RADIO CHANNEL CAPACITY LIMITATIONS 

R. F. Linfield∗

 The amount of information that can be 

transferred over any communications link in a 

unit of time depends upon a number of factors-- 

both technological and physical. This report 

discusses these factors with emphasis on the 

fundamental physical limitations of radio 

channels. The maximum rate of information 

transfer over a number of radio communication 

links is derived and compared with an idealized 

system. The results obtained show that at 

certain signaling rates, dispersion in the 

channel, (which is caused by multipath, for 

example) limits the number of permissible 

signaling characteristics that can be changed. 

This may occur even when only two characteristics are used--the minimum number required 

for information transfer. Thus, dispersive 

propagation mechanisms limit a binary channel's 

capacity, and unlike additive noise, this limit 

cannot be overcome by increasing the signal 

power. 

1. INTRODUCTION 

 There is considerable interest and an increasing need for 

digital data transmission over military tactical networks. As the 

demand for higher and higher data rates increases and communications technology advances, the factors which limit this rate 

tend to shift from technological areas to the fundamental physical 

limitations inherent in the transmission medium itself. 

 This report discusses these physical limitations with 

emphasis on digital systems operating in the microwave region of 

the spectrum and on links between land-mobile to base-station 

terminals operating in urban and suburban environments. 

 The objective of this report is to determine the maximum 

data rates that can be achieved with some practical systems 

currently utilizing the radio channel and to present results in a 

convenient form. 

 

∗ The author is with the Institute for Telecommunication 

Sciences, Office of Telecommunications, U.S. Department of 

Commerce, Boulder, CO 80302.
2

 The report is tutorial in the sense that the presentation is 

designed for readers who may not be working directly in the radio 

propagation or radio communications areas but are at least somewhat familiar with the subject. Mathematical equations and 

derivations have been reduced to an essential minimum. Interested 

readers can find abundant mathematical details in the numerous 

references cited in the report. An extensive bibliography on the 

subject of multipath may be found in Hartman (1974). Additional 

references on data communications via fading channels may be found 

in Brayer (1975). 

1.1 Scope 

 There are several factors which limit the data rate of 

systems operating over any radio channel. These limitations can 

be divided into two main categories--technological and physical---

as noted by Carlson (1975). The technological problems are 

soluable in theory but may not be feasible from a practical 

standpoint. The physical problems, quoting Carlson, "must be 

faced head-on with no recourse--not even in theory." 

 Table 1 provides a breakdown of the technological and physical limitations. The technological limitations include hardware, 

economic, and regulatory constraints. Examples are given in 

Table 1 under each of these. For instance, hardware availability 

is a primary limiting factor on system bandwidth. The practical 

design of antenna systems, band-pass filters and amplifiers with 

negligible distortion require that the operating frequency range 

be small relative to the center operating frequency. A rough 

bound for electronic systems operating anywhere in the radio 

spectrum is a maximum available bandwidth of about 1% to 10% of 

the carrier frequency. This implies that the bandwidth could be 

increased almost indefinitely by going to higher and higher 

carrier frequencies. 

 Economic factors evolve from the user's requirements and 

include such things as system complexity and siting consider-
3
4

ations. Regulatory constraints are set by the spectrum management 

community to maximize utilization of the spectrum resource. For 

example, power emission and bandwidth allocations along with 

frequency assignments also have an impact upon hardware design and 

economics. 

 The fundamental physical limitations are bandwidth and noise. 

The bandwidth limits the number of signaling elements or symbols 

which can be transmitted per unit time. The noise limits the 

number of distinguishable features on each symbol. The combined 

effects limit the capacity of a communication system. 

 It is apparent from Table 1 that a comprehensive review of 

all the factors limiting channel capacity is an extensive undertaking. Therefore the discussion here is restricted to certain 

physical limitations. It largely excludes the limitation due to 

additive noise and interference in the channel. The primary 

emphasis is on multiplicative noise, i.e., the effects of 

unpredictable perturbations of the signal by the channel. In many 

instances results are only qualitative since quantitative measures 

of pertinent channel characteristics are not available. 

 The report discusses only certain types of links and transmission channels. Land mobile systems operating in suburban and 

urban areas are considered quantitatively. However, the factors 

limiting the data rate on these systems are pertinent to other 

types of links and channels. Maximum rates achieved with certain 

other systems have also been estimated and are compared with the 

mobile systems and an idealized system on a common basis. 

1.2 Report Structure. 

 The ultimate goal of the report is to relate radio channel 

characteristics to the maximum data rate achievable in that 

channel and to compare these results with an idealized system 

operating over an undistorted channel against additive noise. 

This is accomplished by presenting the idealized case first 

(Section 2). Then the channel characteristics are defined in 

terms of the channel's effect on the signal (Section 3), the 
5

propagation mechanisms which determine the channel's response 

(Section 4), the mechanisms involved in specific links and their 

response (Section 5), the response relationship to coherent 

bandwidth (Section 6), the coherent bandwidth's effect on signaling rate (Section 7), the signal power requirements as a 

function of signaling rate, (Section 8) and the maximum data 

rates achieved with undistorted signaling rates (Section 9). 

