NTIA Report 00-379

A Frequency- and Time-Domain

Investigation into the Geometric

Optics Approximation for Wireless

Indoor Applications

Michael G. Cotton

Edward F. Kuester

Christopher L. Holloway

U.S. DEPARTMENT OF COMMERCE

William M. Daley, Secretary

Gregory L. Rohde, Assistant Secretary

for Communications and Information

June 2000
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CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. INTRODUCTION ...........................................................1

2. EXPRESSIONS FOR RADIATION IN FREE SPACE OVER GROUND ...............3

2.1. Sommerfeld Formulation ..............................................4

2.1.1. Vertical Electric Dipole (VED) ..................................5

2.1.2. Horizontal Electric Dipole (HED) ...............................6

2.2. Geometric Optics and Norton Surface-Wave Approximation ..................7

2.2.1. Vertical Electric Dipole (VED) ..................................8

2.2.2. Horizontal Electric Dipole (HED) ...............................8

2.2.3. Fresnel Reflection Coefficients ..................................9

3. NUMERICAL INTEGRATION TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1. Romberg Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2. Weighted-Averages Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3. Numerical Evaluation of 2Q/k0

2

, U, V, and W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4. FREQUENCY-DOMAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1. Conductivity Variation to Demonstrate Two-Ray Behavior . . . . . . . . . . . . . . . . . . 16

4.2. Position Variation to Isolate Near-Surface and Near-Field Effects . . . . . . . . . . . . . 17

4.2.1. Surface Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.2. Pseudo-Lateral Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.3. Near-Field Effects on the Direct Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3. Contour Plots to Observe Two-Ray Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5. TIME-DOMAIN ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.1. Windowing Techniques to Optimize Pulse Shape . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2. Practical Bandwidth Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6. TIME-DOMAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.1. Impulse Responses for Wideband Signals Centered at 900 MHz . . . . . . . . . . . . . . 27

6.2. Impulse Responses for Wideband Signals Centered at 2.4 GHz and 5.8 GHz . . . . 31

6.3. Delay Spread Versus Bandwidth and Carrier Frequency . . . . . . . . . . . . . . . . . . . . . 32

7. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8. ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


APPENDIX A: FIELD STRENGTH VERSUS FREQUENCY AND SEPARATION . . . . . . . 47

APPENDIX B: IMPULSE RESPONSES FOR SIGNALS CENTERED AT 2.4 GHz . . . . . . . 53

APPENDIX C: IMPULSE RESPONSES FOR SIGNALS CENTERED AT 5.8 GHz . . . . . . . 57

APPENDIX D: DELAY SPREAD VERSUS BANDWIDTH . . . . . . . . . . . . . . . . . . . . . . . . . . 61
 *

The author is with the Institute for Telecommunication Sciences, National Telecommunications and Information

Administration, U.S. Department of Commerce, Boulder, CO 80303.

 **The author is with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309.

 ***The author is with the National Institute of Standards and Technology, U.S. Department of Commerce, Boulder, CO 80303.

A FREQUENCY- AND TIME-DOMAIN INVESTIGATION INTO THE GEOMETRIC

OPTICS APPROXIMATION FOR WIRELESS INDOOR APPLICATIONS

Michael G. Cotton,*

 Edward F. Kuester,** and Christopher L. Holloway***

In this study we investigated the geometric optics (GO) approximation to the fields

of an incremental electric dipole above a half plane for geometries typical of wireless

indoor communications. This inspection was motivated by efforts to establish a raytrace model to characterize indoor radio propagation channels. Eight canonical

geometries were examined to isolate near-surface and near-field effects that are not

accounted for in the GO approximation. Common building materials and physical

dimensions (i.e., antenna separation and height) as small as 1 cm were investigated

for frequencies up to 8 GHz. Theoretical fields were calculated via numerical

evaluation of Sommerfeld integrals and compared to corresponding GO

approximations. As expected, GO approximations agreed with theoretical results

when the source and observation points were multiple wavelengths above the surface

and relatively far apart. Close to the surface, an interesting interference pattern in the

frequency domain was caused by adjacent fields in the two media propagating at

different speeds. This so-called “pseudo-lateral wave” phenomenon is discussed and

demonstrated in various examples. Next, we emulated system specifications (i.e.,

center frequency and bandwidth), computed time-domain impulse responses, and

used delay spread as a metric to quantify GO error. Results show that mechanisms

exist under certain circumstances which invalidate GO assumptions; conventional

expressions to complement GO approximations are summarized.

Key words: geometric optics; indoor propagation channel; propagation over ground; Sommerfeld

integral; numerical integration; ray-trace model; impulse response; delay spread

1. INTRODUCTION

With the growing emergence of indoor wireless local area networks (WLAN), characterization of

the indoor propagation channel has become more important. This environment can be characterized

in either the frequency or time domain. Frequency-domain modeling takes the form of field-strength
b(t) 
 P



	

a(t	2)g(t,2) d2 , (1)

2

spr 


P



	

(2	2)2 |g(t,2)|

2 d2

P



	

|g(t,2)|

2 d2

, (2)

2 


P



	

2|g(t,2)|

2 d2

P



	

|g(t,2)|

2 d2

. (3)

 predictions over a volume of interest for specified frequencies. In contrast, time-domain modeling

is based on the impulse response of the particular environment. For digital communications, the

impulse response is a quantity used to characterize the multipath propagation environment.

Propagation effects on a transmitted signal are best described by a linear-system representation

where a(t) and b(t) are the symbol waveforms before and after propagation through the channel, g(t,)

is the time-variable impulse response of the propagation channel, and  is the delay variable.

Multipath propagation environments, such as the indoor-propagation channel, exhibit characteristic

impulse responses. For line-of-sight indoor channels, an impulse response is composed of a direct

ray plus numerous rays due to reflections off objects, diffraction around objects, and transmission

through objects. Impulse response data is useful for the analysis and simulation of digital

transmission because it quantifies communication-link degradation within a channel. More

specifically, the delay spread spr of a channel impulse response is a measure of time-dispersion due

to multipath [1, 2] and is defined as 

where  is the first moment or mean delay given by

As a rule of thumb, small spr indicates little degradation, whereas if spr is large then severe distortion

occurs. Therefore, the data rate may need to be decreased in a high-delay-spread multipath

environment in order to maintain or reduce the bit-error rate.

A threshold associated with the calculation of delay spread exists to nullify the noise contribution

(i.e., values of |g(t,)|

2

 below the threshold are set to zero). Throughout this report, a threshold of -30

dB relative to the peak of the impulse is used for all delay spread calculations. Delay spread


quantifies inter-symbol interference and has been the subject of numerous publications such as [1-

10]. In most cases, statistical analyses were necessary to account for stochastic processes within the

time-variant channel. In this report we are not interested in the time-variant nature of the channel,

denoted by the dependent variable t in g(t,), nor statistical procedures to account for stochastic

processes. Hence, we assume that the channel is deterministic and that a single impulse response

is completely representative.

Various approaches are used to determine impulse-response delay spread for indoor applications,

such as ray-trace models [11-24], statistical models based on measurements [25-33], simplifieddecay models [34, 35], and full numerical techniques [36-39] (e.g., finite-difference time-domain).

The most popular of these techniques is ray-tracing, which is based on GO and assumes that nearsurface and near-field effects are negligible and that Fresnel reflection coefficients are valid. For

indoor applications, however, transmit and/or receive antennas are likely to be mounted close to a

wall or ceiling. Therefore, the GO approximation is in question since the antennas are only a few

centimeters from reflecting surfaces. GO errors, relative to the numerical approximation of the

general Sommerfeld formulation, are presented here in frequency-domain field-strength predictions

and time-domain impulse-response results.

Our goal is to investigate the shortcomings of the GO approximation for antennas mounted close to

reflecting surfaces. We focus on a classic problem – dipole radiation above a lossy half space – and

consider the elementary vertical electric dipole (VED) and x-aligned elementary horizontal electric

dipole (HED) as sources. If there exists a set of substantive single-reflection cases which produce

significant near-surface and near-field errors under practical frequency and geometric constraints,

then we conclude that GO is an inadequate approximation. 

This report is organized in the following manner. In Section 2, we summarize Sommerfeld’s

classical integral formulation and provide geometric optics expressions for elementary dipoles above

an infinite ground plane. Numerical techniques to evaluate Sommerfeld integrals are given in

Section 3. Frequency-domain field strength results are given in Section 4. The frequency-domain

results are transformed to a complex-baseband representation of a passband signal in order to

calculate the channel impulse response. A detailed explanation of the time-domain analyses is given

in Section 5, and corresponding results are shown in Section 6.

2. EXPRESSIONS FOR RADIATION IN FREE SPACE OVER GROUND

In order to isolate errors associated with the GO approximation for typical indoor scenarios, we focus

on a single reflection off a planar surface. The classic problem of elementary dipole radiation above

a ground plane has been analyzed extensively [40-46]. The geometry and nomenclature are shown

in Figures 1 and 2. Region 0 is free space; the source dipole and receiver, regardless of nature and

orientation, are located in region 0 at heights h and z above the interface, respectively. Region 1 lies

beneath the interface; its material composition is defined by its relative dielectric constant r and

conductivity . All media are assumed to be isotropic, homogeneous, and non-magnetic.


Figure 1. Parallel polarization geometry for E-fields

above dielectric half-space.

Figure 2. Perpendicular polarization geometry for Efields above dielectric half-space.

/×E3 
 	j&μ0H3 and /×H3 
 J

3  j&0E3 , (4)

E3 
 k 23  /(/#3 ) and H3 
 j&0/×3 . (5)

(/2  k 2 )3 
 	 J

3

j&0 , (6)

The general formulation of the two-media boundary problem is based on the solution of Maxwell’s

equations subject to boundary conditions at the interface. The differential form of Maxwell’s

equations, assuming exp(+jt) harmonic time dependence for the fields, is

where is an electric field, is a magnetic field, is an impressed current source, capital letters E H J



denote time-harmonic fields (in contrast to time-instantaneous field variables, which are in lower

case), and  denotes a vector. The electric-type Hertz vector is useful for solving electromagnetic 

fields generated by a time-harmonic electric current; fields are expressed in terms of as 

An alternative form to Maxwell’s equations is the Helmholtz equation, given by

where k1

2

 = 2μ0(r0 - j/) defines the wave number of medium 1. Boundary conditions enforce

continuity of tangential and components at the interface and allow for a unique solution of . E H 

2.1. Sommerfeld Formulation

The classic Sommerfeld formulation for a dipole above a half plane is a general solution that

contains complex, highly oscillatory integrals. Sommerfeld integrals provide an exact solution, but

have yet to be solved analytically. Numerical schemes for evaluating these integrals are covered in

Section 3. The formulations for VED and HED are given in the following subsections.


z 
 j&μ0 p

4k 2

0

[G0 	 G1  k1

2V ] , (7)

G0 
 exp(	jk0R0 )

R0


 P



0

exp[	l|h	z|]

l

J0( !)  d and

G1 
 exp(	jk0R1 )

R1


 P



0

exp[	l(hz)]

l

J0( !)  d ,

(8)

R0 
 !2  (z	h)

2 and R1 
 !2  (zh)2 , (9)

V 
 P



0

2exp[	l(zh)]

lk1

2  mk02

J0( !)  d , (10)

l 
  2	k0

2 and m 
  2	k12 . (11)

2.1.1. Vertical Electric Dipole (VED)

An incremental vertical current element placed a height h above the half plane in the z direction

produces a z-directed component of the vector potential in regions 0 and 1. For a VED, z(, z) is

independent of  and only the Ez, E!, and H3 field components are radiated. The Helmholtz

equations in both media must be satisfied and boundary conditions apply at the interface. The

solution to the partial differential equation is simplified via the double Fourier transform.

