On the Second-Order Statistics of Correlated Cascaded Rayleigh Fading Channels

Abstract

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.

1. Introduction

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 [4], the mobile-to-mobile transmission channel [5], dual-hop fading channels [8], 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 [13], 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

Figure 1 

The scenarios of keyhole MIMO channels and 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 [20], the second-order statistics, LCR and AFD, are also important for the design of real RFID systems. In [8], the LCR and AFD were analyzed for independent cascaded Rayleigh fading channels. In [16], the LCR and AFD were applied to the burst-error analysis for independent cascaded Rayleigh fading channels. In [17], 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 [13] 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

2.1. Modeling

Let 𝛼_𝑖(𝑡)(𝑖=1,2) be a Rayleigh process with the probability density function (pdf) given by the following: 𝑓_𝑖𝛼_𝑖=𝛼_𝑖𝜎²_𝑖𝑒^{−𝛼2𝑖/(2𝜎2𝑖)},𝛼_𝑖≥0, (1) where 2𝜎²_𝑖=𝐸[𝛼²_𝑖]. Then, two correlated Rayleigh processes in cascade can be modelled by the following: 𝑅(𝑡,𝜏)=𝛼₁(𝑡)𝛼₂(𝑡+𝜏),𝑡,𝜏≥0, (2) where 𝜏 is the time or phase offset between 𝛼₁ and 𝛼₂. The joint pdf of 𝛼₁ and 𝛼₂ is [21] 𝑓₁₂𝛼₁,𝛼₂=𝛼₁𝛼₂𝜎²₁𝜎²₂1−𝜌²𝑒^{−(𝜎22𝛼21+𝜎21𝛼22)/(2𝜎21𝜎22(1−𝜌2))}×𝐼₀𝛼₁𝛼₂𝜌𝜎₁𝜎₂1−𝜌²,𝛼₁,𝛼₂≥0, (3) where 𝐼₀(⋅) is the zeroth order modified Bessel function of the first kind, and the correlation coefficient is in the range −1<𝜌<1. Notice that when 𝜌=1,𝛼₁ and 𝛼₂ have a linear relation, and 𝑅 reduces to the well-known chi-square distribution. With 𝑓₁₂(𝛼₁,𝛼₂) given by (3), it is straightforward to show that the pdf of 𝑅 is 𝑓_𝑅(𝑟)=^∞₀1𝛼₂𝑓₁₂𝑟𝛼₂,𝛼₂𝑑𝛼₂=𝑟𝜎²₁𝜎²₂1−𝜌²𝐼₀𝑟𝜌𝜎₁𝜎₂1−𝜌²𝐾₀𝑟𝜎₁𝜎₂1−𝜌², (4) where 𝐾₀(⋅) 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 𝜌 increases, the cascaded Rayleigh processes has a larger probability of yielding smaller values.

Figure 2 

Probability density function of correlated cascade Rayleigh processes.

2.2. Second-Order Correlation Functions of In-Phase and Quadrature Processes

For 𝑖=1,2, let 𝑋_𝑖 and 𝑌_𝑖 be the underlying in-phase and quadrature components of 𝛼_𝑖(𝑡), and 𝛼²_𝑖(𝑡)=𝑋²_𝑖(𝑡)+𝑌²_𝑖(𝑡). (5) The time derivatives of 𝑋_𝑖,𝑌_𝑖, and 𝛼_𝑖(𝑡) give the second-order information of cascaded Rayleigh processes.

