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We propose an alternative distribution for modelling fading-shadowing wireless channels. This distribution presents certain advantages over the Rayleigh-lognormal distribution and the K distribution and has proved useful in the setting described. We obtain closed-form expressions for the average channel capacity and for the average bit error rate of differential phase-shift keying and of minimum shift keying when the new distribution is used. This distribution can be obtained exactly as the sum of mutual independent Gaussian stochastic processes, because it must represent the simulation of the fading channel; that is, it simulates the signal envelope. Finally, we describe practical applications of this distribution, comparing it with the Rayleigh-lognormal and K distributions.

Mobile communications systems must address various challenges that seriously degrade signal strength. A major problem in this respect is that of the fading signal, i.e., interference to the multiple scattered radio paths between the base station and the vicinity of the mobile receptor. When this occurs, the received signal exhibits rapid signal level fluctuations which are generally Rayleigh distributed.

The multipath channel resembles a physical communication channel and is characterised by bandwidth and gain. For the case of a mobile radio channel with constant gain and a linear phase response over the bandwidth that is greater than the bandwidth of the transmitted signal, the signal received at the terminal will undergo

Figure

Illustration of fading components.

The specular component is a phase-coherent ground-reflected wave that is related to points close to where the receptor (i.e., the vehicle) is dynamically located. This component is responsible for strong fades, with an amplitude comparable to that of the direct component, although its phase is opposite Suh [

Figure

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 [

The above conditions are met by the Slashed-Rayleigh (SR) distribution, a two-parameter distribution that was first proposed by Iriarte et al. [

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.e., taking a statistical approach). Then, as required for a fading distribution, we simulate the distribution from a summation of phasors while accounting for Doppler effects (i.e., a physical approach), by embedding the SR distribution within Clarkes’s model Rappaport [

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

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

As Tse ([

The K distribution (Abdi and Kaveh [

The SR distribution, proposed by Iriarte et al. [

If

Using the following expression, which relates the incomplete gamma function with the Kummer confluent hypergeometric function

Now, from (

Figure

Illustration of the

The cumulative distribution function of

Let

Therefore, from (

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 (

Now, by performing on (

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.e., the signal envelope.

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. [

Following Beckmann ([

Then, expressing (

In the following, we obtain the phase and the amplitude distributions.

The SR distribution satisfies the following:

The phase distribution is uniform, i.e.,

The amplitude distribution is given by (

From (

Then, the unconditional distribution (independent of

The unconditional distribution for the amplitude is given by

Now, in (

Hence, the result follows after identifying a gamma distribution within the integral.

According to result

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 (

For a single-input/single-output system, the amount of fading (see Abdi and Kaveh [

It is well known (see Abdi and Kaveh [

Now, from

The average BERs of DPSK and MSK for the SR distribution are given by

By applying the composite rule we obtain

Now, with the change of variable

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,

The average Shannon capacity of the SR fading channel is given by

Again, from the composite rule and using (

It is known (see Gautschi et al. [

Thus, we have

Figure

Parameter values for the RLN, K, and SR distributions.

Setting | ||||
---|---|---|---|---|

Model | Parameters | A | B | C |

RLN | | 0.63 | 0.51 | –1.57 |

| 0.85 | 1.21 | 1.56 | |

| ||||

K | | 1.00 | 1.00 | 1.00 |

| 0.35 | –0.37 | –0.65 | |

| ||||

SR | | 1.14 | 0.36 | 0.14 |

| 3.45 | 2.80 | 2.50 |

Average BERs of DPSK and MSK for the RLN, K, and SR distributions assuming the parameter values, for different settings, given in Table

The graphs show that the SR distribution is a valuable means of predicting the BER in multipath dispersion fades. Compared with the K distribution, SR also achieves a good fit with the RLN distribution. The third scenario is conclusive in this case. We recall that the RLN distribution is commonly used in DPSK and MSK modulation schemes. It should also be noted that, for the RLN distribution, there is no closed-form expression for the average BER, which must be calculated by numerical integration methods (generally, the Gauss-Hermite method). Cygan [

The analytical expressions of the SR and K distributions both include special functions, namely, the incomplete gamma function and the modified Bessel function, respectively. The two are similar and either can be used as a substitute for the RLN distribution. Therefore, the proposed distribution can be applied efficiently to capture the bleached shading aspects of the wireless channels.

