Electromagnetic Macro Modeling of Propagation in Mobile Wireless ...

Electromagnetic Macro Modeling of Propagation in Mobile Wireless Communication: Theory and Experiment Tapan K. Sarkar1, Walid Dyab1, Mohammad N. Abdallah1, Magdalena Salazar-Palma2, M. V. S. N. Prasad3, Sio Weng Ting4, and Silvio Barbin5 1

Department of Electrical Engineering and Computer Science Syracuse University Syracuse, New York 13244-1240, USA E-mail: [email protected], http://lcs.syr.edu/faculty/sarkar/

Dept. of Signal Theory & Communications Universidad Carlos III de Madrid Avenida de la Universidad, 30, 28911 Leganés, Madrid, Spain E-mail: [email protected] 2

National Physical Laboratory Dr. K. S. Krishnan Road, New Delhi-110012, India E-mail: [email protected] 3

Dept. of Science and Technology University of Macau Av. Padre Tomas Pereira, Taipa, Macau, China E-mail: [email protected] 4

5

Departamento de Engenharia de Telecomunicações e Controle Escola Politécnica da Universidade de São Paulo São Paulo, Brazil E-mail: [email protected]

Abstract The objective of this paper is to illustrate that electromagnetic macro modeling can properly predict the path-loss exponent in mobile cellular wireless communication. This represents the variation of the path loss with distance from the base-station antenna. Specifically, we illustrate that the path-loss exponent in cellular wireless communication is three, preceded by a slow-fading region, and followed by the fringe region, where the path-loss exponent is four. The sizes of these regions are determined by the heights of the base-station transmitting antennas and the receiving antennas. Theoretically, this is illustrated through the analysis of radiation from a vertical electric dipole situated over a horizontal imperfect ground plane, as first considered by Sommerfeld in 1909. To start with, the exact analysis of radiation from the dipole is made using the Sommerfeld formulation. The semi-infinite integrals encountered in this formulation are evaluated using a modified saddle-point method for field points moderate to far distances away from the source point, to predict the appropriate path-loss exponents. The reflection-coefficient method is also derived by applying a saddlepoint method to the semi-infinite integrals, and this is shown to not provide the correct path-loss exponent that matches measurements. The various approximations used to evaluate the Sommerfeld integrals are described for different regions. It is also important to note that Sommerfeld’s original 1909 paper had no error in sign. However, Sommerfeld overlooked the properties associated with the so-called “surface-wave pole.” Both accurate numerical analyses, along with experimental data, are provided to illustrate the above statements. In addition, Okumura’s experimental data, and extensive data taken from seven different base stations in urban environments at two different frequencies, validate the theory. Experimental data revealed that a macro modeling of the environment, using an appropriate electromagnetic analysis, can accurately predict the path-loss exponent for the propagation of radio waves in a cellular wireless communication scenario. IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

AP_Mag_Dec_2012_Final.indd 17

ISSN 1045-9243/2012/$26 ©2012 IEEE

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Keywords: Propagation; imperfect ground; surface wave; ground wave; Norton surface wave; cellular wireless communication; path loss exponent; macro modeling; Sommerfeld formulation; land mobile radio cellular systems; land mobile radio propagation factors

1. Original Sommerfeld Formulation for a Vertical Electric Dipole Over an Imperfect Ground Plane

C

onsider an elementary electric dipole of moment Idz , oriented along the z direction, and located at ( x′, y ′, z ′ ) . The dipole is situated over an imperfect ground plane, characterized by a complex relative dielectric constant, ε , as seen in Figure 1. The complex relative dielectric constant is given by ε= ε r − jσ ωε 0 , where ε r represents the relative permittivity of the medium, ε 0 is the permittivity of vacuum, σ is the conductivity of the medium, ω stands for the angular frequency, and j is the imaginary unit, i.e., j= −1 . It is possible to formulate a solution to the problem of radiation from the dipole operating in the presence of the imperfect ground in terms of a single Hertzian vector, Π z , of the electric type. A time variation of exp ( jωt ) is assumed throughout the analysis, where t is the time variable. The Hertzian vector uˆ z Π z in this case satisfies the wave equation 2 (∇2 + k= 1 ) Π1z

− Édz δ ( x − x′ ) δ ( y − y ′ ) δ ( z − z ′ ) , jω ε 0 (1)

(∇2 + k22 ) Π 2 z =0 ,

(2)

where k12

2

(3)

k22 = ω 2 µ0ε 0ε ,

(4)

= ω µ 0ε 0 ,

and δ represents the delta function in space. The primed and unprimed coordinates are for the source and field points, respectively. Subscript 1 denotes the upper half space, which is air, and subscript 2 denotes the lower half space, which is the imperfectly conducting Earth characterized by a complex relative dielectric constant, ε . The electric and the magnetic field vectors are derived from the Hertzian vector using      Ei =∇ ∇  Π i + ki 2 Π i ,

(

)

(5)

and   = Ç i jωε 0ε i ∇ × Π i ,

(

respectively, with i = 1 , 2.

18


)

(6)

Figure 1. A vertical dipole over a horizontal imperfect ground plane.

In medium 1, ε1 = 1 , and for medium 2, ε 2 = ε . The propagation constants in medium 1 and 2, called k1 and k2 , are thus related by k2 k1 = ε . At the interface z = 0 , the tangential electric- and magnetic-field components must be continuous, conditions which in terms of the Hertzian vector components can be written as ∂ Π1z ∂ Π2z =ε , ∂y ∂y

(7a)

∂Π1z ∂Π 2 z =ε , ∂x ∂x

(7b)

∂  ∂Π1z ∂y  ∂z

 ∂  ∂Π 2 z  = ∂y  ∂z  

 , 

(7c)

∂  ∂Π1z ∂x  ∂z

 ∂  ∂Π 2 z  = ∂x  ∂z  

 . 

(7d)

Since all the boundary conditions must hold at z = 0 for all x and y, the x and y dependence of the fields on either side of the interface must be the same. Therefore, Π1z =ε Π 2 z ,

(8a)

∂Π1z ∂Π 2 z = . ∂z ∂z

(8b)

IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

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The complete solutions for the Hertz vectors satisfying the wave Equations (1) and (2) and the boundary conditions of Equation (8) have been derived by many researchers over the last century. A partial list [1-18] that will be important to our discussions is provided in chronological order, starting with Sommerfeld [1]. The solutions are

Π1z = Π1direct + Π1reflected = P ( g0 + g s ) , z z where Π1direct = P exp ( − jk1R1 ) R1 = P g 0 , z

 exp ( − jk1R1 ) Π1z = P R1 

∞ 0

exp  − λ 2 − k12 ( z + z ′ )  λ d λ  

}

(9)

and ∞ J0

Π2z = 2P ∫

(

( λρ ) exp λ 2 − k22 z − λ 2 − k12 z ′ ε λ 2 − k12 + λ 2 − k22

0

) λ dλ

(10) for Re

(

)

2 λ 2 − k1,2 > 0 . J 0 ( x ) represents the zeroth-order

Bessel function of the first kind of argument x. Here, P=

ρ= R1=

I dz , jω 4 π ε 0

(11)

( x − x ′ )2 + ( y − y ′ )2 2

ρ 2 + ( z − z′) ,

,

(12)

Similarly, the solution for Π 2z can be interpreted as a partial transmission of the wave from medium 1 into medium 2. With these thoughts in mind, Π1z , or equivalently, the potential responsible for the ground wave, can be split up into two terms:


ε λ 2 − k12 + λ 2 − k22

(16) The path of integration for the semi-infinite integral is labeled C2 and is depicted in Figure 2, along with the singularities of the multivalued function, two branch points at k1 and k2 , and a pole, p, arising from the ratio of two functions in Equation (16). A physical explanation to the two components of the Hertz potential, Π1z , can now be given. The first component, Π1direct , can be explained as a spherical wave originating from z the source dipole. This term is easy to deal with. The difficult problem lies in the evaluation of Π1reflected . Therefore, z Π1reflected can be interpreted as a superposition of plane waves z resulting from the reflection of the various plane waves into which the original spherical wave was expanded. This arises from the identity exp ( − jk1R2 ) R2

(13)

and λ is the variable of integration. For Π1z , the first term inside the brackets can be interpreted as the particular solution or the direct line-of-sight (LOS) contribution from the dipole source, and the second term can be interpreted as the complementary solution or a reflection term (reflection from the imperfect ground plane). We will call this potential responsible for the fields of the ground wave, as per IEEE Standard Definitions of Terms for Radio Wave Propagation [19]. Observe that the second term of this potential for the ground wave in Equation (9) is the strongest near the surface of the Earth, and exponentially decays as we go away from the interface. In Appendix 1, we illustrate the different waves and their specific properties that we refer to in this paper, for clarification.


ε λ 2 − k12 − λ 2 − k22

J 0 ( λρ ) exp  − λ 2 − k12 ( z + z ′ )    λ d λ = Pg s . λ 2 − k12

λ 2 − k12 ε λ 2 − k12 + λ 2 − k22

0

(15)

P∫ Π1reflected = z

J 0 ( λρ ) ε λ 2 − k12 − λ 2 − k22

∞

+∫

(14)

=

∞ J0

∫

( λρ ) exp  − λ 2 − k12 ( z + z ′ ) 

λ

0

2

 λ dλ

− k12 (17)

for Re

(

)

λ 2 − k12 > 0

and R2=

2

ρ 2 + ( z + z′) .

(18)

The term under the integral sign in Equation (16) can be recognized as a multiple plane-wave decomposition of the original spherical-wave source. Upon reflection of the plane waves from the dipole source as expressed in Π1reflected , the amplitude of z

each wave must be multiplied by the reflection coefficient, R (λ ) . The complex reflection coefficient, R (λ ) , takes into account the phase change as the wave travels from the source

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where g1 represents the spherical wave originating from the image of the source, and g sV represents the correction factor to accurately characterize the effects of the ground. Equivalently, one can rewrite the same expression as  exp ( − jk1R2 ) Π1reflected =− P z R2  ∞ J0

+2ε ∫ 0



ε λ 2 − k12 + λ 2 − k22

 P [ − g1 + GsV ] . Figure 2. The contour of integration along the real axis from 0 to ∞ in the complex λ plane.

( x′, y ′, z ′ ) to the boundary and then to the point of observation, ( x, y, z ) . The reflection coefficient, R(λ ) , is then defined as R (λ ) =

ε λ 2 − k12 − λ 2 − k22 ε λ 2 − k12 + λ 2 − k22

,

(19)

takes into where the semi-infinite integral over λ in Π1reflected z account all the possible plane waves. As ε → ∞ , i.e., a perfect conductor for the Earth, then g s of Equation (16) reduces to Equation (17), and represents a simple spherical wave originating at the image point. This physical picture will later be applied in the derivation of the reflection-coefficient method. The reflection coefficient takes into account the effects of the ground plane in all the wave decomposition of the spherical wave and sums it up as a ray originating from the image of the source dipole, but multiplied by a specular reflection coefficient, R(θ ) , where θ is interpreted as the angle of the incident wave to the ground. It is now important to point out that there are two forms of that may be used interchangeably, as the two expressions are mathematically identical in nature (but have different asymptotic properties, as we shall see). These are defined as Π1reflected z

 exp ( − jk1R2 ) Π1reflected = P z R2  ∞

2 2  ′  λ 2 − k22 J 0 ( λρ ) exp  − λ − k1 ( z + z ) 

0

λ 2 − k12

−2 ∫

 P [ g1 − g sV ] ,

ε λ 2 − k12 + λ 2 − k22

  λ dλ   

(20)

  λ d λ    

( λρ ) exp  − λ 2 − k12 ( z + z ′ )

(21)

The image from the source now has a negative sign, along with the correction factor. This expansion is useful when both the transmitter and the receiver are close to the ground, and since the reflection coefficient is −1 for a grazing angle of incidence, where θ ≈ π 2 . The direct term, g 0 , then cancels the image term, g1 , leaving only the correction factor, GsV . It is very confusing to appreciate what the IEEE Standard Definitions for Radio Wave Propagation means by the term Norton surface wave, given as “The Norton wave consists of the total field minus the geometrical-optics field.” The Norton surface wave can therefore be associated with both g sV and GsV ! The question is, which one to use? Now, for grazing incidence, the fields are obtained using GsV . The question then is how does the Norton surface wave manifest itself in either of these two representations, as the difference between GsV and g sV is the geometrical-optics field!

