on the statistical properties of mobile-to-mobile fading_los

8/13/2019 On the Statistical Properties of Mobile-To-Mobile Fading_LOS

1/6


2/6

Mobile Relay

(MR)

Destination

Mobile Station

(DMS)

Source Mobile

Station

(SMS)

tm

t1

t2

Fig. 1. The single-LOS double-scattering fading channel.

link LMR reaches the DMS in two steps. First, the signal s (t)arrives through multipath propagation at the MR, and then it is

retransmitted to the DMS. Thus, the signal rMR(t) received by theMR can be expressed as

rMR(t) = (1)(t) s(t) +n1(t) (2)

where (1)(t) is a scattered component that describes the fadingin the SMS-MR link, and n1(t) is an additive white Gaussiannoise (AWGN) process. Here, the scattered component (1)(t) ismodeled as a zero-mean complex Gaussian process having 221variance, i.e.,(1)(t) =

(1)1 (t)+j

(1)2 (t). The MR then amplifies

the signal r MR(t) and retransmits it to the DMS. Thus, the signalrMR-DMS(t) received at the DMS can be written as

rMR-DMS (t) = A(2)(t) rMR(t) +n2(t)

= A(2)(t) (1)(t) s (t) +A(2)(t) n1(t) +n2(t)

= A(t) s (t) +A(2)(t) n1(t) +n2(t) (3)

where A is an amplification factor, (2)(t) is the second scattered

component, (t) corresponds to the doubly scattered component,and n2(t) is a second AWGN process. We have assumed fixedgain relays in our model, meaning that the amplification factor

A is a real constant. The scattered component (2)(t) is a zero-mean complex Gaussian process with variance222 , i.e.,

(2)(t) =

(2)1 (t) + j

(2)2 (t). This process models the fading channel in

the MR-DMS link. The doubly scattered component (t) definesthe overall fading channel in the link LMR. It represents a zero-mean complex double Gaussian process, which is modeled as

the product of two independent, zero-mean complex Gaussian

processes(1)(t) and(2)(t), i.e., (t) = (1)(t) (2)(t). Finally,the total signal rDMS(t) received by the DMS can be expressed asfollows

rDMS(t) = rLOS(t) +rMR-DMS (t)= m(t) s(t) +A(t) s (t) A(2)(t) n1(t) +n2(t)

= A ((t) +m(t)) s (t) +A(2)(t) n1(t) +n2(t)

= A(t) s(t) +A(2)(t) n1(t) +n2(t) (4)

where (t) is a non-zero-mean complex double Gaussian process.The non-zero-mean complex double Gaussian process(t) modelsthe overall fading channel between the SMS and the DMS. It

represents the sum of the doubly scattered component(t) and theLOS componentm(t), i.e.,(t) = 1(t) +j2(t) = (t) + m(t).The absolute value of (t) gives rise to an SLDS process (t),i.e., (t) = |(t)|. Furthermore, the argument of(t) defines thephase process (t), i.e., (t) = arg{(t)}.

III. ANALYSISO FT HE SLDS FADINGC HANNEL

In this section, we present the analytical expressions for the sta-

tistical properties of the SLDS channel introduced in Section II. A

starting point for the derivation of the statistics of the SLDS process

is the computation of the joint PDF p12 1 2(u1, u2, u1, u2) ofthe stationary processes1(t),2(t), 1(t), and 2(t)at the sametimet. Throughout this paper, the overdot indicates the time deriva-tive. Applying the concept of transformation of random variables

[11], we can write the joint PDF p12 1 2(u1, u2, u1, u2) asfollows

p12 1 2(u1, u2, u1, u2) =

dy2dy1dy2dy1 |J|1

p(1)1

(1)2

(1)1

(1)2

(2)1

(2)2

(2)1

(2)2

(x1, x2, x1, x2, y1, y2, y1, y2) (5)

where J denotes the Jacobian determinant, xi (i= 1, 2) isa function of y1, y2, u1, and u2, and xi(i= 1, 2) is afunction of y1, y2, y1, y2, u1, u2, u1, and u2. It isworth mentioning here that the processes i(t), i(t),

(1)i (t),

(1)i (t),

(2)i (t), and

(2)i (t) (i= 1, 2) are uncorrelated in pairs.

