NTIA Report 96-326 

A Study of the Electromagnetic 

Properties of Concrete Block Walls 

for Short Path Propagation Modeling 

Christopher L. Holloway 

Patrick L. Perini 

Ronald R. Delyser 

Kenneth C. Allen 

report series 

u.s. DEPARTMENT OF COMMERCE. National Telecommunications and Information Administration 
NTIA Report 96-326 

A Study of the Electromagnetic 

Properties of Concrete Block Walls 

for Short Path Propagation Modeling 

Christopher L. Holloway 

Patrick L. Perini 

Ronald R. Delyser 

Kenneth C. Allen 

u.s. DEPARTMENT OF COMMERCE 

Ronald H. Brown, Secretary 

Larry Irving, Assistant Secretary 

for Communications and Information 

October 1995 

CONTENTS 

Page 

FIGURES.......................................................................... v 

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

2. EFFECTIVE MATERIAL PROPERTIES OBTAINED FROM HOMOGENIZATION 5 

3. OBLIQUELY INCIDENT PLANE WAVE ......................................... 11 

4. REFLECTION FROM A CINDER BLOCK WALL. . . .. . . . . . . . . .. . . . . . . . . . .. . . . .. 13 

5. REFLECTION FROM A TWO-DIMENSIONAL BLOCK WALL .................. 27 

6. VALIDITY OF THE EFFECTIVE MEDIUM MODEL ............................ 32 

7. DISCUSSION AND CONCLUSION ............................................... 36 

8. REFERENCES ................................................................... 38 

III 

FIGURES 

Page 

Figure 1. Illustration of a concrete block wall. ..................................... 2 

Figure 2. Illustration of a two-dimensional periodic block wall. 3 

Figure 3. Illustration of the rays in the four-ray model. ............................ 4 

Figure 4. Equivalent layered media of a concrete block wall. ....................... 7 

Figure 5. 

Figure 6. 

Figure 7. 

Equivalent layered media of a two-dimensional block wall. 

One-dimensional periodic structure. 

Two-dimensional periodic structure. 

7 

10 

10 

Figure 8. Equivalent anisotropic layer. ............................................ 13 

Figure 9. Reflectivity versus angle of incidence for a perpendicular polarized wave. 15 

Figure 10. Reflectivity versus angle of incidence for a parallel polarized wave. 16 

Figure 11. 

Figure 12. 

Figure 13. 

Figure 14. 

Reflected power off the wall versus antenna separation. 

Reflected power off the wall versus antenna separation. 

Received power versus antenna separation for the four-ray model. 

Received power versus antenna separation for the four-ray model. 

18 

19 

21 

22 

Figure 15. Reflectivity versus angle of incidence for a perpendicular polarized wave. 24 

Figure 16. Reflectivity versus angle of incidence for a parallel polarized wave. 25 

Figure 17. Received power versus antenna separation for the four-ray model. ....... 26 

v 
FIGURES (continued) 

Page 

Figure 18. Reflectivity versus angle of incidence for a perpendicular polarized wave. 28 

Figure 19. Reflectivity versus angle of incidence for a perpendicular polarized wave. 29 

Figure 20. Received power versus antenna separation for the four-ray model. 30 

Figure 21. Received power versus antenna separation for the four-ray model. ....... 31 

Figure 22. Reflectivity versus angle of incidence for a two-dimensional concrete block 

wall. .................................................................. 33 

Figure 23. Received power for the four-ray model versus antenna separation for a twodimensional block wall. ................................................ 34 

VI 
A Study of the Electromagnetic Properties of Concrete Block Walls for Short 

Path Propagation Modeling 

Christopher 1. Holloway! 

Patrick 1. Perini2 

Ronald R. DeLyser3 

Kenneth C. Allen! 

For short propagation paths, correctly representing reflections of electromagnetic energy from surfaces is critical for accurate signal level predictions. In this paper, the 

method of homogenization is used to determine the effective material properties of 

composite material commonly used in construction. The reflection and transmission 

coefficients for block walls and other types of materials calculated with these homogenized effective material properties are presented. The importance of accurately representing the reflections for signal level prediction models is also investigated. It is shown 

that a 5- to lO-dB error in received signal strength can occur if the composite walls are 

not handled appropriately. Such accurate predictions of signal propagation over short 

distance is applicable to micro cellular personal communications services deployments 

in urban canyons as well as indoor wireless private branch exchanges and local area 

networks. 

Key words: composite walls; concrete walls; propagation modeling; reflection coefficient; 

homogenization; effective material properties 

1. INTRODUCTION 

Much work on long path propagation through urban settings has been done in the past 

[1]-[10]. In most of this work, little attention is given to accurate representation of the 

reflection coefficient (r) of a wave striking building surfaces. In this paper the problem of 

electromagnetic wave interaction with composite walls is addressed. Some examples used 

here are concrete block walls and other composite structures depicted in Figures 1 and 2. 

IThe author is with Institute for Telecommunications Sciences, National Telecommunications and Information Administration, U.S. Department of Commerce, Boulder, 80303 

2The author is with US West Advanced Technologies, Boulder, 80303 

3The author is with Department of Electrical Engineering, University of Denver, Denver, 80208 
y 

x lJi 

t ~----++--+--+--\, 

D~DD 

12 

a) 

lJi 

t ~-H----+-t---\.' 

D~DD 

12 

b) 

Figure 1. a) Illustration of a concrete block wall with its slabs oriented along the y-axis, 

b) Illustration of a concrete wall with it slabs oriented along the x - axis. 

In the majority of the published work the reflection coefficients of the buildings are obtained 

by assuming that either the building ·materials are perfect conductors or that the building 

walls are single solid slabs of material with some assumed properties. For the most part this 

may very well be justified for long path propagation in urban canyons. 

In long path propagation, the transmitting and receiving antennas are set a relatively large 

distance apart. The dominant contribution to the total signal for an urban canyon setting 

is waves that make one to two bounces off the building, take a direct path, and make one 

bounce off the ground (see Figure 3). In this case the waves that bounce off of buildings 

are incident at an angle close to grazing, or 900 • Even though the angular dependence of 

2 
y 

Figure 2. Illustration of a two-dimensional periodic block wall. 

the reflection coefficient ofa composite material can behave much differently than that of a 

perfect conductor or a single solid slab, this is not an important issue for long propagation 

paths. Therefore, regardless of the building's material, the actual reflection coefficient for 

large incident angles will approach that of a perfect conductor. 

