2012-tsp

1050 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 3, MARCH 2012

On the Product of Independent Complex GaussiansNicholas ODonoughue, Member, IEEE, and Jos M. F. Moura, Fellow, IEEE

AbstractIn this paper, we derive the joint (amplitude, phase)distribution of the product of two independent non-zero-meanComplex Gaussian random variables. We call this new distribu-tion the complex Double Gaussian distribution. This probabilitydistribution function (PDF) is a doubly infinite summation overmodified Bessel functions of the first and second kind. We analyzethe behavior of this sum and show that the number of terms neededfor accuracy is dependent upon the Rician -factors of the twoinput variables. We derive an upper bound on the truncation errorand use this to present an adaptive computational approach thatselects the minimum number of terms required for accuracy. Wealso present the PDF for the special case where either one or bothof the input complex Gaussian random variables is zero-mean. Wedemonstrate the relevance of our results by deriving the optimalNeymanPearson detector for a time reversal detection schemeand computing the receiver operating characteristics throughMonte Carlo simulations, and by computing the symbol errorprobability (SEP) for a single-channel -ary phase-shift-keying(M-PSK) communication system.

Index TermsComplex double Gaussian, complex Gaussian,detection, time reversal, probability distribution of product ofGaussian variables.

I. INTRODUCTION

T HIS paper presents the probability distribution function(PDF) for the product of two non-zero-mean complexGaussian random variables. This pdf is not known. We refer toit as the complex Double Gaussian PDF. This PDF is usefulin many practical applications, for example, in communicationsystems, the keyhole or pinhole channel model proposed in [1]and [2] describes a system where both the transmitter and thereceiver are surrounded by multipath scattering and all com-munication between them passes through a single waveguide,such as the corner of a building. In this scenario, the channel isthe product of two complex Gaussian random variables.Another class of applications where this distribution can

apply is in time reversal detection [3][5]. In time reversal,a source first probes the channel. The receiver collects thisresponse, performs time reversal and energy normalization,and then transmits the time-reversed signal back through thechannel to the original source. If the channel is random, thenthe aggregate channel becomes the product of two complex

Manuscript received June 18, 2011; revised October 19, 2011; accepted Oc-tober 31, 2011. Date of publication November 23, 2011; date of current versionFebruary 10, 2012. The associate editor coordinating the review of this manu-script and approving it for publication was Prof. Joseph Tabrikian. The work ofN. ODonoughue was partially supported by a National Defense Science andEngineering Graduate Fellowship.The authors are with the Department of Electrical and Computer Engi-

neering, Carnegie Mellon University, Pittsburgh, PA 15217 USA (e-mail:[email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2011.2177264

Gaussian random variables. The optimal design and analysis ofdetectors for this class of problems requires knowledge of theprobability distribution function of the product of two complexGaussian random variables to derive the likelihood ratio testand to compute the corresponding rates of detection and falsealarm.A third example arises when studying the error performance

of -ary phase-shift-keying (MPSK) communication systems.The linear combiner output for a single-channel system can beexpressed by the product of two complex Gaussian random vari-ables [6]. Knowledge of this distribution allows us to analyze thesymbol error probability (SEP) for this class of communicationsystem.This paper addresses the problem of two independent non-

zero-mean complex Gaussian random variables and andtheir product . We derive the joint (amplitude and phase) prob-ability distribution of , where is the am-plitude and is the principal value of the phase of defined inthe interval , i.e., . We show thatis computed via a doubly-infinite summation, whose terms in-clude modified Bessel functions of the first and second kind [7].We call this new distribution the complex Double Gaussian dis-tribution. We analyze this result and show the conditions underwhich the infinite summation can be truncated; we also showthat the number of terms needed for a high degree of accuracydepend upon the statistics of the two input variables. We applyour results to deriving the optimal detector and characterizingthe performance of a time reversal detection scheme, as wellas to perform the error analysis for an M-PSK communicationsystem.In Section II, we review the prior work and outline the

contributions of our work. We provide a detailed formulationof the problem in Section III. The main result is presentedin Section IV, and we provide two example applications inSection V. Finally, we discuss our results in the conclusion,Section VI.Notation: We represent random variables by capital letters,, while lower-case letters, , denote specific instances of a

random variable. Bold capital letters, , denote matrices, whilelower-case bold letters, , denote vectors. The symbol de-scribes the parameters and distribution of a random variable.For example, states that is a Gaussian (orNormal) random variable with mean and variance . Theletter denotes . The functions and are themodified Bessel functions of the first and second kind, respec-tively, with order and argument . The Bessel functions aredefined and discussed in great detail in [7].