Taking into account that the underlying Gaussian processes

and their time derivatives, i.e., (1)i (t),

(2)i (t),

(1)i (t), and

(2)i (t) (i= 1, 2) are statistically independent allows us to write

p(1)1

(1)2

(1)1

(1)2

(2)1

(2)2

(2)1

(2)2

(x1, x2, x1, x2, y1, y2, y1, y2) =

p(1)1

(1)2

(1)1

(1)2

(x1, x2, x1, x2) p(2)1

(2)2

(2)1

(2)2

(y1, y2, y1, y2).

Furthermore, the joint PDFs p(1)1

(1)2

(1)1

(1)2

(x1, x2, x1, x2) and

p(2)1

(2)2

(2)1

(2)2

(y1, y2, y1, y2) can be expressed by the multi-

variate Gaussian distribution (see, e.g., [12, eq. (3.2)]). Thus,

substituting the expressions ofp(1)1

(1)2

(1)1

(1)2

(x1, x2, x1, x2)and

p(2)1

(2)2

(2)1

(2)2

(y1, y2, y1, y2) in (5) and doing some lengthy

algebraic computations results in

p12 1 2(u1, u2, u1, u2) = 1

(2)2 2122

0

v e

1

221

g1(u1,u2,)

v2

e

1

222v2

e 121

h1(u1,u2,)v2

2g1(u1, u2, ) +1v4

e

221

g1(u1,u2,) h1(u1,u2,)

v2(2 g1(u1,u2,)+1v4)

dv (6)

where

g1(u1, u2, ) =u21+u

22+

2 2u1cos(2ft + )2u2sin(2ft+) (7a)

h1(u1, u2, ) = u21+ u

22+(2f)

24fu2cos(2ft+)+4fu1sin(2ft+) (7b)

1 = 2 (1)2

f2max1+f

2max2, 2 = 2 (2)

2

f2max2+f

2max3.

(8a,b)In (8a,b), the quantityi (i= 1, 2)is the negative curvature of theautocorrelation function of the inphase and quadrature components

ofi(t) (i= 1, 2) presented here for the case of isotropic scatter-ing [13]. Furthermore, i (i= 1, 2) is the characteristic quantitycorresponding to M2M fading process [14]. The symbols fmax1 ,fmax2 , andfmax3 appearing in (8a,b) correspond to the maximumDoppler frequency caused by the motion of the SMS, the MR, and

the DMS, respectively.

Starting from (6), the transformation of the Cartesian coordinates

(u1, u2) into polar coordinates(z, ) by means ofz=

u21+u22

and = arctan(u2/u1) results after some lengthy algebraicmanipulations in

pz, z ,, ; t

=

z2

(2)2 2122

0

dvv e 1

221

z

2

+

2

v2

e v

2

222

2g2(z, ,) +1v4

389


3/6

e

z cos(2ft)

21v2 e

12

v2z2+(z)2

+(2fv)2

2 g2(z,,)+1v4

e

2fv2(z sin(2ft)+z cos(2ft))

2 g2(z,,)+1v4

(9)

for z 0,|| ,|z| 1

z

1

212e|z|12

z

po(z) , z 0 (14)

where po(z) represents the Laplace distribution having the meanvalue

and the variance

2

1

2

2.

B. PDF of the Phase Process

The PDF p(; t) of the phase process (t) can be de-rived from (9) by solving the integrals over the joint PDF

p

z, z ,, ; t

according to

p(; t) =

0

p

z, z ,, ; t

d dz dz, || .

(15)

This results in the following final expression

p(; t)= 1

2

0

dx ex 1x

212

2

1+

2R1(x,,f, )

e12R1(x,,f,)

2

1+

R1(x,,f, )

2

,|| (16)

where

R1(n,,) = cos( 2ft )

12

2n. (17)

Furthermore, in (16), () represents the error function [15,eq. (8.250.1)]. From (16), it is obvious that the phase process (t)is not stationary in a strict sense since p(; t) =p(). This timedependency of the PDF p(; t) is due the Doppler frequency f

of the LOS component m(t). However, for the special case thatf = 0 ( = 0), the phase process (t) is a strict sense stationaryprocess.

