Several components are involved; first, in the case of clear line of sight between the stationary source emitter and the mobile vehicle, no scattering mechanism is involved, although Doppler effects will be considered. However, a multipath or diffuse component phase-incoherent wave is created by the multiple random reflections scattering processes of the signal from scattered elements such as mountains and buildings.
This component has little directivity and its magnitude is assumed to be Rayleigh distributed while its phase is uniformly distributed. The specular component is a phase-coherent ground-reflected wave that is related to points close to where the receptor i. This component is responsible for strong fades, with an amplitude comparable to that of the direct component, although its phase is opposite Suh [ 5 ]. Figure 1 also shows the case of blocked line of sight between the stationary source and the mobile vehicle.
In this case, the diffuse component is as before and can be modelled by a Rayleigh distribution , but there is a new element, the shadowed direct component, produced by the scattering of the signal by the leaves, branches, and limbs of trees and by vegetation in general. In consequence, the signal is attenuated, to a degree that depends on the length of its path through the scattering element. The fading produced by this process is called fading-shadowing and can be modelled by the Rayleigh-lognormal RLN distribution Hansen and Meno [ 6 ] , although this has a complicated integral form, or by the K distribution Abdi and Kaveh [ 7 ].
The latter distribution may be viewed as a Rayleigh distribution with a gamma distribution and it has a simpler form than the RLN. This study examines fading-shadowing mechanisms in particular. However, as our proposal also contains the Rayleigh as a general case, it also applies to modelling the diffuse component. To achieve a probability distribution that will efficiently model fading effects, a distribution should be expressed through simple mathematical expressions and subsume the Rayleigh distribution.
Given this, parameter estimation is straightforward and second-order statistics such as fading descriptors level crossing-rate LCR , average fade duration AFD , average bit error rate BER , differential phase-shift keying DFSK , and minimum shift keying MSK can easily be computed see, for instance, Proackis and Salehi [ 8 ], Adachi et al.
The above conditions are met by the Slashed-Rayleigh SR distribution, a two-parameter distribution that was first proposed by Iriarte et al. However, the latter did not address the question of fading applications.
As we discuss below, this distribution is well suited to modelling flat fading effects in wireless communications. Among the benefits offered by the SR distribution, it includes the Rayleigh distribution, thus facilitating the physical modelling of multipath signal propagation through phasors complex signal representation. We complete the description of the SR distribution as follows. First, we simulate it by Monte Carlo analysis i.
Then, as required for a fading distribution, we simulate the distribution from a summation of phasors while accounting for Doppler effects i.
Although fading effects are apparent in general for mobile communications, they are especially noticeable in moving vehicles, where the signal received commonly presents multipath components and not just the direct component. However, for a stationary receiver, too, if the surrounding objects are moving, Doppler shift on the multipath components will affect the quality of the transmitted signal. The models we discuss can be applied to both stationary and moving receptors.
The outline of this paper is as follows. The SR distribution is presented in Section 2. Section 3 then introduces the SR phasor, shows some simulation plots, and discusses specific mathematical measures for the new distribution, together with metrics related to modelling fading effects in wireless communication channels.
In Section 4 , the SR distribution is compared with other distributions that are commonly used to account for the statistics of mobile radio signals. The simulation of the SR distribution through Monte Carlo analysis and the physical modelling of the multipath fading channel are discussed in Section 5. Finally, the main conclusions drawn are summarised in Section 6. The Rayleigh fading model can be used to simulate the situation in which a radio signal is scattered before it arrives at the receiver due to the presence of multiple objects in the environment.
According to the central limit theorem, given sufficient scatter, the channel impulse response will be well modelled as a Gaussian process irrespective of the distribution of the individual components. If there is no dominant component to the scatter, this process will have a zero mean and its phase will be uniformly distributed between and radians.
Therefore, the envelope of the channel response will be Rayleigh distributed, with the following probability density function pdf where is the expected value of. In this case we write. For this reason some alternatives to the Rayleigh fading model have been proposed, such as the RLN distribution with pdf, described by Hansen and Meno [ 6 ] and Stuber [ 13 ]. The K distribution Abdi and Kaveh [ 7 ] , obtained by compounding a Rayleigh distribution with a gamma distribution, is similar to the RLN distribution but it has a simpler structure and its pdf admits a closed form, although due to the Bessel function the estimates of the parameters are not direct.
Its pdf is given by where denotes the modified Bessel function of the second kind of order and argument and which has a complicated integral form. The main advantage of this special function is that it is included in most of the statistical software currently available, such as R, Matlab, and Mathematica see Ruskeepaa [ 14 ].
