On the Second-Order Statistics of Correlated Cascaded Rayleigh Fading Channels
The second-order statistics of two correlated cascaded (double) Rayleigh fading channels are analyzed, where different relevant second-order cross-correlation functions of in-phase and quadrature components of the cascaded Rayleigh channels are derived. The level crossing rate (LCR) and average fade duration (AFD) of the cascaded channels are evaluated, and a single-integral form of the LCR is derived. Numerical results of the LCR and AFD are presented, and the effect of the correlation is illustrated.
Recently, the cascaded fading channels have attracted a lot of research interests [1–12]. The cascaded fading channels can be used to model different wireless communication scenarios, such as the keyhole fading channel , the mobile-to-mobile transmission channel , dual-hop fading channels , and radio frequency identification (RFID) pinhole channels [9–12].
In the published works of [1–9], the cascaded processes are assumed to be independent. For cascaded keyhole or pinhole fading channels, since the faded signals terminate and originate at the same keyhole or pinhole location, the correlation between cascaded fading processes may have some impacts on the performance of wireless transmissions. In [10–12], the bit error rate (BER) of RFID with multiple tags has been examined for correlated cascaded (forward-backscatter) channels. In fact, the completely dependent or correlated cascaded Rayleigh fading channels have been studied in earlier works , where the corresponding BERs of different modulations have been evaluated. The scenarios of keyhole multiple-input multiple-output (MIMO) channels and RFID pinhole channels with a correlation coefficient are illustrated in Figure 1.
(a) Keyhole MIMO channels
(b) RFID pinhole channels
In this paper, in contrast to the evaluation of first-order performance for correlated cascaded Rayleigh channels, such as the bit error rate given in [10–12], second-order statistics of correlated cascaded Rayleigh processes are analyzed and evaluated. The second-order statistics such as the LCR and AFD are useful for the design of practical wireless communication systems [14–19]. The LCR of a faded signal is defined as the average rate at which the signal envelope crosses a given level, and the AFD is the average duration that the signal envelope becomes lower than the given threshold [14, 15]. As addressed in , the second-order statistics, LCR and AFD, are also important for the design of real RFID systems. In , the LCR and AFD were analyzed for independent cascaded Rayleigh fading channels. In , the LCR and AFD were applied to the burst-error analysis for independent cascaded Rayleigh fading channels. In , the LCR and AFD of cooperative selection diversity were studied, where independent cascaded Rayleigh fading channels were assumed. In [18, 19], the LCR and AFD were evaluated for independent cascaded Nakagami-m fading channels.
In the context, two completely correlated Rayleigh processes  in cascaded are modelled, corresponding second-order correlation functions are derived, and the LCR and AFD of the correlated cascaded Rayleigh processes are evaluated. The result of the paper can be applied to the analysis of second-order statistics of correlated cascaded Rayleigh fading channels, such as a path of keyhole or pinhole fading channels shown in Figure 1.
In Section 2, the correlated cascaded Rayleigh random processes and their in-phase and quadrature processes are modelled, where various second-order correlation functions are derived. In Section 3, the LCR and AFD of the correlated cascaded processes are evaluated. In Section 4, numerical results of the LCR and AFD are given and the effect of the correlation is illustrated. Conclusions are drawn in Section 5.
2. Second-Order Correlation Functions of Correlated Cascaded Rayleigh Processes
Let be a Rayleigh process with the probability density function (pdf) given by the following: where . Then, two correlated Rayleigh processes in cascade can be modelled by the following: where is the time or phase offset between and . The joint pdf of and is  where is the zeroth order modified Bessel function of the first kind, and the correlation coefficient is in the range . Notice that when and have a linear relation, and reduces to the well-known chi-square distribution. With given by (3), it is straightforward to show that the pdf of is where is the zeroth order modified Bessel function of the second kind, and [22, (3.478.4)] is used to evaluate the integral. In Figure 2, the pdf with different correlation coefficients is shown. From Figure 2, when increases, the cascaded Rayleigh processes has a larger probability of yielding smaller values.
2.2. Second-Order Correlation Functions of In-Phase and Quadrature Processes
For , let and be the underlying in-phase and quadrature components of , and The time derivatives of , and give the second-order information of cascaded Rayleigh processes.
Le be the maximal Doppler frequency of the cascaded fading channels and be the uniformly distributed phase on . Based on the modeling used in [13, 14], for , we may write in the following forms  where is the corresponding fractional power for the th component, is the frequency, and is the time-delay . In practice, can be modelled by an exponential distribution with mean . The time derivatives of are Let denote the average operator. With (7), we can directly obtain that the second-order autocorrelation functions are Similarly, the second-order intercross-correlation functions of the in-phase and quadrature processes can be obtained as Let and . By using (7) and some manipulations, the second-order cross-correlation function and can be derived as where denotes the th-order Bessel function of the first kind, and [22, (3.715), (8.339), (3.895), and (8.473.1)] are applied to evaluate the integral.