As 0, it follows (t) =|(t) +m(t)| |(t)|, and from(16), we obtain the uniform distribution

p() |=0 = 12

, < . (18)

C. Mean Value and Variance

The expected value and the variance of a stochastic process

are important statistical parameters, since they summarize the

information provided by the PDF. The expected value E{ (t)}of the SLDS process (t) can be obtained using [11]

E{ (t)} =

z p(z) dz (19)

where E{} is the expected value operator. Substituting (12) in(19) results in the following final expression

E{ (t)} = K0()

I1() +

212I0() L1()

2

12I1() L0()

+ 1

2122

I0() C1(z) (20)

where = /(12) and

C1(z) =

z2K0

z

12

dz . (21)

In (20), In() and Kn() denote the nth order modified Besselfunctions of the first and the second kind [15], respectively, and

Ln() designates the nth order modified Struve function [15].The difference of the mean power E{2 (t)} and the squared

mean value (E{ (t)})2 of the SLDS process (t) defines itsvariance Var{ (t)} [11], i.e.,

Var{(t)} =E{2(t)} [E{(t)}]2 . (22)

By using (12) and [11, eq. (5.67)], the mean power E{2

(t)} ofthe SLDS process (t) can be expressed as

E{2(t)} =

z2 p(z) dz

= 22K0()

I2()+

2I3()

+

I0()

(12)2 C2(z) (23)

where the function C2() is defined as follows

C2(z) =

z3K0

z

12dz . (24)

From (20), (23), and by using (22), the variance Var{ (t)} of theSLDS process (t) can easily be calculated.

390


4/6

D. Derivation of the LCR

The LCR of the envelope of mobile fading channels is a measure

to describe the average number of times the envelope crosses a

certain threshold level r from up to down (or vice versa) persecond. The LCR N(r) of the SLDS process (t) can beobtained using [18]

N(r) =

0

zp(r,z) dz (25)

wherep(r,z)is the joint PDF of the SLDS process (t) and itscorresponding time derivative (t) at the same time t. The jointPDF p(z, z) can be derived from (9) by solving the integralsover the joint PDF p

z, z ,, ; t

according to

p(z, z) =

p

z, z ,, ; t

d d, z 0,|z| .

(26)

After doing some lengthy computations, the joint PDF in (26)

results in the following expression

p(z, z)=

2 z

(2)2 2122

0

e

1

221

z2+2

v2

e v2

222e

z cos

v221

2 g3(z, ,) +1v4

e

2(fv sin )2

2 g3(z,,)+1v4e

12

(vz)24fv

2z sin

2 g3(z,,)+1v4

d dv,

z 0,|z| (27)where

g3(z, ,) = z2 +2 2z cos . (28)

Finally, after substituting (27) in (25) and doing some extensive

mathematical manipulations, the LCRN(r) of the SLDS process

(t) can be expressed as follows

N(r)=

2 r

(2)2 2122

0

2 g3(r,,) +1v4

v2 e

v2

222

e

g3(r,,)

2v221 e

12R

22(r,y,,)

1+

2R2(r,v,,)

e12R

22(r,v,,)

1+

R2(r,v,,)

2

d dv (29)

where

R2(r,v,,) = 2f v sin

2g3(r,,) +1v4

. (30)

The quantities 1 and 2 are the same as those defined in (8a,b).Furthermore,g3(, , ) is the function defined in (28).Considering the special case when = 0, then (29) reduces to

the expression of the LCR for the double Rayleigh process given

in [6] as

N(r) |=0= r221

22

0

2r2 +1v4

v2 e

(r/v)2

221 e v2

222dv .

(31)

E. Derivation of the ADF

We will conclude Section III with the discussion on the ADF.

The ADF T(r) of the SLDS process (t) can be defined as

the ratio of the CDF F(r) of (t) and its LCR N(r), i.e.,

T(r) =F(r)

N(r) . (32)

The CDF F(r) of the SLDS process (t) can be expressedusing (12) as follows

F(r) =

r0

p(z) dz

= r12

K0() I1

r12

, r <

1

r

12

I0() K1 r12

, r . (33)

From (33), (29), and by using (32), the ADFT(r) of the SLDSprocess (t) can easily be computed.

It is quite obvious from (33) that as 0, (33) reduces to

F(r) |=0 = 1 r

12K1

r

12

. (34)

This result (34) corresponds to the CDF of the double Rayleigh

process [7]. Thus, substituting (34) and (31) in (32) gives the ADF

of the double Rayleigh process.

IV. NUMERICALR ESULTS

In this section, we will confirm the correctness of the analyticalexpressions presented in Section III with the help of simulations.