Additionally, the specified geometry suggests a transformation to cylindrical coordinates that

introduces a Bessel function into the integral solution. The resulting Sommerfeld formulation for the

vector potential in region 0 is taken from [41] and is given by

where p is the dipole moment. The source and image Green’s functions are defined as

respectively, where J0() is the Bessel function of the first kind and order zero. The distances from

the observation point to the source and image are

respectively. The Sommerfeld integral V is defined as

and the functions l and m are given by

Equation (7) is concise but not optimal for numerical evaluation given the electric properties of the

reflecting surfaces considered in this report. An alternative form was used by van der Pol [45] and


z 
 j&μ0 p

4k 2

0

[G0  G1 	 2Q] , (12)

Q 
 P



0

mk0

2 exp[	l(zh)]

l(lk1

2  mk02 )

J0( !)  d . (13)

Ez 
 j&μ0 p

4k0

2

02

0z 2  k0

2 (G0  G1 	 2Q)

E

! 
 j&μ0 p

4k0

2

02

0!0z

(G0  G1 	 2Q)

H

3 
 p

4

0

0!

(G0  G1 	 2Q) .

(14)

is given by

where

Under this convention, the sum G0 + G1 represents the situation when the ground is perfectly

conducting and the Sommerfeld integral Q represents a correction for the lossy characteristics of

ground. For the material properties considered, the formulation of equation (12) significantly

reduces the magnitude of the integrand in equation (13) and consequently improves the convergence

and accuracy of the numerical integration.

Relevant field components are extracted from equation (5) to form the following field expressions:

These general expressions were derived with no assumptions regarding proximity, wavelength, or

material composition and account for the near-surface and near-field effects we aim to observe. 

2.1.2. Horizontal Electric Dipole (HED)

In the case of an incremental horizontal current element placed a height h above the half plane in the

x direction, Baños [41] employs the x and z components of the electric Hertz vector (i.e.,

 a ). The z components satisfy the homogeneous Helmholtz equation in each medium, x

xazz

and the x components satisfy the inhomogeneous Helmholtz equation in region 0 and the

homogeneous Helmholtz equation in region 1. Imposition of the boundary conditions at the interface

gives vector potential solutions for region 0 that are substituted into equation (5) to give field

expressions. Sommerfeld formulations of the transverse-electric (TE) field components are


E

3 
 j&μ0 p

4k0

2

sin3

1

!

0

0!

(G0 	 G1  k0

2

V)  k0

2 (G0 	 G1  U)

Hz 
 	 p

4

sin3

0

0!

(G0 	 G1  U)

H

! 
 p

4

sin3

0

0z

(G0 	 G1  U) 	 1

!

0W

0! ,

(15)

Ez 
 	 j&μ0 p

4k0

2

cos3

02

0z0!

(G0  G1 	 k1

2V)

E

! 
 	 j&μ0 p

4k0

2

cos3

02

0!2

(G0 	 G1  k0

2V)  k02 (G0 	 G1  U)

H

3 
 p

4

cos3

0

0z

(G0 	 G1  U) 	 02

W

0!2 .

(16)

U 
 P



0

2exp[	l(zh)]

l  m

J0( !)  d

W 
 P



0

2(l	m)exp[	l(zh)]

k1

2 l  k02m

J0( !)  d .

(17)

and the transverse-magnetic (TM) field components are

Sommerfeld integrals U and W are defined by

Note that if one were to compare the VED and HED Sommerfeld formulations given in this report

to the corresponding expressions in [41], then a number of differences would be observed. These

differences are due to an opposite time convention and rotation of the spatial coordinate axes. The

opposite time convention causes the opposite sign in the exponent of G0 and G1. Rotation of the

coordinate axes flips the leading sign of many of the field equations, the sign in front of z in

numerous exponential exponents, and the sign of W.

2.2. Geometric Optics and Norton Surface-Wave Approximation

Special cases exist where closed-form approximations are obtainable. Geometric optics is a far-field

approximation that assumes the source, observation point, and reflecting surface are many

wavelengths apart. Also, the Norton surface-wave term may be used to approximate surface-wave

effects when k0R0 » 1 and |k1| > |k0|.
	

Ez  	 jk0 p

4

μ0

00

[sin2dG0  +sin2rG1  (1	+)F(w)sin2rG1 ]

E

!  jk0 p

4

μ0

00

[sind cosdG0  +sinr cosrG1 	 (1	+)û0F(w)sinrG1 ]

H

3  jk0 p

4

[sindG0  +sinrG1  (1	+)F(w)sinrG1 ] ,

(18)

F(w) 
 1 	 w e 	w erfc(j w) . (19)

w 
 	 jk0R0

2sin2r

(cosrû0 )

2 and û0 
 k0

k1

1	 k0

k1

2

sin2r . (20)

E

3  jk0 p

4

μ0

00

sin3 [G0  +]G1  (1	+] )F(q)G1 ]

Hz  jk0 p

4

sin3 [sindG0  +] sinrG1  (1	+] )F(q)sinrG1 ]

H

!  	 jk0 p

4

sin3 [cosdG0  +] cosrG1 	 (1	+] )F(q)/0G1 ] ,

(21)

2.2.1. Vertical Electric Dipole (VED)

If the observation point is in the far field of the source, then the approximations to fields resulting

from a vertical electric dipole above a half plane are given by

where d and r are shown in Figures 1 and 2. The first two terms make up the GO approximation

which involves the Fresnel reflection coefficient  (see Section 2.2.3). The third term is Norton’s

surface-wave approximation, which was obtained from a high-refractivity, far-field approximation

to V [42]. The Norton term incorporates an attenuation function given by

For a homogeneous half-space the remaining variables are defined as 

The Norton surface-wave terms are valid only when k0R0 » 1 and |k1| > |k0| due to assumptions made

when deriving equations (18), (19), and (20).

2.2.2. Horizontal Electric Dipole (HED)

For observation points in the far field of a horizontal dipole above a homogeneous half plane, the

TE fields are given by



Ez  jk0 p

4

μ0

00

cos3 [sind cosdG0 	 +sinr cosrG1  (1	+)F(w)û0 sinrG1 ]

E

!  	 jk0 p

4

μ0

00

cos3 [cos2dG0 	 +cos2rG1 	 (1	+)F(w)û2

0G1 ]

H

3  	 jk0 p

4

cos3 [cosdG0 	 +cosrG1  (1	+)F(w)û0G1 ] ,

(22)

q 
 	 jk0R0

2sin2r

(cosr/0 )

2 and /0 
 k1

k0

1	 k0

k1

2

sin2r . (23)

+ 
 0 cosi 	 1 cost

0 cosi  1 cost

and +] 
 1 cosi 	 0 cost

1 cosi  0 cost

, (24)

sint 
 1

0

sini and 1 
 μ0

0r00 	 j1/& , (25)

and the TM components in the far field are given by

where d and r are shown in Figures 1 and 2, the sum of the first two terms is the geometric optics

approximation, and the third term is the Norton surface-wave term. For a homogeneous half-space

the remaining variables are defined as 

Similar to the VED case, the Norton surface-wave terms are valid only when k0R0 » 1 and |k1| > |k0|.

The following section provides a summary of Fresnel reflection coefficients to complete the far-field

formulation of the fields radiated by incremental dipoles above a half-space.

2.2.3. Fresnel Reflection Coefficients

Detailed derivations for the expressions given in this section may be found in standard

electromagnetic texts, such as [47]. Electric field polarization is defined relative to the plane of

incidence, which contains the normal to the reflecting surface and the incident propagation vector

(see Figures 1 and 2). Parallel polarized electric fields lie in the plane of incidence and perpendicular

polarized fields are orthogonal. The E-field Fresnel reflection coefficients at a plane boundary are

where the transmitted angle and intrinsic wave impedance of region 1 are complex and defined as

respectively. Figure 3 illustrates  and ] for various material properties and frequencies.


0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

f = 0.9 GHz

θi

 /π

magnitude

0 0.1 0.2 0.3 0.4 0.5

−1

−0.5

0

0.5

1

f = 0.9 GHz

θi

 /π

phase/π

0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

f = 2.4 GHz

θi

 /π

magnitude

0 0.1 0.2 0.3 0.4 0.5

−1

−0.5

0

0.5

1

f = 2.4 GHz

θi

 /π

phase/π

0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

f = 5.8 GHz

θi

 /π

magnitude

0 0.1 0.2 0.3 0.4 0.5

−1

−0.5

0

0.5

1

f = 5.8 GHz

θi

 /π

phase/π

0.00195

0.195

19.5

1950

Figure 3. Parallel (line) and perpendicular (dash) Fresnel reflection coefficients versus angle of incidence for

0r=5. Legend displays conductivity 1 [S/m].

3. NUMERICAL INTEGRATION TECHNIQUES

To evaluate the accuracy of GO, we must analyze near-field interaction and propagation effects near

the interface. General solutions for fields radiated by a VED or an x-aligned HED are obtained via

numerical evaluation of Sommerfeld integrals. In this section, relevant numerical techniques are

presented with reference to [48-52]. Romberg integration for relatively well-behaved functions and

the weighted-averages method for evaluating asymptotically-oscillating slowly-convergent functions

are summarized. Finally, specific details for the numerical evaluation of 2Q/k0

2

, U, V, and W and

the necessary derivatives are given.

3.1. Romberg Quadrature

Romberg integration is an effective means of integrating well-behaved integrands over a finite range

of integration by extrapolating the error associated with a prediction from previous iterations and


P

1

0

( ) d 
 û 1

2

(0 )  1

2

(1 )  O û 3(2) , (26)

SN 
 P

N

0

( ) d 
 û 1

2

(0 )(1 )### 1

2

(N )

	 B2 (N 	0)

2

2!N2

(N )	(0 ) 	 ###

	 B2n (N 	0)

2n

(2n)!N2n (2n	1)(N )	(2n	1)(0 ) 	 ###

(27)

S 
 4

3

S2N 	 1

3

SN (28)

S 
 P



	1

$( )J#(!) d 
 lim

N M

N

n
0 P

n

n	1

$( )J#(!) d 
 lim

N

SN . (29)

subtracting the error to give a higher-order approximation. To demonstrate this, consider the

trapezoidal rule

where  is the distance between abscissae, parenthesized superscripts denote derivative order, and

O defines error order. Equation (26) is a two-point formula and is exact for polynomials up to and

including degree 1. If this equation is used N times for the equally-spaced intervals (0, 1), (1, 2),

..., (N -1, N) and summed, then the extended trapezoidal rule is

where Bn is a Bernoulli number. If one evaluates the integral with N steps and then again with 2N

steps, then the leading error in the second evaluation will be 1⁄4 the size of the error in the first

evaluation. Therefore the combination

subtracts off the leading error term. Similarly, Romberg integration uses Neville’s algorithm to

extrapolate the leading error term for high-order integrands.

