On the Bivariate Nakagami-Lognormal Distribution and Its Correlation Properties

The bivariate Nakagami-lognormal distribution used to model the composite fast fading and shadowing has been examined exhaustively. In particular, we have derived the joint probability density function, the cross-moments, and the correlation coefficient in power terms. Also, two procedures to generate two correlated Nakagami-lognormal random variables are described. These procedures can be used to evaluate the robustness of the sample correlation coefficient distribution in both macroand microdiversity scenarios. It is shown that the bias and the standard deviation of this sample correlation coefficient are substantially high for large shadowing standard deviations found in wireless communication measurements, even if the number of observations is considerable.


Introduction
In wireless communications, the received signal is subjected to fading due to two physical mechanisms.On one hand, the multipath components cause rapid and deep fading in displacements of few wavelengths (small-scale area).This is the well-known short-term fading or fast fading, which has been extensively analyzed in the literature [1,2].This fast fading has been modeled statistically using the Rice, Rayleigh, Nakagami-, and Weibull distributions.The Nakagami- distribution is frequently employed to model the fast fading since it fits better than the other distributions in many measurement campaigns [3,4].On the other hand, the received signal fluctuates slowly around a mean in displacements of hundreds of wavelengths (large-scale area).This variation is known as long-term fading or shadowing.This shadowing is due to the temporal blockage of the direct component between the transmitter and receiver terminals.The shadowing is commonly modeled statistically by a lognormal distribution.Values of standard deviations for shadowing reported in the literature range from 5 to 12 dB in macrocells, from 6 to 10 dB in microcells, and from 3 to 10 dB in indoor environments [5].
In the analysis of wireless propagation channels, these components are traditionally separated to model both the fast fading and shadowing.Nevertheless, this separation is troublesome in scenarios as in indoor and vehicular channels, where the variation of the received signals is considerably high, with few displacements in terms of the wavelength, and thus these effects are severely overlapped.
Therefore, several distributions have been proposed to describe the composite fast fading and shadowing.Initially, the Rayleigh-lognormal distribution was used by Suzuki to model this compound fading [6].Also, the Nakagamilognormal distribution has been employed to model the composite fading [7][8][9].Recently, some other distributions as the - [10], the generalized- [11], or the mixture gamma (MG) distribution [12] have been proposed in the literature.Nevertheless, the results derived from a measurement campaign carried out in a macrocellular urban environment [13] have shown that the Nakagami-lognormal is the bestfit distribution compared with the Rayleigh-lognormal, -, and generalized- distributions.
The effect of the correlated shadowing or correlated fast fading on the performance parameters in wireless communication diversity has attracted the attention of several researches [14][15][16][17][18][19].Nevertheless, all the above works have assumed either correlated fast fading with independent shadowing or correlated shadowing with independent fast fading.