II. PRIOR WORKThe product of two complex Gaussian random variables is

a problem that has been studied to some degree in the past,

1053-587X/$26.00 2011 IEEE

ODONOUGHUE AND MOURA: ON THE PRODUCT OF INDEPENDENT COMPLEX GAUSSIANS 1051

although none of the prior efforts yielded the PDF for theproblem considered here: the product of two independent com-plex Gaussian random variables with non-zero expectation.Recently, Mallik derived the joint (real, imaginary) character-

istic function of the inner product of two independent complexGaussians random vectors [6]. Inversion of this characteristicfunction, however, must be performed numerically, which pre-cludes the analytic derivation of a PDF. The reference handbook[8], written by Simon, provides solutions for a large selection ofproducts involving real Gaussian random variables, as well asRician, Rayleigh, and chi-squared random variables. Reference[8] can be used to construct the marginal PDF of the amplitudeof the product of complex Gaussian random variables, but not

the joint PDF of the amplitude and phase.We derive the joint PDF of two independent complex

Gaussian random variables. We coin the term complex doubleGaussian to refer to this distribution. We also present thecomplex double Gaussian PDF for the special cases where oneor both inputs are zero-mean. The applications considered inSection V, both the design and analysis of the optimal detectorfor a time reversal detection scheme and the error analysisof an M-PSK communication system, serve to illustrate theapplicability of these results.

III. PROBLEM FORMULATIONIn this paper, we consider the product of two indepen-

dent complex Gaussian random variables and . We wishto compute the distribution of . Since is complex, this willbe a bivariate distribution in amplitude and phase

.First, however, we review the complex Gaussian distribu-

tion [9], [10]. Given a complex Gaussian random variable X,with non-zero expectation, the amplitude follows a marginalRician distribution [8], while the phase , when conditioned onthe amplitude, follows the Tikhonov distribution [11]. For com-pleteness, we provide the following prior results. For the com-plex Gaussian random variable with mean value andvariance , the joint distribution of and is [10]

(1)

The marginal distribution of the amplitude is found by inte-grating (1) over the interval , which yields the Riciandistribution [8]

(2)

where

(3)

and is the modified Bessel function of the first kind,with order and argument [7]. The marginal distribution ofthe phase, given the amplitude, follows the Tikhonov distribu-tion [11]. This is computed by dividing (1) by (2):

(4)

is similarly defined. The interested reader is referred to [10],where the complex Gaussian distribution is discussed in greatdetail.For the remainder of this paper, we make use of the Rician-factor [12], which is used extensively in the analysis of Ri-cian fading channels [13][15]. The -factor is defined as theratio of the power between the mean square and the variance.In Rician fading channels, this corresponds to the ratio of thepower between the line-of-sight and the multipath components.Here, we use a similar definition:

(5)

Recall that and are the mean and standard deviation, respec-tively. The -factor modulates the randomness in the under-lying distribution. At its lower limit, , the distribution ofbecomes Rayleigh. As increases, the distribution becomes

dominated by the mean value , and the random variable ap-proaches a deterministic value at .In the next section, we compute the complex distribution of

in polar format, i.e., . We will evaluatethis distribution for one general case and two important specialcases: a) and are non-zero mean; b) is non-zero mean,and is zero mean; and c) and are both zero mean. Inaddition to the complex distribution of , we will also derivethe marginal distribution of its amplitude: . This resultwill be presented for all three scenarios.

IV. DOUBLE GAUSSIAN PDFIn this section, we outline our results and proofs. Our pri-

mary contribution is the complex distribution of . InSection IV-A, we outline the joint distribution (amplitude andphase) of for all three scenarios. In Section IV-B, we outlinethe marginal distribution of the amplitude for all three sce-narios. When necessary, we discuss the convergence of infinitesummations.

