On the Influence of the Power Profile on Diversity Combining Schemes

On the Influence of the Power Profile on Diversity CombiningSchemesD.V.Djonin and V.K.Bhargava,Fellow,IEEEDepartment of Electrical and Computer EngineeringUniversity of VictoriaVictoria,B.C.,Canada V8W3P6Tel:+12507216043,Fax:+12507216048,Email:bhargava,ddjonin@ece.uvic.caAbstractIndex Terms—Diversity Combining,Order Statistics,Majorizations,Schur convex functionsAbstract The usage of multichannel receivers that are followed by diversity combining can sig-nificantly improve the performance of wireless communications links.In this paper we investigate in some detail three popular techniques for diversity combining,namely maximal ratio combining (MRC),equal gain combining(EGC)and selection combining(SC).The main goal is to investi-gate under which circumstances it can be guaranteed that a certain technique does indeed decrease the probability of error of a multichannel receiver.The analyzed fading model is quite general with the only assumption that fading gains are scaled exchangeable random variables.The crucial mathematical tool used to produce the results of this paper is the powerful theory of stochastic majorization.Among other results,it was shown that the average bit error probability of the MRC receiver with equally distributed average powers always decreases with the addition of a new di-versity branch if the total power isfixed.I.I NTRODUCTIONThe performance analysis of digital communication systems in fading channels has been attract-ing attention of the researchers for a long time[1],[2].In recent years,this area has received a renewed interest due to the appearance of the so-called unified approach to the analysis of fading channels[3],[4],[5].This approach relies on the usage of the moment generating function for representation of fading distribution and its main contribution is that it simplifies the calculation of the basic performance measures of communication systems in fading channels.This theory can produce closed form results for different fading distributions,different modulation formats,as well as different receiver structures.It is known that the usage of multichannel receivers followed by certain diversity combining technique in fading channels can significantly improve the performance of the system.The usage of diversity combining techniques is of interest even more in the emerging broadband communi-cation systems(for example Ultra-wide band systems[6]and[7])where the number of diversity branches can be significant due to the strong multipath effects.The diversity combining basically relies on the reception of the replicas of the same information distorted by independent(or near independent)fading paths.The general concepts of diversity combining applies equally to space, polarization,frequency and time diversity.The results of this paper are intended to complement the unified approach[3],[4],[5]for the analysis of the performance of multichannel receivers in generalized fading channels.The fun-damental relations and inequalities presented in this paper can help a system designer compare performances of two fading channels with the same fading statistics but different power delay profiles.These results can instruct the system designer under which conditions adding of one or more diversity branches will improve the performance of a diversity combining system.Further-more,this paper gives asymptotic analysis(as the number of diversity branches goes to infinity)of different diversity combining schemes.The inequalities derived in this paper rely on the powerful theory of stochastic majorization[8],[9],[10],[11]and[12].This theory stems from the extension of the vector majorization to certain stochastic applications and was developed initially primarily having in mind the reliability theory. However,as will be shown in the following material,these results can be easily applied to produce some interesting inequalities regarding performances of diversity combining systems. Previously,the majorization theory have found its application in diverse engineeringfields.In particular,the applications of majorization theory have been reported in solving the task assign-ment problems in computer science[13],queuing algorithms[14],filter design[15]as well as optimal sequence allocation for CDMA systems[16].1Throughout the paper we discuss three basic performance measures of receivers in fading chan-nels[3]:the signal to noise ratio at the receiver,bit error rate of the analyzed scheme and the outage probability.In the section on maximal ratio combining we also discuss the capacity as a possible performance measure.We will denote with upper case letters the random variables while the lower case letters are reserved for sample values.For example if is a discrete random variable with distribution,we have Prob.The multichannel receiver with diversity branches and a general diversity combining scheme is shown in Figure1.We denote with normalized random amplitude channel gains whileare normalized random power channel gains.For simplicity it is assumed that E E. The random fading gain at the receiver is assumed to be equal to and the received power in-th branch is for a radiated power.Vector of average received powers corresponds to the power delay profile,while g will be called gain profile.Note that is the average value of fading gains.Therefore the fading gain of the-th diversity branch is split into the varying random part()andfixed multiplicative part().The model of the-th diversity branch channel is shown in Figure2for easy reference.Let the total average received power be equal to.Then the random signal to noise ratio of-th diversity branch is given with where is the ratio of During the revision of this paper,the authors became aware of an application of the majorization theory to the performance analysis of maximal ratio combining receiver in correlated fading channels[17].the symbol energy per noise spectral density of a certain diversity branch.We will assume that noise spectral densities of all branches are equal but this condition can also be easily relaxed.It is assumed that random delays as well as phases(for coherent reception)are perfectly estimated in the receiver.Several results