Figure

Slopes of the pdf for the RLN, K, and SR distributions for the parameters given in Settings A, B, and C.

The distance or relative information between two probability distributions can be determined by the Kullback-Leibler divergence measure (see Hall [

Table

Numerical values of the JSD and ISE measures for the K and SR distributions in comparison with the RLN distribution.

Setting | ||||
---|---|---|---|---|

Measure | Model | A | B | C |

JSD | K | 0.0013 | 0.0499 | 0.0323 |

SR | 0.1882 | 0.0499 | 0.0246 | |

| ||||

ISE | K | 0.0019 | 0.0902 | 0.3280 |

SR | 0.0052 | 0.0677 | 0.1223 |

To properly model a fading process, a random variate must be generated according to the proposed density distribution. Moreover, the random variate must be generated at low computational cost.

To achieve this, we proceed as in Gómez-Déniz and Gómez [

A Monte Carlo simulation was coded in Matlab MATLAB [

Two datasets were generated through Monte Carlo simulations with 100 000 samples and then compared with the analytical model. Figure

Comparison of the analytic SR (

Table

Comparison of the mean and variance values for the analytic SR and those estimated by statistical simulation, from the simulated data for two sets of parameters.

| | |
---|---|---|

Mean (analytic) | 1.0297 | 3.8375 |

Mean (simulated data) | 1.0491 | 3.8899 |

Relative error | 1.8879% | 1.3671% |

| ||

Variance (analytic) | 0.7397 | 5.2738 |

Variance (simulated data) | 0.7360 | 5.2240 |

Relative error | 0.4974% | 0.9441% |

This kind of simulation is only valid for mathematical analysis. To simulate the fading effects and provide the fading channel characteristics required, the modelling must be performed from the phasor formulation derived above and linked to physical variables (signal carrier frequency, signal sampling, speed of receiver, and Doppler effects). These questions are examined in the next section.

The physical model is completed by reformulating the phasors

Taking into account the above, we first performed a Monte Carlo simulation to obtain the dataset, using 15 scattered random phasors and 20 000 samples. We then compared the resulting pdf with the analytic SR pdf. This comparison is shown in Figure

Samples simulated by using phasors of the SR distribution (

Table

Means and variances for the analytic SR and the values estimated using phasors, from the simulated data for the two parameter sets.

| | |
---|---|---|

Mean (analytic) | 1.9694 | 4.4037 |

Mean (simulated data) | 1.9884 | 4.4491 |

Relative error | 0.9662% | 1.0312% |

| ||

Variance (analytic) | 1.1215 | 5.6075 |

Variance (simulated data) | 1.0633 | 5.3042 |

Relative error | 5.1908% | 5.4086% |

Figure

Simulated samples of the RLN distribution (

To conclude our demonstration of how the SR distribution efficiently models fading effects, we performed a simulation of the physical channel. Algorithm

Figure

Fading signal for the SR distribution (

In the present paper, the physical interpretation of the parameters of the SR distribution is not discussed, but it seems clear from expression (

A simulator block implementing the SR(

In this paper, we present a new distribution, the two-parameter SR distribution, which is competitive with the RLN distribution and the K distribution. The Rayleigh distribution is considered as a special case when one of the parameters tends to infinity. Parameter estimation for the new distribution is also discussed. From a mathematical standpoint, the distribution we propose has certain advantages over the RLN distribution within the framework of fading signal strength. Two methods to obtain the simulated envelope are discussed; the first is based strictly on the pdf of the distribution, and the second is based on a physical model constructed from the Rayleigh physical model, using the classical method described by Clarke. In addition, closed-form expressions are derived for the average channel capacity and the bit error rate (BER) for DPSK and MSK modulations for the proposed distribution.

The data used to support the findings of this work can be reproduced by referring to the mathematical formulas and the algorithms described. In addition, the codes used are available from the corresponding author upon request. These data include Mathematica and Matlab codes and therefore users will require the corresponding licenses.

The authors declare that they have no conflicts of interest.

E. Gómez-Déniz was partially funded by Grant ECO2013-47092, Ministerio de Economía y Competitividad, Spain, and ECO2017-85577-P, Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación. The research of H. W. Gómez was supported by MINEDUC-UA project, Code ANT 1755.