3.2. Weighted-Averages Method

The weighted-averages method is a useful numerical technique for integrating functions that are

periodic yet convergent over a semi-infinite integration range. Sommerfeld integrals are inherently

oscillatory and slowly convergent due to the presence of Bessel functions. We begin our discussion

of the weighted-averages method with the integration then summation technique [52], where the

integral is expressed as a limit of a sequence of partial sums


rN 
 SN 	 S 
 	P



N

$( )J#(!) d

 wN M



n
0

cn N

	n , N   ,

(30)

S 

N 
 WN SN WN1 SN1

WN WN1


 S  WN rN WN1 rN1

WN WN1


 S  r



N . (31)

.N  WN1

WN


 	 rN

rN1

, (32)

S (51)

N 
 S (5)

N  . (5)N S (5)N1

1  . (5)

N

(N  0 and 5  0) , (33)

Note that the Sommerfeld integrand was split into its characteristic factors, () = ()J#(). Also,

-1 is chosen to ensure that no singularities exist in the Sommerfeld tail (| | > |-1 |). The sequence

SN approaches S slowly and the error or remainder complies with the following expression

where wN are remainder estimates specific to (). Series acceleration methods are based on the idea

that information contained in the sequence of partial sums, S0, S1, ..., SN, is extracted and utilized in

a way that is more efficient than the conventional combination techniques. If the weight, WN, is

associated with SN, then a general combination formula is

The second form shows that if

then the remainder of the transformed sequence will be nullified. The difficulty is determining r



N

rN from its asymptotic estimates. Careful scrutiny leads to the generalized weighted-averages

algorithm, given as

where parenthesized superscripts denote transformation order and is formulated in Section 3.3  (5)

N

for the integrands under consideration. Equation (33) is a recursive scheme that produces as the S (5)

0

best approximation to S, given the partial sums S0, S1, ..., S5 , and accelerates the convergence.

3.3. Numerical Evaluation of 2Q/k0

2

, U, V, and W

Efficient numerical integration of the integrals associated with dipole radiation above a half plane

are now considered. Table 1 summarizes the necessary Sommerfeld integrals and Figure 4 illustrates

the common traits among each integrand in the complex plane. Derivatives were pulled inside the
2Q

k0

2

2mexp[	l(zh)]

l( k0

2m  k12 l)

J0 ( !') !

02

0z 2

2Q

k0

2

2lmexp[	l(zh)]

k0

2m  k12 l

J0 ( !') !

02

0z0'

2Q

k0

2

2mexp[	l(zh)]

k0

2m  k12 l

J1 ( !') ! 2

0

0'

2Q

k0

2 	 2mexp[	l(zh)]

l( k0

2m  k12 l)

J1 ( !') ! 2

02

V

0z0'

2lexp[	l(zh)]

k0

2m  k12 l

J1 ( !') ! 2

0V

0'

	 2exp[	l(zh)]

k0

2m  k12 l

J1 ( !') ! 2

02

V

0'2

2exp[	l(zh)]

k0

2m  k12 l

J1 (!')

' 	!J0 (!') ! 2

2exp[	l(zh)]

l  m J0 ( !') !

0U

0z 	 2lexp[	l(zh)]

l  m J0 ( !') !

0U

0' 	 2exp[	l(zh)]

l  m J1 ( !') ! 2

0W

0'

	 2(l	m)exp[	l(zh)]

k0

2m  k12 l

J1 ( !') ! 2

02

W

0'2

2(l	m)exp[	l(zh)]

k0

2m  k12 l

J1 (!')

' 	!J0 (!') ! 2

13

Sommerfeld integral Sommerfeld integrand, 2 asymptotic

1

-1

-1

0

-1

0

0, -1

U 1

0

0

-1

-1, -2

Table 1. Sommerfeld Integrands and Asymptotic Coefficients


Figure 4. Sommerfeld poles and branch points in complex plane.

pole 
 ±k0 k1

k0

2  k12

. (34)

integrals and each resulting Sommerfeld integrand, , contains Bessel functions, exponential terms,

branch cuts, and possible poles. These factors make analytic solutions unlikely, but provide a

commonality which allows for a single numerical integration scheme to evaluate all of the integrals.

In this approach the real-axis integration path is partitioned and specific numerical integration

techniques are used according to the integrand behavior in each subinterval.

Poles may cause strong variations in the integrand, such as those associated with V and W at

These poles are located in the fourth quadrant on the complex plane just below the real axis and

(pole)  k0 as well as in the second quadrant symmetric about the origin. It was confirmed that they

have negligible effect on the real-axis integration path for the electric properties considered.

To avoid branch cuts, the integration path was partitioned into three subintervals [0, k0], [ k0 , k0 r],

and where different techniques were used according to specific difficulties encountered. [ k0 r, ],

In the first subinterval, change of variables  = k0 cosu removed the discontinuity in the derivative

at k0, resulting in


P

k0

0

( ) d 
 P



2

0

(k0 cosu)k0 sinu du . (35)

P

k0 0r

k0

( ) d 
 P

Arccosh 0r

0

(k0 coshv) k0 sinhv dv . (36)

wN 
 (	1)N1

 	1⁄2

N

exp 	 N(zh)

! . (37)

f( )  exp[	 (zh)]

 

CO( 	1)

J#

(!)  2

!

cos !	#



2

	 

4 ,

(38)

.N

(5) 
 exp

(zh)

!

N1

N

	1⁄25

, (39)

In the second subinterval, change in variable  = k0 coshv removed the discontinuity at k0 giving

The integrands in equations (35) and (36) are well behaved and converge when numerically

integrated with Romberg quadrature.

The semi-infinite subintervalwas chosen to begin at  because all singularities, poles, and 	1  k0 r

branchpoints on theright half of thecomplexplanelieeitheron or to theleftofthe line definedby ( )  k0 r.

Change in variable was not necessary and the generalized weighted-averages algorithm was used

to numerically integrate the Sommerfeld integral tail. The break points  were N  k0 r N/

chosen based on the half-period of the Bessel function, /, and the weights were chosen according

to the analytical form of the remainder estimates, given by

A more detailed derivation of equation (37) was given by Michalski in [52]. This equation is based

on the asymptotic behavior of the Sommerfeld integrand, which is characterized by the asymptotic

coefficient  (given in Table 1) and the expressions

where C is a constant. Applying equation (37) to equations (30) and (32) gives

which completes the weighted-averages formulation given by equation (33).


p  j 4

μ0

. (40)

4. FREQUENCY-DOMAIN RESULTS

In this section, field-strength results and corresponding GO errors are illustrated in the frequency

domain. Frequencies ranging from 10 to 8000 MHz (i.e., 30   0.0375 m) were chosen to

encompass the ISM bands at 902 - 928 MHz, 2.4000 - 2.4835 GHz, and 5.725 - 5.850 GHz where

unlicensed wireless LAN products operate [53]. In order for the theoretical received signals to

embody the channel transfer function, we simulate a flat-frequency transmitted spectrum by defining

the electric dipole moment as

First, VED results from a simple two-ray geometry at various ground conductivities are presented

to introduce the behavior of the two-ray model. Next, position dependency is investigated to isolate

near-surface and near-field effects. Lastly, field strength as a function of horizontal separation and

frequency is displayed in order to demonstrate the nature of the fields in a more general sense.

Throughout this report, superscripts (i.e., Ez

S

, Ez

GO, EzGO+N) and colors (i.e., green, red, blue) are used

to distinguish between Sommerfeld, GO, and GO with Norton term results, and all fields are in dB

normalized to 1 V/m for E-fields or 1 A/m for H-fields. Note that in many cases the curves are

indistinguishable because they lie on top of one another. 

4.1. Conductivity Variation to Demonstrate Two-Ray Behavior

Electric properties are defined by the conductivity and relative dielectric constant of the lossy half

space; geometry is specified by height above the interface and horizontal separation. In this section,

we consider { [S/m]} = {0.00195, 0.195, 19.5, 1950}, r = 5, and {h [m], z [m],  [m]} = {1, 1, 10}

in order to minimize the effect of the null in the radiation pattern of the VED source and to ideally

illustrate the nature of the two-ray model. Figure 5 presents the numerically approximated

Sommerfeld solution with the GO approximation for the squared magnitude of Ez radiated by a VED.

As the electrical conductivity gets large, the Fresnel reflection coefficients approach plus or minus

unity and the field expressions approach the sum of the source and image Green’s functions G0 + G1.

This produces a two-ray cancellation effect where equally spaced nulls occur according to the

difference in path lengths, R = R1 - R0. At lower conductivities, the reflected wave is less

influential because the Fresnel reflection coefficient decreases in magnitude and shifts in phase, thus

flattening the cancellation behavior and shifting the nulls in frequency. Geometry associated with

the above case effectively demonstrates the behavior of a two-ray model and its dependence on

material properties. For practical purposes, we limit the scope of the remainder of this analysis to

reflecting surfaces made of concrete. Although the electric properties of concrete vary with

frequency [54], we assume that frequency-independent parameters (i.e., r = 5 and  = 0.00195 S/m)

will provide representative results.


10−2 10−1 100 −45

−40

−35

−30

−25

−20

−15

−10

σ=0.00195 S/m

|Ez

2

| [dB]

f [GHz]

Sommerfeld

geometric optics

10−2 10−1 100 −45

−40

−35

−30

−25

−20

−15

−10

σ=0.195 S/m

|Ez

2

| [dB]

f [GHz]

10−2 10−1 100 −45

−40

−35

−30

−25

−20

−15

−10

σ=19.5 S/m

|Ez

2

| [dB]

f [GHz]

10−2 10−1 100 −45

−40

−35

−30

−25

−20

−15

−10

σ=1950 S/m

|Ez

2

| [dB]

f [GHz]

Figure 5. Magnitude squared of the z component of the E-field for a VED above a half space

(z=1 m, h=1 m, !=10 m, 0r=5) at various conductivities.

4.2. Position Variation to Isolate Near-Surface and Near-Field Effects

In this section, we position the source and observation points to isolate near-surface and near-field

effects. Figures 6 and 7 illustrate fields radiated by a VED above a half plane, Figures 8 and 9 isolate

the TE waves of an x-aligned HED by limiting the observation points to  = 90, and Figures 10 and

11 isolate the TM waves of an x-aligned HED by limiting the observation points to  = 0.

For incremental electric dipoles, simulations show that near-field effects are substantially reduced

for  > 1 m excluding the E! component of the HED-TM case; hence, we isolate near-surface effects

by holding the horizontal separation constant at 5 m and simultaneously reducing the source and

observation heights: {(h [m], z [m],  [m])} = {(10, 10, 5), (1, 1, 5), (0.1, 0.1, 5), (0.01, 0.01, 5)}.

It should be understood that near-field regions depend on antenna size and that ideal infinitesimal

dipole sources are non-realizable. Figures 6, 8, and 10 demonstrate surface effects on field strength

and Sections 4.2.1 and 4.2.2 provide relevant observations and discussion.

It is shown later that for an elementary electric dipole above a concrete half plane, surface effects are

significantly reduced when h  10 m and z  10 m. Therefore, we isolate near-field effects by

holding the source and observation points at a constant height of 10 m and reducing the horizontal

separation: {(h [m], z [m],  [m])} = {(10, 10, 10), (10, 10, 1), (10, 10, 0.1), (10, 10, 0.01)}. Figures

7, 9, and 11 illustrate near-field effects in the frequency domain, and Section 4.2.3 provides analyses.
	

10−2 10−1 100

−21.5

−21

−20.5

−20

−19.5

VED: h=10 m, z=10 m, ρ=10 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−45

−44

−43

−42

−41

−40

−39

VED: h=10 m, z=10 m, ρ=10 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−72.5

−72

−71.5

−71

−70.5

VED: h=10 m, z=10 m, ρ=10 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−10

−5

0

5

10

15

20

25

30

VED: h=10 m, z=10 m, ρ=1 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−61

−60

−59

−58

−57

−56

VED: h=10 m, z=10 m, ρ=1 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−55

−50

−45

−40

−35

VED: h=10 m, z=10 m, ρ=1 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

0

20

40

60

80

100

VED: h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−81

−80

−79

−78

−77

−76

VED: h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−40

−30

−20

−10

0

10

VED: h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

20

40

60

80

100

120

140

160

VED: h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−101

−100

−99

−98

−97

−96

VED: h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−20

−10

0

10

20

30

40

50

60

VED: h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Hφ

|

2 [dB]

Sommerfeld

geometric optics

Figure 7. Near-field effects on field strength of a VED above a concrete half space.