in this paper are applicable when channel fading gains are so-called exchangeable random variables.The exchangeable random variables are defined as:Definition1:Random variables are called exchangeable random variables if the distribution of does not depend on the permutation.This definition can be interpreted such that the joint distribution of is not depen-dent on the permutation of its arguments.An obvious example of exchangeable random variables are identically and independently distributed(i.i.d.)random variables.However,the definition of exchangeable random variables is more general and includes other different classes of mul-tivariate distributions such as multivariate normal distribution with a common mean,a common variance,and common non-negative correlation coefficient,exchangeable gamma and multivariate exchangeable exponential distribution[18].The reader is referred to[12],Section13for detailed list of exchangeable multivariate distributions.Also,the class of exchangeable random variables is equivalent to the class of so-called positively dependent by mixture random variables(see[19])that can be defined asx(1) where is a family of indexed univariate distributions.In the subsequent discussion,the notion of exchangeable fading gains means that multivariate distribution of is exchangeable,or alternatively formulated that normalized received powers are exchangeable.In case of EGC we will consider that are exchangeable. The exchangeable fading model is used for the discussion on bit error probability of MRC and EGC diversity schemes.We note that the results on outage probabilities are however presented only for independent fading gains with Rayleigh(EGC)and exponential distribution(MRC).Howeverthese results might also be extended for a more general class of distributions.The outline of the paper is as follows.Section II introduces mathematical concepts of the theory of stochastic majorization that will be used in course of the paper.Section III gives results on the probability of error and outage probability of the MRC,EGC and SC diversity systems.II.M ATHEMATICAL P RELIMINARIESThe results presented in this paper follow from the theory of majorization and its stochastic applications.Considering that the average reader of this journal might not be familiar with the theory of majorization we here review the notation and the most important definitions of this theory. The following definitions are of crucial interest in proving the results of this paper.Definition2:For any vector x,let x denote the non-increasing rearrangement of the vector x,i.e.where is satisfied.Now we can introduce the definition of vector majorization that will allow us to compare the performances of diversity combining schemes with different power delay profiles.Definition3:For any x y,it is said that y majorizes x,or x y if(2)An important example of majorization between vectors of length is the following(3) Obviously,the previous definition of majorization applies only to vectors that have equal sums of all elements.This condition can be relaxed and the following definition introduces the so-called “weak majorization”.Definition4:For some x,y,x y if(4)andx y if(5) We will give two examples of weak majorization that will be referred to later in the text.Simi-larly to(3),it can be stated that(6) if and only iffor certain vectors x and y.The definition of the decreasing vector function is an obvious extension.The following result (for the proof see3.A.8.in[8])extends the applicability of the Schur convex functions on vectors that are only weakly majorized.Theorem1:A real-valued function defined on a set satisfiesx y on x y(11) if and only if is increasing and Schur convex on.Similarly,satisfiesx y on x y(12) if and only if is decreasing and Schur convex on.III.D IVERSITY C OMBININGThe results on the statistical properties of diversity combining schemes presented in this section rely on the theory of stochastic majorization.The following two general results of this theory can be applied to several different cases.Proposition1:If are exchangeable random variables and is a sym-metric convex function,then function defined asE(13)is symmetric and convex.Therefore,is Schur convex and a a implies a a.For the proof of this Proposition see11.B.2.in[8].This result can be readily extended such that its scope covers weakly majorized vectors(11.B.5.in[8]).Proposition2:Let be exchangeable random variables and functionsand satisfy the following conditions:(a)is convex and increasing while is symmetric,convex and increasing;(b)is convex and decreasing while is symmetric,convex and increasing;then function E is(a)symmetric, convex and increasing;(b)symmetric,convex and decreasing,respectively.This Proposition can be combined with Theorem1to produce inequalities for weakly majorized vectors.Up to now we have presented a very powerful tool that can be used to facilitate comparison be-tween performances of different diversity combining schemes with different power delay profiles. There are,however,two basic limitations of this approach.First,the majorization between two vectors is a partial ordering i.e.it is not possible to compare any two vectors in this manner.We claim that this limitation is not a serious one since many cases of practical interest can be analyzed in this manner.Second limitation is that the results of the next section are applicable to situations where joint distribution of normalized channel fading gains is exchangeable:the case that again models some cases of practical interest.A.Maximal Ratio CombiningMaximal ratio combining uses optimal weighting of the received diversity signal in order to maximize its output signal to noise ratio.The performance of the maximal ratio combining scheme depends on the signal to noise ratio at the output of this combiner.This signal to noise ratio can be expressed as(see for example[3])where is a realization of the exchangeable random variables as discussed in Section II.Now,bit error probabilities of several different modulation schemes with maximal ratio com-bining can be expressed using the same general formfor certain univariate error function that is independent of the number of diversity branches. The long term average error probability can be