A. Joint DistributionsThis section focuses on the joint distribution of the amplitude

and phase of . We discuss the three relevant scenarios: a) bothinputs are non-zero mean; b) one input is zero mean; and c) bothinputs are zero mean.Theorem 4.1 (Product of two non-zero-mean com-

plex Gaussian variables): If andare independent random variables, then

the product is characterized by the polar distribution:

(6)

where .Proof: See Appendix A.

A number of comments regarding (6):


We call this result the complex Double Gaussian distributionand use the shorthand to refer to it:

(7)

It is not immediate from (6) what the first two moments are.However, since the inputs and are independent, computa-tion of those moments is trivial:

(8)(9)

Likewise, higher order moments of are obtained from the firsttwo moments of and .Clearly, by symmetry of the roles of and as factors of ,

the expression (6) is symmetric with respect to the parametersof and . The infinite summations over and arise fromthe Rician -factors and . These terms express the relativestrength of the deterministic components of and , and it istheir presence that leads to the double sum in (6). If one of oris zero, the corresponding sum across or , respectively, be-

comes superfluous and the PDF reduces to a single summationterm (see Section IV-A3). Lemma 4.2 shows that the number ofsummation terms necessary for a high degree of accuracy is re-lated to the values of and .We plot an example of (6) in Fig. 1(a) for the case where

, and . This plotwas generated using the finite sum approximation described in(12) with (100 terms). Fig. 1(b) shows a MonteCarlo simulation of the same scenario, and confirms the validityof our derived result. The plot exhibits a cardioid shape, this isbecause we are plotting the polar form of the two-dimensionalPDF . A simple conversion to Cartesian coordi-nates would remove the hole that appears at the origin. Further-more, the notch at in Fig. 1(b) is an artifact from themethod used to compute the 2-D histogram, and can be ignored.We can see from these plots that, in terms of phase, the energyis centered around , this is a reasonable result, since thetwo inputs were centered at and phase is addi-tive under multiplication. Within the angular range and ,the energy is centered between and . The longtail matches the behavior of Rician distributions. We discuss theaccuracy of these plots below.1) Real-Imaginary Notation: Through a simple transforma-

tion of variables, we can express the joint PDF of the realand imaginary parts of :

(10)

where, , and.

Fig. 1. Plot of the two-dimensional PDF , described in (6), forand . (a) Analytical result. (b) Monte Carlo sim-

ulation. (c) Plot of the maximum error given in (13) versus for varioussummation lengths .


Fig. 2. The infinite summation over two variables (n and p) is depicted graph-ically. The finite summation approximation of (12) is depicted in the lower-leftregion, while the four remaining regions represent error terms.

2) Convergence Analysis: We note that (6) is a doubly-infi-nite summation of modified Bessel functions of the firstand second kinds [7]. To evaluate this summation, wetruncate the series. For simplicity, we define the general sum-mation term such that (6) can be written

(11)

Define the partial sum:

(12)

and the error term

(13)

The numbers of computed terms and in (12) are user se-lected parameters. We simplify by stating that . Thisallows us to break the error term into four distinct summationregions (see Fig. 2) and to rewrite it as .Lemma IV.2 (Upper Bound on (13)): For sufficiently large

number of terms , the truncation error forin (12) is bounded by

(14)

where is the th summation term, explicitly defined in(81), is the upper bound on the decay rate of asincreases, defined in (88), is the upper bound on the decayrate of as increases, defined in (91), and is the upperbound on the decay rate of along the line , definedin (98).

Proof: See Appendix B.The error upper bound (14) allows us to iteratively compute (6)for increasing numbers of terms . At each stage, we compute

TABLE ITABLE OF UPPER BOUNDS FOR , ACCORDING TO (13) WHEN

the upper bound on the error, according to (13). When the erroris guaranteed to be below some desired threshold, the com-

putation is terminated. A set of sample values for (13) is givenin Table I and plotted in Fig. 1(c). Error values below areassumed to be 0. In both Table I and Fig. 1(c), and areexpressed in decibel scale:

(15)

and are swept across the range 6 dB 6 dB . The -axisof Fig. 1(c) is given in decibels, while the -axis plots linearvalues on a log scale. The error clearly increases as andincrease, necessitating more summation terms in order to

maintain a given upper bound on the error . This confirms thatit is and that cause the infinite summations to arise in (6).In other words, the stronger the deterministic components ofand are, the more summations terms are needed for accuratecomputation.From Table I, the number of terms needed for an accuracy of

is (25 terms). Also, using this table, we cansay that each pixel in Fig. 1(a) is accurate to within an error of

