lett., vol. 4, pp. 15–17, Jan. 2000. [28] K. Pearson, Table of Incomplete Beta Functions. London, U.K.: Cambridge Univ. Press, 1932.
Henrik Holm (S’96–M’03) was born in Sarpsborg, Norway, on January 25, 1972. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Department of Telecommunications, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, in 1997 and 2002, respectively. Since May 2002, he has been a Postdoctoral Associate at the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, with funding from the NTNU. His current research interests are in the areas of channel estimation and modulation techniques for wireless communications.
Mohamed-Slim Alouini (S’94–M’99–SM’03) was born in Tunis, Tunisia. He received the “Diplôme d’Ingénieur” degree from the Ecole Nationale Supérieure des Télécommunications (TELECOM Paris), Paris, France, and the “Diplôme d’Etudes Approfondies” (D.E.A.) degree in electronics from the University of Pierre and Marie Curie (Paris VI), Paris, France, both in 1993. He received the M.S.E.E. degree from the Georgia Institute of Technology (Georgia Tech), Atlanta, in 1995, and the Ph.D. degree in electrical engineering from the California Institute of Technology (Caltech), Pasadena, in 1998. He also received the Habilitation degree from the University of Paris VI, Paris, France, in 2003. He joined the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, in September 1998, where he is now an Associate Professor and where his current research interests include statistical characterization and modeling of fading channels, performance analysis of diversity combining techniques and MIMO systems, capacity and outage analysis of multiuser wireless systems subject to interference and/or jamming, and design and performance evaluation of multiresolution, hierarchical, and adaptive modulation schemes. He has published several papers on the above subjects, and he is coauthor of Digital Communication over Fading Channels (New York: Wiley, 2000). Dr. Alouini is a recipient of a National Semiconductor Graduate Fellowship Award, the Charles Wilts Prize for outstanding independent research, corecipient of the 1999 Prize Paper Award of the IEEE Vehicular Technology Conference (VTC’99-Fall), Amsterdam, The Netherlands, and a 1999 CAREER Award from the National Science Foundation. He has also received a McKnight Land-Grant Professorship from the Board of Regents, University of Minnesota in 2001, the Best Instructor Award by the Institute of Technology Student Board, University of Minnesota, in 2001 and 2002, and the Taylor Career Development Award from the Institute of Technology of the University of Minnesota in 2003. He is an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, and for the Wiley Journal on Wireless Systems and Mobile Computing.