2. Properties Related to the Exact Contour of Integration We now look at the properties of Equations (20) and (21). In general, the total field in the upper half-space ( z > 0) consists of the direct field from the source dipole situated at z = z ′ , its image situated at z = − z ′ , and the correction terms of g sV used in Equation (20). The correction term, g sV , takes into account the nature of the imperfect ground, because as ε → ∞ , g sV → 0 . The problem of determining the reflected field amounts to evaluating the integral for g sV , since g 0 and g1 are easy to calculate. In Sommerfeld’s 1909 paper, he changed the limits of the integration from 0 to ∞ in Equation (20) to an integral from −∞ to +∞ by transforming the Bessel function of the first kind and zeroth order to Hankel functions of the first and second kinds and zeroth order through the use of the following identity: J 0 ( x= )

1  (1) 2 Η 0 ( x ) + Η 0( ) ( x )  ,  2 

(22a)

and also utilizing 20



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(

)

1 2 Η 0( ) xe jπ = −Η 0( ) ( x ) ,

(22b)

where H 0(1) and H 0(2) are the Hankel functions of zeroth order and of the first and second kinds, respectively. Sommerfeld thus closed the path integral, C2 , of Figure 2 by using the contour as shown in Figure 3a, plotting the locations of the branch points, k0 and k1 , and the pole, p, for the term Π1reflected . Also shown z in Figure 3a are the two appropriate branch cuts, and a path encircling the pole. The convergence of the integrals in Equations (20) and (21) is assured, even in the presence of the Hankel function, when Im ( λ ) ≤ 0 as ρ → ∞ . Convergence is also assured for

(

Im k12 − λ 2

)

1/2

0 ,

(24)

k1′′ ≤ 0 . If 1

k z′ + j k z′′ , ( k12 − λ 2 ) 2 =

(25)

then k z′′ =

k1′ k1′′ − λ ′λ ′′ k z′

(26)

Figure 3b. The actual location of the pole in the lower complex λ plane.

and

k z′

1/2  2    k1′2 − k1′′ 2 − λ ′2 + λ ′′2  2  + ( k1′k1′′ − λ ′λ ′′ )     2      1/2

k ′2 − k1′′ 2 − λ ′2 + λ ′′2  + 1  2 

>0

(27)

where

λ =λ ′ + jλ ′′ =Re ( λ ) + j Im ( λ ) .

(28)

The positive sign is chosen for k z′ since k z′ > 0 (from Equations (24), (25), and (27)). On the path of integration, k z′′ < 0 , and if k1′′ ≠ 0 , convergence would be assured even if λ ′′ = 0 . If IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012


Figure 3c. The contribution of the pole generating the surface wave is excluded when the branch-cut contour is chosen vertically (Kahan and Eckart [22], Baños [6, pp. 55]). 21

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medium 1 is lossless, k1′′ = 0 , and then from Equation (26), λ ′ > 0 when λ ′′ > 0 , and λ ′ < 0 when λ ′′ < 0 . The first condition, λ ′′ > 0 , conflicts with the convergence requirement for the ( 2) Hankel function, H 0 ( λρ ) . This problem can be avoided if medium 1 is assumed to be lossy (i.e., k1′′ ≠ 0 ), and the “lossless case” then is assumed to be the limiting form of the expression ( 2) k1′′ → 0 . Since H 0 ( λρ ) can be integrated through the origin

(λ = 0)

even if λ ′′ = 0 , the path C2 of Figure 2 can be modified to the path C1 of Figure 3a, following the real axis from −∞ to +∞ . Now, Equation (20) can be rewritten as g sV = − j ∫

C1

by the contours of Figure 3a. As seen in his book [4], this closed contour of integration is equivalent to two integrals around the branch cuts associated with the branch points at + k1 and + k2 , and a contour integration around the pole λ = λP , where k1 k2 λP = . The other branch points, −k1 and −k2 , and k12 + k22 the pole located at −

2 Η 0( ) ( λρ ) exp  − j k12 − λ 2 ( z + z ′ )   

ε k12 − λ 2 + k22 − λ 2

Π1pole z

λ dλ

(29) where the contour of integration, C1 , is shown in Figure 3a, along with the location of the branch points at k1 and k2 , and their associated branch cuts, together with the pole of Equation (29). The presence of the free term, λ , will nullify the singularity of the Hankel function at λ = 0 . The integral from ‒∞ to 0 goes slightly below the negative real axis, as the Hankel function has a branch cut along that line. g sV is now a spectrum of plane waves traveling away from the ground plane, with the vertical component of the propagation constant given as k12 − λ 2 . The integral in Equation (29) also contains doublevalued functions

k12 − λ 2 z ∞ z = −∞

|

= 0 , (30)

is satisfied. This can occur only if Im  k12 − λ 2  > 0 , i.e.,   g sV → 0 as ρ → ∞ for a fixed z , and g sV → 0 as z → ∞ for a fixed ρ .

3. Sommerfeld’s Original Formulation Had No Error in Sign, But... Historically, in 1909, Sommerfeld computed the integral along the positive real axis of Figure 2. He did this by first applying the Cauchy principal-integral method to close the contour by a large semicircle at infinity, given by the semicircular contour with indentations of Figure 3a, lying in the third and the fourth quadrants. The result was the integral given 22


  2 (2) 2 2  k2 H 0 ( λP ρ ) exp − z λP − k1 = −2π jP  k22 k12  +  λP 2 − k12 λP 2 − k 2 2  (31)

(

Limit 2 Η 0( ) ( λρ ) →

λρ → ∞

)

   ,   

2

πλρ

exp [ − jλρ + j π 4] , (32)

resulting in

valued functions are those on which the radiation condition,  −j e 

, are of no concern, as they are

where Π1pole is part of the solution from the pole contribution. z Next, a large-argument approximation was made for the Hankel function by following the path of integration of Figure 3a. For large values of ρ , the asymptotic representation for the Hankel function was used:

2 k1,2 − λ 2 . The proper sheets of the double-

 ∂g sV 2 2  ∂z − j k − λ g sV 

k12 + k22

located in the upper half-plane where the contour is not closed, as seen in Figure 3a. Sommerfeld then evaluated the residue at the pole, and showed that it has the form of a surface wave (this wave is defined in Appendix 1). Using Equation (21), Sommerfeld then showed that

k2 2 − λ 2 k12 − λ 2

k1 k2

Π1pole z

  2 2 2  2π k2 exp − jλP ρ − z λP − k1 = P k22 k12  j λP ρ +  λP 2 − k12 λP 2 − k 2 2  (33)

(

)

   .   

As Sommerfeld, in his book, then pointed out: this formula bears all the marks of surface waves [a true surface wave is a slow wave, and the fields become concentrated on the interface as the frequency increases]: It was the main point of the author’s work of 1909 to show that the surface wave fields are automatically contained in the wave complex. This fact has of course, not changed. What has changed is the weight which we attached to it. At that time it seemed conceivable to explain the overcoming of the Earth’s curvature by radio signals with the help of the character of the surface waves; however we know now that it is due to the ionosphere. In any case, the recurrent discussion in the literature on the IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

12/9/2012 3:49:56 PM

reality of the Zenneck waves seems immaterial to us. Sommerfeld revisited the problem in 1926, redeveloped the same equations by using a different approach, and referred to the saddle-point method of Ott [20] “which he says is best represented by the one contour integral that goes near the pole and the second saddle point at k2 ” [20]. At this point, it is important to note that Sommerfeld never referred to an error in the sign in his original work. In a companion paper [21], we discussed the various problems associated with the original Sommerfeld formulation, and what were the controversies. Here, we summarize our conclusions from [21]. The main problem with Sommerfeld’s 1909 paper was that he overlooked the actual location of the surface-wave pole. The actual location of the pole is not as shown in Figure 3a, taken from Sommerfeld’s work, but it is located as shown in Figure 3b. For example, the specific locations for the pole for different dielectric constants are given by

ε= λp 2.5 − j 5 ; = r ε r = 9 − j5 ; = λp

( 0.95 − j 0.07 ) k1

( 0.96 − j 0.02 ) k1

ε= λp r 15 − j 5 ; =

( 0.97 − j 0.001) k1

ε= λp r 81 − j 5 ; =

( 0.99 − j 0.0004 ) k1

These values indicate that the location of the pole is as shown in Figure 3b, and not as originally given by Sommerfeld in Figure 3a. As pointed out by Kahan and Eckart [22], Sommerfeld did not notice while computing his asymptotic development of the branch cut integral that this contains besides the space wave, the surface wave with a negative sign and so cancels the residue of the pole, and that therefore, the path taken primitively by Sommerfeld is the correct one and not the one proposed by Epstein [23]. This is clear in Figure 3c when one changes the path of the branch cut. Furthermore, Kahan and Eckart [22] pointed out that the pole should not come into the picture, as the singularity does not meet the radiation condition, and it appears only through an inadvertency in Sommerfeld’s calculations. It was also illustrated by Baños [6, pp. 55-61] that the pole is not located on the right plane of the branch cut. This will be much clearer when the saddle-point method is applied, as the saddlepoint path never crosses the pole, and so the surface-wave contribution as envisaged by Sommerfeld never arises. This was the difference in the solutions of Sommerfeld and Weyl [24], as also pointed out by Baños [6, pp. 55-61].



In summary, the real cause of the source of disagreements among various authors that Sommerfeld made an error in the sign in his 1909 paper has no basis, as the error in the sign is a myth. We will further discuss this in a companion paper [21]. The most succinct and clear explanation of this controversy is available in Schelkunoff [25], which interestingly is not referenced in any of the papers. In his book, Schelkunoff [25, p. 430] categorically states “that the denominator equation in Equation (16) can have no roots, the integrals can have no poles, and there are no surface waves. This conclusion is contrary to that reached by early writers on the subject.” Schelkunoff then proves his point by substituting the solution into the equation for the pole. When we substitute k1k2 λP = ± in the equation k12 + k2 2 k22 λP2 − k12 + k12 λP2 − k22 = 0, it then becomes clear that k22 λ 2 − k12 + k12 λ 2 − k22 = k22

k12 k22

k12 + k2 2

− k12 + k12

k12 k22

k12 + k2 2

− k2 2

= k22 − k14 + k12 − k2 4 ≠ 0 , so there does not appear to be any surface wave in the final solution, a conclusion that everybody agrees with, and what Schelkunoff pointed out about 70 years ago [25]!

4. Properties of the Integration Path Related to the Saddle-Point Method of Integration We next are going to follow Ott’s formulation [20], which is more complete than that of Weyl’s [24], as Ott deals with the presence of the pole near the saddle-point path of integration. The integral in Equation (29) can be simplified by making the following substitutions:

λ = k1 sin β ,

(34)

ρ = R2 sin θ ,

(35)

z + z′ = R2 cos θ .