When the two cascaded channels are using the same carrier frequency, we have . In Figure 3, this second-order cross-correlation of in-phase and quadrature processes with is plotted for . From Figure 2, the vibration range of the cross-correlation may increase when becomes higher.
Other non-zero second-order correlation functions can be derived in a similar way. Following manipulations similar to those used for deriving (10), we can obtain the related second-order correlation functions as follows: The above second-order correlation functions can be applied to the derivation of the cross-correlation between and that will be useful for the LCR and AFD derivation in Section 3.
2.3. Cross-Correlation Function of and
For the time derivative of has the form The correlation function between and under fixed and can be written as follows: where the variables in and in are omitted for simplification. To derive , we need to obtain the four expectations in the numerator of (13). Based on the model of central limit theorem considered in [14, 15], [24, (8.111)], all , and have a zero-mean Gaussian distribution. On the other hand, it is well known that if the random variables have a jointly zero-mean Gaussian distribution, can be evaluated by [19, (8.61)] the following:
Thus, we can use the property given by (14) to analyze the numerator of (13). The four expectations in the numerator of (13) can be derived by using (14) and the expectations of the forms and . For or in (14), we have
Using the result obtained in Section 2.2, and (13)–(15), we can simplify (13) into the form To evaluate (16), we employ the following relation derived in [15, (1.5.20)] and [23, (4.31b)] for and From (17), for , the correlation coefficient is a function of the maximum Doppler frequency and the phase offset between the cascaded processes. Substituting (17) with , we can simplify (16) into the form where we use the recursive relations of Bessel functions given by [22, (8.473.1), (8.473.4)], and is defined by For the derivation of the LCR and AFD of the correlated cascaded Rayleigh channels, in the next section, without loss of generality, is considered.
3. Derivation of LCR and AFD
The average LCR of a random process at can be evaluated by [14, 15] where denotes the time derivative of , and is the pdf of the time derivative conditioned on .
To evaluate the LCR given by (20), we first characterize the probability distribution of below, where Based on , is Gaussian distributed with zero mean and variance . Consequently, conditioned on fixed is Gaussian distributed with zero mean and the variance where denotes the covariance of and under fixed , and is given by (18) for . Let denote the conditional pdf of under fixed . Thus,
Let denote the conditional pdf of under fixed . Then, for the correlated cascaded processes, the LCR given by (20) can be rewritten in the form where is the Dirac delta function, has the form given by (3) with replaced by , and is given by (23) also with replaced by . Substituting (18) into (22) and (3) into (24), we can express the normalized LCR in the single-integral form that can be easily evaluated with a computing software, such as MATLAB. If , it can be easily check that (25) will reduce to the form given in [8, 17, 18] for independent cascaded Rayleigh channels.
Let be the cumulative distribution function (cdf) of . With the normalized LCR given by (25), the normalized AFD can be evaluated by  By using the pdf of given by (4), the cdf in the above AFD can be evaluated as where is the first-order modified Bessel function of the first kind, and a formula modified from [22, (6.521.4)] with () is used to write the integral into a closed-form.
4. Numerical Results of LCR and AFD
From (17), due to the property of Bessel functions, is not a monotone function of . However, as increases, the vibration range of tends to be smaller, and becomes smaller.
To demonstrate the impact of the correlation on the LCR and AFD, the normalized LCR of the correlated cascaded Rayleigh channels for different values of is plotted in Figure 4. The result in Figure 4 implies that when correlation increases, the cascaded envelope has more chances to enter a deep fading area. For example, as shown in Figure 4, the correlated cascaded channels may have a more crossing rate than independent cascaded channels to go into deep fade below −10 dB.
In Figure 5, the numerical AFD is plotted. According to Figure 5, as the correlation increases, the average duration when temporarily stays in the deep fading area can be shorter although enters the area more frequently.
Thus, the correlation between cascaded channels not only affects the BER as shown in [10–13] but also changes the second-order statistics which are more serious in a deep fading area.
Different second-order correlation functions of completely correlated cascaded fading channels have been derived and the corresponding LCR and AFD are analyzed, which can facilitate the design of wireless communication or RFID systems on keyhole or pinhole channels. Under the consideration of the channel correlation, the LCR and AFD will depart from those obtained for independent cascaded channels. Numerical results show that the correlation has a higher impact on the second-order statistics in the deep fading region. The evaluation of LCR and AFD will also be helpful for the design of interleaving transmission and encoding schemes over correlated cascaded fading channels. Further research on related topics may consider the second-order statistics of diversity schemes over correlated cascaded channels.
This paper was supported by the National Science Council, Taiwan, under Grant nos. NSC100-2220-E-155-001 and NSC100-2220-E-155-005.
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