Furthermore, for a detailed analysis the results for the SLDS

process are compared with those of the classical Rayleigh, clas-

sical Rice, double Rayleigh, and double Rice processes. It is

important to note that the double Rice process is defined as the

product of two independent classical Rice processes, i.e., (t) =(1)(t) +1 (2)(t) +2. To simplify matters, the amplitudes1 and 2 of the LOS components of the double Rice process areconsidered to be equal, i.e., 1 = 2 = . The concept of sum-of-sinusoids (SoS) [13] is used to simulate uncorrelated complex

Gaussian processes that make up the overall SLDS process. The

numbers of sinusoids (N1 and N2) considered to generate these

Gaussian processes were selected to be 20. For the computation

of the model parameters, we selected the generalized method of

exact Doppler spread (GMEDSq) proposed in [19] for q= 1. Thevalues for the maximum Doppler frequencies fmax1 , fmax2 , andfmax3 were set to 91 Hz, 75 Hz, and 110 Hz, respectively.

The results presented in Figs. 2 4 show an excellent fitting

of the analytical and the simulation results. In Fig. 2, the envelope

PDFp(z)of the SLDS process (t)is being compared with thoseof the classical and the double Rice processes for different values of

, wheref was set to zero. For the sake of comparison, the PDFsof the classical and the double Rayleigh processes are also included

in Fig. 2. It can be observed that the maximum value of the PDF

p(z)of the SLDS process (t)is higher than that of the classicaland the double Rice process for same value of. On the other hand,

the spread of the PDF p(z) of the SLDS process (t) followsthe same trend as that of the classical Rayleigh, the classical Rice,

and the double Rayleigh processes. However, the PDF p(z) ofthe SLDS process (t) has a narrower spread when comparedto the spread of the double Rice process for the same value of

. It should also be noted that with increasing values of thePDFp(z) of the SLDS process (t) approaches the symmetricalLaplace distribution. Similarly, Fig. 3 presents a comparison of the

PDF p() of the phase process (t) with that of the classicaland the double Rice processes. Figure 4 shows that the maximum

value of the LCRN(r) of the SLDS process (t) increases withincreasing , keeping f constant. Furthermore, the LCR N(r)corresponding to the SLDS process (t) has higher maximumvalues and narrower spread as compared to that of the double Rice

process for the same value of . Similarly, Fig. 5 compares theADF T(r) of the SLDS process (t) with that of the doubleRice process for different values of keeping f equal to zero.

391


5/6

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

z

Probability

density

function,p

(z) Theory

Simulation

= 1 (SLDS Process)

= 2 (SLDS Process)

= 1 (Classical Rice)


= 1 (Double Rice)

= 2 (Double Rice)

Classical Rayleigh

= 0 (Double Rayleigh)

f= 0

Fig. 2. The PDFp(z) of the SLDS process (t).

3.14 0 3.140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability

densityfunction,p

()

TheorySimulation


= 1 (Double Rice)

= 1 (SLDS Process)

= 2 (Double Rice)

= 2 (SLDS Process) = 2 (Classical Rice)

(Double Rayleigh)

= 0

f= 0

Fig. 3. The PDFp() of the phase process (t).

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250

Level, r

Level-crossing

rate,

N

(t)

TheorySimulation

= 1 (SLDS process)

= 2 (Double Rice)

= 0 (Double Rayleigh)

= 2 (SLDS process)

= 1 (Double Rice)

f= fmax 1+fmax 3

Fig. 4. The LCR N(r) of the SLDS process (t) for various values of.

V. CONCLUSION

In this paper, we have studied the statistics of M2M fading

channels in cooperative wireless networks under LOS conditions.

Considering the amplify-and-forward relay type system, the exis-

tence of an LOS component in the direct transmission link between

the SMS and the DMS results in the SLDS fading channel. Here,

we have modeled the NLOS link of the system as a zero-mean

complex double Gaussian channel. Thus, the overall SLDS fading

channel is modeled as the superposition of a deterministic LOS

component and the zero-mean complex double Gaussian channel.

Statistical properties of the SLDS fading channel are thoroughly

investigated in this paper. We have derived analytical expressions

for the mean, the variance, the PDFs, the LCR and the ADF.

Furthermore, we have verified our analytical expressions using nu-

0 1 2 3 4 5 6 7 8 9 1010

5

104

10

3

102

101

100

101

Level, r

Average

durationoffades,

T

(r)

TheorySimulation

f = 0

= 1 (SLDS Process)

= 2 (SLDS Process)

= 1 (Double Rice)

= 2 (Double Rice)

Fig. 5. The ADF T (r) of the SLDS process (t) for various valuesof .

merical techniques in simulations. The close fitting of the presented

theoretical and the simulation results proves correctness of our

analytical expressions. It has been shown that the properties of the

SLDS process are quite different from both the double Rayleighand the double Rice processes. For example, the envelope PDF of

the SDLS process approaches the symmetrical Laplace distribution

when the amplitude of the LOS component increases. Furthermore,

we have provided sufficient evidence in this paper that the SLDS

process reduces to the double Rayleigh process in the absence of

the LOS component.