There is a growing need to predict signal levels for short propagation paths, in the range 

of 2-100 meters. Business campuses utilizing wireless private branch exchanges (PBXs) and 

wireless local area networks (LAN s) to provide mobile voice and data communications, vehicular communications through urban canyons to nearby relays, and micro cellular personal 

communications services (PCS) deployment in malls and airport are just a few examples. 

For short propagation paths like these, the accurate behavior of waves reflecting off walls 

can be very important. 

Calculating the fields interaction (i.e. the reflection and transmission coefficient) of the composite structures similar to those shown in Figures 1 and 2 is a classic problem (see [11]-[27]). 

These techniques range from analytical techniques like Floquet analysis and mode matching to full numerical approaches like method of moments (MOM), finite element, and finite 

3 
Figure 3. Illustration of the rays in the four-ray model. 

difference methods. These techniques are capable of high accuracy, but are computationally intensive and hence do not lend themselves to ready use in signal prediction models. 

For short path signal prediction models (like ray tracing and other geometric optic models), efficient closed form expressions for calculating both the the reflection and transmission 

coefficients for composite structures are desired. 

In this paper, we introduce expressions for the effective material properties for some commonly used composite building materials. These expressions for the effective material properties can be used to efficiently obtain the reflection and transmission coefficients for walls 

composed of composite materials. With these reflection and transmission properties we investigate the importance of accurately representing the interaction of the composite walls for 

the prediction of received signal strength (RSS) over short distances. The paper is organized 

as follows: following the introduction, Section 2 discusses the technique of homogenization, 

showing how the equivalent material properties for composite walls can be obtained. Ex4 
presslOns are then given for the effective material properties for both singly and doubly 

periodic structures. In Section 3 the equations needed for calculating the reflection and 

transmission coefficient for an obliquely incident plane wave (either perpendicular or parallel 

polarizations) are introduced. In Section 4 results from a singly periodic structure for different orientations and polarizations are presented. In Section 5 results for doubly periodic 

structures are presented. Also in Section 4 and 5, the importance of accurately representing 

the reflection properties of walls for signal level prediction models is investigated. Finally, 

in Section 6 a discussion on the validity of the expressions presented here is given. 

2. EFFECTIVE MATERIAL PROPERTIES OBTAINED FROM 

HOMOGENIZATION 

The problem at hand is to determine the reflection and/or transmission coefficient for a 

field incident onto the composite periodic structures illustrated in Figures 1 and 2. The 

concrete block wall in Figure 1 is equivalent to a five layer medium depicted in Figure 4. 

Layers 1 and 5 are free space, layers 2 and 4 have the material properties of the concrete, 

and layer 3 represents a one-dimensional periodic structure. The two-dimensional composite 

wall in Figure 2 is equivalent to a four-layer medium depicted in Figure 5. Layers 1 and 4 

are free space, layer 3 has the material properties of the concrete, and layer 2 represents a 

two-dimensional periodic structure. In order to calculate the reflection and/or transmission 

coefficient for these composite structures the field's interaction with the periodic sections 

labeled as layer 3 (for the one-dimensional structure .of Figure 4) and layer 2 (for the twodimensional structure of Figure 5) must be determined. 

Recently, a method for analyzing periodic structures known as homogenization has been 

used to solve problems of this type when the period of the structure is small compared to 

5 
the wavelength. Only a few of these published results are applicable to electromagnetic 

problems: [28J and [29J for a corrugated impedance surface, [30J and [31 J for a wire grid and 

conducting strips, [32J for a rough perfectly and non-perfectly conducting rough surfaces, 

and [33J-[35J for analyzing pyramidal electromagnetic absorbers. 

Even though the homogenization technique is based on the period of the structure being small 

compared to a wavelength, results given in [34J and [36J-[38J indicate that the homogenization 

models are accurate for periods at least as large as 1/2-1 free space wavelength and possibly 

even higher for lossy periodic structures. This is discussed in more detail in Section 6. 

Homogenization is a technique utilized in the early 1970's, primarily by a group of French 

mathematicians (see [28J and [39J-[45]). This asymptotic technique is based on the method 

of multiple-scales associated with the microscopic and macroscopic field variations due to 

the periodic structure. In most situations, only the averaged (slowly varying) fields are of 

interest, and not the microstructure of the fields. Homogenization allows the separation of 

the average field from the microstructure. It is then possible to show that the averaged fields 

satisfy Maxwell's equations for some homogeneous media. The equivalent material properties 

of these homogeneous media are related to the properties of the composite structure. 

With homogenization, the periodic layers of the composite structures (layer 3 in Figure 4 

and layer 2 in Figure 5) can be replaced with a medium with an equivalent material property. 

Once the equivalent material property of the medium is determined, then the reflection and 

transmission coefficients of the composite structures can be efficiently obtained with either 

classical layered media approaches or by classical transmission line methods. 

Homogenization uses asymptotic expansions and the concept of multiple-scales to expand 

the E and· H fields in an asymptotic power series with both slow and fast variations. These 

slow and fast variations are associated with the microscopic and macroscopic field variations. 

With asymptotic power series of both the E and H fields, Maxwell's equations can be grouped 

6 
Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 

Figure 4. Equivalent layered media of a concrete block wall. 

Layer 1 Layer 2 Layer 3 Layer 4 

Figure 5. Equivalent layered media of a two-dimensional block wall. 

7 
in terms of different powers of the period (p) of the structure. The details of this procedure 

are found in [33]. 

Following this type of analysis, the zeroth order averaged fields [(EO)avg and (HO)avg] are 

related by the following: 

\7 x (EO)avg 

\7 x (HO)avg 

-jw [flh] . (HO)avg 

-jw [ch ] . (EO)avg . 