(36)

The interpretation of the angle θ is shown in Figure 1. Hence, the application of Equations (32), (34), (35), and (36) to Equation (29) yields

23

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1/2

g sV

 2k sin β  ≈ ∫ 1  π R2 sin θ  Γ1 

{

Since Im ( = k1 cos β ) Im  k1 sin (π= 2 ± β )  0 , these branch

ε − sin 2 β ε cos β + ε − sin 2 β

}

exp j  − π 4 − k1 R2 cos ( β − θ )  d β , (37) where Γ1 is a path in the complex β plane, as shown in Figure 4. There is one obvious weakness in the arguments presented to derive Equation (37), namely, that there are points on the ( 2) path for which the argument of the Hankel function, H 0 ( λρ ) , used in Equation (37) through Equation (32) is not large and may even be zero, so that the asymptotic expansion for large arguments cannot be used. However, as argued by Brekhovskikh [9], the arguments will be rigorous if the large-argument approximation is used only after the path of integration has been changed to the path of steepest descent, Γ 0 , of Figure 4. The result will then be the same. Assuming medium 1 to be lossless, then k1 in the transformation of Equation (34), λ = k1 sin β , implies, for a complex β ( β= β ′ + j β ′′ ) , that the complex value of λ is = λ ′ + jλ ′′ k1 ( sin β ′ cosh β ′′ + j cos β ′ sinh β ′′ ) . (38)

cuts will run parallel to the path Γ1 ,  Im ( k1 sin β= ) λ=′′ 0 , but shifted by ± π 2 from the origin along the real axis. The transformation λ = k1 sin β has thus transformed the upper and lower sheets associated with the branch points ± k1 into one sheet, where certain strips on the sheet belong to the previous upper (U ) and lower ( L ) Riemann sheet on the λ plane. The remaining branch points, λ = ± k2 , are transformed into sin β B2 = ± ε , which has solutions

β B2 = j ln  ± j ε ± 1 − ε  .

(42)

The branch cuts Im  k22 − λ 2  = 0 are transformed into   Im  ε − sin 2 β  = 0 . In the β plane, there are then two   Riemann sheets, connected along the branch cuts 1/2   Im  ε − sin 2 β 0 of the branch points β B2 . =  

(

)

Finally, the poles in the λ plane are now given by

ε cos β P + ε − sin 2 β P = 0 . Hence,

Equivalently,

λ ′ = k1 sin β ′ cosh β ′′ ,

(39)

λ ′′ = k1 cos β ′ sinh β ′′ .

(40)

Hence, the mapping in Equation (34), λ = k1 sin β , transforms the quadrants of the λ plane in parallel strips of width π 2 radians, and the path of integration from λ ′ = −∞ to ∞ is = transformed to the path Γ1 = , where λ ′′ Im ( k1 sin β ) 0 , as shown in Figure 4. The requirement Im

(

)

k12 − λ 2 < 0

amounts to Im ( k1 cos β ) < 0 on the path of integration, or, for k1 real, sin β ′ sinh β ′′ > 0 ,

(41)

+ j β ′′ ) cos β ′ cosh β ′′ − j sin β ′ sinh β ′′ . The script as cos ( β ′ = U in Figure 4 denotes the strips of the β plane on which the above inequality, Equation (41), is satisfied (the upper Riemann sheet). The other strips are denoted by L (the lower Riemann sheet). The path Γ1 then totally lies on U . The location of the branch points at λ = ± k1 in the λ plane are now transformed into sin β B1 = ±1 in the β plane, and are situated at ± π 2 , ± 3π 2 , and so on. The branch cuts along Im

(

)

k12 − λ 2 = 0

are now transformed into Im ( k1 cos β ) = 0 , and begin at the branch points β = ± π / 2. 24


Figure 4. The complex β plane, showing possible branch points, branch cuts, poles, and the path of steepest descent for an imperfect ground plane with the material parameter 2

= ε ε ′ (1 − j ) and ε > 1 . IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

12/9/2012 3:49:58 PM

sin β P = ±

ε

Application of Equations (32), (34), (35), and (36) to Equation (29) yields

with cos β P = −

5. Analysis of the Reflected Field

(43)

1+ ε

1 , 1+ ε

(44)

g sV ≈

∫e

− jπ

ε − sin 2 β 2k1 sin β π R2 sin θ ε cos β + ε − sin 2 β

4

Γ1

exp  − jk1 R2 cos ( β − θ )  d β ,

since Im ( k1 cos β ) < 0 . The possible locations for the poles can be approximated by

βP ± sin=

ε 1+ ε

≈ 1−

1 π  ≈ cos   β P  , 2ε 2 

which results in 1  π βP ≈ ±    . 2 ε 

(45)

For the parameters of a highly conducting ground, 2

= ε r ε ′ (1 − j ) , the locations of the poles and the branch cuts are pictorially depicted in Figure 4 (not to scale). Out of the

possible locations of the branch points and poles, B23 , B24 , P2 , and P3 are situated on the upper Riemann sheet of the branch points β B2 , on which Im ( k1 cos β ) < 0 . It is also important to

note that none of the poles ( P2 , P3 ) are situated between the original path of the integration, Γ1 , and the path of the steepest descent, Γ 0 . However, when the path of steepest descent, Γ 0 , lies in close proximity of the pole P2 , special precautions must be taken in the evaluation of the integral of Equation (37), as carried out by Ott [20]. The pole P1 is of no concern, since it lies on the second Riemann sheet of the branch point β B2 , on which Im

(

2

)

(46) where Γ1 is a path in the complex β plane as shown in Figure 4. The path of steepest descent never crosses any of the poles. The contributions along the borders of the second branch cut associated with the branch point k2 – particularly for low values of the dielectric constant, ε – are not necessary, as they would be quickly decreasing exponentials [7] that can be neglected in comparison to the contribution from the saddlepoint integration. Hence, by application of the method of steepest descent [Equations (65) and (70), as explained in Appendix 2] to Equation (46), for θ < π 2 (i.e., when the pole is not near the saddle point), one obtains g sV ≈

2 exp [ − jk1R2 ] R2

  1 − 1  2 jk1R2   +

In summary, the saddle-point method clearly shows that the pole contribution is in no way contained in the final solution, and Sommerfeld simply overlooked the actual location of the pole when he transformed the contour of the branch cuts from Figure 3a to Figure 3c. When the branch-cut contour is modified for the true location of the pole, illustrated in Figure 3b, the contribution from the surface-wave-pole term would then have cancelled, as illustrated in Figure 3c. This was pointed out by Kahan and Eckart [22]. IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012


ε cos θ + ε − sin 2 θ

  ε ( ε − 1)  2ε ( ε − 1) + ε cos 2 θ 3 − cos 2 θ      2 2  ε − sin 2 θ ε cos θ + ε − sin 2 θ    

(

(

)

)

(

)

ε ( ε − 1) cos θ ε − sin 2 θ 2ε + sin 2 θ  

(ε − sin 2 θ )

ε − sin β > 0 . The presence of the branch point

B23 should ordinarily be taken into account when ε is close to unity, when deforming the path Γ1 into the path of steepest descent, Γ 0 . Often, the contribution along the borders of the branch cut would be a quickly decreasing exponential that could be neglected in comparison to the contribution from the saddle-point integration. It is just important to note that when the saddle-point path crosses the branch cut, one would get a lateral wave (as defined in Appendix 1), which we do not discuss in this paper [9; 26, p. 508].

ε − sin 2 θ

2

ε cos θ + ε − sin 2 θ   

2

   −  . 4sin 2 θ   1

(47) Hence, Π1reflected of Equation (20) can now be written as z Π1reflected ≈P z

  ε ε − 1)  2ε ( ε − 1) + ε cos 2 θ 3 − cos 2 θ   (    3 3/2  ε − sin 2 θ ε cos θ + ε − sin 2 θ     ε ( ε − 1) cos θ 2ε + sin 2 θ ε − sin 2 θ   

1 + jk1R2

+

exp ( − jk1R2 )  ε cos θ − ε − sin 2 θ  R2  ε cos θ + ε − sin 2 θ 

(

(

) (

(ε − sin 2 θ )

3/2

)

)

ε cos θ + ε − sin 2 θ   

  −  . 2sin 2 θ ε cos θ + ε − sin 2 θ     

ε − sin 2 θ

3

(48) 25

12/9/2012 3:49:58 PM

The first term of Equation (48) represents a spherical wave originating from the image, and can be rewritten as Π1reflected ≈ P ΓTM z

exp ( − j k1R2 ) R2

,

(49)

where ΓTM can be recognized as the TM reflection coefficient associated with the spherical wave [7, 10]. This is given by

ε cos θ − ε − sin 2 θ ΓTM = . ε cos θ + ε − sin 2 θ

(50)

then we observe that the fields will be solely determined by the second- and higher-order terms of Equation (52). Also, there is no surface-wave term in the expression, and the dominant term behaves as 1 R 2 . The reason for this poor convergence in the vicinity of θ ≈ π 2 is that the effect of the pole at π 2 becomes important. Also, it is important to point out that the saddle-point path does not cross any of the poles. We will address this point in a companion paper [21]. The bottom line is that since it is the higher-order terms that are responsible for the calculation of the fields along the interface in Equation (52), we need to carry out a different asymptotic expansion, starting using Equation (21), and not Equation (20). This is the topic of the next section.

The name reflection-coefficient method is derived from Equation (49), since Π1reflected is now obtained as the reflection z

coefficient times the potential from the image of the source. The method represents a good approximation, as long as the fields are computed far away from the ground plane, and also away from the source dipole to ensure θ < π 2 . This implies that the use of the reflection coefficient in the computations of the reflected fields are not valid near the ground, where θ ≈ π 2 [10, 12]. The total Hertz potential in medium 1, when the conductivity of the relative permittivity of the lower medium is large, i.e., ε > 1 , is given by  exp ( − jk1R1 ) exp ( − jk1R2 ) Π1z ≈ P  + R1 R2  3  ε cos θ − 1   2ε  1    . + + ...   jk1R2  ε cos θ + 1   ε cos θ + 1   (51)

Note that when ε → ∞ , Π1z of Equation (51) goes properly into the form of a source plus an image term due to a vertical electric dipole located above a perfectly conducting ground plane.

6. Fields Near the Interface In order to solve for the total fields near the interface [7], a modified saddle-point method, as explained in Appendix 3, is applied to take into account the effect of the pole β P near the saddle point. In the expressions for both g sV and GsV in Equations (20) and (21), there is a pole, β P , which is seen from [27] 1 ε cos β + ε − sin 2 β   

and cos β P = −

(54)

1

ε +1

ε ε +1

. Applying Equations (54) and (108) to

Equation (21), we obtain 1/2

 π   2k sin β  GsV ε exp  − j  ∫  1 =  4  Γ  π R2 sin θ   1

exp  − jk1 R2 cos ( β − θ )  cos β

ε cos β + ε − sin 2 β

Π1z ≈

≈ε

 exp ( − jk1R1 ) exp ( − jk1R2 )  2ε P − + exp ( − jk1R2 ) + ... 2 R1 R2 jk1R2   (52)

It is now important to recognize from Equation (52) that the sum of the first two terms may be smaller than the third term. As a matter of fact, when both the transmitter and the receiver are near the ground, i.e., R= 1 R2 ≈ ρ ,

(53a)

z ≈ 0 ≈ z′ ,

(53b)


ε − sin 2 β − ε cos β , ε 2 − 1 sin ( β + β P ) sin ( β − β P ) 1

where ε cos β P + ε − sin 2 β P = 0 , with sin β P = ±

However, when θ ≈ π 2 , this becomes

26

=

dβ

4π k1 j cos θ ε − sin 2 θ − ε cos θ R2 cos θ − 1 ε 2 −1 ε +1  exp − jk1R2 − W 2  erfc ( jW )   , 1+

cos θ ε sin θ + ε +1 ε +1 (55)

where


12/9/2012 3:49:58 PM

 θ − βP W 2 = − jk1R2 2sin 2   2  cos θ = − jk1R2 1 + − ε +1 

The total Hertz potential in medium 1, which is valid near the interface for ε > 1 and θ ≈ π 2 , thus becomes

  

ε sin θ  , ε +1 

(56)

and W was called the numerical distance by Sommerfeld [1]. If ε > 1 and θ ≈ π 2 , and if W is very small, then we have exp  −W 2  erfc ( jW ) ≈ 1 . Under this assumption, applying   Equation (36) to Equation (55), one gets a simplified expression for GsV ≈ −

( z + z′) ε 2π k1 j exp [ − jk1R2 ] R2 R2 ε 2 −1

≈ − 2π k1 j

( z + z ′ ) exp [ − jk1R2 ] R1.5 2

.