The theoretical analysis presented in this paper is useful for

the researchers and designers of the physical layer for mobile-

to-mobile communication systems. Our study provides an insight

into the dynamics of SLDS fading channels, which can be ex-

ploited to develop robust modulation and coding schemes for such

fading environments. Furthermore, with the help of the designed

channel simulator, the overall performance of the system can also

be evaluated by simulation for different kinds of SLDS fadingenvironments.

REFERENCES

[1] A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversity Part I: System description, IEEE Trans. Commun., vol. 51, no.11, pp. 19271938, Nov. 2003.

[2] A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversity Part II: Implementation aspects and performance analysis, IEEETrans. Commun., vol. 51, no. 11, pp. 19391948, Nov. 2003.

[3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative di-versity in wireless networks: efficient protocols and outage behavior,

IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 30623080, Dec.2004.

[4] A. S. Akki and F. Haber, A statistical model of mobile-to-mobileland communication channel, IEEE Trans. Veh. Technol., vol. 35,no. 1, pp. 27, Feb. 1986.

[5] M. Dohler, Virtual Antenna Arrays. Ph.D. dissertation, KingsCollege, London, United Kingdom, 2003.

[6] C. S. Patel, G. L. Stuber, and T. G. Pratt, Statistical propertiesof amplify and forward relay fading channels, IEEE Trans. Veh.Technol., vol. 55, no. 1, pp. 19, Jan. 2006.

[7] I. Z. Kovacs, P. C. F. Eggers, K. Olesen, and L. G. Petersen, Investi-gations of outdoor-to-indoor mobile-to-mobile radio communicationschannels, in Proc. IEEE 56th Veh. Technol. Conf., VTC02-Fall,Vancouver BC, Canada, Sept. 2002, pp. 430434.

[8] J. Salo, H. M. El-Sallabi, and P. Vainikainen, Impact of double-Rayleigh fading on system performance, in Proc. 1st IEEE Int. Symp.on Wireless Pervasive Computing, ISWPC 2006, Phuket, Thailand,Jan. 2006, 0-7803-9410-0/10.1109/ISWPC.2006. 1613574.

[9] B. Talha and M. Patzold, On the statistical properties of doubleRice channels, in Proc. 10th International Symposium on WirelessPersonal Multimedia Communications, WPMC 2007, Jaipure, India,Dec. 2007, accepted for publication.

392


6/6

[10] J. Salo, H. M. El-Sallabi, and P. Vainikainen, Statistical analysis ofthe multiple scattering radio channel, IEEE Trans. Veh. Technol., vol.54, no. 11, pp. 31143124, Nov. 2006.

[11] A. Papoulis and S. U. Pillai, Probability, Random Variables andStochastic Processes. New York: McGraw-Hill, 4th edition, 2002.

[12] M. K. Simon, Probability Distributions Involving Gaussian RandomVariables: A Handbook for Engineers and Scientists. Dordrecht:Kluwer Academic Publishers, 2002.

[13] M. Patzold, Mobile Fading Channels. Chichester: John Wiley &Sons, 2002.

[14] A. S. Akki, Statistical properties of mobile-to-mobile land commu-nication channels, IEEE Trans. Veh. Technol., vol. 43, no. 4, pp.826831, Nov. 1994.

[15] I. S. Gradshteyn and I. M. Ryzhik, Tables of Series, Products, andIntegrals, vol. I and II. Frankfurt: Harri Deutsch, 5th edition, 1981.

[16] G. N. Watson, A Treatise on the Theory of Bessel Functions. BentleyHouse, London: Cambridge University Press, 2nd edition, 1966.

[17] S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribu-tion and Generalizations: A Revisit with Applications to Communi-

cations, Economics, Engineering, and Finance. Boston: Birkhauser,2001.

[18] S. O. Rice, Mathematical analysis of random noise, Bell Syst. Tech.J., vol. 24, pp. 46156, Jan. 1945.

[19] M. Patzold and B. O. Hogstad, Two new methods for the generationof multiple uncorrelated Rayleigh fading waveforms, in Proc.

IEEE 63rd Semiannual Veh. Tech. Conf., VTC06-Spring, Melbourne,

Australia, May 2006, vol. 6, pp. 27822786.

393

on the statistical properties of mobile-to-mobile fading_los

Documents