.As described in Appendix B, this error bound is not always

valid. Depending on the parameters , and theamplitude that is considered, smaller numbers of termsmay produce erroneous bounds. This stems from the decay ratebounds applied and occurs when the parameters and ,which are defined in (88) and (91), respectively, are greater than1. If this error is encountered, then should be incremented,and their values retested.3) One or More Inputs Are Zero Mean: If we let ,

then the joint distribution in (6) reduces to

(16)

If, in addition to , we allow , then the jointdistribution in (6) reduces to

(17)


Fig. 3. Plot of the 2-dimensional PDF , depicted in (16), when, and .

These results can be quickly verified by first noting theapproximation to a modified Bessel function of the first kindfor small arguments and non-negative integer orders

[7, Sec. (10.30.1)]:

(18)

and then taking the limit of (6) first as for (16) and thenagain as for (17).We first note that, while the general joint PDF in (6) is a

doubly-infinite summation, this simplifies to an infinite summa-tion over one term when one of the inputs is zero mean, as in(16), and reduces further to a closed form solution when bothinputs are zero mean, as in (17).We note that both PDFs are independent of . Thus,

is uniformly distributed and is independent of in both sce-narios. This result is intuitive, since the input is uniformly dis-tributed across phase (as well as , in the second case). Thus,when and are added, the result is also uniformly dis-tributed over . We plot an example of (16) in Fig. 3 forthe case where , and . The plotshows circular symmetry about the origin, caused by the inde-pendence of . This is in stark contrast to Fig. 1(a), where astrong dependence on is seen. Further, we note that the en-ergy in Fig. 3 is concentrated between and ,as opposed to the window of to for Fig. 1(a).The difference comes from the fact that was reduced from 1to 0. A reduction of from 1 to 0 (as seen in the case whereboth and are 0) would further reduce the radius of the ex-pectation.

B. Marginal Distribution

In addition to the joint distribution , we presentthe marginal distribution of the amplitude . Thisresult is useful in many applications, and is a direct extension ofthe joint distribution. We present the results for all three targetscenarios and note that prior work corroborates all three of thederivations in this section. This section is provided for com-pleteness, tying these prior results into the contribution pre-sented in Section IV-A.Theorem IV.3 (Product of two non-zero-mean com-

plex Gaussian variables): If andare independent random variables, then

the amplitude is characterized by the distribution

(19)

Proof: We first note that can be rewritten

(20)

We can, thus, begin with the joint distribution of and in(61) and apply the transformation

(21)

which results in the distribution

(22)

Next, we integrate over , recall the identity in(70) and convert both of the modified Bessel functions to theirinfinite summation representations

(23)

We follow the same approach as in (74) and arrive at the solution

(24)

This result appears in [8] under the product of independent Ri-cian random variables. See Fig. 4(a) for a plot of (19) for variousvalues of , when is fixed to 1. The solid lines correspondto the analytical results computed from (19), while the dottedlines follow a weighted histogram from Monte Carlo trials. For


Fig. 4. (a) Plot of the probability distribution computed using (19) forvarious values of , when is set to 1. These plots are compared against theRician distribution . The solid lines plot the analytical results, whilethe dotted lines follow a weighted histogram from Monte Carlo trials used toverify our results. (b) Plot of the maximum error given in (27) versus ,for various number of terms .

this test, . As the Rician -factor increases,the PDF approaches the Rician distribution . This trendmakes sense. As approaches , the variance approaches0 (since is held constant at 1). The result is that the randomvariable becomes less random and starts to behave as the de-terministic variable . In this limiting case .1) Convergence Analysis: The result (19) is an infinite sum-

mation over the terms . In order to evaluate it,we truncate the summations. We define the general term of thesummation in , such that (19) can be written

(25)

TABLE IITABLE OF REPRESENTATIVE VALUES FOR , ACCORDING TO (27) WHEN

We approximate (19) with the partial sum

(26)

and define the truncation error

(27)

Lemma IV.4 (Upper Bound on (27)): For sufficiently largenumber of terms , the truncation error for in (26)is upper bounded by