(57)

Equation (57) thus illustrates that when θ ≈ π 2 , the dominant 1 term of the potential Π1z ∝ 1⋅5 , and therefore the leading R2 term for the fields will also vary as

1

ρ1⋅5

, if ( z + z ′ ) is small

compared to ρ in Equation (18). It is interesting to observe that Equation (57) is not a function of the ground parameters. The path-loss exponent factor in mobile urban cellular communication should thus be three near the ground and the reflection-coefficient method is not applicable, under those circumstances. This should approximately hold for any types of ground parameters, such as urban, suburban, or even lakes and oceans. One possible reason for this is given by Stratton [3]: the reflection coefficient is approximately +1 for a perfectly conducting ground when the fields are observed far from the ground, and it transforms to −1 when the fields are observed near the ground, when θ ≈ π 2 . This particular variation of the field near an imperfect ground will be verified by experimental data and by a more-accurate numerical analysis in the next sections. However, as W becomes large, then −j exp  −W 2  erfc ( jW ) ≈   W π

1   1 +   2W 2  3π for W → ∞ and arg W < , 4

and for ε > 1 , W 2 ≈

(58)

− jk1R2 . Under this condition, 2ε

GsV ≈ 2 ε exp [ − jk1 R2 ]

( z + z′)  R22

ε  1 − . jk1 R2  

(59)



Π1z

  exp ( − jk1R1 ) exp ( − jk1R2 ) − P  R1 R2     − j 2π k z + z ′ exp ( − jk1 R2 )  , W 1  jk1 R2   R22  

The above simplified expressions illustrate that a Norton surface wave decays asymptotically as 1 R 2 , and this applies only in the far-field region, where W > 1 , as the first two terms cancel in the second expression. It is also interesting to note that the third term for W > 1 provides the so-called height-gain for the transmitting and receiving antennas. However, this heightgain again only applies to the far-field regions. In the intermediate region, the fields decay as approximately 1 ρ1.5 . This represents a two-dimensional radiation field often associated with a surface wave, as we shall see in a companion paper [21], which will discuss the physical characteristics of the fields that arise in a mobile cellular wireless communication system. Observe also that for W < 1 , the above expression is independent of the ground parameters. This will be confirmed in the next section using a more-accurate numerical analysis. In summary, Sommerfeld characterizes W as the numerical distance. When the large-argument approximation is invoked for W, then the fields decay as 1 R 2 . Interestingly, this is one of the confusing parts in all the discussions, as Sommerfeld stated: ...for small values of the numerical distance, the spatial-wave type predominates in the expression for the reception intensity; in this case the ground peculiarities have no marked influence and we can make computations using an infinite ground conductivity without introducing great errors. For larger values of W the rivalry between the space and the surface waves are apparent. However, our interests in a cellular wireless communication system are for small to intermediate values of W, for which very little information is available. Our observations for the path-loss exponent in a cellular wireless communication is that the value is three for moderate distances from the base-station antenna. In the fringe region (i.e., further away from the basestation antenna) the value is four, which will be further illustrated by a more-accurate numerical analysis and experimental results! In addition, in this region, the ground parameters have little effect, as seen by Equation (57).

27

12/9/2012 3:49:59 PM

At this point, it is important to point out the novelty of our solution, which is not available in the popular literature, because for the intermediate region, we have used two different procedures that deviated from the classical formulations. First, we used the second form of the Green’s function, as shown in Equation (21), to observe the fields near the interface. Second, we used a different saddle-point method of integration in handling the pole near the saddle point in evaluating the integrals of Equation (21), which was outlined by Clemmow [27] and also used by Hill and Wait [13]. If one applies the modified saddle-point method of evaluating the integral – as explained in Appendix 3 – to the Green’s function given by Equation (20), one then obtains g sV ≈  − jπ  2k1 exp    4  π R2

∫

Γ1

≈

( ε − sin θ )( ε − sin θ − ε cosθ ) 2

(ε 2 − 1) ( cosθ − 1

exp  − jk1R2 cos ( β − θ ) 

( β − βP ) sin

2



)

ε + 1  2sin

(θ + β P )  2

 

dβ

2

4π k1 j R2

ε − sin 2 θ − ε cos θ ε − sin 2 θ cos θ − 1 ε + 1 ε 2 −1 exp  − jk1R2 − W 2  erfc ( jW )   . 1+

cos θ ε sin θ + ε +1 ε +1

(60a)

For small values of W, and when the fields are desired close to the interface, then we also require θ ≈ π 2 . In this case, for ε >1 we obtain g sV ≈ −

2π k1 j exp [ − jk1R2 ] . R2 ε +1

(60b)

By incorporating Equation (60b) into Equation (20), it is seen that the space-wave term dominates, and the additional contribution of the surface-wave term, given by Equation (60b) and as predicted by Sommerfeld, is small. However, when W is large, then g sV ≈

2 exp [ − jk1R2 ]  ε  1 − . R2 jk  1 R2 

(60c)

Substituting this expression into Equation (20), it is seen that the dominant terms for the space waves cancel each other, and the Hertz potential is given by the higher-order terms. Therefore, Π1z ≈

2 Pε exp [ − jk1R2 ]

28


jk1R22

.

(61)

A similar asymptotic form was previously obtained, as seen in Equation (52). That is why we expanded the Hertz potential in a different form, given by Equation (21), which cancelled the space waves and provided the dominant ground-wave term. The rationale for doing this was explained by Stratton, as the reflection coefficient is +1 for a perfect ground when the fields are evaluated far from the interface, but it then transforms to −1 when the fields are evaluated near the interface. This second form was also originally used by Sommerfeld. In short, there are two unique features of this presentation as it differs from other researchers’ work. First, there is the use of Equation (21) in the modified saddle-point method to calculate the fields in the regions both near and far from the base-station antenna. Second, there is the use of a different mathematical form when applying the modified saddle-point method when there is a pole near the saddle point, as explained in Appendix 3. Almost all researchers factored out the pole term and then had a remainder term that did not contain the pole, and applied the saddle-point method to that expression. We decided to follow a different route, which was outlined by Clemmow [27] and is presented in Appendix 3. That is why our expressions are different than, for example, those of Tyras [7] and Collin [18], who used the conventional method of extracting the pole, even though they made use of Equation (21). It is also true that for small values of the numerical distance W, the variation of the fields is quite different than from the results when the value of W is large. However, in all fairness, there have been so many approximations made – and also approximations made further to approximating those approximations – that it is very difficult to gauge what the degree of accuracy is of the expressions that we have developed so far. That is why it is extremely important to complement this analysis with a more-rigorous purely numerical computation, which involves integrating the Sommerfeld integrals in a very accurate way, as explained in [11] and [28]. In addition, it is always necessary to further supplement all the analysis with experimental data, as presented in [29-31]. This is exactly what we propose to present in the next two sections.

7. A More-Accurate Analysis of the Fields Near an Earth-Air Interface This section provides an accurate computation of the fields in an urban environment, using an accurate computer program [28]. This is done using the Sommerfeld formulation and was described in [31]. It is followed by measured experimental data for the path loss for various base stations in an urban environment at two different frequencies of 900 MHz [30] and 1800 MHz [29]. More experimental results will be presented in the accompanying paper [21] for suburban and coastal environments, which, interestingly, have similar behaviors. So far, various types of approximations have been made to compute the variation of the electric field with distance in order to study the nature of its decay as a function of the disIEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

12/9/2012 3:49:59 PM

tance along with the height of transmitters and receivers from matrix elements for any electromagnetic problem. In fact, in the ground. Some of the approximations that have been made in [19] the gradient operator was interchanged with the integral the earlier sections were often contradictory. The bottom line is following it. This is plain wrong, as the gradient operator that after all these various approximations, and after expanding cannot be taken under the integral in Equation (62b)! It is seen the solutions in terms of asymptotic series, it is difficult to that if Equation (16) is differentiated with respect to either ρ or judge with confidence the validity of these conclusions, from z, then the resultant integral will not converge! a scientific point of view. At times, this type of approximate  The mutual impedance, Z BA , between two z -directed analysis has raised lots of controversy and debate [18]. We will discuss that in [21]. At this point, to avoid these in-depth current elements, I A and I B , of lengths  A and  B , is discussions or speculations, we will take recourse to a purely expressed as numerical methodology, which evaluates the Sommerfeld   integrals in an essentially very accurate manner. There have Z BA = − ∫ E A  I B dz B′ B been several methodologies and user-friendly codes that have been published in the literature to numerically compute the jωµ0 fields from a transmitting to a receiving antenna using the exact = I B dz B′ ∫ I A ( g o + g s ) dz ′A A Sommerfeld formulation, which is presented next. 4π ∫ B   The electric field, E A , in medium 1 due to a z -directed current element, I A , of length  A is given by   ∂  = E A  zk 2 + ∇  Π1z ∂z   =

− jωµ0 4π

∫

 zI A ( g o + g s ) dz ′A A  = Z BA ∇ ∂ ′ + I A ( g o + g s ) dz A , (62a) jω 4πε 0 ∂z ∫ A

 − jωµ0  = EA zI A ( g o + g s ) dz ′A ∫  4π A  ∂I A ∇ + ( go − g s ) dz ′A . ∫  jω 4πε 0 A ∂z ′A

1 jω 4πε 0

∫

B

IB

 ∂  ∂I A g o − g s ) dz ′A  dz B (  ∫ ∂z B  A ∂z ′A  (63a)

Transferring the derivative operation on I B in the second term of the above expression, and assuming I B to go to zero at the open ends of the wires, by integration by parts one obtains

where g o and g s are defined in Equations (15) and (16), dg 0 dg dg s dg s respectively. Since = − 0 and = , then assumdz dz ′A dz dz ′A ing that the current goes to zero at the ends of the open wires, ∂ derivative operation on ( g o + g s ) can be transformed the ∂z ∂ to now operating on I A instead by applying integration ∂z ′A by parts. It is important to note that only one of the derivatives can be interchanged with the integral in the second term in Equation (62a). Hence,

(62b)

At this point, it is important to remember that because of the nature of the singularity of the Green’s function, the gradient operator on the second term cannot be interchanged with the integral sign. This is because if the gradient operator is interchanged with the integral sign and it operates on the Green’s functions, then it would result in a divergent integral. This point is important. In the IEEE Standard Definitions of Terms for Radio Wave Propagation [19], this subtle point has been overlooked, resulting in an incorrect expression for the electricfield integral equation, which cannot be applied to the problem at hand, and also when computing the self-impedance of the IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012


−

jωµ0 I B dz B′ ∫ I A ( g o + g s ) dz ′A A 4π ∫ B  ∂I B  ∂I A 1 + ( go − g s ) dz ′A  dzB  jω 4πε 0 ∫ B ∂z B  ∫ A ∂z ′A  (63b)

In this presentation, observe that the Green’s function in Equation (63b) is never differentiated, and therefore one can obtain a stable, accurate solution using this methodology. This procedure has been implemented in a general-purpose computer code to analyze arbitrarily shaped and oriented wire antennas over an imperfect ground plane. The methodology chosen was also implemented in the commercially available software package Analysis of Wire Antennas and Scatterers (AWAS, V. 2) [28], which accurately computes the fields from a dipole source over an imperfectly conducting ground. AWAS is a complete electromagnetic field simulator for wire-like structures, utilizing the accurate Sommerfeld formulation for treating an imperfect ground plane. We used AWAS to calculate the variation of the vertical component of the fields from a halfwavelength dipole located above an imperfectly conducting ground. We will use it as a confirmation tool using a similar scenario that was used by Okumura et al. [32] in their classic propagation measurements in the city of Tokyo. Okumura et al. [32] placed the transmitting antenna on Mount Fuji at various heights, but we will pick a height of 140 m for our comparison. Okumura et al. received the signal 3 m above the ground by placing the receiving antenna – which was a five-element Yagi, having a gain of approximately 11 dB – on top of a van, and drove the van in the city of Tokyo from 1 km to 100 km from the transmitting antenna. Here, we consider only the measurements done at 453 MHz. As a transmitting antenna, Okumura et al. used a five-element Yagi 29