10−2 10−1 100

−14.8

−14.6

−14.4

−14.2

−14

−13.8

VED: h=10 m, z=10 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−48

−47

−46

−45

−44

−43

VED: h=10 m, z=10 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−66

−65.5

−65

−64.5

−64

−63.5

−63

−62.5

−62

VED: h=10 m, z=10 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−16

−15

−14

−13

−12

−11

−10

−9

−8

VED: h=1 m, z=1 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−60

−50

−40

−30

−20

−10

VED: h=1 m, z=1 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−68

−66

−64

−62

−60

−58

VED: h=1 m, z=1 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−30

−25

−20

−15

−10

−5

VED: h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−45

−40

−35

−30

−25

−20

−15

VED: h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−85

−80

−75

−70

−65

−60

−55

VED: h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−50

−40

−30

−20

−10

0

VED: h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−70

−60

−50

−40

−30

−20

−10

VED: h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−110

−100

−90

−80

−70

−60

−50

VED: h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

Sommerfeld

GO + Norton term

geometric optics

Figure 6. Near-surface effects on field strength of a VED above a concrete half space.



10−2 10−1 100

−22

−21.5

−21

−20.5

−20

−19.5

−19

−18.5

−18

HED(TE): h=10 m, z=10 m, ρ=10 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−72.5

−72

−71.5

−71

−70.5

−70

HED(TE): h=10 m, z=10 m, ρ=10 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−88

−87.5

−87

−86.5

−86

−85.5

−85

−84.5

−84

HED(TE): h=10 m, z=10 m, ρ=10 m

f [GHz]

|Hρ

|

2 [dB]

10−2 10−1 100

−10

−5

0

5

10

15

20

25

30

HED(TE): h=10 m, z=10 m, ρ=1 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−55

−50

−45

−40

−35

HED(TE): h=10 m, z=10 m, ρ=1 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−86

−85.5

−85

−84.5

−84

HED(TE): h=10 m, z=10 m, ρ=1 m

f [GHz]

|Hρ

|

2 [dB]

10−2 10−1 100

0

20

40

60

80

100

HED(TE): h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−40

−30

−20

−10

0

10

HED(TE): h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−86

−85.5

−85

−84.5

−84

HED(TE): h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Hρ

|

2 [dB]

10−2 10−1 100

0

50

100

150

HED(TE): h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−20

−10

0

10

20

30

40

50

60

HED(TE): h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−86

−85.5

−85

−84.5

−84

HED(TE): h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Hρ

|

2 [dB]

Sommerfeld

geometric optics

Figure 9. Near-field effects on field strength of an x-aligned HED above a concrete half space.

Observation points are restricted to 3=90(.

10−2 10−1 100

−16

−15.5

−15

−14.5

−14

−13.5

−13

HED(TE): h=10 m, z=10 m, ρ=5 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−66

−65.5

−65

−64.5

−64

−63.5

−63

−62.5

−62

HED(TE): h=10 m, z=10 m, ρ=5 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−86.5

−86

−85.5

−85

−84.5

−84

HED(TE): h=10 m, z=10 m, ρ=5 m

f [GHz]

|Hρ

|

2 [dB]

10−2 10−1 100

−25

−20

−15

−10

−5

HED(TE): h=1 m, z=1 m, ρ=5 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−75

−70

−65

−60

HED(TE): h=1 m, z=1 m, ρ=5 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−78

−76

−74

−72

−70

−68

−66

HED(TE): h=1 m, z=1 m, ρ=5 m

f [GHz]

|Hρ

|

2 [dB]

10−2 10−1 100

−45

−40

−35

−30

−25

−20

−15

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−100

−95

−90

−85

−80

−75

−70

−65

−60

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−100

−95

−90

−85

−80

−75

−70

−65

−60

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Hρ

|

2 [dB]

10−2 10−1 100

−70

−60

−50

−40

−30

−20

−10

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Eφ

|

2 [dB]

10−2 10−1 100

−140

−130

−120

−110

−100

−90

−80

−70

−60

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Hz

|

2 [dB]

10−2 10−1 100

−120

−110

−100

−90

−80

−70

−60

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Hρ

|

2 [dB]

Sommerfeld

GO + Norton term

geometric optics

Figure 8. Near-surface effects on field strength of an x-aligned HED above a concrete half

space. Observation points are restricted to 3=90(. 


10−2 10−1 100

−45

−44

−43

−42

−41

−40

−39

HED(TM): h=10 m, z=10 m, ρ=10 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−50

−40

−30

−20

−10

0

HED(TM): h=10 m, z=10 m, ρ=10 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−89

−88.5

−88

−87.5

−87

HED(TM): h=10 m, z=10 m, ρ=10 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−61

−60

−59

−58

−57

−56

HED(TM): h=10 m, z=10 m, ρ=1 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−50

−25

0

25

50

HED(TM): h=10 m, z=10 m, ρ=1 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−86

−85.5

−85

−84.5

−84

HED(TM): h=10 m, z=10 m, ρ=1 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−81

−80

−79

−78

−77

−76

HED(TM): h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−50

0

50

100

HED(TM): h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−86

−85.5

−85

−84.5

−84

HED(TM): h=10 m, z=10 m, ρ=0.1 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−101

−100

−99

−98

−97

−96

HED(TM): h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−50

0

50

100

150

200

HED(TM): h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−86

−85.5

−85

−84.5

−84

HED(TM): h=10 m, z=10 m, ρ=0.01 m

f [GHz]

|Hφ

|

2 [dB]

Sommerfeld

geometric optics

Figure 11. Near-field effects on field strength of an x-aligned HED above a concrete half

space. Observation points are restricted to 3=0(.

10−2 10−1 100

−48

−47

−46

−45

−44

−43

HED(TM): h=10 m, z=10 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−50

−40

−30

−20

−10

0

HED(TM): h=10 m, z=10 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−87

−86.5

−86

−85.5

−85

−84.5

HED(TM): h=10 m, z=10 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−60

−50

−40

−30

−20

−10

HED(TM): h=1 m, z=1 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−80

−70

−60

−50

−40

−30

−20

−10

0

HED(TM): h=1 m, z=1 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−105

−100

−95

−90

−85

−80

−75

HED(TM): h=1 m, z=1 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−45

−40

−35

−30

−25

−20

−15

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−80

−70

−60

−50

−40

−30

−20

−10

0

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−100

−95

−90

−85

−80

−75

−70

−65

−60

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

10−2 10−1 100

−70

−60

−50

−40

−30

−20

−10

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Ez

|

2 [dB]

10−2 10−1 100

−120

−100

−80

−60

−40

−20

0

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Eρ

|

2 [dB]

10−2 10−1 100

−120

−110

−100

−90

−80

−70

−60

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

f [GHz]

|Hφ

|

2 [dB]

Sommerfeld

GO + Norton term

geometric optics

Figure 10. Near-surface effects on field strength of an x-aligned HED above a concrete half

space. Observation points are restricted to 3=0(.


U 
 2

k1

2	k02

02

G1

0z 2 	 P



0

mexp[	l(zh)] J0( !)  d . (41)

P



0

mJ0( !)  d 
 









02

0z 2

exp[	jk1R1]

R1 z
h
0

, (42)

U 
 2

k1

2	k02

02G1

0z 2 	 









02

0z 2

exp(	jk1R1 )

R1 z
h
0

	 P



0

m exp[	l(zh)]	1 J0(!)  d . (43)

4.2.1. Surface Wave

Figures 6, 8, and 10 illustrate near-surface effects as the source and observation points are drawn

close to the interface. Norton surface-wave terms provide a means to isolate surface-wave effects.

Near the interface, deviation from geometric optics due to surface-wave propagation peaks near d.c.

and decreases continuously with increasing frequency. As expected, surface-wave propagation is

more influential in TM fields than in TE fields radiated by HED. In comparison to the more general

Sommerfeld solution, GO + Norton term approximations seem adequate for predicting the radiation

of a VED above a half-plane; significant discrepancies, however, appear in the HED case where

near-field effects are influential, as discussed in the following sections.

4.2.2. Pseudo-Lateral Wave

Sommerfeld results in Figure 8 show strong oscillations in the TE fields radiated by an x-aligned

HED over a wide range of frequencies when the source and observation point are near the interface

(i.e., h  0.01 m and z  0.01 m). In this section, we provide alternative expressions for U, V, and

W to help explain this oscillation.

Notice that the HED-TE fields in equation (15) are strongly influenced by the integral U. With

reference to [55], rationalization of the denominator in U and use of the integral representation of

G1 in equation (8) allows for it to be rewritten as

If we add and subtract 1 within the integrand and make use of 

then U can be expressed as

Close to the interface the integral term is relatively small and the first two terms depict two waves

traveling adjacent to one another but at different speeds due to the different propagation media. The

two waves destructively interfere when -k0R1  -k1R1 + 2n, where n is an integer. For the material

properties chosen and R1 = 5 m, the interference occurs approximately every 48.5 MHz which agrees
P

k1 /k0

k0 /k1

v dv

(  2	pole

2 ) v 2	pole2


 k1

4	k04

k1 k0

1

k1

2 lk02m , (44)

H (1)

0 ( k1 /k0 , !) 
 2

j P

k1 /k0

0

exp( j ! 1u 2 )

1u 2

du , (45)

V 
 2k1 k0

k1

4	k04

k1

k0

exp(	jk1 !)

! 	 k0

k1

exp(	jk0 !)

! 	 2

pole H(1)

0 (k1 /k0 , pole!)	H(1)0 (k0 /k1 , pole!)

 P



0

2 exp[	l(zh)]	1

lk1

2mk02

J0(!)  d ,

(46)

W 
 	 0

0z

2G1 	 ( k0

2k12 )V , (47)

with the oscillation observed. The interference caused by the adjacent fields dies off as the source

and observation point are moved away from the surface, which suggests near-field excitation for

generating the pseudo-lateral wave.

By the same motivation, an alternative form of the Sommerfeld integral V is derived with reference

to [55]. As in the approach used in reformulating U, we add and subtract 1 within the integrand in

equation (10). The denominator of V cannot be rationalized as with U, but the integral identity

allows for integration by parts, which produces two additive exponential terms. Finally, change in

variable and identification of the integral form of the incomplete Hankel function, given by

completes the derivation and yields

where the positive form of equation (34) is used for pole. Values for incomplete Hankel functions

may be attained via numerical expansion. A useful relation given by

provides the means for expressing the last Sommerfeld integral W in terms of V.

The leading terms of equations (43) and (46) show evidence of a pseudo-lateral wave, which

originates in the less dense media and propagates along the surface; it travels inside the dielectric

and is excited via near-field coupling. This phenomenon resembles the lateral wave that has

application in geophysical exploration of the lithosphere [56] and propagation modeling in highly

vegetated environments [57]. Lateral waves are excited along a boundary between two media by a

source either at the interface or in the dense medium and travel atop the interface in the less dense

medium. In Section 5, time-domain results will also illustrate the pseudo-lateral wave phenomenon,

where a pulse arrival time corresponding to the wave velocity of the dense media is shown.