(28)

where the summation term is defined as

(29)

and the various decay rate bounds are defined as

(30)

(31)

(32)

For brevity, we have used the term and inthe definitions above to signify the quantity

(33)

Proof: Repeat the steps taken in Appendix B, replacing (6)with (19).A set of sample values for (27) is given in Table II and Fig. 4(b).We see from the table that (225 terms) are required for


Fig. 5. Plot of the probability distribution computed using (34) forvarious values of when is set to 0. These plots are compared against theRayleigh distribution . The solid lines plot the analytical results, whilethe dotted lines follow a weighted histogram from Monte Carlo trials used toverify our results.

the error bound . This is much larger than in LemmaIV.2, which required only 25 terms. This increase is becausethe error here represents the integral of the error in Lemma IV.2across phase.2) One or More Inputs Are Zero Mean: If we let ,

then the marginal distribution in (19) reduces to

(34)

If, in addition, we let , then the marginal distributionin (19) reduces to

(35)

These results appear in [8], the former under the product of in-dependent Rician and Rayleigh random variables and the latterunder the product of independent Rayleigh random variables.These equations can be verified by using the same approach asin Section IV-A3.See Fig. 5 for a plot of (34) for various values of . For this

test, . As the Rician -factor increases, the pdf tendstowards the Rayleigh distribution .

V. APPLICATIONSIn this section, we present two example applications. The

first, in Section V-A, is a blind time reversal detection strategywhere the complex Double Gaussian distribution is used toderive the likelihood ratio test and NeymanPearson detector.The second, in Section V-B, is an -ary phase-shift-keying(M-PSK) communication system where the complex DoubleGaussian distribution is used to derive the SEP.

A. Time Reversal DetectionFor our first example, we consider the detection of a target in

the presence of clutter using a single transmitter and a single re-ceiver. We will utilize a time reversal detection strategy similarto the ones outlined in [3][5] with one notable difference. In thedetection system we consider here, the time reversal mirror willnot communicate with the detector. Thus, the detection systemmust operate without knowledge of the result of the forwardtransmission. The meaning of this statement will be clarifiedbelow.We define the frequency samples . We

model the target as a point target with a deterministic response. The clutter is drawn from a zero-mean complex Gaussiandistribution with power spectral density . This channelmodel is discussed in detail in [16]. We transmit the probingsignal and write the response

(36)

where is the clutter response, andrepresents additive noise. For simplicity,

we use a white probing signal

(37)

for some transmit power . The time reversal probing signalis generated using a scaled, phase-conjugated version of the re-ceived signal

(38)

where is the energy normalization factor defined by

(39)

We assume that is approximately deterministic, as was arguedin [17]. The time reversal probing signal is distributedas a complex Gaussian:

(40)

The signal is then transmitted from the receiver backto the source, where the received signal is

(41)

where is the clutter channel for the second transmission,and is the noise signal for the second transmission. Weassume that the clutter and noise signals are independent of eachother and independent from one transmission to the next. If weignore the noise term, then (41) is distributed according to theproduct of independent complex Gaussians:

(42)

where , and. At this stage, we set up the binary hy-

pothesis test. The detectors in [3][5] were designed to use both


and . In this application, however, we considerthe case where is not available to the detector. In the nullhypothesis, the case where no target is present, . In thealternative hypothesis, . Thus

(43)

1) Likelihood Ratio Test: The received signalfollows the complex double Gaussian distribution presentedin Section IV. Under is distributed accordingto the special case (17). Under is distributedaccording to the general result in Theorem IV.1. To computethe likelihood ratio, we divide (6) by (17):

(44)

where , and. It would be possible to

construct the likelihood ratio numerically, through a MonteCarlo simulation, but that approach would be much morecomputationally complex, and would not yield the analyticalrepresentation in (44). From the likelihood ratio , we canconstruct the test [18]

(45)

for some threshold . Ideally, we would use the distributionof the test statistic to determine the appropriate thresholdfor some desired false alarm rate. However, its distribution isunknown, so we must rely on Monte Carlo simulations to deter-mine the appropriate threshold.2) Monte Carlo Simulations: We simulated a scenario where

the clutter channel follows a Gaussian power spectral densityin the band 24 GHz, and the target (a point target) has a con-stant value . We confine our transmit power to thesame level and vary the total clutter power

. We sample frequencies and conductedMonte Carlo trials for each scenario.