12/9/2012 3:49:59 PM

antenna radiating 150 W of power. In our simulations, we used a simple half-wave dipole antenna, center-fed with 1 V. We now compare Okumura et al.’s measured field-strength data with our electromagnetic-field computations. Since the power levels in the measurement and the theory were different, we would not expect the same field-strength levels for both of the plots. Our goal was to observe the nature of the decay of the fields as a function of distance from the transmitting antenna, to see if a macro modeling of the environment could yield meaningful results for carrying out accurate prediction of the measured path-loss exponent. Figure 5 plots Okumura et al.’s experimental data, using the blue triangles, from 1 km to 100 km. Superimposed on the same plot is Hata’s prediction [33], illustrated by the solid blue line passing through the experimental data. The red curve on the top in Figure 5 displays the decay of the field strength if the ground plane was absent. Clearly, the Okumura et al. measurements did show that the ground had an effect. The bottom black curve, representing the dots, is the prediction obtained using AWAS from a macro model. This implied that the effects of trees, buildings, and the terrain effects were not considered to be essential in propagation modeling. We only assumed a flat urban ground. It was clear from Figure 5 that the blue curve representing the experimental data and the black curve representing the electromagnetic-analysis data showed remarkable agreement in their slopes. It is important to point out that for both the theoretical and experimental data, the slope between 1 km and 10 km was about 30 dB per decade, which we expected from our earlier theoretical analysis using the saddlepoint method. The slope between 10 km and 100 km was 40 dB per decade, as predicted by the Norton surface wave for the far field. This illustrated that similar results could be obtained from all three computations, using an approximate theoretical analysis, an accurate numerical analysis of the antennas over an imperfect ground plane using a numerical electromagnetics code, and from measurements. The three separate results confirmed that a macro modeling of the environment was sufficient to accurately predict the path loss, and a micro modeling may be overkill. These data raised an important question, having to do with the phenomenon of slow fading. Why did neither the theoretical data obtained from a numerical code, nor the experimental data, show the slow-fading characteristics, namely, an oscillatory variation of the field strength with distance? It is important to point out that all these data represented phasor quantities, and therefore represented the complex voltages in the frequency domain. The time variation does not show up in the phasor notation in electrical engineering! In our plots of the phasor voltages, for all the cases, the decay was monotonic! To search for the origin of fading, we generated an expanded version of the theoretical plots, as shown in Figure 6. In this figure, we considered the fields radiated by a half-wave dipole antenna, transmitting at a frequency of 1 GHz. The height of the transmitting dipole from the ground was varied. We chose to analyze the scenario for different heights of the transmitting antenna above the imperfectly conducting ground 30


Figure 5. The variation of the magnitude of the Ez component of the electric field from a half-wavelength dipole located 140 m from the ground as a function of distance, at an operating frequency of 453 MHz. The height of the observation point was 3 m from an urban ground with a permittivity of ε r = 4 and a conductivity of σ = 2 × 10−4 .

Figure 6. The variation of the magnitude of the Ez component of the electric field from a half-wavelength dipole as a function of distance, at an operating frequency of 1 GHz. The height of the observation point was 2 m from an urban ground with a permittivity of ε r = 4 and a conductivity of

σ = 2 × 10−4 .

of 5 m, 10 m, 20 m, 100 m, and 500 m. A relative permittivity of ε r = 4.0 and a conductivity of σ = 2 × 10−4 mhos/m, representing the properties of a typical urban ground as described in [34], were chosen for the ground plane. Electric fields were calculated at a height of 2 m from the ground. Figure 6 shows the variation of the received field due to different heights of the transmitter above the urban ground. We saw that when the field was computed at a distance of approximately 8 HTX H RX λ from the transmitting antenna [34], the magnitude of the field increased with the height of the transmitting antenna above the ground, propagating the dictum of height-gain in deploying antennas, as prevalent in the literature. In addition, the decay of the vertical component of the field was monotonic. However, for distances less than 8 HTX H RX λ from the base station, the field strength actually IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

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decreased with the height. In addition, in the near-field regions, there appeared to be an interference pattern between the direct wave and its image (due to the imperfect ground), leading to slow fading. After this interference pattern, the field decayed quite smoothly, which generally started around 4 HTX H RX λ . Initially, the decay was 30 dB per decade, increasing to 40 dB per decade in the far field. Therefore, the path-loss exponent inside the cell was three, and in the fringe regions it went to four, as expected. As illustrated in [34], the higher the height of the antenna from the ground, the farther was the starting point of the far field. In summary, what we observed is that when the fields decayed as R −1.5 , then we were in the intermediate region, as predicted by the theory. In the fringe regions, the fields decayed as R −2 , as predicted. Let us now focus our attention on the situation when the transmitting antenna was located at a height of 5 m from the ground. If we look at the decay of the fields from a distance of 200 m to 400 m, we approximately observed a decay of the fields in amplitude as a function of distance by a factor of three, which was close to the expected value of 21.5 = 2.83 . For the decay of the amplitudes of the fields from 4 km to 8 km, we observed that the reduction in the amplitude of the fields was about 4.1, which closely followed the 22 = 4 law. We now considered another case, when the transmitting antenna was located 20 m from the ground. We looked at the decay of the fields from a distance of 1 km to 2 km from the transmitter. We approximately observed a decay in the amplitude of the fields by a factor of 3.19, which was close to 21.5 = 2.83 . If we focused our attention from 5 km to 10 km in the same plot, the decay of the fields was about 3.5 in magnitude, approaching the 22 = 4 law.

height of 2 m, as a function of the separation between the two antennas. We considered different scenarios for the electrical parameters of the ground. We considered a poor urban ground ( ε r = 4 , σ = 0.001 ), an average ground ( ε r = 15 , σ = 0.005 ), a good ground ( ε r = 25 , σ = 0.02 ), fresh water ( ε r = 81 , σ = 0.01 ), and sea water ( ε r = 81 , σ = 5.0 ). These data were taken from [35, Table 1, p. 83]. The vertical component of the electric field was plotted in Figure 7 as a function of the separation of the horizontal distance between the transmitting and the receiving antennas. It was seen that there was not much difference among these five plots, indicating that the nature of the ground has very little effect inside a cell. These results were also predicted from Equation (57). We next look at some experimental data.

8. Experimental Data for the Propagation Path Loss for Wireless Transmission in an Urban Environment Here, we present experimental data from seven cellular wireless base-station antennas, operating in an urban environment. The measurements were carried out for various base stations in both dense and sparse urban environments. Experimental data for the path loss as a function of distance was available for various stations operating at frequencies of 1800 MHz [29] and 900 MHz [30, 31]. First, the results are presented for data taken at 1800 MHz for three base stations located in dense urban, urban, and suburban environments of the Indian capital city of Delhi. The special feature of the Delhi urban environment is that the houses are not uniformly spaced.

Finally, we considered a transmitting dipole 500 m above the imperfectly conducting ground, and observed the fields from 20 km to 40 km from the transmitting antenna. We observed that the decay in the fields was approximately a factor of 3.06, which was close to 21.5 = 2.83 . From these sparse samples, it appears that once the slowfading region disappears, then the fields manifest a decay of 30 dB per decade, then going to 40 dB per decade. It appears then that for mobile urban communication, there is a region in the propagation path where the field behaves as if being emanated from a two-dimensional source (the physics of this phenomenon is explained in a companion paper [21]), and this seems to be the main propagation mechanism in cellular wireless communication. However, as we go far away from the transmitter, the field decays as the square of the distance. The illustrations of Figure 6 show that inside a cell, the wave has a decay rate of 1 R1.5 , which increases to 1 R 2 in the fringe regions. We considered the variation of the fields for a fixed transmitter-antenna height of 5 m, with the receiver located at a IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012


Figure 7. The variation of the magnitude of the Ez component of the electric field from a half-wavelength dipole as a function of distance, at an operating frequency of 900 MHz. The height of the observation point was 2 m above different types of ground. 31

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The carrier signals of 1800 MHz GSM base-station transmitters, located in the dense urban/urban environments of the New Delhi area belonging to Idea cellular network, were monitored with a Nokia GSM receiver (model 6150), generally used as a drive-in tool for planning cellular networks, along with a GPS receiver to provide the latitude and longitude of the mobile. The sensitivity of the receiver was −102 dBm. The transmitting powers of all of the base stations were +43 dBm. The receiving equipment was installed in a vehicle, along with the data-acquisition system. The vehicle moved on a normal road at a permissible speed in the traffic, and the downlink signal strength was monitored by the receiver. The observed signal levels were converted into path-loss values, based on the received signal level, the transmitter power, and the transmitting and receiving antenna gains, for further analysis. The gain of the transmitting antenna was 18 dB. The estimated measurement rms (root mean square) error was around 1.5 dB. The height of the mobile receiving antenna was 1.5 m from the ground. In the case of the 900 MHz results, the experiment was carried with the help of Aircom International Limited, a UK company based in India. The transmitting power of all the base stations used in that study was 43.8 dBm, and the transmitting antenna gain was 2 dBi for all the base stations. The gain of the receiving antenna was 0 dB, and the height from the ground was 1.5 m. The receiver was standard Nokia equipment used in drive-in tools used for field trials.

Figure 8. A Google map illustrating the locations of the various base-station transmitters used in this study.

For all the measurements, the position of the mobile was determined from the GPS receiver. This information, with the coordinates of the base station, was utilized to deduce the distance traveled by the mobile from the base station. The signalstrength information recorded in dBm was converted into pathloss values utilizing the gains of the antenna. The data were recorded with 512 samples in one second on a laptop computer, and the number of samples collected for each site varied from

1× 105 to 2 × 105 . The measured rms (root mean squared) error was around 1.5 dB. The data were averaged over a conventional range of 40λ .

All of the seven base stations used in this study were located in the national capital region of Delhi. They are depicted on a Google environmental map, as seen in Figure 8. In this map, the various stations denoted by abbreviations OM1, MAM, and BJV came under the 1800 MHz category. The base stations denoted by UA, SNT, FBD, and VKH were under the 900 MHz category. Figures 9a and 9b depict photographs of the environmental features of some of the base stations where measurements were conducted, to give an idea about the clutter. Figures 10a and 10b depict the clutter diagrams generated from Aircom radio planning tool ASSET 3G (http://www. aircominternational.com), based on the digital terrain data for the BJV, VKH, and FBD base stations. The FBD base station was situated in an industrial area in an urban environment. Similarly, VKH was located in an urban environment, whereas BJV was located in a low-density urban-zone environment. These are shown as representative diagrams to illustrate the utility of the radio-planning tool for checking the validity of prediction of the results in the presence of environmental clutter. 32


Figure 9a. A photograph of a typical urban environment used in the study.

Figure 9b. A photograph of a typical urban environment used in the study. IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

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Figure 10a. Clutter diagrams for the BJV and VKH base stations, generated from the Aircom radio planning tool.

Figure 12. The variation of the path-loss exponent with distance for the MAM base station (1800 MHz).

Figure 13. The variation of the path-loss exponent with distance for the BJV base station (1800 MHz).

Figure 12. The variation of the path-loss exponent with distance for the MAM base station (1800 MHz).

The first base station, called OM-1, was located in a dense urban environment, and the height of the transmitting basestation antenna was 24 m from the ground. The variation of the path loss is shown in Figure 11. It was seen that in the cell, the path-loss exponent settled down approximately to a factor of 4 HTX H RX 4 × 24 × 1.5 = 864 m , = three after a distance of 1/ 6 λ and that was what the experimental results showed. In Figure 12, the second base station, called MAM, was located in an urban environment. Again, it was seen that the propagation path-loss exponent inside the cellular region was close to three after approximately a distance of 864 m from the base-station antenna. Finally, the base station called BJV was located in a suburban environment, and the path-loss exponent in Figure 13 seemed to settle down to a factor of approximately three, again after a distance of approximately 864 m from the base station.