(1) 

This equation states that the average fields satisfy Maxwell's equations in an anisotropic 

homogeneous medium characterized by the tensors [ch] and [flh]. These effective material 

properties are referred to as the homogenized permittivity [ch] and permeability [flh], and 

are defined by the following: 

(cEO) 

(flHo) 

avg 

avg 

[ c

h

] . (Eo tVg 

[flh] . (Ho tVg (2) 

The average zero order fields [(EO)avg and (HO)avg] see the properties of the medium in terms 

of a single tensor quantity. The values of these tensors can be obtained from the solutions 

of the two-dimensional static source-free field problems that govern EO and HO (see [33]): 

\7e x EO 0 

\7e x HO 0 (3) 

and 

\7c (cEO) 0 

\7c (flHO) 0 (4) 

where e is the so-called fast variable and is defined as the following: 

(5) 

where p is the period of the structure. 

Now that it has been shown that the averaged field sees the periodic medium as an effective 

8 
anisotropic homogeneous region with tensor permittivity [Eh] and [/1 h]: 

(6) 

we now need to determine the effective material properties of this region. There has been a 

great deal of attention in the past towards determining the effective properties of composite 

regions. For a survey of this work see [46]. 

For the one-dimensional periodic structure, the effective properties of layer 3 (see Figure 4) 

is needed. If the period (p) of the slab structure shown in Figure 6 is small compared to a 

wavelength in either medium, and also small compared to the skin depth, then the effective 

properties are given by [33], [17] and [47]-[50] as: 

-1 

Ex 

-1 

/1x 

lOy = Ez 

/1y = /1z 

(1 - 9 )E-;l + g(;l 

(1 - g)/1-;l + gl1";/ 

(1 - g) Eo + 9 lOa 

(1 - 9)/10 + g/1a 

(7) 

where 9 = a/p (p and a are defined in Figure 1) is the relative volume of space occupied by 

the material, lOa and /1a are the complex parameters of the bulk material, and Eo and /10 are 

the free space values. 

For the two-dimensional block wall shown Figure 7 (where E2 = Eo and E1 = lOa), the longitudinal permittivity and permeability are known exactly ([46], [28] and [39]) as: 

/1z 

(1 - g)Eo + gEa 

(1- g)/1o + g/1a (8) 

where 9 = a2/p2 (a and p are defined in Figure 2) is the volume fraction of space occupied 

by the material, and lOa and /1a are the complex parameters of the bulk material. 

For this type of symmetric two-dimensional periodic structure, lOt = Ex = lOy, and /1t = 

/1x = /1y. Reference [33] indicates that the transverse permittivity (lOt) and permeability 

9 
z 

H 

a 

Figure 6. One-dimensional periodic structure. 

Figure 7. Two-dimensional periodic structure. 

10 
(f.lt) are not simple spatial averages as had often been assumed. There are no exact closed 

form expressions for the transverse material properties; however, upper and lower bounds 

for these properties can be found in [46]. 

Nakamura and Hirasawa [51] have done a numerical study of a similar periodic structure 

and they showed that the Hashin-Shtrikman upper bound given in [46] and [52] correlates 

very well to the effective material properties of this type of periodic structure. Thus, the 

transverse material properties can be approximated by the following: 

l-g (9) 

l-g 

fa + 1 9 -+- Ea -Eo 2ta 

where once again 9 = a2/p2 is the volume fraction of space occupied by the material. 

In [33], a structure converse to the one shown in Figure 7, i.e., a dielectric surrounded by 

air (where f2 > fd, was analyzed. If the roles of the material properties in equation (9) are 

interchanged with Keller's scaling theorem [53] (such that f1 < (2), then the results given in 

equation (25) of [33] are obtained. 

In general, the permittivity given in equations (7)-(9) is complex, resulting from the possibility that the material has a conductivity (Y. For this scenario, the complex permittivity in 

these equations is expressed as: 

(10) 

3. OBLIQUELY INCIDENT PLANE WAVE 

We want to investigate the problem of a plane wave incident onto the medium that behaves 

like a uniaxially anisotropic, but homogeneous, material. The periodic structures shown in 

11 
Figures 6 and 7 can be replaced with an equivalent layer (Figure 8). The material properties 

of this effective layer can be given by either equation (7) (for the one-dimensional periodic 

structure) or by equations (8) and (9) (for the two-dimensional periodic structure). If the 

plane of incidence is the xz-plane (Figure 8), then we can assume 0/ oy == O. Maxwell's 

equations can now be decoupled into two independent sets of equations; one set for the 

perpendicular polarization (referred to as E-polarization): 

oHx oHz 8z"" jWEyEy - axoEy 

ox -jw/izHz 

oEy -jwfJxHx oz 

and one set for the parallel polarization (referred to as H-polarization): 

oEx oEz EiZ"" - ox 

with an assumed time factor of ejwt . 

oHy 

ox 

oHy . E 

oz -JWEx x 

(11) 

(12) 

The x-dependent factor of the fields is given by: e-jkoxsin() , which means that the derivatives 

with respect to x can be replaced by: 

-o ----+ -J 'k' sm e Ox 0 

(13) 

With this, the following general set of equations for the electromagnetic fields is obtained in 

which the z component is eliminated: 

dE(z) 

dz 

dH(z) 

dz 

where for the perpendicular polarization: 

E(x) = Ey(z) 

Eeff 

fJeff 

-jWfJeJJH(z) 

-jwEeJJE(z) 

. 2() E _ /loEoSln 

Y /lz 

fJx 

12 

(14) 

(15) 

( 16) 
L 

z 

Figure 8. Equivalent anisotropic layer. 

and for the parallel polarization: 

E(x) = Ex(z) H(z) = Hy(z) 

Eeff Ex 

f.Lef f 

. 28 J.1 - J.LoEosln 

y Ez 

(17) 

(18) 

With these expressions for the angular dependence on the effective material properties, we 

can now calculate reflection and transmission coefficients for composite structures. 

4. REFLECTION FROM A CINDER BLOCK WALL 

In this section, results for a one-dimensional periodic structure resembling a cinder block 

wall will be given (see Figures 1 and 4). Four block walls are analyzed: two different 14.5-cm 

(5.71-in) walls, one 7.2-cm (2.83-in) wall, and one 19.6-cm (7.72-in) wall. The dimensions of 

these different walls are shown in Table 1. 