Using these Monte Carlo trials, we were able to compute thereceiver operating characteristics for the detector presented in(45), for varying levels of signal-to-noise ratio, defined as

SNR (46)

We also define the signal-to-clutter ratio SCR :

SCR (47)

Fig. 6. Receiver operating characteristics for the likelihood ratio test (LRT)given in (44) and the energy detector (ED) given in (48) for various SNR valueswhen (a) SCR 5 dB and (b) SCR 0 dB.

For comparison, we also computed the receiver operating char-acteristics of an energy detector (ED):

(48)

For the first test, we set the SCR and plot the re-ceiver operating characteristics ( versus ) for both thelikelihood ratio test (LRT) and the energy detector (ED) forSNR and 10. The results are shown in Fig. 6(a). Forthe second test, we set the SCR and repeated the firsttest. The results are shown in Fig. 6(b). In both plots, the solidlines depict the results for the case where SNR , and thedotted lines depict the results for the case where SNR . It


is clear from these results that the LRT derived from the distri-butions we have presented in this paper outperforms the ED inthis application.

B. M-Ary Phase Shift Keying Communications

A second potential application is the error analysis forM-PSK communication systems (a system where the con-stellation points are equally spread about the unit circle). Weconsider a system similar to the one analyzed in [6], with theexception that we limit the number of diversity branches to

and, since there is only one receiver, do not considerMaximal Ratio Combining. Given the transmission of a symbol, we write the received signal as

(49)

where is the channels complex random gain andis an additive noise term. The symbol belongs

to an M-PSK constellation , given by ,where

(50)

and is the transmit energy. In order to estimate the channelgain, we first transmit pilot symbols:

(51)

and write the received signal vector

(52)

with the IID noise vector . The least-squaresestimate of is given by

(53)

Following the pilot symbols each data symbol is trans-mitted, and the received data signal is run through a linearcombiner

(54)

This correlation output is distributed according to the complexDouble Gaussian distribution:

(55)

We can compute the SEP with [6]

(56)

where the product is distributed. We compute this probability

with the integral

(57)

Fig. 7. Symbol Error Probability against SNR for an M-PSK CommunicationSystem with (Binary PSK), (quadrature PSK), and .Solid lines show analytical results computed from the complex Double Gaussiandistribution, while dotted lines show average error results from Monte Carlosimulations.

These analytical results are shown as the solid lines in Fig. 7,which plots the probability of error against SNR. To verify theseanalytical results, we compute the error probability term in (57)by simulating the transmission of pilot symbols and100 data symbols through a channel . This experi-ment is repeated for independentMonte Carlo trials.These are shown as the open circles in Fig. 7. We define theSNR:

SNR (58)

To compute the integral in (57), we evaluate the PDFover a grid of 50 range bins and 100 phase

intervals and numerically integrate over the region of interest.At each range bin, we use the error bound in (14) to determinethe minimum number of terms in order to guarantee atruncation error of . There is a strong correlationbetween the Monte Carlo (circular marks) and analytical (solidlines) results.In order to study the effects of the number of terms on the

analytical computation of symbol error probability, we repeatedthe simulation with 25, 100, 225, 400, and 625 terms.The results are shown in Fig. 8 for the case where the numberof constellation points is . It is clear from Fig. 8 that,for the range of SNR and parameters considered,is sufficient for high accuracy. However, if we are interestedin higher noise scenarios, then fewer terms are required. Fig. 8also shows that, for example, at SNR termsis sufficient. This dependence on the parameters illustrates thevalue of the error bound in (14), as it can be used to determinethe appropriate number of terms needed adaptively.


Fig. 8. Symbol Error Probability versus SNR for an M-PSK CommunicationSystem with constellation points. Solid lines show analytical resultscomputed from the complex Double Gaussian distribution with varying numberof terms , while dotted lines show average errorresults from Monte Carlo simulations.