Figure 11. The variation of the path-loss exponent with distance for the OM-1 base station (1800 MHz). IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012


Next, the results are presented for data taken at 900 MHz, for four base stations located in a medium urban environment 33

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of the Indian capital city of Delhi. The first data set was plotted for base station UA, where the base station antenna was located at a height of 24 m. From Figure 14, it appeared that the pathloss exponent factor in the cell had settled down to a value of 4 HTX H RX 4 × 24 × 1.5 = 432 m . three at around = 1/ 3 λ The second base station was called SNT, and the transmitting antenna was located at 18 m from the ground. The pathloss-exponent data is plotted in Figure 15. It was clear that it had settled down to a value of three at approximately 4 HTX H RX 4 × 18 × 1.5 = 324 m from the base-station = 1/ 3 λ antenna.

Figure 14. The variation of the path-loss exponent with distance for the UA base station (900 MHz).

The third base station was called FBD, where the transmitting antenna was located at 10 m from the ground. The pathloss-exponent data is plotted in Figure 16. It was also clear that it had settled down to a value of three at approximately 4 HTX H RX 4 × 10 × 1.5 = = 180 m from the base-station 1/ 3 λ antenna. The fourth base station was called VKH, where the transmitting antenna was located at 13m from the ground. From Figure 17, it appeared that the path-loss exponent factor in the cell had settled down to a value of three at approximately 4 HTX H RX 4 × 13 × 1.5 = = 234 m from the base-station 1/ 3 λ antenna. It was seen for all the four base stations that within the coverage of the cell, the path-loss exponents at 900 MHz had settled down approximately to a factor of three, as predicted by the theory.

Figure 15. The variation of the path-loss exponent with distance for the SNT base station (900 MHz).

It may be concluded that for propagation modeling, it is sufficient to carry out physics-based macro modeling, instead of using a finely detailed micro model of the environment. The final point is that within the cell, it was the path-loss exponent of 30 dB per decade with distance that was predominant. The resultant field strength inside a cell therefore cannot be predicted by using ray tracing, diffraction theory, or by using the reflection-coefficient method to find the fields in the neighborhood of the transmitter of a base station in a cellular wireless communication system operating in an urban environment. The free-space fields provide an asymptotic rate of decay of 20 dB per decade with distance. When the antenna was placed over a perfect ground plane, the rate of decay of the fields was still 20 dB per decade with distance from the base-station antenna. Furthermore, when the antenna was located over an imperfect ground plane and the reflection-coefficient method was used to predict the scattered fields, the rate of decay of the fields with distance was still 20 dB per decade. It was the use of the accurate Sommerfeld formulation that provided a rate of decay of 30 dB per decade – and in the fringe area, of 40 dB per decade – that was also verified by experimental data. At 34


Figure 16. The variation of the path-loss exponent with distance for the FBD base station (900 MHz). IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

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at 900 MHz and 1.8 GHz for various transmitters located in different urban environments, then the path-loss exponent factor at moderate distances from the base-station antenna settles down to a value of three.

Figure 17. The variation of the path-loss exponent with distance for the VKH base station (900 MHz).

least both an accurate theory and experiments carried out for different base stations in an urban environment confirmed this observation. This analysis also showed that a macro model of the environment is sufficient to correctly predict the propagation path loss in a cellular environment, and a micro model is not necessary!

9. Conclusion The objective of this article has been to achieve the following goals: 1.

2.

To illustrate that if the Hertz potentials are appropriately written as originally envisaged by Sommerfeld for fields at the interface, and the various expansions using appropriate approximations are carried out in a mathematically meaningful way, it may be concluded that the path-loss exponent in a cellular wireless communication system is three, and in the fringe areas, is four. To illustrate that experimental data also demonstrates that the path-loss exponent in a cellular wireless communication system is three.

3.

To illustrate that if one uses an accurate numerical electromagnetics code such as AWAS, then it is seen that in a cellular urban environment, the path-loss exponent factor is three, and in the fringe areas it is four, and therefore it is sufficient to carry out a macro modeling of the environment.

4.

To illustrate that if a modified method of steepest descent is utilized to treat a pole near the saddle point for the field near the interface, then this characteristic field can be derived when using the appropriate form of the Green’s function.

5.

To illustrate that if experimental verification is carried out to measure the path-loss exponent factor



6.

To demonstrate that there was no error in the sign in Sommerfeld’s 1909 paper: the defect was that Sommerfeld overlooked some mathematical subtleties associated with the pole while computing his asymptotic development of the branch-cut integral.

7.

To state that physics-based macro modeling is sufficient to predict the propagation path loss in a cellular urban environment, and a detailed micro modeling of the environment is overkill.

10. References 1. A. N. Sommerfeld, “Propagation of Waves in Wireless Telegraphy,” Ann. Phys., 28, March 1909, pp. 665-736; and 81, December 1926, pp. 1135-1153. 2. K. A. Norton, “The Propagation of Radio Waves Over the Surface of the Earth,” Proceedings of the IRE, 24, 1936, pp. 1367-1387; and 25, 1937, pp. 1203-1236. 3. J. A. Stratton, Electromagnetic Theory, New York, McGrawHill Book Company, 1941, p. 573. 4. A. N. Sommerfeld, Partial Differential Equations in Physics, New York, Academic Press, 1949. 5. J. R. Wait, Electromagnetic Waves in Stratified Media, New York, McGraw Hill, 1962. 6. A. Baños, Jr., Dipole Radiation in the Presence of a Conducting Half-Space, Oxford, Pergamon Press, 1966, pp. 151158. 7. G. Tyras, Radiation and Propagation of Electromagnetic Waves, New York, Academic Press, 1969. 8. E. K. Miller, A . J . Poggio, G . L. Burke and E. S. Selden, “Analysis of Wire Antennas in the Presence of a Conducting Half Space: Part I. The Vertical Antenna in Free Space,” Canadian Journal of Physics, 50, 1972, pp. 879-888. 9. L. Brekhovskikh, Waves in Layered Media, New York, Academic Press, 1973. 10. T. K. Sarkar, Analysis of Arbitrarily Oriented Thin Wire Antenna Arrays Over Imperfect Ground Planes, PhD Thesis, Syracuse University, 1975. 11. T. K. Sarkar, “Analysis of Radiation by Arrays of Parallel Vertical Wire Antennas over Plane Imperfect Ground 35

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(Sommerfeld Formulation),” IEEE Transactions on Antennas and Propagation, AP-24, 4, July 1976, pp. 544-545. 12. T. K. Sarkar, “Analysis of Arbitrarily Oriented Thin Wire Antennas over a Plane Imperfect Ground,” AEU, 31, 11, 1977, pp. 449-457. 13. D. A. Hill and J. R. Wait, “Excitation of the Zenneck Surface Wave by Vertical Apertures,” Radio Science, 13, 1978, pp. 969-977. 14. G. K. Karawas, Theoretical and Numerical Investigation of Dipole Radiation over a Flat Earth, PhD Dissertation, Case Western Reserve University, Cleveland, OH, 1985. 15. R. W. P. King, “Electromagnetic Field of a Vertical Electric Dipole over an Imperfectly Conducting Half-Space,” Radio Science, 25, March-April 1990, pp. 149-160. 16. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering, Englewood Cliffs, New Jersey, Prentice Hall, 1991, Chapter 15 and Appendix to Chapter 15. 17. J. R. Wait, “The Ancient and Modern History of EM Ground Wave Propagation,” IEEE Antennas and Propagation Magazine, 40, 5, October 1998, pp. 7-24. 18. R. E. Collin, “Hertzian Dipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20th-Century Controversies,” IEEE Antennas and Propagation Magazine, 46, 2, April 2004, pp. 64-79. 19. IEEE, IEEE Standard Definitions of Terms for Radio Wave Propagation, IEEE Std. 211-1997, 1998. 20. H. Ott, “Die Sattelpunktsmethode in der Umgebung eines Pols mit Anwendungen auf die Wellenoptic und Akustik,” Ann. Phys., 43, 1943, pp. 393-404. 21. T. K. Sarkar, W. Dyab, M. N. Abdallah, M. SalazarPalma, M. V. S. N. Prasad, S. Barbin, and S. W. Ting, “Physics of Propagation in a Cellular Wireless Communication Environment,” Radio Science Bulletin, No. 343, December 2012 (to be published). 22. T. Kahan and G. Eckart, “On the Existence of a Surface Wave in Dipole Radiation over a Plane Earth”, Proc. of the IRE, 38, 7, 1950, pp. 807-812. 23. P. S. Epstein, “Radio Wave Propagation and Electromagnetic Surface Waves,” Physics, 33, 1947, pp. 195-199. 24. H. Weyl, “Propagation of Electromagnetic Waves over a Plane Conductor,” Ann. Phys., 60, 1919, pp. 481-500. 25. S. A. Schelkunoff, Electromagnetic Waves, New York, D. Van Nostrand Company Inc., 1943. 26. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Upper saddle River, NJ, Prentice Hall, 1973. 36


27. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, New York, Pergamon Press, 1966, pp. 46-58. 28. A. R. Djordjevic, M. B. Bazdar, T. K. Sarkar, and R. F. Harrington, AWAS Version 2. 0: Analysis of Wire Antennas and Scatterers, Software and User ‘s Manual, Norwood, MA, Artech House, 2002. 29. M. V. S. N. Prasad, S. Gupta, and M. M. Gupta, “Comparison of 1.8 GHz Cellular Outdoor Measurements with AWAS Electromagnetic Code and Conventional Models Over Urban and Suburban Regions of Northern India,” IEEE Antennas and Propagation Magazine, 53, 4, August 2011, pp. 76-85. 30. M. V. S. N. Prasad, K. Ratnamala, P. K. Dalela, C. S. Misra, “Evaluation of Cellular Prediction Models Using 900 MHz Outdoor Measurements and Tuning of Lee Model over Indian Urban and Suburban Regions,” Int. J. Wireless and Mobile Computing, 5, 1, 2011, pp. 77-87. 31. M. V. S. N. Prasad, P. K. Dalela, C. S. Misra, “Experimental Investigation of GSM 900 MHz Results over Northern India with AWAS Electromagnetic Code and other Prediction Models,” PIER, 125, 2012, pp. 559-581. 32. T. Okumura, E. Ohmori, and K. Fukuda, “Field Strength and its Variability in VHF and UHF Land Mobile Service,” Rev. Elect. Commun. Lab., 16, 9-10, 1968, pp. 825-873. 33. M. Hata, “Empirical Formula for Propagation Loss in Land Mobile Radio Service,” IEEE Transactions on Vehicular Technology, 29, 3, 1980, pp. 317-325. 34. A. De, T. K. Sarkar, and M. Salazar-Palma, “Characterization of the Far-Field Environment of Antennas Located over a Ground Plane and Implications for Cellular Communication Systems,” IEEE Antennas and Propagation Magazine, 52, 6, 2010, pp. 1940. 35. W. C. Jakes, Microwave Mobile Communications, New York, IEEE Press, 1974, p. 101.

Appendix 1: Definitions of the Various Waves Used in This Paper In this appendix, we present the definitions for the various types of waves that we have used in this paper. Our definitions are slightly different from [19], and are included here for completeness. Ground wave: A wave from a source in the vicinity of a flat Earth that would exist in space in the absence of the ionosphere. NOTE: A ground wave near the surface of the Earth can be identified with a Norton surface wave for a grazing angle of incidence and when both the transmitter and the receiver are far away from each other on the Earth’s surface. IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

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Lateral wave: A wave guided along the interface between two media. For sufficiently large distances from the source, the field decays as the square of the distance. This wave is usually present when a wave is incident from a denser to a less-dense medium undergoing total internal reflection. In that case, it displays the Goos-Hänchen effect of a lateral shift in the reflected wave. Norton surface wave: A propagating electromagnetic wave on the surface of the Earth when both the transmitters and the receivers are close to the ground. Asymptotically, this wave also decays as the square of the distance. It consists of the total field minus the geometrical-optics field, and hence does not satisfy Maxwell’s equations. Space wave: A wave that propagates through the atmosphere from a transmitting antenna to a receiving antenna. These waves can travel directly, or they can travel after reflecting from the Earth’s surface to the troposphere surface of Earth. Surface wave: A wave guided by a boundary of two dissimilar media and that has a phase velocity smaller than the velocity of light. Its field perpendicular to the direction of propagation is evanescent in nature. As the frequency increases, the wave is more confined to the interface. There are no radiative fields associated with a surface wave. Zenneck wave: This is a wave that is a solution of Maxwell’s equations, and decays exponentially both in the transverse plane and along the direction of propagation. It has a phase velocity faster than the speed of light, and the propagation constants of this wave are generally not highly dependent on frequency. In addition, a cylindrical Zenneck surface wave decays as 1 R . According to Schelkunoff, this is strictly not a true surface wave. A detailed analysis of these related waves is given in [21].

