Rejection Methods for Nakagami-m Fading Simulation

CHEN Chao, ZHU Qiuming, WANG Chenghua College of Electronic Information Engineering

Nanjing University of Aeronautics and Astronautics Nanjing, China

Abstract—Nakagami-m distribution has gained more and more attention in wireless fading channel modeling. Much work has been done to generate the Nakagami-m variables. In this letter three rejection models for Nakagami-m fading are analyzed and compared. Simulation results show that Polynomial and Gaussian method have better performance while Uniform method is simpler. These results are helpful to balance the accuracy and complexity in channel simulation.

Keywords-Nakagami; simulation; rejection method; channel model;

I. INTRODUCTION Rayleigh and Rician distributions are commonly used to

describe the fading features of wireless channels. However, as the distance of communication gets longer the channel fading gets more severe so that the measured data manifest some deviation, which is firstly observed by Nakagami. Then he presented an empirical distribution Nakagami-m to fit the measured results[1]. With the different values of m, this distribution can model various channels from severe to moderate fading ones including Rayleigh and Rician channels.

For simulating the Nakagami-m distribution, which yields a satisfactory fit for the fading channels over a wide range of distance and frequency, several methods have been raised. Typical methods involve the brute force method[2], sum of sinusoids method[3], CDF inverse method[4] and rejection method.

Among the methods rejection method is preferred because of its good performance on simplicity, accuracy and universality. In this letter three rejection methods for implementing the Nakagami-m simulation are analyzed and compared.

II. NAKAGAMI-M DISTRIBUTION Nakagami-m distribution is proposed by Nakagami (1960)

with pdf (probability density function) showed by the following formula:

22 12( ) , 0, 0

( )

mrm m

R m

m rf r e rm

− −Ω= Ω ≥ ≥

Ω Γ (1)

where Ω is the second moment of R , ( )Γ ⋅ is the Gamma function and m is the shape parameter that controls the

severity of channel fading. In the range of 0.5 1m< < , the pdf models a channel with more severe fading than Rayleigh channel ( 0.5m = equals to half-Gaussian channel, 1m = equals to Rayleigh channel ). In the range of 1m > , the pdf models a channel similar to Rician channel. When m = ∞ the channel is ideal with no fading.

As discussed aforementioned, Nakagami-m channel performs better than the traditional Rayleigh and Rician channel in long-distance and wide-band communication, and therefore it is appropriate for some communication environment such as suburban area or moon surface.

III. THREE REJECTION METHODS Nakagami-m random variables should be simulated before

put into practice and the methods of Nakagami-m simulation gain wide interest of research.

There are two general methods to simulate arbitrary random variables: CDF inverse method and rejection method. Compared to the CDF inverse method, rejection method is easier to implement and gets considerate accuracy.

Rejection method can generate a specific random variable from an easy-generating random variable (so-called hat function). According to the diverse selections of hat function rejection methods can be divided to three types. This section will demonstrate the general calculation process of three rejection methods.

A. Uniform Hat Function This method generates the Nakagami-m samples out of

uniform samples, which can be directly generated in Matlab.

• Find the border of x-axis and y-axis in ( )Rf r ( a and b separately).

• Generate two uniform random variable V and Y within the range of [0, ]a and [0, ]b .

• Use the inequality ( )i R iY f V≤ to sift the samples and get the Nakagami-m samples.

The key point in Uniform method is the calculation of b, which finally determines the efficiency of the method.

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B. Gaussian Hat Function Literature [5] shows how to generate the Nakagami-m

samples out of folded-Gaussian and Gaussian samples.

• When 0.5 1m≤ ≤ , we generate a folded-Gaussian random variable V .

• Else when 1m ≥ we generate a Gaussian random variable V .

• Use the inequality below to sift the samples and get the Nakagami-m samples.

( ) ( )W i X iCf V f V≥ (2)

where ( )Wf w is the pdf of the folded-Gaussian or Gaussian distribution.

The key point in Gaussian method is the calculation of the Gaussian parameters xμ and 2σ , as well as the calculation of the constant list of C .

C. Polynomial Hat Function Literature [6] proposes a new method to generate the

Nakagami-m samples out of uniform samples.

