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A Statistical Theory of Mobile-Radio

The statistical characteristics of the fields and signals in the reception

of radio frequencies by a moving vehicle are deduced from a scattering propa-

gation model. The model assumes that the field incident on the receiver

antenna is composed of randomly phased azimuthal plane waves of arbi-

trary azimuth angles. Amplitude and phase distributions and spatial

correlations of fields and signals are deduced, and a simple direct rela-

tionship is established between the signal amplitude spectrum and the

product of the incident plane waves' angular distribution and the azimuthal

antenna gain.

The coherence of two mobile-radio signals of different frequencies is

shown to depend on the statistical distribution of the relative time delays

in the arrival of the component waves, and the coherent bandwidth is shown

to be the inverse of the spread in time delays.

Wherever possible theoretical predictions are compared with the experi-

mental results. There is sufficient agreement to indicate the validity of the

approach. Agreement improves if allowance is made for the nonstationary

character of mobile-radio signals.

I. INTRODUCTION

In a typical mobile-radio situation one station is fixed in position

while the other is moving, usually in such a way that the direct line

between transmitter and receiver is obstructed by buildings. At ultra-

high frequencies and above, therefore, the mode of propagation of the

electromagnetic energy from transmitter to receiver will be largely

by way of scattering, either by reflection from the flat sides of build-

ings or by diffraction around such buildings or other man-made o r

natural obstacles.

i.i The Model

It therefore seems reasonable to suppose that at any point the

received field is made up of a number of generally horizontally trav-

By R. H. CLARKE

957

958 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 196 8

eling free-space plane waves whose azimuthal angles of arrival occur

at random for different positions of the receiver, and whose phases

are completely random such that the phase is rectangularly distributed

throughout 0 to 2w. The phase and angle of arrival of each component

wave will be assumed to be statistically independent. The probability

density function ρ (a) which gives the probability ρ {a) da that a com-

ponent plane wave will occur in the azimuthal sector from a to a + da

will not be specified, since it will b e different for different environ-

ments, and is also likely to vary from region to region within one

environment; but the assumption that the phase φ has a rectangular

probability density function throughout 0 to 2* will be made in all

cases.

For simplicity, it will be assumed that at every point there are

exactly Ν component waves and that these Ν waves have the same

amplitude. In addition it will be assumed that the transmitted radia-

tion is vertically polarized, that is, with the electric-field vector di-

rected vertically, and that the polarization is unchanged on scattering

so that the received field is also vertically polarized.

The model described so far gives what might be termed the "scat -

tered field," since the energy arrives at the receiver by way of a

number of indirect paths. Another term for this scattered field is the

"incoherent field," because its phase is completely random. Some-

times a significant fraction of the total received energy arrives by

way of the direct line-of-sight path from transmitter to receiver . The

phase of the "direct wave" is nonrandom and it may therefore be

described as a "coherent wave." It will be seen later that the field

in a heavily built-up area such as New York City is entirely of the

scattered type, whereas the field in a suburban area with the trans-

mitter not more than a mile or two distant is often a combination o f

a scattered field with a direct wave.

1.2 Comparison With Other Proposed Models

J. F. Ossanna

was the first to attempt an explanation of the sta-

tistical character of the received mobile-radio signal in terms of a set

of interfering waves. He was concerned with measurements taken in

a suburban environment, and assumed that reflection occurred at the

flat sides of houses and that the incident and reflected waves form an

interference pattern through which the receiver moves. He the n as-

sumed that all orientations of the sides of houses are equally likely,

and hence obtained spectra for the randomly fading signal wit h the

MOBILE RADIO

959

angle between the direction of vehicle motion and the direction to

the transmitter as a parameter.

There is quite good agreement between Ossanna's theoretical spectra

and those derived from measurements on several suburban streets

situated within 2 miles of the transmitter. There i s marked disagree-

ment, however, at very low frequencies and at frequencies i n the

region of the sharp cut-off associated with the maximum Doppler

frequency shift. At very low frequencies the spectral energy is al-

ways observed to be higher than that predicted by theory, whether

Ossanna's or the one we use in this paper. The reason for this is that

neither theoretical model takes into account the large-scale varia-

tions in total energy which result from the changing topography

between transmitter and mobile receiver.

The basic difference between Ossanna's theoretical model and the

model used here is that the former is essentially a reflection model

whereas the latter is essentially a scattering model and so include s

the former as a special case. An example of the limitations of the

reflection model can be seen from the experimental spectra plotted

in Ossanna's paper. The spectra are derived from signal-fading rec-

ords made on several streets whose inclination to the transmitter

direction ranged from 15 degrees to 84 degrees, and in each case

there is evidence of a shelf which cuts off at twice th e maximum

Doppler frequency shift. Ignoring the higher harmonics generated in

the detection process, the reflection model predicts a spectral cutoff

which depends on the direction of the street with respect to the trans-

mitter, ranging from the maximum Doppler frequency shift itself

when the street is at right angles to the transmitter direction to twice

that value when the street is in line with the transmitter.

With the scattering model, on the other hand, the angular distribu-

tion ρ (a) of scattered waves can be chosen to predict the existence

of a spectral shelf out to twice the Doppler frequency shift for any

street direction. Another feature of the reflection model which makes

it rather inflexible is that for every randomly oriented reflected wave

there exists a direct wave incident on the mobile receiver and carry-

ing the same power. Thus the ratio of coherent to incoherent power

in the received signal is fixed, whereas in the scattering model this

ratio is arbitrary and may be adjusted according to the environment.

In his study of energy reception in mobile radio, Ε. N. Gilbert*

examined several models of the scattering typ e and established a

number of important relationships between them. One feature com-

960 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968

mon to all of them, however, was the uniform distribution of waves

in angle, although he briefly mentioned the effect of a single strong

component arriving directly from the transmitter. The first model

Gilbert considered was that of Ν waves arriving from fixed directions,

equally spaced in angle. The phases of the waves were assumed to be

independent and uniformly distributed throughout 0 to 3ir; their

amplitudes were assumed to be Rayleigh distributed and independent,

but with the same variance. In a second model the angles o f arrival

were allowed to occur at random with equal probability for any

direction; the phases were again completely random but the ampli-

tudes were assumed to be constant. (This model is the same as the

one we use in this paper, with the restriction that p[a] = [2π·]

-1

.) A

third model was an extension of the second to include the case of an

arbitrary distribution of the amplitudes. Gilbert showed that the

second and third models were equivalent to the first for sufficiently

large N.

1.3 Scope

This paper shows that the scattering model ca n be used to predict

the statistical characteristics of the signal received at the antenna

terminals, hence at the output of a square-law or envelope detector,

of the mobile receiving vehicle. These characteristics include the

probability distributions of amplitude and phase, spatial correlations,

amplitude spectra, and frequency correlations.

A simple relationship is established betwee n the spectrum of the

signal input and the product of the azimuthal power gain g[a) of the

antenna and the probability distribution function ρ (a) of the angle

of arrival of the component waves. This relationship will be particu-

larly useful in analyzing mobile-radio systems with directional an-

tennas on the mobile unit.

Other topics discussed are the use of space and frequency diversity,

coherent bandwidth, and random frequency modulation. Some com-

ments also are made on the nonstationary aspects of mobile-radio

fields and on the consequent need for their characterization in terms

which will be useful to the mobile-radio system designer. Whenever

possible the theory is discussed in the light of available experiments.

II.

FIRST-ORDER STATISTICS OF THE FIELD

2.1

Theory

Under the assumption that the total field at any receiving point

is vertically polarized and is composed o f the superposition of Ν

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