Appendix 2: Asymptotic Evaluation of the Integrals by the Method of Steepest Descent The method of steepest descent (or the saddle-point method) deals with the approximate evaluation of integrals of the form = I (ρ)

∫ F (ξ ) exp − ρ f (ξ ) dξ

(64)

C

for large values of ρ , where the contour C in the complex ξ plane is such that that integrand goes to zero at the ends of the contour. The functions f (ξ ) and F (ξ ) are arbitrary analytic functions of the complex variable ξ . The basic philosophy of the method of steepest descent is as follows: A path is selected in the complex ξ plane in such a IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012


way that the entire value of the integral is determined from a comparatively short portion of the path. Within certain limits, the contour of integration C may be altered to such a path without affecting the value of the integral. The integral is then replaced by another, simpler function, which closely approximates the integrand over the essential portions of the path. The behavior of the new integrand outside the important portion of the path is of no concern. For real and positive values of ρ , and for a general contour C , the quantity ρ f (ξ ) is positive on some parts of the path, and there are other regions where it is negative. The latter regions are more important since the integrand is larger. In those regions, where the negative of Re  ρ f (ξ )  is the largest, it is important to reduce oscillations. A contour is chosen along which the imaginary part of  ρ f (ξ )  is constant in the region where the negative of its real part is largest. The path in the region where Re  ρ f (ξ )  is greatest

may be chosen so that Im  ρ f (ξ )  varies, if this turns out to be necessary to complete the contour. In this way, the oscillations of the integral cause the least trouble. Since the path of integration must pass along the line of most rapid increase and decrease of Re  f (ξ )  , the path must coincide with the line Im  f (ξ )  = constant , which may be a line of constant phase. The point of the path at which Re  f (ξ )  is an extremum is

called the saddle point, and the derivative of Re  f (ξ )  must be zero at this point. Since Im  f (ξ )  is a constant on this path, then its derivative must also be zero, and therefore df =0 dξ

(65)

at the saddle point. The most-advantageous path of integration must thus go through the saddle point along the line of the most-rapid decrease of the function Re  f (ξ )  , which coincides with the line Im  f (ξ )  = constant . This path then is called the path of steepest descent. If the saddle point occurs at ξ = ξ0 , then it follows that the path of integration will be determined from = f (ξ ) f (ξ 0 ) + s 2 ,

(66)

where s is real and −∞ ≤ s ≤ ∞ . The saddle point corresponds to the point s = 0 . Now, going back to the integral of Equation (64) and using Equation (66), ∞

I SD = exp  − ρ f (ξ0 )  ∫ F (ξ ) exp  − ρ s 2  d ξ . (67)   −∞

If

37

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F (ξ ) Φ (s) = then

dξ , ds

I SD = exp  − ρ f (ξ0 ) 

(

(

∞

∫ Φ ( s ) exp − ρ s

2



−∞

ds . (69)

off rapidly with an increasing value of s, the distance from the

saddle point. Only small values of s will thus significantly contribute, and therefore we can expand Φ ( s ) in a Taylor series about the saddle point, s = 0 . We can therefore write s2 Φ ′′ ( 0 ) + ... . 2

I SD = exp  − ρ f (ξ0 ) 

∫ exp − ρ s

−∞

 2   

Φ (0) +

s 2 = Ax 2 + Bx3 + Cx 4 + ... ,

A=

B=

∫ exp − ρ s

−∞

π ( 2n ) ! 1 s ds = ,  n ! 22n ρ n + 0.5

2  2n

f ′′ (ξ0 ) 2

(72)

f ′′′ (ξ0 ) 6 f ′′′′ (ξ0 ) 24

)

B 2 A3/2

,

(82)

F= (ξ ) F (ξ0 ) 1 + Px + Qx 2 + ... ,

(83a)

and P=

F ′ (ξ 0 ) F (ξ 0 )

and

(

)

2 x= a0 2 s 2 1 + 2a1s + 2a2 + a12 s 2 + ... ,  


,

We next expand the function F (ξ ) in a Taylor series around the saddle point, ξ0 , and with ξ − ξ0 = x , we get

(75)

and therefore

38

)

C 5 B2 − 2+ a2 = . 8 A3 2A

(74)

The goal here is to relate x to a power series of s. To this end, we get

A

a1 = −

Taylor series around the saddle point f (ξ0 ) . If ξ − ξ0 = x, then one obtains x2 x3 x4 f (ξ ) = f (ξ0 ) + f ′′ (ξ0 ) + f ′′′ (ξ0 ) + f ′′′′ (ξ0 ) + ... 2! 3! 4!

1

a0 =

tion (68). To make the connection, we first expand f (ξ ) in a

(

.

from which

Now we need to relate Φ ( s ) to F (ξ ) as described in Equa-

x= a0 s 1 + a1s + a2 s 2 + a3 s 3 + ... ,

(80)

= s 2 A  a02 s 2 + 2a1a0 2 s3 + a12 + 2a2 a02 s 4    3 3 3 4 4 + B  a0 s + 3a1a0 s + ... + C  a0 s 4 + ... , (81)    

  π 1 Φ ′′ ( 0 ) + ... . (73) exp  − ρ f (ξ0 )  Φ ( 0 ) + ρ 4 ρ  

= f (ξ 0 ) + s 2 .

,

Now, substituting Equations (75)-(78) into Equation (79), we get

and substituting Equation (72) into Equation (71) yields = I SD

(79)

,

(

as the odd powers of s will integrate to zero. Since ∞

C=

2

(71)

(78)

where

(70)

 s Φ ′′ ( 0 ) + ... ds 2 

(77)

Rewriting Equation (74), we get

Substituting Equation (70) into Equation (69), one obtains ∞

)

4 x= a04 s 4 1 + 4a1s + 2 2a2 + 3a12 s 2 + ... .  

Now, if ρ is large, then the integrand in Equation (69) will fall

Φ ( s ) =Φ ( 0 ) + s Φ ′ ( 0 ) +

)

3 x= a03 s3 1 + 3a1s + 3 a12 + a2 s 2 + ... ,  

(68)

(76)

Q=

F ′′ (ξ0 )

2 F (ξ0 )

, ....

(83b)


12/9/2012 3:50:06 PM

Using Equations (74), (76)-(78), (82), and (83), we get F (ξ ) Φ (s) = = F (ξ )

The interesting point regarding the result of Equation (87) is that it is a divergent series for a fixed ρ as the number of terms in this expansion increases for a fixed value of ρ . Such a series is called an asymptotic series, as introduced by Poincaré.

dξ , ds dx ds

A divergent series is one such as

(

)

= F (ξ )  a0 + s ( 2a0 a1 ) + s 2 a0 a2 2 + 2a2 a0 + ...  

g ( ρ ) = A0 +

= F (ξ0 ) 1 + Px + Qx 2 + ...    a + 2a a s + s 2 a a 2 + 2a a + ... 0 1 0 2 2 0   0 (84) = F (ξ0 ) 1 + a0 Ps + a0 a1Ps 2 + a0 a2 Ps 3  + a02Qs 2 + 2a02 a1Qs3 + ...   a0 + 2a0 a1s + 3a0 a2 s 2 + ...  

(

(

= F (ξ0 )  a0 + s a02 P + 2a0 a1 

(

Sn ( ρ ) =

,

(88)

n

∑ Ai ρ −i .

(89)

L =0

(90)

lim Rn ( ρ ) = 0 for n fixed

(91)

lim Rn ( ρ ) = ∞ , ρ − fixed.

(85)

 F ′′ 5 ( f ′′′ )2 f ′′′ F ′ f ′′′′   2Φ ( 0 )  Φ ′′ ( 0 ) = + − −  Ff ′′ 12( f ′′)3 ( f ′′ )2 F 4 ( f ′′ )2    (86) Hence, 2π F (ξ0 ) exp  − ρ f (ξ0 )  ρ f ′′ (ξ0 )  F ′′ 5 ( f ′′′ )2 f ′′′ F ′ f ′′′′      + ... + − −  Ff ′′ 12 ( f ′′ )3 ( f ′′ )2 F 4 ( f ′′ )2     (87)


(92)

n →∞

When this is the case, one can make Rn ( ρ ) =

Therefore,


ρn

ρ →∞

F (ξ 0 )  B   P − 3/2  1 + s  A   A A 

 1  1 + 2 ρ 

An

even though

  Q 15 B 2 3BP 3C  +s2  + − − + ...  A 8 A3 2 A2 2 A2    

I sd

+ ... +

= Rn ( ρ ) ρ n { g ( ρ ) − Sn ( ρ )}

)

F (ξ 0 )  B   P − 3/2  1 + s  A   A A 

2 F (ξ 0 ) , Φ (0) = f ′′ (ξ0 )

ρ

2

satisfies the condition

 Q  C 5 B 2  3 BP  +s2  − 3 2 − 3  − + ...    2A A 2 A3/2  8 A   A   =

A2

This is said to be an asymptotic expansion of a function g ( ρ ) for a given range of argument ρ if the expression

+ s 2 a02 a1P + 2a0 2 a1P + 3a0 a2 + a03Q + ... 

=

ρ

+

in which the sum of the first ( n + 1) terms is Sn ( z ) , given by

)

)

A1

ρ n { g ( ρ ) − Sn ( ρ )} < ε ,

(93)

where ε is arbitrarily small, by making ρ sufficiently large. Some of the properties of this definition are: 1.

Asymptotic expansions can be multiplied unconditionally.

2.

Asymptotic expansions unconditionally.

3.

An asymptotic expansion of a function is unique.

4.

One asymptotic expansion may represent several functions.

5.

Asymptotic expansions can be divided providing the divisor contains at least one non-zero coefficient.

can

be

integrated

The point about the series of Equation (88) is that for sufficiently large values of ρ , the terms of the series at least initially decrease, and that if the series is truncated before the smallest term, the error is of the order of magnitude of the first discarded term. Thus, if Φ ( s ) is any function in Equation (69) 39

12/9/2012 3:50:06 PM

∞

for which

∫ Φ ( s ) exp − ρ s

−∞

2



ds converges for sufficiently

large values of the parameter ρ , then the asymptotic expansion of Equation (69) in descending powers of ρ can be given by replacing Φ ( s ) by a Taylor series in ascending powers of s in Equation (70), and then integrating term by term. In that case, Equation (87) is the asymptotic expansion of Equation (69). In terms of the problem for our case, the integrals that we will be dealing with are of the following form: I ( kR )=

∫ F ( β ) exp − jkR cos ( β − θ ) d β ,

near the saddle point, θ . For large values of kR , the pole can F (β ) be factored out from F1 ( β ) by writing F1 ( β ) =  β − βP  sin    2  . It is then argued that since F ( β ) has no singularities in the vicinity of the saddle point, it may be removed from under the integral sign with β equated to θ , as presented by Clemmow [26]. The integral of Equation (98) can thus be written as I SD ( kR ) = F (θ ) ∫

(94)

(95)

= f ( β ) cos ( β − θ ) .