expressed asE(16)It should be noted here that function is convex for several different modulation formats including binary phase shift keying(BPSK),QAM,M-PSK,coherent FSK and Minimum Shift Key(MSK) in additive Gaussian noise channels.A.1Average BER of BPSK ModulationWe discuss here only the case of BPSK modulation but the discussion can be applied also for other mentioned modulation formats.For BPSK modulation,we have[20])and coherent BFSK with minimum correlation().It is easy to see that is a convex function since its derivative with respect to,An interesting consequence of the previous equation can be produced when majorization exam-ple(3)is used and constant is interpreted as thefixed normalized total power through all diversity branches.Then,if powers are equally distributed through all the diversity branches,the average probability of error is decreasing with the number of branches.Furthermore,in case of independent identically distributed fading gains with probability density function,asymptot-ically,as increases to infinity,the average bit error probability with equal distribution of powers in diversity branches is non-increasing and converges toE(20)where thefirst equality follows from the weak law of large numbers.Note that E E. The above discussion can be extended to the comparison of two power delay profiles that do not have equal total powers.In this case we resort to the definition of the weak majorization and apply Proposition2on the function defined in(18).The function that would correspond to the statement of the Proposition2would be which is an increasing and convex function. Therefore,case(a)of this Proposition is applicable and we conclude that function is convex, symmetric and increasing or alternatively,using the Theorem1g g g g(21) This analysis carries on to different modulation formats(like QAM,MPSK,FSK...)whenever bit error probability can be expressed as a convex function of.A.2Outage ProbabilityThe long term average bit error probability analysis is not applicable in all cases.For example, if fading is very slow or when there are restriction on instantenous bit error probabilities,the knowledge of average bit error probability is not useful.In these cases,the appropriate measure is outage probability defined asg Prob Probfor a certain threshold.The results in this section are valid for independent Rayleigh distributed fading gains.We start with the following Proposition that follows from[11]and[8].Proposition3:Let be independent random variables with the same exponential distributionProb(23) Let p where for all.Also denote with p the ordered(from the weakest to the strongest)version of the random vector p.Then for certain vectors(24)it follows thatProb p Prob p(25)for any increasing function and vectors p and p. Proof:The proof follows by application of12.H.1.a and the generalization of stochastic ordering of section17.A(Equation(3’))in[8].In order to apply this Proposition to the analysis of the outage probability given in(22),increas-ing function has to be the function that sums all its arguments and multiplies the result by. Therefore,applying directly Proposition3,we can state that(26)for certain gain profiles g and g.The outage probability inequality of this section is applicable for independently and expo-nentially distributed power channel gains but this analysis can also be extended to distribution functions that fall into the category of so-called proportianaly hazard function(see for example 12.H.2.[8]).These details will not be analyzed any further in this section.Regarding the behavior of outage probability of the MRC with equal power distribution in case of increase in the number of branches,we note the ly,we compare a system with branches and power delay profilep(27)and system with branches and power delay profilep(28)where fading gains in two cases are g p and g p.Now,according to(26)and (5)the outage probability is guaranteed to decrease with the increase in the number of branches ifor(30)and where ergodic capacity for fast changing fading channels is defined as g EEof stochastic majorization.Noting that functioncan be expressed as E.Now,in order to show that stochastic majorization results of Section III.A carry over to the case of equal gain combining,it is necessaryto explore if function(34)is convex for certain error function.Note that if is convex,then function as defined in (33)is also convex.In the case of BPSK/BFSK modulation(35) This function is always convex since its derivative is always increasing.Note that function as defined in(33)is also increasing function of with respect tothe Definition6.Therefore,using the same argumentation that was previously used in Section III.A.(namely Proposition2and Theorem1),the most general conclusion on the behavior of the average bit error probability of the equal gain combiner with diversity branches is that(36)for two modified gain profiles defined as and. It is straightforward to show that these results also hold for M-PSK and M-AM signals.Now we compare branch EGC combiner having equal gain profile elements withbranch EGC combiner having equal gain profile.In that case condition(36)that guaranteesthe decrease in the probability of error with the addition of a diversity branch is given with(37) which satisfied if and only ifThis condition is satisfied if the total power isfixed i.e.when(39)which follows from the weak law of large numbers.Here E.To obtain better insight in the difference of asymptotic performance of the MRC and the EGC we use the example of Rayleigh fading with.In this case the average asymptotic BER of MRC is while for EGC it is(40) wherefor orthogonal non-coherently detected FSK,and for binary DPSK.The total SNR is given as the sum of individual branch signal to noise ratios,similarly to the expression for total SNR of MRC receiver(14).It has been observed that is convex function in for every integer valued .This observation is based on numerical calculation of the second derivative of(40)and showing that it is positive for.Maximal number of diversity branches of explored in our numerical observations should be sufficient for most practical applications.In ad-dition we here state a conjecture that function(40)is convex for any integer,but no formal proof is