The 14.5-cm (5.71-in) block wall labelled block # 1 in Table 1 is represented as the layered 

structure shown in Figure 4, and has the following geometry: layers 1 and 5 are free space; 

13 
Table 1. Geometries of the concrete blocks 

Block # 1 Block # 2 Block # 3 Block # 4 

[14.5-cm wall] [14.5-cm wall] [7.2-cm wall] [19.6-cm wall] 

12 2.25 cm 2.25 cm 1.7cm 3.4 cm 

13 10.0 cm 10.0 cm 3.8 cm 12.8 cm 

a 2.6 cm 2.6 cm 2.8 cm 2.7 cm 

d 14.3 cm 14.3 cm 9.5 cm 15.3 cm 

Cr 3.0 6.0 3.0 3.0 

a 1.95 . 10-3 1.95.10-3 1.95. 10-3 1.95 . 10-3 

layers 2 and 4 are a solid medium with 12 = 2.25 em, /-l = /-lo, Cr = 3 and a = 1.95 . 10-3 ; 

and layer 3 is a medium with effective material properties given by equations (8) and (9) 

where a = 2.6 em, d = 14.3 em, 13 = 10.0 em, Ca = 3 and a = 1.95.10-3 . Once the material 

properties of these layers are determined, the reflection coefficient can be obtained by using 

either classical layered media methods or classical transmission line methods. Figure 9 shows 

results for the reflectivity (defined as the magnitude squared of the reflection coefficient) of 

a block wall oriented both along the y-axis and x-axis (see Figure 1) for a perpendicularly 

polarized E field with a frequency of 900 MHz. Figure 10 shows results for a parallel 

polarization of the E field. These results were obtained using equations (16) and (18). Also 

shown in these figures are results for a solid layer of concrete (cr = 3 and a = 1.95 . 10-3

) 

14.5-cm thick. 

From these figures it is apparent that the resonant behavior of the reflectivity cannot be 

achieved if a composite wall (block wall) is approximated by a solid layer. Correctly representing this resonant behavior is important for short path propagation. For large path 

propagation the reflection coefficient needed will correspond to angles approaching grazing 

(90°), and from Figures 9 and 10 show that the reflection coefficient for the composite wall 

and the solid wall approach one another. This point is further illustrated in the following 

example. 

14 
>. 

..., 

1.00 

0.80 

oJ 0. 60 

> 

..., 

o 

CD 

If-

&0.40 

0.20 

-- y-ox ts or len to t ton 

---- x-oxts orlentotlon 

- - So ltd W 0 I I 

/ 

,/ 

,/ 

--------------..-<:-

- - -

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

0.00~~~-r-r~~-'~--~r-~~.-~'-~~~ 

0.00 15.00 30.00 45.00 60.00 75.00 90.00 

(1 (Degrees) 

Figure 9. Reflectivity versus angle of incidence for a perpendicular polarized wave. These 

results are for block # 1 (see Table 1) with slabs oriented along both the y-axis 

and x - axis and with f = 900 MHz. The large dashed curve represents the 

results for a single layered slab of thickness equal to 212 + l3, the -solid curve 

represents the actual concrete block wall with the slabs oriented along the y -

axis, and the small dashed curve represents the results for the actual concrete 

block wall with the slabs oriented along the x - axis. 

15 
>. 

.,J 

1 .00 

0.80 

-' 0 60 > . 

.,J 

o 

CD 

If-

&0.40 

0.20 

-- y-ox ls or len to t lon 

- - - - x-ox ls or ten to t lon 

- - So I l d W 0 I I 

..... .... ..... .... 

- ..... ..... -' "- , , 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

0.00~~~~~~~~~~~~~~~~~ 

0.00 15.00 30.00 45.00 60.00 75.00 90.00 

6 ( Degrees) 

Figure 10. Reflectivity versus angle of incidence for a parallel polarized wave. These results 

are for block # 1 (see Table 1) with slabs oriented along both the y-axis and 

x - axis and with f = 900 MHz. The large dashed curve represents the results 

for a single layer slab of thickness equal to 212 + 13 , the solid curve represents the 

actual concrete block wall with the slabs oriented along the y-axis, and the 

small dashed curve represents the results for the actual concrete block wall with 

the slabs oriented along the x - axis. 

16 
Assume that a transmitting and receiving antenna are located 1 m from a wall. The power 

received (Pw ) due to the wall reflection only, is given by: 

(19) 

where Po (in dB) is the transmitted power, k is the wavenumber; rand 4;[' are the magnitude 

and phase of the reflection coefficient from the wall, respectively; and x is the total distance 

that the wave travels and is given by: 

(20) 

where w is the distance between the wall and the transmitting and receiving antenna and d 

is the distance between the two antennas. 

Figure 11 shows results for the received power as a function of antenna separation for a 

frequency of 900 MHz and for both antennas placed 1 m from the wall. The four plots in 

this figure correspond to a solid wall, a block wall oriented along the y-axis, a block wall 

oriented along the x-axis, and a perfect conductor. All these results were calculated assuming 

a perpendicularly polarized wave. These results indicate that for a long path length (d > 500 

m) there is little difference between a solid wall, a composite wall, or a perfectly conducting 

wall. However, for short path lengths the power received by the solid wall or a perfectly 

conducting wall cannot reproduce the resonance behavior that is present in the composite 

walls. This figure illustrates that for short propagation paths, 10- to 20-dB inaccuracies can 

occur if one assumes the composite wall is treated as a solid wall. 

As one might expect, the further the distance between the wall and the two antennas, the 

larger the separation distance between the antennas must be before the three curves coincide. 

Figure 12 illustrates results of the received power versus distance for a frequency of 900 MHz 

and for both antennas placed 4 m from the wall. For this example, the four curves correlate 

for d> 1 km. 