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International Journal of Antennas and Propagation Therefore, to the best of the authors' knowledge an analysis of the general bivariate distribution Nakagami-lognormal with correlated both fading and shadowing and arbitrary fading parameters is novel in the literature.
Thus, in this paper the joint probability density function (PDF) and the cross-moments of the bivariate Nakagamilognormal distribution are derived.Also, we have focused our attention on calculating the correlation coefficient between the composite signals.This correlation coefficient is particularized for two wireless scenarios of interest: macro-and microdiversity cases.
Two procedures to generate two correlated Nakagamilognormal random variables (RVs) are proposed and several examples for the sample correlation coefficient are evaluated in both macro-and microdiversity scenarios.
This paper is organized as follows.First, the joint PDF, the central moments, and the correlation coefficient of the Nakagami-lognormal distribution are derived in Section 2. Second, in Section 3 we describe two procedures for generating Nakagami-lognormal RVs and several results are analyzed.Finally, the conclusions are discussed in Section 4.
The correlation coefficient between   1 and   2 in power terms can be expressed as where var(⋅) denotes the variance.We can define   1 and   2 as correlated lognormal RVs corresponding to shadowing processes, whose marginal PDFs can be written as where being the mean of the associated Gaussian process of the shadowing, expressed in dBV/m,   the standard deviation of the associated normal distribution corresponding to the lognormal shadowing, expressed in dB.
where Ω  is a lognormal RV which represents the shadowing process.Note that the PDF given by ( 6) corresponds to the Nakagami-lognormal PDF in power terms given by [8, (8)].
The th moment of   ,  = 1, 2, is calculated as where (⋅) denotes expectation.Note that ( 7) is equivalent to [21, (2.59)] by substituting 2 instead of  in (7), which corresponds to the th moment of the Gamma-lognormal distribution.
Assuming   = 1 in ( 8) and ( 10), the infinite summation is not convergent.In order to overcome it, we can alternatively express the joint PDF for
Next, we analyze the correlation coefficient between the composite fast fading shadowing of the received powers in two wireless communication scenarios.
Case 1 (macrodiversity).The macrodiversity is the technique used to combat the effects of the shadowing, by combining the signals received in several antennas separated by hundreds of wavelengths.For instance, in the macrodiversity scenario depicted in Figure 1, the mobile terminal is connected simultaneously to two base stations (BSs), BS1 and BS2.The gain obtained with macrodiversity in this scenario depends strongly on the cross-correlation coefficient of the received field strengths at BS1 and BS2.
The behavior of the site-to-site shadowing correlation coefficient has been extensively analyzed in the literature [5,26,27].In [5], a model of the shadowing correlation has been proposed where the cross-correlation coefficient is given by where  1 is the smallest of the two path lengths  1 and  2 ;  is a parameter of adjustment which depends on the size and height of terrain and clutter, and according to the height of the BS antenna relative to them; and being the shadowing correlation distance.
In this situation, it is reasonable to assume that the short-term variations of the signal envelopes at BS1 and BS2 are independent, and consequently   = 0. Hence, the correlation coefficient between the instantaneous powers received at BS1 and BS2 can be easily simplified from (13) as where   is given by (14).
Case 2 (microdiversity).The microdiversity has been extensively used to reduce the harmful effects of the fast fading.
The most common methods employed in microdiversity are spatial, polarization, frequency, angular, and temporal [2].
In order to obtain a high degree of improvement using microdiversity, two conditions should be fulfilled: (i) low cross-correlation coefficients between individual diversity branches and (ii) similar mean power available from each branch of the combiner [5].For instance, the correlation coefficient using spatial microdiversity at the BS depends basically on the antenna separation, the direction of arrival of the main contribution arriving from the mobile station (MS), and the angular spread corresponding to the scatterers surrounding the MS [28].
It is reasonable to assume the following hypotheses in a microdiversity scenario: (i) total shadowing correlation, that is,   = 1; (ii) identical distributions for shadowing processes, that is,  =  1 =  2 ; and (iii) identical fading parameters for the short-term distributions, that is,  =  1 =  2 .Accordingly, the correlation coefficient between the composite distributions, whose joint PDF is given by (11), is simplified from (13) to In this case, the correlation coefficient in power terms is limited to

Numerical Results
In this section, we compare the values of the correlation coefficient given by ( 13) to those obtained by generating two correlated Nakagami-lognormal RVs using two methods.
3.1.Nakagami-Lognormal RV Generation.First, two correlated Gaussian variables will be generated following the next method.Let  1 be a Gaussian RV with mean  1 and standard deviation  1 , denoted as  1 ∼ N( 1 ,  1 ).
Second, we generate two correlated lognormal variables using the following transformations: Hence, Ω 1 and Ω 2 are lognormal distributed, denoted as {Ω 1 , Ω 2 } ∼ L( 1 ,  1 ;  2 ,  2 ;   ).Note that the mean, the variance, and the correlation coefficient between Ω 1 and Ω 2 are given by Mostafa and Mahmoud [22] where   and   are related to the logarithmic parameters,   and   , in (4) and ( 5), respectively.Finally, we generate two correlated Nakagami- variables with arbitrary fading parameters, whose mean powers are correlated lognormal RVs.These RVs, denoted as { 1 ,  2 } ∼ M( 1 , Ω 1 ;  2 , Ω 2 ;   ), can be generated by following the method described in [29].Each RV,   ,  = 1, 2, follows a Nakagami- distribution, whose PDF is given by where   is the fading parameter; Ω  is the mean power calculated in this case from (19); and   is the power correlation coefficient.Note that  2 ≥  1 and   < √ 1 / 2 .Estimated values of the fading parameter typically range from 1 to 3. For example, a mean of 1.56 and a standard deviation of 0.34 for the estimated fading parameters were reported in [4].