Er  p

2R0

2

μ0

0

cosd 1 1

jk0R0

exp(jk0R0 )

E  jk0 p

4R0

μ0

0

sind 1 1

jk0R0

 1

(k0R0 )

2 exp(jk0R0 )

H

3  jp

4R0

sind 1 1

jk0R0

exp(jk0R0 ) .

(48)

4.2.3. Near-Field Effects on the Direct Ray

Electromagnetic fields in the near-field region were computed via numerical evaluation of

Sommerfeld integrals and compared to GO approximations in Figures 7, 9, and 11. Note that Norton

surface wave approximations are not included in these plots because they are valid only in the far

field. As separation decreases to  < 	, the direct ray is observed in the near field of the source, and

field components influenced by the direct ray (i.e., Ez and H3 for VED and E3 and Hz for HED-TE)

demonstrate deviation from the GO approximation at low frequencies due to near-field effects. At

h = z = 10 m, the reflected ray travels well into the far field of the source before being reflected and

observed; consequently, field components influenced by the reflected ray and not the direct ray (i.e.,

E! for VED, H! for HED-TE, and Ez and H3 for HED-TM) show little near-field error.

The fields of the direct ray may be expressed in spherical coordinates as

Notice that Er is strictly a near- and intermediate-field expression with an phase exp(jk0R0 )

dependence that is not accounted for in the geometric optics expressions. If d = 0, then Er is

equivalent to the direct-ray of E! in the HED-TM case and near-field influence is maximized in the

null of the antenna pattern because the cosd factor is equal to 1. In Figures 10 and 11, the E!

component of the HED-TM radiated field displays an oscillatory deviation from the GO + Norton

term approximation when locations are many wavelengths from the interface and from each other.

In this scenario, the near-field direct ray is comparable in magnitude to the reflected ray, hence

causing a near-field direct ray plus a GO reflected ray cancellation pattern. Further discussion is

given in the time-domain results section where the results are more intuitive.

4.3. Contour Plots to Observe Two-Ray Behavior

We assigned the field strength and corresponding GO error to colors and plotted those colors versus

frequency and horizontal separation for constant h and z; this allows us to visualize the results in a

more general sense. Appendix A displays numerical solutions to Sommerfeld formulations, residual

errors (i.e. ||Ez

S

|

2

 - |Ez

GO|

2

| [dB]), and dashed lines on the residual error plots representing the R0 = 10	

contour to provide a geometric guideline. Note that these results have low resolution in frequency,

so highly oscillatory behavior may not be seen.


Figure 12. Signal representations.

Half the figures in Appendix A present cases where the source and observation points lie on the same

horizontal plane {(h [m], z [m])} = {(10, 10); (1, 1); (0.1, 0.1); (0.01, 0.01)}. In these plots, observe

the decrease in two-ray cancellation frequency with increasing R (i.e., constant  and decreasing

height) when the direct and reflected rays are both present. Also notice the error magnification near

the Brewster angle. The remaining figures in Appendix A give representative contour plots for

source and observation points at different heights {(h [m], z [m])} = {(1, 10); (0.1, 10); (0.1, 1);

(0.01, 1)}. Notice the effect of the source antenna pattern; that is, if the observation point is not at

a null in an antenna pattern, then the two-ray cancellation effect becomes visible.

5. TIME-DOMAIN ANALYSIS

Field strength plots in the frequency domain are descriptive, but the practical significance of the

results are difficult to interpret. Of more importance to the digital communications engineer is how

the errors translate to the time domain. In order to assess the practical significance of GO error for

indoor scenarios, we compute the delay spread of the channel impulse response and quantify the

cumulative error seen in the frequency domain. This section provides a detailed explanation of the

time-domain analyses used in this report.

Digital signals are typically transmitted by some type of carrier modulation. The transmitted signal

is limited in bandwidth to an interval of frequencies centered around the carrier or center frequency

(i.e., c = 2fc) and must be real-valued in the time domain in order to have a physical interpretation;

consequently, it is complex-conjugate symmetric about d.c. in the frequency domain. For

mathematical convenience with no loss in generality, the passband signal may be expressed as a

complex-baseband representation. This is accomplished by first filtering out the negative

frequencies to produce an analytic signal. In the time domain, the real and imaginary parts of the

analytic signal are a Hilbert transform pair. Next, we shift the analytic signal down to baseband to

give a complex-baseband representation of the original passband signal. A pictorial representation

of real-time, analytic, and complex-baseband signals is given in Figure 12 and more extensive

discussions can be found in [58, 59].


g(t,)  v · e(t,  )

h(t,  ) , (49)

e(t,  )  FT 	1

[E(t,  c )
(c )]

h(t,  )  FT 	1

[H(t,  c )
(c )] , (50)

The impulse response of the channel can be given by

where is a unit direction vector corresponding to the receive antenna. The complex-baseband v

representation of the applicable vector field components, and , is given by e(t,  ) h(t,  )

where FT-1 is the inverse Fourier transform operator,  is the transform variable, and 
() is a scalar

window function that filters out negative frequencies and defines pulse shape and bandwidth.

We seek an expression for the squared magnitude of the impulse response |g(t,)|

2

 in terms of electric

and magnetic field variables in order to compute delay spread via equations (2) and (3). Depending

on the type and orientation of the receive antenna, one or some combination of the field components

will be the dominant coupling mechanism. Orientation, radiation pattern, and efficiencies of the

receive antenna are fairly arbitrary and only clutter the field effects we wish to observe. For

simplification purposes, we examine each field component at the observation point separately. This

augments the influence of the propagation mechanisms, generalizes the results, and allows for

flexible receive antenna specification to be easily realized. In the following subsections, we discuss

optimal pulse shapes and frequency considerations for indoor scenarios.

5.1. Windowing Techniques to Optimize Pulse Shape

In this section we address time-domain pulse shaping issues associated with the choice of 
(). It

is desirable to choose a window that produces a well-behaved pulse shape for large dynamic ranges.

The single-ray scenario (i.e., source and observation points existing in infinite space) allows us to

analyze delay spread sensitivity to threshold for a given pulse shape. We computed a single-ray

impulse response and its corresponding delay spread as a function of threshold (i.e., 0 to 40 dB

below the peak) for four windows (i.e., flat, Hanning, Hamming, and Blackman). Results are shown

in Figure 13. As expected, the behavior of the flat window is undesirable because of its high

sidelobes in the time-domain and sensitivity to threshold. If one considers threshold levels between

20 and 40 dB then the Hamming window is optimal because it has the most narrow well-behaved

pulse shape. Henceforth, the Hamming window is applied to all results. Use of non-rectangular

windows allows us to speak in terms of bandwidth BW (i.e., 3-dB bandwidth). For a Hamming

window, bandwidth is about 35% of the total frequency content.


0 1 2 3 4 5 6 −40

−30

−20

−10

0

f

c

=0.9 GHz

f [GHz]

|Λ(ω) exp(−jk0

R0)|2 [dB]

 0 2 4 6 8 −80

−60

−40

−20

 0

h=10 m, z=10 m, ρ=1 m

τ [ns]

|g(t,τ)|

2 [dB]

0 10 20 30 40 0

0.1

0.2

0.3

0.4

0.5

threshold

τ

spr

 [ns]

flat

Hanning

Hamming

Blackman

Figure 13. Windows, pulse shapes, and single-ray delay spread

versus threshold.

5.2. Practical Bandwidth Limitations

We aim to describe how radio-propagation channels limit communication system performance.

Simulation and measurement of broader frequency bands produce narrower pulses and more

effectively distinguish multipath contributors as well as subtle near-surface and near-field effects.

Consequently, the resulting impulse response contains more information and is a more general

description of the channel. Here we consider a pragmatic upper limit to bandwidth.

Current WLAN systems operate within allocated bands with less than 100 MHz bandwidth. Broader

bandwidths are achievable and are in demand. In order to extend the relevance of this study, we do

not limit our focus to allocated operational bandwidths. It is reasonable, however, to confine

simulation bandwidths to those achievable by wideband measurement systems, which are hardware

limited. Typically, impulse response measurement systems are limited by the digitizer’s analog

bandwidth, which is on the order of 1 GHz [60]. Henceforth, we limit our simulation bandwidth to

1 GHz, impose the frequency constraint 0 < f < 2fc to avoid overlap into negative frequencies, and

limit the center frequency to 900  fc  5800 MHz. Notice that if fc  1.35 GHz, then the bandwidth

is limited by the 0 < f < 2fc constraint (e.g., the 900-MHz signal shown in Figure 13).

It should also be noted that results are computed at discrete frequencies, and if too few points are

used then the noise floor rises in the time domain due to the correlating nature of the inverse Fourier

transform in equation (50). The frequency step dictates the pulse repetition rate of the transformed

signal according to PRR = 1/fstep, and should be chosen so that influential multipath components

arrive before the repetition rate elapses. The maximum dimension considered in this report is 10 m;

therefore, a frequency step of 10 MHz (i.e., PRR = 100 ns) is sufficient for two-dimensional

scenarios because the maximum delay is approximately max = (202

 + 102 )

1⁄2 ÷ 3x108  75 ns.


6. TIME-DOMAIN RESULTS

Time-domain results corresponding to previously specified geometries are presented in this section.

First, we consider the worse-case scenario (i.e., fc = 900 MHz, BW  600 MHz) by maximizing the

influence of low frequencies. This illustrates propagation effects in the time domain and their

influence on delay spread. Next, we shift the center frequency up into the next relevant bands to

demonstrate fc and BW dependence. Delay spread versus center frequency and bandwidth is

presented in order to assess the practical significance of errors associated with GO approximations

for indoor scenarios.

6.1. Impulse Responses for Wideband Signals Centered at 900 MHz

Impulse responses for a wideband signal centered at 900 MHz (i.e., 	c = 0.33 m) are given in Figures

14 - 19. Since low frequencies have the most influence, this is the worst scenario considered here.

The eight geometries used in the frequency-domain analysis were maintained in order to isolate the

near-surface and near-field effects. Given the large data set, a summary of basic observations is

given to avoid unnecessary confusion:

1) Near the interface (i.e., h  0.1 m and z  0.1 m), the direct and reflected rays overlie each other

if the time resolution of the pulse cannot resolve the rays.

2) The presence of a specific ray depends on the observed field component (e.g., if h = z, then the

direct ray is below the noise floor in E! for VED, H! for HED-TE, and Ez and H3 for HED-TM).

3) Individual rays may be affected by the antenna pattern, by transmission into the dielectric half

plane, and by free-space loss (e.g., if h = z and  is small, then the reflected ray from a VED is

strongly attenuated by the source antenna pattern).

Near-surface effects are best observed at h = z = 0.01 m in Figures 14, 16, and 18. The blue curve

(i.e., GO + Norton term) isolates surface-wave effects. In the frequency domain, the surface wave

causes a sharp increase in field strength concentrated at d.c.; in the time domain, this corresponds

to a raised noise floor. Also, the surface wave has an exp(-jk0R1) phase dependence, arrives at the

observation point along with the reflected ray, and enhances signal strength at that delay. The green

curve (i.e., numerical evaluation of Sommerfeld integrals) demonstrates the more influential pseudolateral wave, which causes a delayed pulse near 37 ns. A simple calculation shows that a wave

propagating 5 m through concrete at (μ0)

-1⁄2  1.34×108

 m/s arrives at the observation point with a

37.4 ns delay.

As shown in Figures 15, 17, and 19, the reactive nature of the near field dominates for   0.1 m.