The SR distribution, proposed by Iriarte et al. Observe that the pdf given in 1 is obtained from 4 when tends to. Furthermore, this distribution tends to the Dirac delta function at 0 when tends to infinity.
Henceforth, when a random variable follows this distribution it will be represented as. If with scale parameter and kurtosis parameter , then 4 can be represented as where and the uniform distribution are independent and. Using the following expression, which relates the incomplete gamma function with the Kummer confluent hypergeometric function the pdf given in 4 can be rewritten as.
Figure 2 shows slopes of the pdf of the distribution, revealing the dependency of the scale parameter and the shape parameters the fading. The cumulative distribution function of is given by which can be used to obtain the hazard function of the random variable , given by. Then, for and , it follows that the - th moment of the proposed distribution is given by. Therefore, from 11 we obtain the mean and the variance of the distribution, which are given by respectively. Moment and maximum likelihood estimation is studied in detail in Iriarte et al.
The pdf of the SR distribution can be written as an infinite convex sum of Nakagami distributions. To do so, observe that from the series expansion of the Kummer confluent hypergeometric function in 8 this pdf can be rewritten as.
Now, by performing on 13 the change of variable , where and taking , 13 can be written as where which is the Nakagami distribution with parameters and. Thus, we conclude that the SR distribution studied here can be written as an infinite convex sum of Nakagami distributions.
In this section, we demonstrate that the SR distribution can be obtained in an exact form as the sum of mutual independent Gaussian stochastic processes, as is required in order to simulate the fading channel, i. It is known that Rayleigh fading envelopes can be generated from zero-mean complex Gaussian random variables. Other fading distributions see, for instance, Yacoub et al. Hence, in line with these precedents, we must now prove that the phase and the amplitude of a given propagating signal are distributed according to a uniform pdf in the interval and the SR distribution, respectively.
Following Beckmann [ 18 ], p. When is large, and assuming that is not correlated with the , both and will be distributed normally with mean 0 and variance. Goldsmith [ 19 ], see page 69, pointed out that under some conditions this is also true for small. Now, let , where represents the uniform distribution in.
Then, the joint distribution of and is. Then, expressing 18 as polar coordinates. Proposition 1. The SR distribution satisfies the following: i The phase distribution is uniform, i.
From 19 it is straightforward to obtain the conditional distributions of and given , which are given by Then, the unconditional distribution independent of for the phase is The unconditional distribution for the amplitude is given by Now, in 22 we perform the change of variable and obtain Hence, the result follows after identifying a gamma distribution within the integral.
According to result in Proposition 1 , if and , then , i. Moreover, this representation of the distribution facilitates parameter estimation via the expectation-maximisation EM algorithm. The SR distribution can readily be obtained as a scale mixture of the Rayleigh distribution and the uniform distribution, which facilitates the computation of some measures of interest in the framework of the fading channel, such as the amount of fading AF also known as the strength of intensity fluctuations , and the BER for DPSK and MSK when the SR distribution is employed as that of the fading channel.
First, in 8 we perform the change of variable to obtain which is a generalisation of the exponential distribution proposed in Bhattacharya [ 20 ]. Thus, when , the received signal power is distributed according to 24 with mean. We recall that the value of AF for the Rayleigh, , distribution is. Proposition 2. Here , where is the transmitted energy per bit and is the noise power spectral density and is the hypergeometric function. By applying the composite rule we obtain Now, with the change of variable expression 29 reduces to which is immediately identified with 28 after simple algebraic manipulation.
The expression for is obtained in a similar way. The average channel capacity for fading channel is a useful metric, in that it provides an estimation of the information rate that the channel can support, with little probability of error. Channel capacity, see, for instance, Li et al.
Typical scenarios include cellular telecommunications where there are large number of reflections from buildings and the like and also HF ionospheric communications where the uneven nature of the ionosphere means that the overall signal can arrive having taken many different paths. The Rayleigh fading model is also appropriate for for tropospheric radio propagation because, again there are many reflection points and the signal may follow a variety of different paths. The Rayleigh fading model is particularly useful in scenarios where the signal may be considered to be scattered between the transmitter and receiver.
In this form of scenario there is no single signal path that dominates and a statistical approach is required to the analysis of the overall nature of the radio communications channel.