VI. DISCUSSION AND CONCLUSION

We derived a new distribution, which we refer to as the com-plex double Gaussian distribution, to describe the product of twoindependent complex Gaussian random variables. We also de-rived the special cases where one or more of the inputs is zeromean. We analyzed the convergence of this result, which con-tains a doubly infinite summation, and derived an upper boundon the truncation error. We showed that the number of termsneeded for accurate results varies with the Rician -factors (theratio of mean-squared to variance) for both inputs. We adap-tively compute the PDF by specifying a desired accuracy anduse the error bound to adaptively determine how many termsare needed. We applied this result to derive the optimal filter,in a NeymanPearson sense, for a time reversal based detectionsystem, wherein the received signal is distributed according tothe complex Double Gaussian distribution and verified that thedetector outperforms the standard energy detector for that sce-nario. We also presented a single-channel MPSK communica-tion system for which the linear combiner output follows thecomplex Double Gaussian distribution. Using this fact, we de-rived the SEP and verified our results using Monte Carlo trials.This distribution is a novel result and is useful in a wide arrayof applications.

APPENDIX APROOF OF THEOREM 4.1

To begin, we recall the polar representationsand . The product is written

(59)

where , and isthe principal value of , defined over the interval .

Recall that the desired distribution is . Asan intermediate step, we will compute the joint distribution

, using Bayes theorem [19]:

(60)

and then perform the transformation of random variables. Since and are independent, the joint dis-

tribution is simply the product of two Riciandistributions:

(61)

In order to compute the distribution of conditioned onand , we convolve two Tikhonov distributions circularly overthe interval [19]:

(62)

We make the substitution

(63)

(64)

where

(65)

The integral solution for (64) can be found in [20][22]. Wesubstitute (61) and (64) into (60):

(66)

Next, we define the nonlinear transformation

(67)

which results in the distribution

(68)

noting that is now a function of . We integrate out the dummyvariable :

(69)

We first note the identity [22, Sec., (8.445)]

(70)


for integer values . We recall the definition for in (65) anddefine and according to (3). Thus, the modified Besselfunction can be written

(71)

We make use of the trinomial expansion, which can be easilyderived from the binomial expansion:

(72)

Thus

(73)

We insert the expanded form of into (69):

(74)

If we make the substitution , then the inte-gral portion of (74) becomes

(75)

From the hyperbolic cosine identity [22, Sec. (1.311.3)] and theintegral solution [22, Sec. (3.337.1)], (75) reduces to

(76)

where is the modified Bessel function of the second kindwith order and argument [7]. From (76), we see that (74)reduces to

(77)

Now, we reorder the summation indices to place on the inside,and isolate all of the terms dependent upon :

(78)

We recall the Bessel function identity presented in (70); thisleads to the solution

(79)

where is defined by

(80)

APPENDIX BPROOF OF LEMMA 4.2

We recall the definition of the summation term in (11) and thePDF in (6). From those equations, can be written

(81)

We also recall the truncation error in (13); we can thus writewith

(82)

for large (83)


We assume that the number of terms is sufficiently large sothat for all four error regions (see Fig. 2), we can apply theapproximation [7]. Thus,

(84)

A. Error Region 1

We turn first to Error Region 1 (see Fig. 2), and wish to showthat the infinite summation over is upper bounded by the geo-metric series:

(85)

In order to define , we look at the ratio between successiveterms of , as increases

(86)

Note that we negated the order of the modified Bessel func-tion of the second kind. This operation is permitted because

[7]. For the ratio of modified Bessel func-tions, we cite the inequality [23, Sec. (1.13)]:

(87)

This can be used to create a relaxation such that ,which is defined as

(88)

This equation is monotonically decreasing with (simplifica-tion yields a polynomial with principal order of ). Thus, wecan provide an upper bound with the smallest value of in theseries, ; thus: . This leads to the upper boundgiven in (85), as long as In Table I, we demonstratethat this is not always true for small N. However, the values forwhich this is not valid are also the values of for which theerror bound is large. Therefore, whenever , we cansay that should be incremented. Inserting (85) into the fullsummation yields the bound:

(89)

B. Error Region 2

We turn to error region 2 (see Fig. 2) and note that its struc-ture is similar, so the approach will be the same. We begin bydefining the ratio between successive terms of , this time asincreases:

(90)

We apply (87) to define an upper bound , where

(91)

This equation is monotonically decreasing with (the sim-plification yields a polynomial with principal order of ),thus . Thus, we have the upper bound for errorregion 2:

(92)

As with , this bound is only valid when .