17 
0.00 

-- y-ox ts or len to t lon 

---- x-ox ts or len tot lon 

- - So ltd J..l 0 I I 

-10.00 lrlrlrlrl:.Perfect conductlng J..loll 

-20.00 

\ 

\ r->. \ rn -30.00 \ '""0 \ 

\ 

" \ 

CL \ 

\ 

-40.00 \ 

\ 

-50.00 

-60.00~--~~~~~~--~~~~~~--~-.-.~ 

1 10 100 

d (m) 

Figure 11. Reflected power off the wall versus antenna separation. These results are for 

block # 1 (see Table 1) with slabs oriented along the y-axis and f = 900 MHz. 

The antennas are 1 m away from the wall. 

18 
-1121.121121 

-- y-ax ls or len ta t lon 

----x-axls orlentatlon 

- - So I ld 101 a I I 

-2121.121121 ~Perfect conductlng wall 

-3121.121121 

'I 

" \ 

\ ,..... , 

m -4121.121121 \ 

""C \ 

\ 

\ :r \ 0... \ 

-5121.121121 

\ 

\ 

\ 

\ I 

\ I J 

-6121.121121 

-7121.121121;---.-~~~~--~~-.~~.---.-~~~~ 

1 1121 1121121 1121121121 

Figure 12. Reflected power off the wall versus antenna separation. These results are for 

block # 1 (see Table 1) with slabs oriented along the y-axis and f = 900 MHz. 

The antennas are 4 m away from the wall. 

19 
In Figure 13 we show results for the total received power as a function of antenna spacing for 

an antenna placed above a perfectly conducting ground and between two walls. The results 

in Figure 13 are for a transmitter and receiver antenna spaced 1 m off of the ground and 

1 m out from each wall. These two walls are assumed to be either a perfectly conducting 

wall, a single layer slab wall, or a concrete block wall. The received power is calculated by 

assuming that the total power is comprised offour different rays (Figure 3): the direct path, 

the ground reflection, and one reflection off each of the two walls. The predicted signal level 

is given by the following: 

[ 

ejkd ejks ej(kr-¢w) 1 

PT = 20 loglO d + rG - s

- + 2 rw r + Po [dB] (21) 

where r G is the reflection coefficient of the ground and for horizontal polarization r G = -1 

and for vertical polarization r G = 1, r wand cPw are the magnitude and phase of the reflection 

coefficient of the walls; s (the reflection path off the ground) and r (the reflection path off 

the walls) are given by: 

(22) 

where h is the distance the antennas are above the ground, w is the distance the antennas 

are from the walls, and d is the separation of the two antennas. 

The results are consistent with those shown in Figure 11. For small distance (d < 1 km) 

the results for the composite wall show a difference in received signal of 8 to 10 dB from 

the results for the other two types of walls. Figure 14 shows results for a transmitter and 

receiver antenna spaced 4 m away from each wall. This figure shows that between 40 and 

500 m, the results for the block wall indicate an average of about 8- to 10-dB difference in 

received signal than that obtained from the solid or perfectly conducting wall. The nulls for 

the block wall are not as deep as those for the other two walls. 

Depending upon frequency, the bulk material properties of the blocks, or the dimensions of 

the blocks, the composite wall mayor may not behave like either a single slab wall or a 

20 
2121.121121 

-- y-ox la or len tot lon 

- - x-ox la or len to t lon 

----Solld .... 011 

121.121121 ~Perfect conductlng .... 011 

-2121.121121 

m -4121.121121 

"U 

.... (L 

-6121.121121 

-8121.121121 

-1121121.1211214---~~~~~--~~~~~--~~~~~ 

1 1121 1121121 1121121121 

Figure 13. Received power versus antenna separation for the four-ray model. These results 

are for block # 1 (see Table 1) with slabs oriented along the y-axis and f = 900 

MHz. The antennas are 1 m off the ground and are spaced 1 m from each of the 

two walls. 

21 
20.00 

-- y-ox ls or len to t lon 

----x-oxls orlentotlon 

- - So I ld 1-10 I I 

Irlrlrlrl:. Per fee t eon due t In 9 1-10 I I 

0.00 

-20.00 

n:' - 40 . 00 

-60.00 

-80.004---~~~~n---~~~nln---'-II-rnon 

1 10 100 1000 

Figure 14. Received power versus antenna separation for the four-ray model. These results 

are for block # 1 (see Table 1) with slabs oriented along the y-axis and f = 900 

MHz. The antennas are 1 m off the ground and are spaced 4 m from each of the 

two walls. 

22 
perfectly conducting wall. Figures 15 and 16 show the results of the reflectivity for a wall 

composed of block # 2 (see Table 1). This block is identical to block # 1 with the exception 

that the dielectric constant (Er) is different. Figure 15 shows that for the perpendicular 

polarization (E-polarization), the block wall has very large values for the reflectivity for small 

incident angles, whereas the results for the solid slab have small values for the reflectivity 

for these small angles. For the perpendicular polarization, we should expect that if the total 

power received from the four-ray model (equation (21)) was calculated, then the results for 

the block wall would correlate fairly well with the perfectly conducting walls. 

Figure 16 shows that for the parallel polarization (H-polarization) the reflectivity for block 

# 2 exhibits deep nulls, and further more, the results for the y-axis orientation composite wall 

corresponds very closely for large incident angles to the results for the solid wall. Therefore, 

the total predicted power from the composite wall is expected to correlate more closely to 

the solid wall than to the perfectly conducting wall. This is the case, and the result? for 

the total received power for the four-ray model are shown in Figure 17. For an antenna 

separation between 5 and 60 meters, the signal predicted for a block wall is about 5 dB to 

10 dB less than for a perfectly conducting wall. 

Figures 18 and 19 show the reflectivity for a 7.2-cm (4-in) and 19.6-cm (8-in) block wall, 

respectively. The 7.2-cm (4-in) block wall corresponds to block # 3 in Table 1, and the 

19.6-cm (8-in) block wall corresponds to block # 4 in Table 1. The results shown in these 

figures are for a perpendicular polarized wave. Results for the predicted signal levels of the 

four-ray model for these two block walls are shown in Figure 20 and 21. 