available at this time.If function(40)is convex it is also convex and symmetric in the individual signal to noise ratios.We note here that(40)is also increasing in the individual signal to noise ratios. Therefore,the conclusions of the Section III.A.1carry over to the case of non-coherent EGC.The majorization between power delay profiles implies the relationship between average bit-error rates of the non-coherent receiver with branches i.e.g g g g(42)where g is the average bit-error probability of the branch non-coherent receiver with power delay profile g.The previous result is also valid only for fading gains of individual branches having an exchangeable joint distribution.Note that due to the postdetection performed in the non-coherent EGC combiner,(42)allows only the comparison of the average bit error probabilities of the receiver havingfixed number of diversity branches with different diversity profiles.In order to compare receivers with different numbers of diversity branches we have to use different expressions for the probability of error. Note that with the increase of the number of diversity branches and the same total SNR,the bit error probability of non-coherent EGC can increase(for smaller values of total SNR)and the receiver can suffer from a combining loss,as was noted in([20],Section12).B.1Outage ProbabilityAs is the case with the discussion on outage probability of the MRC,the outage probability of coherent EGC will be discussed only in the case of independent Rayleigh distributed amplitude fading gains.Wefirst note that Rayleigh distribution falls into the class of proportionally hazard distributions(see Section12.H.in[8])and that it is equivalent to the Weibull distribution withparameter.The following Proposition that follows from12.H.4.in[8]will be needed to characterize the behavior of the outage probability of coherent EGC with Rayleigh distributed fading gains.Proposition4:Let be independent identically distributed random variables with Weibull distribution given(for some)byProb(43) If thenProb Prob(44)for certain univariate increasing function.Proof:The proposition is straightforward extension of12.H.4.in[8]to weak majorization. Now,the previous Proposition can be directly applied to the outage probability of the EGC with Rayleigh distributed amplitude fading gains.By comparing the expression for the signal to noise ratio at the output of the EGC(32)with(44)and using,we can state that for branch coherent EGC diversity scheme(45)for certain power delay profiles and.The previous equation follows from Proposition4 having for and.Note,that the same conclusion was derived for the outage probability of the MRC diversity scheme.C.Selection CombiningSelection combining is less complex than the other two linear diversity combining schemes MRC and EGC,since it processes only one of the diversity branches.In selection combining approach,the receiver processes only the signal with the largest signal to noise ratio,or in case of equal noise powers in branches,the signal with largest received power.The signal to noise ratio atthe output of the selection combiner is then(46)where are considered realizations of exchangeable random variables as discussed in Section II. In this section we will be only able to produce results on the average signal to noise ratioE.By observing that the function is convex,symmetric and increasing(see for example11.B.2.b.in[8]),and invoking Proposition2and Theorem1we can state thatg g g g(47) In the case offixed average total received power,the previous relation implies that the increase in the number of equitably power distributed diversity branches decreases the average signal to noise ratio of the selection combiner.In fact,it can be demonstrated using the asymptotic order statistics approach(see[22],[23]and[24]),that the average signal to noise ratio converges in probability to0as the number of equal power distributed branches increases to infinity!This fact is a consequence of the decrease in the average power per branch which is dominating the effect of the selection of the branch with the strongest power.The proof of this claim will be omitted due to its simplicity.Similarly,in the asymptotic case,the average error probability of the selection combiner converges to0.5.If the selection combiner is used,we can not state results similar to(36)and(21)since the error function of parameters is not necessarily convex.This is due to the fact that the composition of two convex functions is always convex only if the outer function is increasing.This is obviously not true in the case of error function which are always decreasing with the increase of signal to noise ratios.Therefore we can expect that the increase in the number of equitably power distributed diversity branches for smaller number of branches can both increase and decrease the probability of error of SC.However,taking into account the asymptotic results,for large number of diversity branches,adding a new one will definitely increase the error probability.C.1Outage ProbabilitySimilar to the Section III.A.2,the outage probability of the Selection combiner is defined asg Prob(48) where is defined with(46).Now,in order to produce comparisons similar to that of the Section III.A.2we again apply Proposition3.It is assumed that channel gains are independent and exponentially distributed.The increasing function is now chosen to be the function that returns its last argument(i.e.the maximal argument).In that case,Proposition3can be applied straightforwardly and the outage probabilities of the selection combiner also satisfy(26).IV.C ONCLUSIONSThis paper discusses some fundamental relations that characterize the diversity combining sys-tems.For exchangeable fading gains,certain partial ordering conditions between power delay profiles were shown to guarantee the decrease of the average bit error probability.Starting from one such relation,it was shown that bit error probability of a MRC decreases with the addition of a diversity branch for the case offixed equally distributed total average received power.Similarly for the coherent EGC receiver,the increase in number of equal power diversity branches withfixed total power and 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