Rayleigh fading is a model that can be used to describe the form of fading that occurs when multipath propagation exists. In any terrestrial environment a radio signal will travel via a number of different paths from the transmitter to the receiver. The most obvious path is the direct, or line of sight path. However there will be very many objects around the direct path. These objects may serve to reflect, refract, etc the signal. As a result of this, there are many other paths by which the signal may reach the receiver.
Other parameters are the same as the single-time channel prediction system. The dimension of input layer sample is. The number of hidden layer neurons is. Output layer neurons are and , respectively. For multi-time channel prediction system, as the prediction time increases, the corresponding error increases exponentially. We compare the prediction of two sampling methods, i. Figure 7 shows that prediction accuracy generally improves as the number of epochs increases.
Owing to the early stopping strategy, it is shown that the normal sample construction scheme stops after epochs and the sparse sample prediction scheme stops after 42 epochs. Figure 8 is the NMSE performance of two-sample construction schemes.
Moreover, in order to achieve the same target of NMSE, the latter has less epochs than the former, which is more practical. The SSCS effectively reduces the resource consumption without degrading system performance.
Figure 9 shows that the normal samples construction scheme stops after 86 epochs. The sparse sample prediction scheme stops after 61 epochs. Figure 10 is the NMSE of two-sample construction schemes. The multi-time prediction, just like the input and output prediction system, error increases exponentially with the different timeslot. With the weakening of the time domain correlations, the prediction errors of different time slot increase exponentially.
Figure 12 compares the prediction performance of three-layer neural networks and two-layer neural networks. It reveals that the three-layer neural network outperforms the two-layer neural network. However, its effectiveness is not obvious since the channel information is not very complicated.
The BP neural network with multi-hidden layer is introduced into the channel prediction application. The proposed prediction scheme can perform effectively with a short pilot overhead, which is suitable for resource-constrained communication scenes.
Meanwhile, we proposed two significant sample construction methods, which extremely improves the prediction performance and reduces the computing complexity. Wide experiences verified the effectiveness of our proposed scheme. The data used to support the findings of this study are available from the corresponding author upon request. This research was supported by NSFC no. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Article of the Year Award: Outstanding research contributions of , as selected by our Chief Editors. Read the winning articles. Journal overview. Special Issues. Academic Editor: Dajana Cassioli. Received 25 Jan Revised 26 Apr Accepted 04 Jun Published 25 Jul Abstract This paper presents a multi-time channel prediction system based on backpropagation BP neural network with multi-hidden layers, which can predict channel information effectively and benefit for massive MIMO performance, power control, and artificial noise physical layer security scheme design.
Introduction The future wireless communications 5G put forward the demands of high-speed transmission, quick access, high reliability, and strong security communications [ 1 ]. Figure 1. Figure 2. The weight matrices , are initialized randomly from 0 to 1. The threshold vectors , initialized to 0.
Set the training goal and learning rate a reasonable value, respectively; 2. Input the channel information training set. Calculate , and the , of the loss function and cost function according to equation 7 ; 5. According to equation 8 , the gradient of the output layer weight matrix and the gradient of the threshold are calculated respectively; 6. The weight matrix of the hidden layer and the gradient of the threshold vector are calculated are calculated according to 9 ; 7.
Update the weight matrix of the hidden matrix and the output layer and the threshold vectors ; 8. End while 9. Input the channel information test set and calculate the NMSE according to Algorithm 1.
The threshold vectors are initialized to 0. Set the training goal and the learning rate to a reasonable value, respectively. The intermediate variable is initialized to 1; 2. Input the channel information training set and verification set to train the neural network. For : 4. Calculate the hidden layer , output layer data and cost function of training set and verification set according to equation 7 , respectively; 5. Quit 7. Else do: 8.
According to 8 , calculate the gradient of the output layer weight matrix and threshold vector , respectively; According to 9 , calculate the gradient of hidden layer weight matrix and threshold vector , respectively; Update the weight matrix of hidden layer and output layer , and the threshold vector ; End for Input the channel information test set , and calculate the NMSE according to Algorithm 2.
Figure 3. Figure 4. Simulated channel and the predicted channel amplitude, phase. Figure 5. The prediction accuracy, BPNN versus existing methods.
Figure 6. The prediction accuracy under different channel models. Figure 7. Figure 8. Figure 9. Figure Two-layer neural network and three-layer neural network performance comparison. References M.
Agiwal, A. Roy, and N. Cheng, X. Zhang, and H. View at: Google Scholar L. Hu, H. Wen, B. Wu, J. Tang, and F. Wu et al. Duel-Hallen, H. Hallen, and T. Sharma and K. Arredondo, K. Dandekar, and G. Sternad and D.
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