The results in Figure 18 indicate that the reflectively of the composite wall is very similar 

to the results for the solid wall. Figure 20 shows that for the 7.2-cm (4-in) block wall, 

the predicted signal level correlates more closely with the solid wall than to the perfectly 

conducting wall. However, from Figure 20 it is shown that the total received power for the 

23 
~ 

o 

CD 

q.... 

1.00 

0.80 

&0.40 

0.20 

-- ------ -------

-- y-ox ls or len to t lon 

- - - - x-ox ls or len to t lon 

- - Sol ld wo I I 

- - / 

/ 

/ 

/ 

/ 

/ 

I 

I 

I 

I 

I 

I 

I 

I 

15.00 30.00 45.00 60.00 75.00 90.00 

fJ (Degrees) 

Figure 15. Reflectivity versus angle of incidence for a perpendicular polarized wave. These 

results are for block # 2 (see Table 1) with slabs oriented along both the y-axis 

and x-axis and with f = 900 MHz. The large dashed curve represent the results 

for a single layered slab of thickness equal to 212 + l3, the solid curve represents 

the actual concrete block wall with the slabs oriented along the y-axis, and 

the small dashed curve represents the results for the actual concrete block wall 

with the slabs oriented along the x - axis. 

24 
...> 

o 

CD 

Ii--

1.00 

0.80 -------

---- y-axts ortentatton 

---- x-axts orlentatlon 

- So I td wa I I 

-.. .... .... .... .... .... .... , 

, 

, 

, 

, 

, 

\ 

\ 

\ 

\ 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

&0.40 I 

I 

\ 

\ 

\ 

\ 

\ 

\ 

0.20 

----

\ 

, 

.... .... 

I 

/ 

I 

I 

I 

I 

0.00~~~~~~~~~~r-~~~~~~~~~ 

0.00 15.00 30.00 45.00 60.00 75.00 90.00 

(I (Degrees) 

Figure 16. Reflectivity versus angle of incidence for a parallel polarized wave. These results 

are for block # 2 (see Table 1) with slabs oriented along both the y-axis and 

x - axis and with f = 900 MHz. The large dashed curve represent the results 

for a single layered slab of thickness equal to 212 + 13 , the solid curve represents 

the actual concrete block wall with the slabs oriented along the y-axis, and 

the small dashed curve represents the results for the actual concrete block wall 

with the slabs oriented along the x - axis. 

25 
20.00 

---- y-oxts ortsntotlon 

----x-oxts ortsntotton 

- - So I td wo I I 

lrlrlrlr6 Psrfsc t con duc t tn 9 wo I I 

0.00 

-20.00 

,..... 

CD \ 

"0 \ 

\ 

\ .. -40.00 0... 

-60.00 

-80.004-----~~~~~~~----~--~~~~~ 

1 10 100 

d (m) 

Figure 17. Received power versus antenna separation for the four-ray model. These results 

are for the parallel polarization for block # 2 (see Table 1) with slabs oriented 

along the y-axis and f = 900 MHz. The antennas are 1 m off the ground and 

are spaced 1 m from each of the two walls. 

26 
solid and composite wall are not as close as might be expected upon examining the results in 

Figure 18, especially between 2 and 20 m. The total received power is a function of both the 

magnitude and phase of the reflection coefficient (f). The results in Figure 18 only depict the 

magnitude of f, the phase of f for the composite and solid walls can behave quit differently 

from each other depending upon the geometry and material properties of the walls, and this 

characteristic is equally important in determining the total received power. 

These examples illustrate how the predicted signal level can vary for block walls with different 

geometries and material properties. Depending on the block wall parameters, the predicted 

signal level for short path propagation can correlate to either a solid slab wall, or to a wall 

composed of a perfect conductor. It can also behave differently from either of these two 

types of walls. 

5. REFLECTION FROM A TWO-DIMENSIONAL BLOCK WALL 

The two-dimensional composite structure shown in Figure 2 is replaced by the four-layer 

medium shown in Figure 5. Layers 1 and 4 are free space, layer 3 is a solid medium with 

tr = 6.1, 0' = 1.95.10-3 and l3 = 4.75 em, and layer 2 is a periodic medium with effective 

material properties given by equations (8) and (9). For this medium it is assumed that 

tr = 6.1,0' = 1.95.10-3 , a = 2.7 em, d = 15.3 em and l2 = 12.8 em 

Figure 22 shows results for the reflectivity of this composite structure for both perpendicular 

and parallel polarizations. Also shown in this figure are the results of a solid wall 17.55-cm 

thick, where tr = 6.1 and 0' = 1.95 . 10-3 . Notice that the resonance behavior of the solid 

wall for the perpendicular polarization is different than that for the composite structure. 

27 
..,J 

o 

CD 

IJ1. 121121 

121.8121 

---- y-oxls ortentotlon 

---- x-oxls ortentotlon 

- - So I ld 1010 I I 

/ 

/ 

/ 

/ 

/ 

/ 

/ I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

& 121. 4121 / I 

I 

I 

I 

121.2121 

-- ::J----::-:--- - -------

/' 

/ 

,., ,., .-

/ 

.- .- .- .- .-

.- / 

.- I 

I 

12J.12J12J4-~~~~~~~~~~~~~~~~~~~ 

121. 121121 1 5 • 121121 3121. 121121 45 • 121121 6121 • 121121 75 • 1313 9121 • 121121 

8 (Degrees) 

Figure 18. Reflectivity versus angle of incidence for a perpendicular polarized wave. These 

results are for a 7.2-cm concrete block wall (block # 3 in Table 1) with slabs 

oriented along both the y-axis and x - axis and with f = 900 MHz. The large 

dashed curve represents the results for a single layer slab of thickness equal to 

212 + 13 , the solid curve represents the actual concrete block wall with the slabs 

oriented along the y-axis, and the small dashed curve represents the results 

for the actual concrete block wall with the slabs oriented along the x - axis. 