C. Error Region 3

For error region 3 (see Fig. 2), we have the boundsand . The inner summation is similar to that

in error region 1. Thus, we begin with (89), and adjust the sum-mation limits:

(93)

Recalling their definitions in (84) and (88), we have

(94)

(95)

We define as the ratio between successive terms of the sum-mation, as increases:

(96)

We note that is a monotonically decreasing function forlarge (as increases, the ratio approaches 1and becomes a noncontributing factor). For this reason, we cansay that

(97)


This allows us to define the upper bound

(98)

For large , the ratio is approximately 1, thus, we canwrite

(99)

This is a monotonically decreasing function of and can thus beupper-bounded by its limiting value (at ): . Thisleads to the upper bound on the summation for error region 3:

(100)

Once again, this bound is only valid when , which wecan easily show to be true for all that satisfy the inequality

(101)

D. Error Region 4Finally, we turn to error region 4 (see Fig. 2) and note its

similarity to error region 2. We start with (92) and adjust thesummation limits

(102)

Recalling their definitions in (84) and (91), we have

(103)

(104)

We define the ratio

(105)

We note that is a monotonically decreasing function forlarge (as increases, the ratio approaches 1and becomes a noncontributing factor). For this reason, we cansay that

(106)

This allows us to place an upper bound on the ratio

(107)

For large , the ratio is approximately 1; we can write

(108)

This is a monotonically decreasing function of and can beupper-bounded by its limiting value (at ):

(109)

We can see that . This leads to the upper bound on thesummation for error region 3:

(110)

E. Synthesis

We take the bounds calculated in (89), (92), (100), and (110),and insert them into the error term defined in (13), to obtain theupper bound on the error

(111)

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Nicholas ODonoughue (S03M12) received theB.S. degree in computer engineering from VillanovaUniversity, Villanova, PA, in 2006 and the M.S. andPh.D. degrees in electrical and computer engineeringfrom Carnegie Mellon University, Pittsburgh, PA, in2008 and 2011, respectively.His general research interests lie in the area of sta-

tistical signal processing, with applications in radar/sonar and communications and a special interest inthe technique of time reversal signal processing.Dr. ODonoughue is a recipient of the 2006 Na-

tional Defense Science and Engineering Graduate (NDSEG) Fellowship, the2006 Dean Robert D. Lynch Award from the Villanova University EngineeringAlumni Society, and the 2006 Computer Engineering Outstanding StudentMedallion from Villanova University. He has published more than 20 technicaljournal and conference papers, including two that were chosen as Best StudentPaper. He is a member of several IEEE societies, Tau Beta Pi, and Eta Kappa Nu.

Jos M. F. Moura (S71M75SM90F94)received the E.E. degree from Instituto SuperiorTcnico (IST), Lisbon, Portugal, and the M.Sc., E.E.,and D.Sc. degrees in Electrical Engineering andComputer Science from the Massachusetts Instituteof Technology (MIT), Cambridge.He is University Professor at Carnegie Mellon

University (CMU), Pittsburgh, PA, with the De-partments of Electrical and Computer Engineeringand, by courtesy, BioMedical Engineering. He wasa Professor of IST and has been a visiting professor

at MIT. He is a founding co-director of the Center for Sensed Critical Infra-structures Research (CenSCIR) and manages a large education and researchprogram between CMU and Portugal, www.icti.cmu.edu. His research interestsinclude statistical and algebraic signal and image processing. He has publishedover 450 technical journal and conference papers and has eight issued patentswith the USPTO.Dr. Moura is the IEEE Division IX Director Elect (2011), to serve as

IEEE Division IX Director in 20122013. He was President of the IEEESignal Processing Society (SPS) (20082009), Editor-in-Chief for the IEEETRANSACTIONS IN SIGNAL PROCESSING, and was on the Editorial Boardof several other journals and steering and technical committees of severalconferences. He is a Fellow of the American Association for the Advancementof Science (AAAS) and a corresponding member of the Academy of Sciencesof Portugal. He received several awards, including the IEEE SPS TechnicalAchievement Award, the IEEE SPS Meritorious Service Award, the IEEEMillennium Medal, the CMUs College of Engineering Outstanding ResearchAward, and the Philip L. Dowd Fellowship Award for Contributions to Engi-neering Education. In 2010, he was elected University Professor at CMU.

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