• generate the polynomial random variable V from the formula below:

2

2 4( 4 tan( ) )2 2

t BV B BA

Ω −= − + (3)

where

2(2 )( )

mmmA B e

m−= −

Γ (4)

2 (2 1) / 2B m m= − (5)

where t is the uniform samples in min max[ , ]t t .

where

1

2

min 2

2 tan ( )4

4

BABt

B

− −−=

− (6)

2max

4

A

Bt π

−= (7)

• Use the second-order inverse polynomial inequality

below to sift the samples and get the Nakagami-m samples:

2

( ) ( )Ap x f xB x x

Ω= ≥Ω − Ω +

(8)

where ( )p x is the pdf of the polynomial distribution, and ( )f x is the pdf of Nakagami-m distribution.

The key point in Polynomial method is the selection of polynomial function, which results in the fitness between the theoretical and the actual parameters.

IV. ANALYSIS AND COMPARISON OF REJECTION METHODS This section will analyze and compare the three rejection

methods mentioned in last section. The viewpoints cover the accuracy of parameters, the fitness of pdfs, the complexity of calculation and the rejection efficiency.

A. Accuracy of Parameters In channel models the accuracy of parameters affects the

final performance of communication systems, so it is important to make research on it. There are three main parameters in Nakagami-m distribution: Ω , m and μ (mean value). Figure 1. , Figure 2. and Figure 3. show the diverse results in accuracy of parameters separately with relative error rule shown in(9).

ˆ

RE( )p p

pp−

= . (9)

where p is the designed parameter, and p̂ is the simulated parameter.

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

m

RE

(m)

Uniform Hat function

Gaussian Hat function

Polynomial Hat function

Figure 1. Relative Error of m ( 1Ω = )

0 2 4 6 8 10-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Ω

RE

( Ω)

Uniform Hat function

Gaussian Hat function

Polynomial Hat function

Figure 2. Relative Error of Ω ( 1m = )

0 0.5 1 1.5 2 2.5 3 3.5 40.045

0.05

0.055

0.06

0.065

0.07

0.075

μ

RE

( μ)

Uniform Hat FunctionGaussian Hat functionPolynomial Hat function

Figure 3. Relative Error of μ ( 1Ω = , 1m = )

B. Fitness of pdfs The fitness of pdf is the most convincing specification in a

simulation method, since it is the only identification that can tell a specific distribution from others. Figure 4. illustrates the fitness of pdf of the three methods with absolute error rule shown by (10).

ˆ( )AE p p p= − (10)

0 2 4 6 8 100.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

m

AE

(pdf

)

Uniform Hat FunctionGaussian Hat functionPolynomial Hat function

Figure 4. Absolute Error of pdf

C. Complexity of Calculation The complexity of calculation is very important for a

simulation method if it ought to be put into practice. The lower the complexity is, the less hardware resource is occupied. In Matlab environment, complexity is represented by the time of simulation. TABLE I. shows the mean norm-time of simulation with the three methods (by 20 simulations).

TABLE I. TIME OF SIMULATION

Method Mean simulation time(normalized)

Uniform Hat Function 1

Gaussian Hat Function 1.1163 Polynomial Hat

Function 1.2786

D. Rejection Efficiency The simulation efficiency of rejection method is defined by

the expression 1/fE C= , which shows the probability that the original data is rejected. The bigger C is, the less data will be rejected, which means faster simulation speed and less RAM space required, and also more efficiency the rejection method gets. The comparison of the rejection efficiency with the three methods is shown in Figure 5.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m

reje

ctio

n ef

ficie

ncy

Uniform Hat functionGaussian Hat functionPolynomial Hat function

Figure 5. Rejection Efficiency of three methods（ 1Ω = ）

V. CONCLUSION In this letter, three rejection methods are introduced,

analyzed and compared. Judging from the comparison, Gaussian method and Polynomial method achieve more rejection efficiency. Furthermore, Polynomial method performs better than Gaussian method on the accuracy of parameters and fitness of pdf. Moreover, Polynomial method obtains a simpler calculation process, while the calculation process and the efficiency of Gaussian method is diverse in two parts ( 0.5 1m≤ ≤ and 1m ≥ ). However, Gaussian method gets a steadier efficiency in all range of m , while the efficiency of Polynomial method is better in the range of interest ( 0.5 1m≤ ≤ ). Although Uniform method gets less efficiency, it is simplest in the calculation process. With all the discussion above it is figured out that Polynomial method is the better schema for Nakagami-m simulation in severe fading channel while Gaussian method is a better selection for Nakagami-m simulation in moderate fading channel. In some cases that call for high speed of simulation Uniform method is worth a try.

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