This is expected, as the separation is well within the carrier wavelength. When the direct ray is

observed in the near field, GO error increases because the magnitude of the direct pulse and the noise

floor rise. For this frequency band, the geometric optics approximation fails for   0.1 m if the

direct ray is significant.
	

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.3 ns

0.29 ns

0.29 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0.3 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

12.28 ns

0.32 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0.3 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.28 ns

0.27 ns

0.29 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|2 [dB]

0.44 ns

0.31 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|2 [dB]

0.26 ns

0.27 ns

0.29 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.59 ns

0.33 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

1.42 ns

0.35 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.43 ns

0.33 ns

0.3 ns

Figure 14. Near-surface effects on impulse responses of a VED above a concrete half space

( fc=900 MHz, BW660 MHz, threshold=-30 dB). Legends display Sommerfeld,

GO + Norton term, and GO approximations to delay spread.

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.3 ns

0.29 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|hφ(t,τ)|

2 [dB]

2.69 ns

2.69 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.35 ns

0.29 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.29 ns

0.29 ns

0 20 40 60 80

−80

−60

−40

−20

0

20

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

24.19 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

2.41 ns

0.3 ns

0 20 40 60 80

−50

0

50

100

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

24.2 ns

0.3 ns

0 20 40 60 80

−200

−180

−160

−140

−120

−100

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hφ(t,τ)|

2 [dB]

11.57 ns

0.3 ns

Figure 15. Near-field effects on impulse responses of a VED above a concrete half space

(fc=900 MHz, BW660 MHz, threshold=-30 dB). Legends display Sommerfeld

and GO approximations to delay spread.



0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

4.5 ns

4.5 ns

4.5 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|h

z

(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.29 ns

0.29 ns

0.29 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.66 ns

0.66 ns

0.66 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.65 ns

0.65 ns

0.65 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.29 ns

0.29 ns

0.3 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.32 ns

0.31 ns

0.31 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.82 ns

0.31 ns

0.31 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

3.12 ns

0.33 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

13.42 ns

2.64 ns

0.29 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

15.3 ns

2.63 ns

0.29 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

12.07 ns

3.25 ns

0.3 ns

Figure 16. Near-surface effects on impulse responses of an x-aligned HED above a concrete

half space (fc=900 MHz, BW660 MHz, threshold=-30 dB, 3=90(). Legends

display Sommerfeld, GO + Norton term, and GO approximations to delay spread.

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

7.47 ns

7.47 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hz(t,τ)|

2 [dB]

3.37 ns

3.37 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.29 ns

0.29 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.35 ns

0.29 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.29 ns

0.29 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−80

−60

−40

−20

0

20

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

24.19 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hz(t,τ)|

2 [dB]

2.41 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−50

0

50

100

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

24.2 ns

0.3 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hz(t,τ)|

2 [dB]

11.57 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

Figure 17. Near-field effects on impulse responses of an x-directed HED above a concrete

half space (fc=900 MHz, BW660 MHz, threshold=-30 dB, 3=90(). Legends

display Sommerfeld and GO approximations to delay spread.


0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.29 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

4.71 ns

0.29 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.29 ns

0.29 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

35.12 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−100

−50

0

50

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

29.45 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−200

−180

−160

−140

−120

−100

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.3 ns

0 20 40 60 80

−100

−50

0

50

100

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

29.46 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.3 ns

0.3 ns

Figure 19. Near-field effects on impulse responses of an x-aligned HED above a concrete

half space (fc=900 MHz, BW660 MHz, threshold=-30 dB, 3=0(). Legends

display Sommerfeld and GO approximations to delay spread.

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.3 ns

0.29 ns

0.29 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

16.27 ns

0.29 ns

0.29 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.29 ns

0.29 ns

0.29 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

12.28 ns

0.32 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

4.89 ns

0.41 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.28 ns

0.32 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.44 ns

0.31 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

3.01 ns

0.42 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.29 ns

0.31 ns

0.3 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

1.42 ns

0.35 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

−60

−40

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

3.38 ns

0.37 ns

0.3 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.34 ns

0.35 ns

0.3 ns

Figure 18. Near-surface effects on impulse responses of an x-aligned HED above a concrete

half space (fc=900 MHz, BW660 MHz, threshold=-30 dB, 3=0(). Legends

display Sommerfeld, GO + Norton term, and GO approximations to delay spread.


0 2 4 6 8

−50

−25

0

25

50

HED(TM): h=10 m, z=10 m, ρ=1 m

f [GHz]

|Eρ

|

2 [dB]

direct + reflected

near−field direct ray

GO reflected ray

0 20 40 60 80

−120

−100

−80

−60

−40

−20

f

c

=900 MHz, BW=658.6 MHz

τ [ns]

|e

ρ(t,τ)|

2 [dB]

35.12 ns

0.3 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

f

c

=2400 MHz, BW=658.6 MHz

τ [ns]

|e

ρ(t,τ)|

2 [dB]

24.37 ns

0.3 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

f

c

=5800 MHz, BW=658.6 MHz

τ [ns]

|e

ρ(t,τ)|

2 [dB]

31.42 ns

0.3 ns

Figure 20. Combination of a near-field direct ray and far-field reflected ray for an x-aligned

HED above a concrete half space (3=0().

Interesting observations are found in the HED-TM results in Figures 18 and 19, in which the

observation points lie directly in the null of the x-aligned HED antenna pattern. As mentioned

earlier, the transverse field components (i.e., ez and h3) corresponding to the direct pulse are

attenuated due to the orientation of the dipole. The radial component (i.e., e!) corresponding to the

direct ray, however, is influential due to near- and intermediate-field propagation.

Geometric optics predictions may be improved by including near-field terms when computing the

direct-ray field components. When considering an x-aligned HED at z = h and  = 0, the analytic

expression for E! corresponding to the direct ray is equivalent to Er of equation (48) with d = 0.

For the geometry {(h [m], z [m],  [m])} = {(10, 10, 1)}, the direct ray was calculated via equation

(48), the reflected ray was calculated via the second term of equation (22), and the magnitude of the

individual rays along with the magnitude of their sum are displayed as a function of frequency in

Figure 20. Notice that the near-field direct ray is comparable in magnitude to the reflected ray. The

two rays destructively interfere, as shown in grey, which agrees with the Sommerfeld solution in

Figure 11. The magnitude of the direct ray is frequency sensitive, as shown in the impulse responses

in Figure 20. At fc = 900 MHz, the spread of the impulse response is dominated by the high noise

floor, which corresponds to the strong near fieldss near d.c. At higher frequencies the high noise

floor is avoided, but the near-field direct ray produces high delay spreads as its magnitude

approaches the magnitude of the reflected ray. Given the close confinement of most indoor channels,

it seems logical to use the general expressions to account for near-field effects on the direct ray.

6.2. Impulse Responses for Wideband Signals Centered at 2.4 GHz and 5.8 GHz

Appendices B and C show impulse response results for carrier frequencies at 2.4 GHz and 5.8 GHz,

respectively, at maximum bandwidth (i.e., BW = 1 GHz). The wider frequency range produces more

narrow pulses in the time domain. As shown by the delay spreads, near-surface and near-field errors

were reduced significantly at higher center frequencies because low frequencies had less influence.

These errors, however, are not negligible. Appendix D provides delay spread curves as a function

of bandwidth for center frequencies at 900 MHz, 2.4 GHz, and 5.8 GHz. These plots demonstrate

that GO accuracy depends on frequency content and show cases where considerable GO error occurs

for signals as high as 5.8 GHz.


% error 

v

spr  100 |

S,v

spr  

GO,v

spr |



S,v

spr

, (51)

6.3. Delay Spread Versus Bandwidth and Carrier Frequency

Thus far, we have demonstrated significant GO error in practical indoor scenarios and its dependence

on material composition, geometry, and center frequency. This section helps to determine under

what frequency conditions GO is valid.

Figures 21 - 26 are contour plots of delay spread (from the numerical evaluation of Sommerfeld

integrals) versus bandwidth and carrier frequency. White areas indicate where negative frequencies

would be encountered. Figures 27 - 32 are contour plots of percentage error (associated with the GO

approximation) versus bandwidth and carrier frequency. Percentage error is formulated as

where v stands for the relative field component. From these plots we draw the following conclusions

based on delay spread results of elementary electric dipoles above a concrete half space for typical

indoor dimensions and frequencies:

1) Surface-wave effects are significant for heights on the order of 0.1m or less at BW > 0.6 fc for

TM fields radiated by VED and HED sources.

2) Pseudo-lateral-wave effects are significant for heights on the order of 0.01 m or less at fc < 1.8

GHz for E! fields radiated by a VED source, fc < 3.8 GHz for TE fields radiated by a HED

source, fc < 1.8 GHz for Ez fields radiated by a HED source, and fc < 2.5 GHz for E! fields

radiated by a HED source.

3) Near-field effects are significant for separation on the order of 0.1 m or less at BW > 0.6 fc for

transverse fields radiated by VED and HED sources.

4) Near-field effects are significant for radial fields observed in the null of the antenna patterns of

VED and HED sources.

7. CONCLUSION

In this report, we evaluated the error associated with GO predictions for indoor propagation models.

Our conclusions were based on results from the classical problem of elementary dipoles above a

lossy half space. Exact (i.e., Sommerfeld) and approximate (i.e., GO and GO + Norton surface-wave

term) formulations for vertical and horizontal dipole sources were summarized, and numerical

schemes for evaluating the complex integrals in the Sommerfeld formulation were given. Eight

canonical geometries were chosen to isolate individual propagation effects on relevant field

components. Most scenarios were limited to material properties appropriate for concrete and

physical dimensions up to 10 m and as small as 1 cm. Results were given in both the frequency and

time domains.


0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=10 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=1 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.1 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.01 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=10 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=1 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.1 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.01 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=10 m

τ

spr

hφ

0

0.5

1

1.5

2

2.5

3

3.5

4

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=1 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.1 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.01 m

τ

spr

hφ

0

0.5

1

1.5

2

Figure 22. Near-field effects on delay spread versus BW and center frequency for a VED

above a concrete half space (Hamming window, threshold=-30 dB).

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=1 m, z=1 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=5 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=1 m, z=1 m, ρ=5 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=1 m, z=1 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

Figure 21. Near-surface effects on delay spread versus BW and center frequency for a VED

above a concrete half space (Hamming window, threshold=-30 dB).


0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=10 m

τ

spr

eφ

0

2

4

6

8

10

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=1 m

τ

spr

eφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ

spr

eφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ

spr

eφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=10 m

τ

spr

hz

0

2

4

6

8

10

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=1 m

τ

spr

hz

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ

spr

hz

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ

spr

hz

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=10 m

τ

spr

hρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=1 m

τ

spr

hρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ

spr

hρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ

spr

hρ

0

0.5

1

1.5

2

Figure 24. Near-field effects on delay spread versus BW and center frequency for a HED

above a concrete half space (Hamming window, threshold=-30 dB, 3=90().

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=5 m

τ

spr

eφ

0

1

2

3

4

5

6

7

8

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=1 m, z=1 m, ρ=5 m

τ

spr

eφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

eφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

eφ

0

0.5

1

1.5

2

2.5

3

3.5

4

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=5 m

τ

spr

hz

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=1 m, z=1 m, ρ=5 m

τ

spr

hz

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

hz

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

hz

0

0.5

1

1.5

2

2.5

3

3.5

4

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=5 m

τ

spr

hρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=1 m, z=1 m, ρ=5 m

τ

spr

hρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

hρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

hρ

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 23. Near-surface effects on delay spread versus BW and center frequency for a HED

above a concrete half space (Hamming window, threshold=-30 dB, 3=90().