Le 𝑓_𝑚 be the maximal Doppler frequency of the cascaded fading channels and 𝜃_𝑘 be the uniformly distributed phase on [0,2𝜋]. Based on the modeling used in [13, 14], for 𝑖=1,2, we may write (𝑋_𝑖,𝑌_𝑖) in the following forms [23] 𝑋_𝑖(𝑡)=lim_𝑛→∞2𝜎²_𝑖^𝑛_𝑘=1𝑎_𝑘cos𝜔_𝑘𝑡−𝜔_𝑐,𝑖𝑇_𝑘,𝑌_𝑖(𝑡)=lim_𝑛→∞2𝜎²_𝑖^𝑛_𝑘=1𝑎_𝑘sin𝜔_𝑘𝑡−𝜔_𝑐,𝑖𝑇_𝑘, (6) where 𝜔_𝑘=2𝜋𝑓_𝑚cos𝜃_𝑘,𝑎_𝑘 is the corresponding fractional power for the 𝑘th component, 𝜔_𝑐,𝑖 is the frequency, and 𝑇_𝑘 is the time-delay [15]. In practice, 𝑇_𝑘 can be modelled by an exponential distribution with mean 𝑇 [15]. The time derivatives of (𝑋_𝑖,𝑌_𝑖) are 𝑋̇_𝑖(𝑡)=−lim_𝑛→∞2𝜎²_𝑖^𝑛_𝑘=1𝑎_𝑘𝜔_𝑘cos𝜃_𝑘sin𝜔_𝑘𝑡−𝜔_𝑐,𝑖𝑇_𝑘,𝑌̇_𝑖(𝑡)=lim_𝑛→∞2𝜎²_𝑖^𝑛_𝑖=1𝑎_𝑘𝜔_𝑘cos𝜃_𝑘cos𝜔_𝑘𝑡−𝜔_𝑐,𝑖𝑇_𝑘. (7) Let ⟨⋅⟩ denote the average operator. With (7), we can directly obtain that the second-order autocorrelation functions are 𝑋_𝑖(𝑡)𝑋̇_𝑖(𝑡+𝜏)=𝑌_𝑖(𝑡)𝑌̇_𝑖(𝑡+𝜏)=0,𝑖=1,2. (8) Similarly, the second-order intercross-correlation functions of the in-phase and quadrature processes can be obtained as 𝑋_𝑖(𝑡)𝑌̇_𝑖(𝑡+𝜏)=𝑋̇_𝑖(𝑡)𝑌_𝑖(=𝑋̇𝑡+𝜏)_𝑖(𝑡)𝑌̇_𝑖(𝑡+𝜏)=0,𝑖=1,2. (9) Let Δ𝜔=𝜔_𝑐,2−𝜔_𝑐,1 and 𝜔_𝑚=2𝜋𝑓_𝑚. By using (7) and some manipulations, the second-order cross-correlation function ⟨𝑋̇₁(𝑡)𝑋̇₂(𝑡+𝜏)⟩ and ⟨𝑌̇₁(𝑡)𝑌̇₂(𝑡+𝜏)⟩ can be derived as 𝑋̇₁(𝑡)𝑋̇₂(𝑡+𝜏)=𝑌̇₁(𝑡)𝑌̇₂(𝑡+𝜏)=𝜎₁𝜎₂₀^2𝜋∞₀𝑒^{−𝑇𝑘/𝑇}2𝜋𝑇×cos𝜔_𝑚𝜏cos𝜃_𝑘−Δ𝜔𝑇_𝑘𝑑𝜃_𝑘𝑑𝑇_𝑘+𝜎₁𝜎₂𝜔²_𝑚0^2𝜋∞₀𝑒^{−𝑇𝑘/𝑇}4𝜋𝑇×cos2𝜃_𝑘cosΔ𝜔𝑇_𝑘×cos𝜔_𝑚𝜏cos𝜃_𝑘𝑑𝜃_𝑘𝑑𝑇_𝑘+𝜎₁𝜎₂𝜔²_𝑚0^2𝜋∞₀𝑒^{−𝑇𝑘/𝑇}4𝜋𝑇×cos2𝜃_𝑘sinΔ𝜔𝑇_𝑘×sin𝜔_𝑚𝜏cos𝜃_𝑘𝑑𝜃_𝑘𝑑𝑇_𝑘=𝜎₁𝜎₂𝜔²_𝑚21+Δ𝜔𝑇²×𝐽₀𝜔_𝑚𝜏−𝐽₂𝜔_𝑚𝜏, (10) where 𝐽_𝑖(⋅)(𝑖=0,1,2,…) denotes the 𝑖th-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 Δ𝜔=0. In Figure 3, this second-order cross-correlation of in-phase and quadrature processes with Δ𝜔=0 is plotted for 𝜎²₁=𝜎²₂=1. From Figure 2, the vibration range of the cross-correlation may increase when 𝑓_𝑚 becomes higher.

Figure 3 

Cross-correlation function of in-phase and quadrature processes of the cascaded fading channels.