Sample Correlation Coefficient.
We have generated  observations of correlated Nakagami-lognormal RVs following the methods described above.Hence, the sample coefficient correlation, , is calculated. samples of the sample correlation coefficient are obtained in order to estimate the bias and the variance of .The variance of  depends strongly on the skewness of the marginal Nakagami-lognormal distributions as it was shown for different examples of the lognormal distribution in [30].The skewness of the marginal Nakagami-lognormal distributions can be evaluated from (7) as where   is the mean of   ,  = 1, 2. The skewness,   , increases with   .For instance, the skewness oscillates from 3.55 to 12.11 with   = 1 for shadowing standard deviations from 6 to 10 dB, respectively.Nevertheless, the skewness decreases slightly with the fading parameter,   .For instance,   = 3.55 for   = 6 dB and   = 1 and   = 3.15 for   = 6 dB and   = 3.   .The results correspond to spatial microdiversity, with   = 1,  =  1 =  2 , and   =  1 =  2 .In particular, the effect of the sample size, , and the number of samples, , on the variation of the standard deviation is illustrated in Figure 2. In this figure, the parameters of the distribution are  =  1 =  2 = 1.2 and   =  1 =  2 = 8 dB.The sample size, which corresponds to the number of observations used in simulations, is  = 1000 and 100000, and the number of samples is  = 25, 50, and 100.Obviously, the higher the number of observations, , the smaller the standard deviation.Also, the standard deviation reduces as the fast fading correlation coefficient,   , grows.Nevertheless, the standard deviation of  is scarcely affected by the number of samples calculated; that is,  equals 25, 50, and 100.
In Figure 3, the effect of both the fading parameter, , and the shadowing standard deviation,   , on the standard deviation of the sample correlation coefficient is analyzed for  = 100000 and  = 50.The standard deviation of the sample correlation coefficient depends strongly on the shadowing standard deviation.Also, it can be observed that the standard deviation of  for   = 10 dB decreases as  grows.In general, the standard deviation of  tends to reduce for high correlation coefficient values.
Figure 4 shows the mean of the sample correlation coefficient and the correlation coefficient  given by (17), that is, the analytical prediction, as a function of the fast fading correlation coefficient.The simulations are performed for  =  is higher for small fast fading correlation coefficients and for  =  1 =  2 = 3.2;   =  1 =  2 = 8 dB.
In Figure 5, the correlation coefficient given by ( 16) and the mean of the sample correlation coefficient are plotted as a function of the angle between the MS and the base stations, , for  values.This bias is considerably high for a small number of observations, that is,  = 10000 and  = 100000, specially for  close to 0 ∘ .Also, from the results shown in Figure 5, the higher the shadowing standard deviation is, the larger the values of the bias are using the same number of observations and .This bias is large for   =  1 =  2 = 10 dB and low values of  even for a high number of observations; for example, the bias is around 0.035 for  =  1 =  2 = 1.2,  ≤ 10 ∘ , and  = 1000000.

Conclusions
In this paper, we have analytically derived the PDF of bivariate Nakagami-lognormal distribution with arbitrary distribution parameters of the marginal distributions and arbitrary correlation.Also, the correlation coefficient between the compound signals has been obtained as a function of the distribution parameters and both the fast fading and shadowing correlation coefficients.This correlation coefficient has been particularized for specific scenarios of wireless communications: micro-and macrodiversity.In microdiversity, the correlation coefficient between the composite signals is substantially higher than the fast fading correlation coefficient.Otherwise, the composite correlation coefficients are small (in the examples lower than 0.32) in the macrodiversity scenario.
Two procedures to generate two correlated Nakagamilognormal variables have been described.In examples of both microdiversity and macrodiversity scenarios, we have evaluated the sample correlation coefficient between the composite Nakagami-lognormal signals, comparing it to the analytical correlation coefficient derived previously.In general, this sample correlation coefficient is positive biased.
From those examples, it can be observed that the bias and the standard deviation of the sample correlation coefficient depend substantially on the shadowing standard deviation.
The results reported in this work can be of interest in the assessment of performances (outage probabilities, bit error rates, etc.) in diversity receivers with highly dynamical environments, such as vehicular or indoor wireless channels.In these scenarios, the separation of the fast fading and shadowing processes is cumbersome, and the received signals at each branch of the diversity combiner can be accurately modeled as Nakagami-lognormal composite distributions.

Figure 3 :Figure 4 :
Figure 3: Standard deviation of the sample correlation coefficient as a function of the fast fading correlation coefficient for different parameters of the marginal distributions.The number of observations and number of samples are  = 100000 and  = 50, respectively.
The power correlation coefficient between the associated Gaussian processes,  1 = 20 log   1 and  2 = 20 log   2 , is denoted by   and it is defined in the same way as (2), by substituting  1 and  2 for   1 and   2 , respectively.If we calculate   =    ⋅    ,  = 1, 2, then   is Nakagamilognormal distributed, with PDF given by