28 
-oJ 

o 

CD 

IfLee 

13.813 

-- y-ox ls or len lo llon 

---- x-oxls orlenlolton 

- - So I l d W III I I 

---

., ., ., 

., '" 

., '" ., '" '" 

'" ,-

,- ., ., ., 

.- .- .- .- .- .- .- " 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/' 

/ 

&13.413 

/ 

/ 

/ 

/ 

13.213 / 

/ 

/ 

13.13134=~-r~~~~~~~~~~~~~~~~~ 

13.1313 15.1313 313.1313 45.1313 613.1313 75.1313 913.1313 

e ( Dsgr sss) 

Figure 19. Reflectivity versus angle of incidence for a perpendicular polarized wave. These 

results are for an 19.6-cm concrete block wall (block # 4 in Table 1) with slabs 

oriented along both the y-axis and x - axis and with f = 900 MHz. The large 

dashed curve represents the results for a single layer slab of thickness equal to 

212 + l3, the solid curve represents the actual concrete block wall with the slabs 

oriented along the y-axis, and the small dashed curve represents the results 

for the actual concrete block wall with the slabs oriented along the x - axis. 

29 
1121.121121 

---- y-oxls orlentotlon 

---- x-ox ls or len tot lon 

- - So I ld wo I I 

121.121121 ~Perfect conductlng 1-1011 

-1121.121121 

m -2121.121121 

"U 

... 0... 

-30.121121 

-4121.121121 

-5121.1211214-----.--.-.-.-.11,,----,---,-.-,,"11 

1 1121121 

Figure 20. Received power versus antenna separation for the four-ray model. These results 

are for a 7.2-cm concrete block wall (block # 3) with slabs oriented along the 

y-axis and f = 900 MHz. The antennas are 1 m off the ground and are spaced 

1 m from each of the two walls. 

30 
213.1313 

---- y-oxla orlentotlon 

---- x-ox la or len tot lon 

- - So I ld 1-10 I I 

IrtrlrlrlJ.Perfect c.onductlng 1-1011 

13.1313 

ED -213.1313 

-0 

... D.... 

-413.1313 

-613.13134-----~~~~~~~----~--.-.-II-.rn 

1 113 11313 

d (m) 

Figure 21. Received power versus antenna separation for the four-ray model. These results 

are for an 19.6-cm concrete block wall (block # 4) with slabs oriented along the 

y-axis and f = 900 MHz. The antennas are 1 m off the ground and are spaced 

1 m from each of the two walls. 

31 
Figure 23 illustrates the results of the received power for the four-ray model with a frequency 

of 900 MHz and the antenna placed 1 m from the wall. The results shown in this figure 

assume a solid wall, a composite two-dimensional structure, and a perfectly conducting wall. 

Here again, as for the concrete block wall, the results from the three different walls approach 

one another for long propagation paths. However, for short propagation paths « 1 km) a 

difference of about 5 dB can be predicted. This again illustrates the importance of properly 

representing the wall reflections for short propagation paths. 

6. VALIDITY OF THE EFFECTIVE MEDIUM MODEL 

The underlining assumption in the effective material properties model used in this paper, is 

that the period of the structure is small compared to a wavelength. For how large of a period 

compared to a wavelength can we expect valid results? This question can be answered by 

referring to homogenization results for a similar problem. 

In earlier work, one-dimensional wedges and two-dimensional pyramidal absorber structures 

were analyzed [33]-[35] using the same techniques presented here. Reference [34] and [35] 

illustrate that with the effective properties of the periodic absorbing structures, the reflection 

coefficients can be obtained by solving a classic inhomogeneous layered media problem. The 

theoretical reflection coefficient obtained with these effective material properties have been 

compared to both experimental results, [36] and [37], and to results obtained from a full 

numerical simulation of the absorbing materials, [34] and [38]. Experimental results from 

Ellam [36] and Pues [37] have indicated that the effective material properties model used to 

analyze the absorbing material were valid for a period as large as 1-3 free space wavelengths. 

32 
>. 

...., 

1.00 

0.80 

-' 0 60 > • 

-' 

...., 

o 

CD 

I+-

&0.40 

0.20 

---- E-polorlzotlon 

- - - H-po I or lzo t lon 

eeeee E-po lor lzo t lon (so I ld wo I I) 

trt:..~l!rAH-polorlzotlon (sol td woll) 

I 

.. 

I 

I 

I 

1 

J. 

I 

I 

I 

I 

I 

.. 

I I. 

-- f,/ --I --- I 

I 

0.00~~--~~-'--~-r-'--~-r~r-~-r~rc~~--.-~-, 

0.00 15.00 30.00 45.00 60.00 75.00 90.00 

(} (Degrees) 

Figure 22. Reflectivity versus angle of incidence for a two-dimensional concrete block wall 

with 12 = 4.75 cm, h = 12.8 cm, d = 15.3 cm, a = 2.7 cm, Cr = 6.05, (J" = 

1.95.10-3 , and with f = 900 GHz. The solid curve represents the results for the 

actual concrete block wall for the perpendicular polarization, the dashed curve 

represents the actual concrete block wall for the parallel polarization, the squares 

represent the the results for a single layer slab of thickness equal to 212 + 13 for 

the perpendicular polarization, and the triangles represent the the results for a 

single layer slab of thickness equal to 212 + 13 for the parallel polarization. 

33 
20.00 

---- Composte wol I 

- - - So I ld wa I I 

6-6-6-A-APerfect conductlng wall 

0.00 

-20.00 

~ 

(L -40.00 

-60.00 

-80.004---~~~~~~--.--r'-llno'---'-'-Ti 

1 10 100 

Figure 23. Received power for the four-ray model versus antenna separation. These results 

are for a two-dimensional block wall with 12 = 4.75 cm, 13 = 12.8 cm, d = 15.3 

cm, a = 2.7 cm, Cr = 6.05, (J = 1.95.10-3 , and with f = 900 MHz. The antennas 

are 1 m off the ground and are spaced 1 m from each of the two walls. 

34 
In [34] reflection coefficients calculated by using the effective material properties model were 

compared to results obtained from a moment-method calculation and excellent agreement 

was demonstrated. The results in this comparison were for a period of the structure equal to 

half a free space wavelength, and excellent agreement was demonstrated for incident angles 

as large as 90°. Using a finite difference time domain technique, Holloway, Mckenna, and 

DeLyser [38] have indicated that excellent agreement is achieved for a period as large as a 

free space wavelength for normal incidence. Agreement was also achieved for a period of 1/2 

of a wavelength for incident angles as large as 90°. 