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: 𝑋̇₁(𝑡)𝑌̇₂(𝑡+𝜏)=−𝑌̇₁(𝑡)𝑋̇₂(𝜎𝑡+𝜏)=−¹𝜎₂𝜔²_𝑚Δ𝜔𝑇21+Δ𝜔𝑇²×𝐽₀𝜔_𝑚𝜏−𝐽₂𝜔_𝑚𝜏,𝑋̇₁(𝑡)𝑋₂(𝑡+𝜏)=−𝑋₁(𝑡)𝑋̇₂(𝑌̇𝑡+𝜏)=−₁(𝑡)𝑌₂(𝑌𝑡+𝜏)=−₁(𝑡)𝑌̇₂(𝜎𝑡+𝜏)=−¹𝜎₂𝜔_𝑚𝐽₁𝜔_𝑚𝜏1+Δ𝜔𝑇²,𝑋₁(𝑡)𝑌̇₂(𝑡+𝜏)=𝑌̇₁(𝑡)𝑋₂(𝑋̇𝑡+𝜏)=−₁(𝑡)𝑌₂(𝑌𝑡+𝜏)=−₁(𝑡)𝑋̇₂(=𝜎𝑡+𝜏)₁𝜎₂𝜔_𝑚Δ𝜔𝑇𝐽₁𝜔_𝑚𝜏1+Δ𝜔𝑇². (11) The above second-order correlation functions can be applied to the derivation of the cross-correlation between 𝛼̇₁(𝑡) and 𝛼̇₂(𝑡+𝜏) that will be useful for the LCR and AFD derivation in Section 3.

2.3. Cross-Correlation Function of 𝛼̇₁(𝑡) and 𝛼̇₂(𝑡)

For 𝑖=1,2 the time derivative of 𝛼_𝑖(𝑡) has the form 𝛼̇_𝑖(𝑡)=𝑋_𝑖(𝑡)𝑋̇_𝑖(𝑡)+𝑌_𝑖(𝑡)𝑌̇_𝑖(𝑡)𝑋²_𝑖(𝑡)+𝑌²_𝑖(𝑡)=𝑋_𝑖(𝑡)𝑋̇_𝑖(𝑡)+𝑌_𝑖(𝑡)𝑌̇_𝑖(𝑡)𝛼_𝑖(𝑡). (12) The correlation function between 𝛼̇₁(𝑡) and 𝛼̇₂(𝑡+𝜏) under fixed 𝛼₁(𝑡) and 𝛼₂(𝑡+𝜏) can be written as follows: 𝛼̇₁(𝑡)𝛼̇₂(=𝑋𝑡+𝜏)₁𝑋̇₁𝑋₂𝑋̇₂+𝑌₁𝑌̇₁𝑌₂𝑌̇₂+𝑋₁𝑋̇₁𝑌₂𝑌̇₂+𝑋₂𝑋̇₂𝑌₁𝑌̇₁𝛼₁(𝑡)𝛼₂(),𝑡+𝜏 (13) where the variables 𝑡 in (𝑋₁,𝑌₁) and (𝑡+𝜏) in (𝑋₂,𝑌₂) are omitted for simplification. To derive ⟨𝛼̇₁(𝑡)𝛼̇₂(𝑡+𝜏)⟩, 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 𝑋_𝑖,𝑋̇_𝑖,𝑌_𝑖, and 𝑌̇_𝑖 have a zero-mean Gaussian distribution. On the other hand, it is well known that if the random variables (𝑍₁,𝑍₂,𝑍₃,𝑍₄) have a jointly zero-mean Gaussian distribution, ⟨𝑍₁𝑍₂𝑍₃𝑍₄⟩ can be evaluated by [19, (8.61)] the following: 𝑍₁𝑍₂𝑍₃𝑍₄=𝑍₁𝑍₂𝑍₃𝑍₄+𝑍₁𝑍₃𝑍₂𝑍₄+𝑍₁𝑍₄𝑍₂𝑍₃. (14)

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 ⟨𝑍_𝑖𝑍_𝑗⟩ and ⟨𝑍_𝑖𝑍̇_𝑗⟩(𝑖,𝑗=1,2). For 𝑍=𝑋 or 𝑌 in (14), we have 𝑋_𝑖(𝑡)𝑌_𝑖(𝑋𝑡+𝜏)=0,₁(𝑡)𝑋₂(𝑡+𝜏)=𝑌₁(𝑡)𝑌₂(=𝜎𝑡+𝜏)¹𝜎₂𝐽₀𝜔_𝑚𝜏1+Δ𝜔𝑇²,𝑋₁(𝑡)𝑌₂(𝑡+𝜏)=−𝑋₂(𝑡)𝑌₁(𝑡+𝜏)=−Δ𝜔𝑇𝜎₁𝜎₂𝐽₀𝜔_𝑚𝜏1+Δ𝜔𝑇². (15)