(96)

= F (θ )

I SD ( kR ) = F (θ )

f (θ ) = 1 , f ′ (θ ) = 0 ,

exp ( − jkR cos α ) dα ,  α + θ − βP  Γ0 sin   2  

∫

I SD ( kR ) = 2sin

f ′′′ (θ ) = 0 ,

exp − jkR cos α dα ,  θ − α − βP  Γ0 sin   2  

2π j j  F ′′(ϑ ) 1    F (θ ) e − jkR 1 + +   + ...  kR  2kR  F (ϑ ) 4   (97)

Appendix 3: Asymptotic Evaluation of the Integrals when there Exists a Pole Near the Saddle Point The asymptotic expansion given by Equation (97) is not valid if there is a pole near the saddle point, θ . However, the method of steepest descent can be modified in such a way that the presence of poles is taken into account from the very beginning in evaluating these integrals. Of special interest in the analysis will be an integral of the form (98)

Γ1

where Γ1 is a path of integration in the complex β plane, as discussed in the previous sections. F1 ( β ) now has a pole, β P ,


(101)

γ 2

F (θ )

exp ( − jkR cos α )

∫

cos α − cos γ

Γ0

cos

α

dα

2

(102)

where γ= θ − β P . Now, by changing the variable of integration from α to τ such that

Substituting these values into Equation (87), we get

40

(100)

∫

f ′′′′ (θ ) = 1 .

∫ F1 ( β ) exp − jkR cos ( β − θ ) d β ,

(99)

adding Equation (100) to Equation (101) and then dividing by two will convert Equation (99) into

f ′′ (θ ) = −1 ,

I ( kR= )

dβ

where α= β − θ . By reversing the sign of α as

The saddle point then occurs at β = 0 , and we get

I SD ( kR ) =

 β − βP  sin    2 

Γ1

so that

ρ = jkR ,

exp  − jkR cos ( β − θ ) 

= τ

2 exp [ − j π 4] sin

α 2

,

(103)

the path Γ 0 is now transformed into an integral from −∞ to +∞ . Hence, I SD ( kR= ) 2b exp [ − jkR + j 3π 4] F (θ )

∞

e − kRτ

2

∫ τ 2 + jb2

dτ

−∞

(104) where b = 2 sin

γ 2

,

(105)

since ∞

2

e − kRτ π dτ exp  j b 2 kR − π 4  erfc ∫= 2 2   b −∞ τ + jb (106)

(

)

(

jkRb 2

)


12/9/2012 3:50:06 PM

journal articles and numerous conference papers, 32 chapters in books, and fifteen books.

and  θ − βP  W2 = − jkRb 2 = − j 2kR sin 2   .  2 

(107)

Equation (104) then becomes = I SD ( kR ) 2π jF (θ ) exp  − jkR − W 2  erfc ( jW ) . (108)   This completes the derivation. It is important to stress the point that only Clemmow [26] and Wait and Hill [13] followed the developments given by Equation (99). Others, such as Tyras [7] and Collin [18], followed a different procedure for handling the pole.

Introducing the Feature Article Authors

Dr. Sarkar is a Registered Professional Engineer in the State of New York. He received the College of Engineering Research Award in 1996 and the Chancellor’s Citation for Excellence in Research in 1998, at Syracuse University. He was an Associate Editor for Feature Articles of the IEEE Antennas and Propagation Society Newsletter (1986-1988), Associate Editor for the IEEE Transactions on Electromagnetic Compatibility (1986-1989), Chair of the Inter-Commission Working Group of URSI on Time-Domain Metrology (1990-1996), a Distinguished Lecturer for the Antennas and Propagation Society from 2000-2003, a member of the AP-S AdCom (2004-2007), a member of the IEEE Electromagnetics Award Board (2004-2007), on the Board of Directors of ACES (2000-2006), and Vice President of the Applied Computational Electromagnetics Society (ACES). He is currently an Associate Editor for the IEEE Transactions on Antennas and Propagation. He is on the editorial board of Digital Signal Processing – A Review Journal, Journal of Electromagnetic Waves and Applications, and Microwave and Optical Technology Letters. He is a member of Sigma Xi and USNC-URSI Commissions A and B. He received the Docteur Honoris Causa both from the Universite Blaise Pascal, Clermont Ferrand, France, in 1998, and from the Politechnic University of Madrid, Madrid, Spain in 2004. He received the Medal of the Friend of the City of Clermont Ferrand, France, in 2000.

Tapan K. Sarkar received the BTech from the Indian Institute of Technology, Kharagpur, in 1969; the MScE from the University of New Brunswick, Fredericton, NB, Canada, in 1971; and the MS and PhD from Syracuse University, Syracuse, NY, in 1975. From 1975 to 1976, he was with the TACO Division of the General Instruments Corporation. He was with the Rochester Institute of Technology, Rochester, NY, from 1976 to 1985. He was a Research Fellow at the Gordon McKay Laboratory, Harvard University, Cambridge, MA, from 1977 to 1978. He is now a Professor in the Department of Electrical and Computer Engineering, Syracuse University. His current research interests deal with numerical solutions of operator equations arising in electromagnetics and signal processing, with applications to system design. He obtained one of the “best solution” awards in May 1977 at the Rome Air Development Center (RADC) Spectral Estimation Workshop. He received the Best Paper Award of the IEEE Transactions on Electromagnetic Compatibility in 1979, and at the 1997 National Radar Conference. He has authored or coauthored more than 300



Walid M. Dyab received the BSc and MSc from University of Alexandria, Alexandria, Egypt, in 2003 and 2007, respectively, both in Electrical Engineering. Currently, he is a doctoral candidate at Syracuse University, Syracuse, NY. From 2003 until 2005, he was a teaching assistant at Alexandria Institute of Technology (AIT), Alexandria, Egypt. From 2005 until 2006, he worked as a Technical Support Engineer in the Radio Network Department at Alcatel, Egypt. From 2006 until 2009, he was a Teaching and Research Assistant at the German University in Cairo (GUC), Cairo, Egypt. His research interests include the fields of antennas and electromagnetic wave propagation, antenna measurements, adaptive antenna systems, adaptive signal processing, and the design and analysis of microwave passive circuits.

41

12/9/2012 3:50:06 PM

Mohammad N. Abdallah received the BSc in Electrical Engineering from the University of Jordan, Amman, Jordan in 2006, and the MSc in Electrical Engineering from Syracuse University, Syracuse, NY, in 2011. Currently, he is a PhD student at Syracuse University, Syracuse, NY. Since 2007, he has been with ICM controls company, Syracuse, NY, where he works as pre-production engineer. His research interests include electromagnetic wave propagation, antenna array design and analysis, and computational methods in electromagnetics.

Magdalena Salazar-Palma was born in Granada, Spain. She received the MS and PhD in Ingeniero de Telecomunicación (Electrical and Electronic Engineer) from Universidad Politécnica de Madrid (UPM), Spain. She has been Profesor Colaborador and Profesor Titular de Universidad at the Department of Signals, Systems and Radio communications, UPM. Since 2004 she has been with the Department of Signal Theory and communications (TSC), College of Engineering, Universidad Carlos III de Madrid (UC3M), Spain, where she is a full Professor, co-director of the Radiofrequency Research Group, and served for three years as Chair of the TSC Department. She has been a member of numerous university committees, both at UPM and UC3M. She has developed her research in the areas of electromagnetic field theory; advanced computational and numerical methods for microwave and millimeter-wave passive components and antenna analysis and design; advanced network and filter theory and design; antenna array design and smart antennas; the use of novel materials and metamaterials for the implementation of devices and antennas with improved performance (multi-band, miniaturization, and so on) for the new generation of communication systems; design, simulation, optimization, implementation, and measurement of microwave circuits, both in waveguide and integrated (hybrid and monolithic) technologies; millimeter, sub-millimeter, and THz frequency-band technologies; and the history of telecommunications. 42


She has authored a total of 490 scientific publications. She has coauthored two European and USA patents, which have been extended to other countries, and several software packages for the analysis and design of microwave and millimeter-wave passive components, antennas and antenna arrays, as well as computer-aided design (CAD) of advanced filters and multiplexers for space applications, which are being used by multinational companies. She has delivered numerous invited presentations and seminars. She has lectured in more than 50 short courses, some of them in the framework of Programs of the European Community, and others in conjunction with IEEE International AP-S Symposium, the IEEE MTT-S Symposium, and other IEEE symposia. She has participated at different levels (principal investigator or researcher) in a total of 81 research projects and contracts, financed by international, European, and national institutions and companies. She is a registered engineer in Spain. She has received two individual research awards. She has assisted the Spain National Agency of Evaluation and Prospective and the Spain CICYT in the evaluation of projects and research-grant applications. She is member of the Accreditation Committee of Full Professors in the field of Engineering and Architecture of the Spanish Agency of Quality Evaluation and Accreditation (ANECA). She has also served on several evaluation panels of the Commission of the European Communities. She has been a member of the editorial board of three scientific journals. She has been associate editor of several scientific journals, among them the European Microwave Association Proceedings and IEEE Antennas and Wireless Propagation Letters. She has been a member of the Technical Program Committees of many international and national symposiums, and reviewer for different international scientific journals, symposiums, and editorial companies. Since 1989, she has served the IEEE in different volunteer positions: Vice Chair and Chair of IEEE Spain Section AP-S/MTT-S Joint Chapter, President of IEEE Spain Section, Membership Development Officer of IEEE Spain Section, member of IEEE Region 8 Committee, member of IEEE Region 8 Nominations and Appointments Subcommittee, Chair of IEEE Region 8 Conference Coordination Subcommittee, member of IEEE Women in Engineering (WIE) Committee, liaison between IEEE WIE Committee and IEEE Regional Activities Board, Chair of IEEE WIE Committee, member of IEEE Ethics and Member Conduct Committee, member of IEEE History Committee, member of IEEE MGAB (Member and Geographic Activities Board) Geographic Unit Operations Support Committee, and member of IEEE AP-S Administrative Committee. Presently she is serving as a member of the IEEE Spain Section Executive Committee (officer for Professional Development), a member of IEEE MTT-S Subcommittee #15, and a member of IEEE AP-S Transnational Committee. In December 2009, she was elected the 2011 President of the IEEE Antennas and Propagation Society, acting as President Elect during 2010.


12/9/2012 3:50:07 PM

Dr. M. V. S. N. Prasad is currently working as chief scientist in National Physical Laboratory, New Delhi, India, and is actively involved in the area of radio-channel modeling and measurements for fixed and mobile communications. He has interacted with various user organizations in the area of telecommunications during his research career, conducted experiments and raised data sets extensively over various parts of India, published several papers, and also rendered consultancy services. He has conducted user researcher interaction seminars and workshops in the above areas. He is a member of several assessment committees, and has reviewed papers for many journals. He is a Fellow of the Institute of Electronics and Telecommunication Engineers (India). He has delivered many lectures and invited talks in the above areas and developed many active collaborations with institutes. He was awarded the URSI Young Scientist Award in 1990, and Broadcast Engineering Society (India) awards for best papers.

Sio-Weng Ting received the BSc, MSc, and PhD in Electrical and Electronics Engineering from University of Macau, Macao, China in 1993, 1997, and 2008, respectively. From 2009 to 2010, he was a Post Doctorate Research Associate at Syracuse University, Syracuse, NY. He is now an Assistant Professor in the Department of Electrical and Computer Engineering at University of Macau. His current research interests include passive microwave circuits and electromagnetic simulation of wireless-communication systems.

Silvio Ernesto Barbin was born in Campinas, SP, Brazil in 1952. He did undergraduate work on Electric Engineering at EPUSP – Politechnical School of São Paulo University in 1974, getting his master’s degree and PhD at the same institution. He was a visiting researcher at the University of California, Los Angeles, CA, and a professor at the University of New Mexico, in Albuquerque, NM. He worked at AEG –Telefunken for four years, in Backnang and Ulm, Germany, and for nine years in São Paulo, SP Brazil. He was technical director of Microline Multiplexadores de RF. Nowadays, he is a professor in the Engineering of Communications and Control Department of the Politechnical School of São Paulo University, and General R&D and Innovation Projects Coordinator and Substitute Director of CTI – Centro de Tecnologia da Informação Renato Archer of the Brazilian Science and Technology Ministry.

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43

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