These numerical and experimental results indicate that the effective material properties 

model presented here for the composite structure are accurate for periods at least as large 

as 1/2 to 1 free space wavelength and possibly even higher. The upper frequency limit for 

which the effective material properties of periodic structures can be used is currently being 

investigated [38]. 

The only results in the literature analyzing electromagnetic wave interaction with concrete 

block walls are those of Honcharenko and Bertoni [19]. We have made some comparison 

to their results, and for 900 MHz we get quantitatively similar, but not identical, results. 

Because of the error in Figure 4 of [19] we have to question if their other results are correct. 

Honcharenko and Bertoni [19] make no comparisons in their paper that suggest that their 

results are correct. We have compared our model to results from Bertoni's earlier paper [18] 

(the basis of [19]) and for a incident angle of 45°, agreement is achieved for a period of 1/2 

wave length ().). 

Bertoni, Cheo, and Tamir [18] and Pinello, Lee, and Cangellaris [26] show that for a period 

greater than ),/2 for incident angles of 45°, higher order Floquet modes begin propagating. 

The homogenization model presented here cannot represent these higher order Floquet modes 

for large periods and the reflection coefficient based on homogenization would no longer be 

valid. 

35 
Watters [54] has investigated the problem of acoustical wave interactions with masonry block 

walls, but his results are not applicable here. Watters does however, suggest that some type 

of area weighted average model can be used to calculate transmission loss through these 

types of walls (see Figure 10 of [54]). 

1. DISCUSSION AND CONCLUSION 

We have presented a model for analyzing the reflections and transmissions of electromagnetic waves from periodic composite structures. With this model we have investigated the 

importance of correctly predicting electromagnetic field interaction with walls for short path 

propagation channels. For short path propagation « 1 km), differences of 5-10 dB in received power can be predicted by modelling composite walls as either a single layer structure 

or as a perfectly conducting wall. This illustrates the importance of properly representing 

the wall reflections for short propagation paths. 

For large propagation paths (> 1 km) the results for either the solid or the composite 

structures approach one another. This is expected because for large propagation paths, 

the angle of incidence of the wave on a wall approaches grazing (900 ), thus regardless of the 

type of wall, the magnitude of the reflection coefficient approaches one. For long propagation 

paths, the magnitude of the reflected energy from the walls in the vicinity of the propagation 

path can be treated as if the walls behave as perfect conductors with little loss of generality. 

The one exception to this assumption, is for a periodic structure in which the period is large 

enough to guide energy in the structure. For this situation the periodic structure acts like a 

waveguiding structure, and energy can be carried away as waveguide modes, and so is not 

reflected off the surface in a spectral direction. 

36 
Some building materials consist of metal rods periodically spaced in concrete. One might 

first attempt to use the homogenization concepts discussed here to analyze such a problem. 

However, since there is a large material property contrast between the metal rods and the 

concrete, standard homogenization fails (see [55] and [38]) and the so-called stiff homogenization must be used. This approach has been used to tackle scattering from dense periodic 

space scatterers embedded in a dielectric [56], and details will be published later. 

Even though the model presented for the reflection coefficient breaks down for large periods, the results still illustrate the importance for accurately predicting reflection for short 

distance propagation. Unfortunately once the period of the structure becomes large, simple 

expressions for the reflection coefficient are not available. Under these conditions one must 

resort to more complicated means, such as a Floquet type of analysis or a full numerical 

approach [12], [18], [19], [26], [27] and [57], to calculate the reflection coefficients. 

37 
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40 
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41 

FORM NTIA-29 U.S. DEPARTMENT OF COMMERCE 

(4-80) NAn. TELECOMMUNICATIONS AND INFORMATION ADMINISTRATION 

BIBLIOGRAPHIC DATA SHEET 

1. PUBLICATION NO. 2. Gov't Accession No. 3. Recipient's Accession No. 

96 - 326 

4. TITLE AND SUBTITLE 5. Publication Date 

A Study of the Electromagnetic Properties of Concrete Block Walls October 1995 

for Short Path Propagation Modeling 6. Performing Organization Code 

NTIAIITS 

7. AUTHOR(S) Christopher L. Holloway, Patrick L. Perini, 9. ProjectlTaskIWork Unit No. 

Ronald R. Delyser, and Kenneth C. Allen 5910 5105 

8. PERFORMING ORGANIZATION NAME AND ADDRESS 

National Telecommunications & Information Administration 

Institute for Telecommunication Sciences 10. ContracVGrant No. 

325 Broadway 

Boulder CO 80303-3328 

11. Sponsoring Organization Name and Address 12. Type of Report and Period Covered 

same as above NTIA 

13. 

14. SUPPLEMENTARY NOTES 

15. ABSTRACT (A 2OO-word or less factual summary of most significant information. If document includes a significant bibliography or literature 

survey, mention it here.) 

For short propagation paths, correctly representing reflectionsof electromagnetic energy from surfaces is critical for 

accurate signal level predictions. In this paper, the method of homogenization is used to determine the effective 

material properties of composite material commonly used in construction. The reflection and transmission 

coefficients for block walls and other types of materials calculated with these homogenized effective material 

properties are presented. The importance of accurately representing the reflections for signal level prediction 

models is also investigated. It is shown that a 5- to lO-dB error in received signal strength can occur if the 

composite walls are not handled appropriately. Such accurate predictions of signal propagation over short distance 

is applicable to microcellular personal communications services deployments in urban canyons as well as indoor 

wireless private branch exchanges and localarea networks. 

16. Key Words (Alphabetical order, separated by semicolons) 

Composite walls, concrete walls, effective material properties, homogenization, propagation modeling, reflection 

coefficient 

17. AVAILABILITY STATEMENT 18. Security Class. (This report) 20. Number of pages 

0 UNLIMITED. Unclassified 41 

19. Security Class. (This page) 21. Price: 

0 FOR OFFICIAL DISTRIBUTION. 

Unclassified 

U.S, GOVERNMENT PRINTING OFFICE:1995-673-018/00137