Using the result obtained in Section 2.2, and (13)–(15), we can simplify (13) into the form 𝛼̇₁(𝑡)𝛼̇₂(=𝜎𝑡+𝜏)₂₁𝜎²₂𝜔²_𝑚𝛼₁(𝑡)𝛼₂(𝑡+𝜏)1+Δ𝜔𝑇²×𝐽²₀𝜔_𝑚𝜏−2𝐽²₁𝜔_𝑚𝜏−𝐽₀𝜔_𝑚𝜏𝐽₂𝜔_𝑚𝜏. (16) To evaluate (16), we employ the following relation derived in [15, (1.5.20)] and [23, (4.31b)] for 𝜌 and Δ𝜔𝑇𝜌²𝜔_𝑚𝜏=𝐽²₀𝜔_𝑚𝜏1+Δ𝜔𝑇². (17) From (17), for Δ𝜔=0, the correlation coefficient is a function of the maximum Doppler frequency and the phase offset between the cascaded processes. Substituting (17) with Δ𝜔=0, we can simplify (16) into the form 𝛼̇₁(𝑡)𝛼̇₂(𝑡+𝜏)=−𝜎²₁𝜎²₂𝛼₁(𝑡)𝛼₂()𝑑𝑡+𝜏𝑑𝜏²²𝐽²₀𝜔_𝑚𝜏=−𝜎²₁𝜎²₂𝜔²_𝑚𝜌^2′′𝛼₁(𝑡)𝛼₂()=𝑡+𝜏−𝜎²₁𝜎²₂𝜔²_𝑚𝜌^2′′𝑅(),𝑡,𝜏 (18) where we use the recursive relations of Bessel functions given by [22, (8.473.1), (8.473.4)], and (𝜌²)^′′ is defined by 𝜌^2′′=𝑑²𝜌²(𝑥)𝑑𝑥²_{𝑥=𝜔𝑚𝜏}. (19) For the derivation of the LCR and AFD of the correlated cascaded Rayleigh channels, in the next section, without loss of generality, Δ𝜔=0 is considered.

3. Derivation of LCR and AFD

3.1. LCR

The average LCR of a random process 𝑅 at 𝑅=𝑟 can be evaluated by [14, 15] 𝑁_𝑅(𝑟)=^∞₀𝑟𝑓̇_̇𝑅𝑅,(𝑟,𝑟)̇𝑑𝑟̇=𝑓_𝑅(𝑟)^∞₀𝑟𝑓̇_{̇𝑅∣𝑅}(𝑟̇∣𝑟)𝑑𝑟,̇ (20) where 𝑟̇ denotes the time derivative of 𝑟, and𝑓_{̇𝑅∣𝑅}(𝑟̇∣𝑟) is the pdf of the time derivative 𝑅̇ conditioned on 𝑅=𝑟.

To evaluate the LCR given by (20), we first characterize the probability distribution of 𝑅̇ below, where 𝑅̇(𝑡,𝜏)=𝛼₂(𝑡+𝜏)𝛼̇₁(𝑡)+𝛼₁(𝑡)𝛼̇₂(𝑡+𝜏). (21) Based on [15], 𝛼̇_𝑖 is Gaussian distributed with zero mean and variance 𝜎²_̇𝛼𝑖=2𝜋²𝑓²_𝑚𝜎²_𝑖. Consequently, conditioned on fixed (𝛼₁,𝛼₂),𝑅̇ is Gaussian distributed with zero mean and the variance 𝜎̇²_𝑅=𝛼²₂𝜎²_̇𝛼1+𝛼²₁𝜎²_̇𝛼2+2𝛼₁𝛼₂cov𝛼̇₁,𝛼̇₂, (22) where cov(𝛼̇₁,𝛼̇₂) denotes the covariance of 𝛼̇₁(𝑡) and 𝛼̇₂(𝑡+𝜏) under fixed (𝛼₁,𝛼₂), and is given by (18) for Δ𝜔=0. Let 𝑓_{̇𝑅∣𝑟,𝛼}(𝑟̇∣𝛼₁,𝛼₂) denote the conditional pdf of 𝑅̇ under fixed (𝛼₁,𝛼₂). Thus, 𝑓_{̇𝑅∣𝑟,𝛼}𝑟̇∣𝛼₁,𝛼₂∼Gaussian0,𝜎̇²_𝑅. (23)