0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=10 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=1 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=10 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=1 m

τ

spr

eρ

0

5

10

15

20

25

30

35

40

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=10 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=1 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ

spr

hφ

0

0.5

1

1.5

2

Figure 26. Near-field effects on delay spread versus BW and center frequency for a HED

above a concrete half space (Hamming window, threshold=-30 dB, 3=0().

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TM): h=1 m, z=1 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

ez

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=5 m

τ

spr

eρ

0

1

2

3

4

5

6

7

8

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TM): h=1 m, z=1 m, ρ=5 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

eρ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=10 m, z=10 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TM): h=1 m, z=1 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ

spr

hφ

0

0.5

1

1.5

2

Figure 25. Near-surface effects on delay spread versus BW and center frequency for a HED

above a concrete half space (Hamming window, threshold=-30 dB, 3=0().


0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=10 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=1 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.1 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.01 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=10 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=1 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.1 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.01 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=10 m

% error τ

spr

hφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=1 m

% error τ

spr

hφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.1 m

% error τ

spr

hφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=0.01 m

% error τ

spr

hφ

0

1

2

3

4

5

Figure 28. Near-field effects on % error delay spread versus BW and center frequency for a

VED above a concrete half space (Hamming window, threshold=-30 dB).

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=5 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=1 m, z=1 m, ρ=5 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.1 m, z=0.1 m, ρ=5 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.01 m, z=0.01 m, ρ=5 m

% error τ

spr

ez

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=5 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=1 m, z=1 m, ρ=5 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.1 m, z=0.1 m, ρ=5 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.01 m, z=0.01 m, ρ=5 m

% error τ

spr

eρ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=10 m, z=10 m, ρ=5 m

% error τ

spr

hφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=1 m, z=1 m, ρ=5 m

% error τ

spr

hφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.1 m, z=0.1 m, ρ=5 m

% error τ

spr

hφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

VED: h=0.01 m, z=0.01 m, ρ=5 m

% error τ

spr

hφ

0

1

2

3

4

5

Figure 27. Near-surface effects on % error delay spread versus BW and center frequency for

a VED above a concrete half space (Hamming window, threshold=-30 dB).


0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=10 m

% error τ

spr

eφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=1 m

% error τ

spr

eφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.1 m

% error τ

spr

eφ

0

1

2

3

4

5

0.2 0.4 0.6 0.8

1

2

3

4

5

BW [GHz]

f

c

 [GHz] HED(TE): h=10 m, z=10 m, ρ=0.01 m

% error τ

spr

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 [GHz]

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 [GHz]

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% error τ

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% error τ

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% error τ

spr

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0

1

2

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Figure 30. Near-field effects on % error delay spread versus BW and center frequency for a

HED above a concrete half space (Hamming window, threshold=-30 dB, 3=90().

0.2 0.4 0.6 0.8

1

2

3

4

5

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f

c

 [GHz]

HED(TE): h=10 m, z=10 m, ρ=5 m

% error τ

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% error τ

spr

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30

40

50

Figure 29. Near-surface effects on % error delay spread versus BW and center frequency for

a HED above a concrete half space (Hamming window, threshold=-30 dB, 3=90().
	

0.2 0.4 0.6 0.8

1

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% error τ

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% error τ

spr

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% error τ

spr

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Figure 32. Near-field effects on % error delay spread versus BW and center frequency for a

HED above a concrete half space (Hamming window, threshold=-30 dB, 3=0().

0.2 0.4 0.6 0.8

1

2

3

4

5

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f

c

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% error τ

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Figure 31. Near-surface effects on % error delay spread versus BW and center frequency for

a HED above a concrete half space (Hamming window, threshold=-30 dB, 3=0().



As expected, GO predictions agreed with the numerical approximations to Sommerfeld integrals

when the source and observation points were multiple wavelengths above the surface and multiple

wavelengths apart. By drawing the points close to the surface and close together, we were able to

isolate classic surface-wave and near-field effects and observe an additional propagation mechanism

we call the pseudo-lateral wave. Frequency-domain observations are summarized as follows:

Surface-wave observations in the frequency domain

1) Surface-wave effects peak at d.c. and decrease sharply with increasing frequency.

2) Surface-wave propagation is more influential in TM fields than in TE fields.

Pseudo-lateral-wave observations in the frequency domain

3) Pseudo-lateral-wave effects are influential over a broad range of frequencies.

4) Pseudo-lateral-wave propagation is more influential in TE fields than in TM fields.

Near-field observations in the frequency domain

5) Near-field effects peak at d.c. and decrease sharply with increasing frequency.

6) Near-field error depends on the individual propagation paths and field components.

7) Near-field effects are significant for radial fields observed in the null of the dipole antenna

pattern at observation points many wavelengths away.

These observations demonstrated frequency dependency and interesting electromagnetic behavior,

but they were fairly inconclusive; hence, we transformed the results to the time domain in order to

assess the practical significance of the propagation mechanisms.

Frequency domain results were translated into band-limited impulse responses. The frequency

content was specified by center frequency and bandwidth, and time dispersion was quantified via

delay spread. Time-domain observations are summarized as follows:

Surface-wave observations in the time domain

1) Surface waves create a pulse at the reflected-ray delay and increase the noise floor.

2) Surface-wave effects are significant for heights on the order of 0.1 m or less at BW > 0.6 fc for

TM fields radiated by VED and HED sources.

Pseudo-lateral-wave observations in the time domain

3) Pseudo-lateral-wave pulses are delayed according to the electric properties of the dielectric.

4) Pseudo-lateral-wave effects are significant for heights on the order of 0.01 m or less at

a) fc < 1.8 GHz for E! fields radiated by a VED source,

b) fc < 3.8 GHz for TE fields radiated by a HED source,

c) fc < 1.8 GHz for Ez fields radiated by a HED source, and

d) fc < 2.5 GHz for E! fields radiated by a HED source.

Near-field observations in the time domain

5) Near-field effects increase the noise floor.

6) Near-field effects are significant for separation on the order of 0.1 m or less at BW > 0.6 fc for

transverse fields radiated by VED and HED sources.

7) Near-field effects can be predominant for radial fields observed in the null of the antenna

patterns of VED and HED sources.


Conditions which cause the GO approximation to be invalid depend on the transmit and receive

antennas and their position, on the material composition of the reflecting surface, and on the

operational frequency band. General expressions for incremental electric dipoles above a half space

have been provided to improve the accuracy of GO predictions by accounting for surface-wave and

near-field effects. More specifically, the GO approximation may be complemented by the Norton

surface-wave terms in equations (18), (21), and (22) when k0R0 » 1 and |k1| > |k0| and by the near-field

terms in equation (48) to model the direct ray. These expressions, however, apply only to elementary

dipoles above a dielectric half space. Care should be taken when modeling actual antennas and

finite-thick reflection surfaces.

As demonstrated, pseudo-lateral wave effects can be severe. The leading terms of equations (43)

and (46) show evidence of the pseudo-lateral wave; these equations, however, are cumbersome and

require numerous derivatives and substitutions in order to acquire field equations. A more

mathematically rigorous derivation is necessary in order to provide a closed-form asymptotic

approximation to the pseudo-lateral wave.

8. ACKNOWLEDGMENTS

The authors thank Dr. George A. Hufford, J. Randy Hoffman, Paul McKenna, and Robert J. Achatz

for insights shared.


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46

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2 4 6 8 10

2

4

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ρ [m]

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VED: h=10 m, z=10 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=10 m, z=10 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=10 m, z=10 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=0.1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=0.1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

1

2

3

4

5

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=0.1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.01 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.01 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

5

10

15

20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.01 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

2

4

6

8

10

Figure A-2. Field strength residual error of a VED above a concrete half space (h=z) versus

frequency and horizontal separation.

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=10 m, z=10 m

|Ez

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=10 m, z=10 m

|Eρ

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=10 m, z=10 m

|Hφ

S

|

2

 [dB]

−80

−75

−70

−65

−60

−55

−50

−45

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=1 m

|Ez

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=1 m

|Eρ

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=1 m

|Hφ

S

|

2

 [dB]

−80

−75

−70

−65

−60

−55

−50

−45

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=0.1 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=0.1 m

|Eρ

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=0.1 m

|Hφ

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.01 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.01 m

|Eρ

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.01 m

|Hφ

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

Figure A-1. Sommerfeld field strength of a VED above a concrete half space (h=z) versus

frequency and horizontal separation.

APPENDIX A: FIELD STRENGTH VERSUS FREQUENCY AND SEPARATION
	

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=10 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=10 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=10 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=10 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=10 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=10 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

1

2

3

4

5

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

5

10

15

20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

1

2

3

4

5

Figure A-4. Field strength residual error of a VED above a concrete half space (hgz) versus

frequency and horizontal separation.

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=10 m

|Ez

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=10 m

|Eρ

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=1 m, z=10 m

|Hφ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=10 m

|Ez

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=10 m

|Eρ

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=10 m

|Hφ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=1 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=1 m

|Eρ

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.1 m, z=1 m

|Hφ

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.1 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.1 m

|Eρ

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

VED: h=0.01 m, z=0.1 m

|Hφ

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

Figure A-3. Sommerfeld field strength of a VED above a concrete half space (hgz) versus

frequency and horizontal separation.



2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=10 m, z=10 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=10 m, z=10 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=10 m, z=10 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=1 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=1 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=1 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=0.1 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=0.1 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=0.1 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.01 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.01 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.01 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

2

4

6

8

10

Figure A-6. Field strength residual error of a HED above a concrete half space (h=z, 3=90()

versus frequency and horizontal separation.

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=10 m, z=10 m

|Eφ

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=10 m, z=10 m

|Hz

S

|

2

 [dB]

−80

−75

−70

−65

−60

−55

−50

−45

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=10 m, z=10 m

|Hρ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=1 m

|Eφ

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=1 m

|Hz

S

|

2

 [dB]

−80

−75

−70

−65

−60

−55

−50

−45

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=1 m

|Hρ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=0.1 m

|Eφ

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=0.1 m

|Hz

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=0.1 m

|Hρ

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.01 m

|Eφ

S

|

2

 [dB]

−80

−70

−60

−50

−40

−30

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.01 m

|Hz

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.01 m

|Hρ

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

Figure A-5. Sommerfeld field strength of a HED above a concrete half space (h=z, 3=90()

versus frequency and horizontal separation.


2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=10 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=10 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=10 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=10 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=10 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=10 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=1 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=1 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=1 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.1 m

||Eφ

S

|

2 − |Eφ

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.1 m

||Hz

S

|

2 − |Hz

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.1 m

||Hρ

S

|

2 − |Hρ

GO|

2

| [dB]

0

2

4

6

8

10

Figure A-8. Field strength residual error of a HED above a concrete half space (hgz, 3=90()

versus frequency and horizontal separation.

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=10 m

|Eφ

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=10 m

|Hz

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=1 m, z=10 m

|Hρ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=10 m

|Eφ

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=10 m

|Hz

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=10 m

|Hρ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=1 m

|Eφ

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=1 m

|Hz

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.1 m, z=1 m

|Hρ

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.1 m

|Eφ

S

|

2

 [dB]

−80

−70

−60

−50

−40

−30

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.1 m

|Hz

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TE): h=0.01 m, z=0.1 m

|Hρ

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

Figure A-7. Sommerfeld field strength of a HED above a concrete half space (hgz, 3=90()

versus frequency and horizontal separation.