Let 𝑓_𝑅∣𝛼(𝑟∣𝛼₁,𝛼₂) denote the conditional pdf of 𝑅 under fixed (𝛼₁,𝛼₂). Then, for the correlated cascaded processes, the LCR given by (20) can be rewritten in the form 𝑁_𝑅(𝑟)=^∞₀𝛿𝑟−𝛼₁𝛼₂𝑟𝑓̇_{̇𝑅∣𝑟,𝛼}𝑟̇∣𝑟,𝛼₁,𝛼₂𝑑𝑟̇×𝑓_𝑅∣𝛼𝑟∣𝛼₁,𝛼₂𝑓₁₂𝛼₁,𝛼₂𝑑𝛼₁𝑑𝛼₂=^∞₀^∞₀𝑟𝑓̇_{̇𝑅∣𝛼}𝑟𝑟̇∣𝛼₂,𝛼₂1𝑑𝑟̇𝛼₂𝑓₁₂𝑟𝛼₂,𝛼₂𝑑𝛼₂, (24) where 𝛿(⋅) is the Dirac delta function, 𝑓₁₂(𝑟/𝛼₂,𝛼₂) has the form given by (3) with 𝛼₁ replaced by 𝑟/𝛼₂, and 𝑓_{̇𝑅∣𝛼}(𝑟̇∣𝑟/𝛼₂,𝛼₂) is given by (23) also with 𝛼₁ replaced by 𝑟/𝛼₂. Substituting (18) into (22) and (3) into (24), we can express the normalized LCR in the single-integral form 𝑁_𝑅(𝑟)𝑓_𝑚=√𝜋𝑟𝜎²₁𝜎²₂1−𝜌²𝐼₀𝑟𝜌𝜎₁𝜎₂1−𝜌²×^∞₀𝜎²₁−4𝜎²₁𝜎²₂𝜌^2′′𝛼²₂+𝑟²𝜎²₂𝛼⁴₂×𝑒^{−((𝜎21𝛼22+𝜎22𝑟2/𝛼22)/2𝜎21𝜎22(1−𝜌2))}𝑑𝛼₂ (25) that can be easily evaluated with a computing software, such as MATLAB. If 𝜌=0, it can be easily check that (25) will reduce to the form given in [8, 17, 18] for independent cascaded Rayleigh channels.

3.2. AFD

Let 𝑃(𝑅≤𝑟) be the cumulative distribution function (cdf) of 𝑅. With the normalized LCR given by (25), the normalized AFD can be evaluated by [14] 𝑇_𝑅(𝑟)=𝑓_𝑚⋅𝑃(𝑅≤𝑟)𝑁_𝑅(𝑟). (26) By using the pdf of 𝑅 given by (4), the cdf in the above AFD can be evaluated as 𝑃(𝑅≤𝑟)=^𝑟₀𝑓_𝑅𝑦=1𝑑𝑦𝜎²₁𝜎²₂1−𝜌^2𝑟₀𝑦𝐼₀𝑦𝜌𝜎₁𝜎₂1−𝜌²×𝐾₀𝑦𝜎₁𝜎₂1−𝜌²𝑟𝜌𝑑𝑦=1−𝜎₁𝜎₂1−𝜌²𝐼₁𝑟𝜌𝜎₁𝜎₂1−𝜌²×𝐾₀𝑟𝜎₁𝜎₂1−𝜌²−𝑟𝜎₁𝜎₂1−𝜌²𝐼₀𝑟𝜌𝜎₁𝜎₂1−𝜌²×𝐾₁𝑟𝜎₁𝜎₂1−𝜌², (27) where 𝐼₁(⋅) is the first-order modified Bessel function of the first kind, and a formula modified from [22, (6.521.4)] with 𝐼_𝑖(𝑧)=𝑗^−𝑖𝐽_𝑖(𝑗𝑧) (𝑗=√−1) 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, 𝜌² is not a monotone function of 𝜏. However, as 𝜏 increases, the vibration range of 𝐽²₀(𝜔_𝑚𝜏) tends to be smaller, and 𝜌² 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 𝜏 is plotted in Figure 4. The result in Figure 4 implies that when correlation increases, the cascaded envelope 𝑅 has more chances to enter a deep fading area. For example, as shown in Figure 4, the correlated cascaded channels may have a 5%∼10% more crossing rate than independent cascaded channels to go into deep fade below −10 dB.

Figure 4 

Normalized LCR for 𝜎21=𝜎22=1.

In Figure 5, the numerical AFD is plotted. According to Figure 5, as the correlation increases, the average duration when 𝑅 temporarily stays in the deep fading area can be shorter although 𝑅 enters the area more frequently.

Figure 5 

Normalized AFD for 𝜎21=𝜎22=1.

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.

5. Conclusions

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.

Acknowledgment

This paper was supported by the National Science Council, Taiwan, under Grant nos. NSC100-2220-E-155-001 and NSC100-2220-E-155-005.