2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=10 m, z=10 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=10 m, z=10 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

1

2

3

4

5

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=10 m, z=10 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

1

2

3

4

5

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

1

2

3

4

5

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=0.1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=0.1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

5

10

15

20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=0.1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.01 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

5

10

15

20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.01 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

20

40

60

80

100

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.01 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

5

10

15

20

Figure A-10. Field strength residual error of a HED above a concrete half space (h=z, 3=0()

versus frequency and horizontal separation.

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=10 m, z=10 m

|Ez

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=10 m, z=10 m

|Eρ

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=10 m, z=10 m

|Hφ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=1 m

|Ez

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=1 m

|Eρ

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=1 m

|Hφ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=0.1 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=0.1 m

|Eρ

S

|

2

 [dB]

−80

−70

−60

−50

−40

−30

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=0.1 m

|Hφ

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.01 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.01 m

|Eρ

S

|

2

 [dB]

−80

−70

−60

−50

−40

−30

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.01 m

|Hφ

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

Figure A-9. Sommerfeld field strength of a HED above a concrete half space (h=z, 3=0()

versus frequency and horizontal separation.


2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=10 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=10 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=10 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=10 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=10 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=10 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

1

2

3

4

5

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

0.5

1

1.5

2

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.1 m

||Ez

S

|

2 − |Ez

GO|

2

| [dB]

0

2

4

6

8

10

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.1 m

||Eρ

S

|

2 − |Eρ

GO|

2

| [dB]

0

10

20

30

40

50

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.1 m

||Hφ

S

|

2 − |Hφ

GO|

2

| [dB]

0

2

4

6

8

10

Figure A-12. Field strength residual error of a HED above a concrete half space (hgz, 3=0()

versus frequency and horizontal separation.

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=10 m

|Ez

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=10 m

|Eρ

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=1 m, z=10 m

|Hφ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=10 m

|Ez

S

|

2

 [dB]

−60

−55

−50

−45

−40

−35

−30

−25

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=10 m

|Eρ

S

|

2

 [dB]

−40

−35

−30

−25

−20

−15

−10

−5

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=10 m

|Hφ

S

|

2

 [dB]

−100

−95

−90

−85

−80

−75

−70

−65

−60

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=1 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=1 m

|Eρ

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.1 m, z=1 m

|Hφ

S

|

2

 [dB]

−100

−90

−80

−70

−60

−50

−40

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.1 m

|Ez

S

|

2

 [dB]

−60

−50

−40

−30

−20

−10

0

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.1 m

|Eρ

S

|

2

 [dB]

−80

−70

−60

−50

−40

−30

−20

2 4 6 8 10

2

4

6

8

ρ [m]

f [GHz]

HED(TM): h=0.01 m, z=0.1 m

|Hφ

S

|

2

 [dB]

−120

−110

−100

−90

−80

−70

−60

Figure A-11. Sommerfeld field strength of a HED above a concrete half space (hgz, 3=0()

versus frequency and horizontal separation.


0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|hφ(t,τ)|

2 [dB]

2.69 ns

2.69 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−80

−60

−40

−20

0

20

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−50

0

50

100

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−200

−180

−160

−140

−120

−100

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.19 ns

0.2 ns

Figure B-2. Near-field effects on impulse responses of a VED above a concrete half space

(fc=2.4 GHz, BW1 GHz, threshold=-30 dB). Legends display Sommerfeld and

GO approximations to delay spread.

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

Figure B-1. Near-surface effects on impulse responses of a VED above a concrete half space

(fc=2.4 GHz, BW1 GHz, threshold=-30 dB). Legends display Sommerfeld, GO

+ Norton term, and GO approximations to delay spread.

APPENDIX B: IMPULSE RESPONSES FOR SIGNALS CENTERED AT 2.4 GHz


0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

7.47 ns

7.47 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hz(t,τ)|

2 [dB]

3.37 ns

3.37 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−80

−60

−40

−20

0

20

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−50

0

50

100

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.19 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

Figure B-4. Near-field effects on impulse responses of an x-directed HED above a concrete

half space (fc=2.4 GHz, BW1 GHz, threshold=-30 dB, 3=90(). Legends display

Sommerfeld and GO approximations to delay spread.

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

4.47 ns

4.47 ns

4.47 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|h

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.62 ns

0.62 ns

0.62 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.6 ns

0.6 ns

0.6 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

1.76 ns

0.19 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

3.81 ns

0.19 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

2.46 ns

0.19 ns

0.2 ns

Figure B-3. Near-surface effects on impulse responses of an x-aligned HED above a concrete

half space (fc=2.4 GHz, BW1 GHz, threshold=-30 dB, 3=90(). Legends display

Sommerfeld, GO + Norton term, and GO approximations to delay spread.


0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.88 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

23.96 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−50

0

50

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−200

−180

−160

−140

−120

−100

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−50

0

50

100

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

Figure B-6. Near-field effects on impulse responses of an x-aligned HED above a concrete

half space (fc=2.4 GHz, BW1 GHz, threshold=-30 dB, 3=0(). Legends display

Sommerfeld and GO approximations to delay spread.

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

4.62 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.65 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

−60

−40

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.35 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

Figure B-5. Near-surface effects on impulse responses of an x-aligned HED above a concrete

half space (fc=2.4 GHz, BW1 GHz, threshold=-30 dB, 3=0(). Legends display

Sommerfeld, GO + Norton term, and GO approximations to delay spread.
56

This Page Intentionally Left Blank

This Page Intentionally Left Blank


0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=1 m, z=1 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|h

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

Figure C-1. Near-surface effects on impulse responses of a VED above a concrete half space

(fc=5.8 GHz, BW1 GHz, threshold=-30 dB). Legends display Sommerfeld, GO

+ Norton term, and GO approximations to delay spread.

0 20 40 60 80

−120

−100

−80

−60

−40

−20

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=10 m

τ [ns]

|hφ(t,τ)|

2 [dB]

2.69 ns

2.69 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

VED: h=10 m, z=10 m, ρ=1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−80

−60

−40

−20

0

20

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

VED: h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−50

0

50

100

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−200

−180

−160

−140

−120

−100

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

VED: h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

Figure C-2. Near-field effects on impulse responses of a VED above a concrete half space

(fc=5.8 GHz, BW1 GHz, threshold=-30 dB). Legends display Sommerfeld and

GO approximations to delay spread.

APPENDIX C: IMPULSE RESPONSES FOR SIGNALS CENTERED AT 5.8 GHz
	

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

7.47 ns

7.47 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hz(t,τ)|

2 [dB]

3.37 ns

3.37 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−80

−60

−40

−20

0

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−80

−60

−40

−20

0

20

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−50

0

50

100

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

Figure C-4. Near-field effects on impulse responses of an x-directed HED above a concrete

half space (fc=5.8 GHz, BW1 GHz, threshold=-30 dB, 3=90(). Legends display

Sommerfeld and GO approximations to delay spread.

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

4.47 ns

4.47 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|h

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=10 m, z=10 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.62 ns

0.62 ns

0.62 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.6 ns

0.6 ns

0.6 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

φ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hz(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

Figure C-3. Near-surface effects on impulse responses of an x-aligned HED above a concrete

half space (fc=5.8 GHz, BW1 GHz, threshold=-30 dB, 3=90(). Legends display

Sommerfeld, GO + Norton term, and GO approximations to delay spread.



0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=10 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−160

−140

−120

−100

−80

−60

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

31.44 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−50

0

50

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.1 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−200

−180

−160

−140

−120

−100

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−100

−50

0

50

100

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=0.01 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

Figure C-6. Near-field effects on impulse responses of an x-aligned HED above a concrete

half space (fc=5.8 GHz, BW1 GHz, threshold=-30 dB, 3=0(). Legends display

Sommerfeld and GO approximations to delay spread.

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−120

−100

−80

−60

−40

−20

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

1.54 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=10 m, z=10 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.57 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=1 m, z=1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−140

−120

−100

−80

−60

−40

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

z

(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

−60

−40

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|e

ρ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

0 20 40 60 80

−180

−160

−140

−120

−100

−80

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

τ [ns]

|hφ(t,τ)|

2 [dB]

0.2 ns

0.2 ns

0.2 ns

Figure C-5. Near-surface effects on impulse responses of an x-aligned HED above a concrete

half space (fc=5.8 GHz, BW1 GHz, threshold=-30 dB, 3=0(). Legends display

Sommerfeld, GO + Norton term, and GO approximations to delay spread.
60

This Page Intentionally Left Blank

This Page Intentionally Left Blank


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-2. Near-field effects on delay spread versus bandwidth for a VED above a concrete

half space at fc=900 MHz (Hamming window, threshold=-30 dB).

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-1. Near-surface effects on delay spread versus bandwidth for a VED above a concrete

half space at fc=900 MHz (Hamming window, threshold=-30 dB).

APPENDIX D: DELAY SPREAD VERSUS BANDWIDTH


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

7

8

9

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

φ [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

ρ [ns]

Figure D-4. Near-field effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=900 MHz (Hamming window, threshold=-30 dB, 3=90().

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

16

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

Figure D-3. Near-surface effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=900 MHz (Hamming window, threshold=-30 dB, 3=90().


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30

35

40

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-6. Near-field effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=900 MHz (Hamming window, threshold=-30 dB, 3=0().

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-5. Near-surface effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=900 MHz (Hamming window, threshold=-30 dB, 3=0().


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-8. Near-field effects on delay spread versus bandwidth for a VED above a concrete

half space at fc=2.4 GHz (Hamming window, threshold=-30 dB).

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-7. Near-surface effects on delay spread versus bandwidth for a VED above a concrete

half space at fc=2.4 GHz (Hamming window, threshold=-30 dB).


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

Figure D-9. Near-surface effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=2.4 GHz (Hamming window, threshold=-30 dB, 3=90().

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

7

8

9

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

φ [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

ρ [ns]

Figure D-10. Near-field effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=2.4 GHz (Hamming window, threshold=-30 dB, 3=90().


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30

35

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-12. Near-field effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=2.4 GHz (Hamming window, threshold=-30 dB, 3=0().

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

7

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-11. Near-surface effects on delay spread versus bandwidth for a HED above a

concrete half space at fc=2.4 GHz (Hamming window, threshold=-30 dB, 3=0().


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

VED: h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-14. Near-field effects on delay spread versus bandwidth for a VED above a concrete

half space at fc=5.8 GHz (Hamming window, threshold=-30 dB).

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

VED: h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-13. Near-surface effects on delay spread versus bandwidth for a VED above a

concrete half space at fc=5.8 GHz (Hamming window, threshold=-30 dB).
	

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

7

8

9

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

φ [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

ρ [ns]

Figure D-16. Near-field effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=5.8 GHz (Hamming window, threshold=-30 dB, 3=90().

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τhz

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TE): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

ρ [ns]

Figure D-15. Near-surface effects on delay spread versus bandwidth for a HED above a

concrete half space at fc=5.8 GHz (Hamming window, threshold=-30dB, 3=90().



0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30

35

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=10 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.1 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=0.01 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-18. Near-field effects on delay spread versus bandwidth for a HED above a concrete

half space at fc=5.8 GHz (Hamming window, threshold=-30 dB, 3=0().

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

Sommerfeld

geometric optics

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τez

spr

 [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

espr

ρ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=10 m, z=10 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=1 m, z=1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.1 m, z=0.1 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

HED(TM): h=0.01 m, z=0.01 m, ρ=5 m

BW [GHz]

τ

hspr

φ [ns]

Figure D-17. Near-surface effects on delay spread versus bandwidth for a HED above a

concrete half space at fc=5.8 GHz (Hamming window, threshold=-30 dB, 3=0().