Estimates of the maximum rates which can be obtained over various 

types of links are illustrated (Section 10) and compared with 

some wideband tactical networks using FM carriers (Section 11). 

The report is summarized and then, using the results obtained, 

the principal conclusions are listed (Section 12) and a recommendation is made for additional measurements to confirm these 

conclusions (Section 13). 

2. FUNDAMENTAL PHYSICAL LIMITATIONS 

 The transmission of information, i.e., telecommunications, 

involves signals or symbols changing with time in a manner which 

is unpredictable to the recipient. There are two fundamental 

physical limitations on the amount of information that can be 

transferred per unit of time. They are bandwidth and noise.

 The bandwidth limitation is caused by energy storage 

elements in the system which prevent instantaneous responses to 

signal changes. The noise limitation prevents the system from 

distinguishing between differences in the symbols by constraining 

the number of information bits that can be conveyed by each symbol 

with an acceptable error rate. 

 Symbols changing with time determine the signaling rate. 

The number of distinguishable symbols available determines their 

information content. A minimum of at least two is required to 

transfer information. If the signaling rate is l/τ symbols per 

second and there are H information bits per symbol, then the 

received data rate, R is given by 

τ

H R = b/s. (2-1)
6

 If there are m distinguishable and equally probable symbols 

at some acceptable error rate, then, by definition, the information content is 

H = log2 m b/symbol. (2-2) 

For binary systems, m=2, and there is 1 bit/symbol. 

 The signaling rate is inversely related to the bandwidth. 

For an ideal filter whose amplitude vs. frequency response is 

constant over a bandwidth B, the maximum signaling rate is given 

by Nyquist, (1928), assuming no intersymbol interference as 

2B 1 = τ symbols/s. (2-3) 

This Nyquist restriction applies only to the symbol or baud rate - 

not the data rate in b/s. For this condition, the maximum data 

rate is 

R = 2B log2 M (2-4) 

 The maximum number of symbols m which can be distinguished 

with arbitrarily low error probability in additive Gaussian noise 

was shown by Shannon (1948) to be 

1 / 2

1 ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = +

N B

S

m

o

, (2-5) 

where S is the average received signal power, with no restriction 

on signal shape or duration, No is the noise power spectral density 

and B is the ideal filter bandwidth. 

 The capacity of this idealized channel thus becomes 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = +

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

≤ = +

N B

S B

N B

S R C B

o

o

log 1

2 log 1

2

1 / 2

2

b/s. (2-6) 

 Unfortunately, the existence proof for this upper bound on 

the information rate provides no insight into the signal encoding 

process. Also there are very few instances of real communication 

channels where Equation 2-6 applies. 
7

 The actual data rate can only approach the channel capacity 

using an extremely complicated encoder. The statistical properties of the signal approach those of Gaussian noise and the 

coding time increases indefinitely. 

 Rice (1950) has shown that in order to achieve a data rate 

which is 96 percent of the channel capacity with S/NoB = 10 and an 

error probability of 10-5 the number of block encoded channel 

signals required is 210,000. This is just as impractical as the 

infinite coding delay required. 

 An upper bound for the average probability of error, Pe, can 

be written in the form 

( ) TsE Rc Pe− ≤ 2 (2-7) 

for random codes where the signals are constrained only in average 

power and bandwidth. TS is the signal duration and E(Rc) is called 

the error exponent. An exact expression for E(Rc) is extremely 

complicated. In the limiting case where T → ∞ and the channel has 

unity gain over a bandwidth B then the E(Rc) expression is 

simplified to the expression for capacity given by Equation (2-6). 

Thus, the capacity is the highest rate where the error exponent is 

positive. 

 Equation (2-7) shows that the upper bound of the average 

error rate decreases exponentially with increasing signal 

duration, Ts. Even when more practical (from an implementation 

standpoint) digital signals are used with distinguishable levels 

constrained to be relatively small and with codes composed of 

sequences of such signals, the exponential decrease in error rate 

with code length is retained, but at a lower rate and the capacity 

is reduced. 

 The capacity equation (2-6) can be solved for a lower bound 

on S, thus 

S ( )NoB C B 2 1 / = − watts. (2-8) 

 In terms of decibels (dB) relative to one watt 

( ) o

C B 10 log S 10 log 2 1 BN /

10 = 10 − dBW. (2-9)
8

 The noise power spectral density is given by 

21 4 10− N = kT = x o watts/Hz, (2-10) 

where k=1.38 x 1023 joules per degree Kelvin is Boltzmann's 

constant, and T = 290 degrees Kelvin, a typical temperature seen 

by a radio system's antenna. 

 Thus, 

 10 log10No = -204 dB, (2-11) 

and the average receive signaling power becomes 

10 log 10 log (2 1) 204 /

10 S = 10 − B − C B dB. (2-12) 

 A plot of the capacity, C, versus bandwidth, B, and parametric in signal power in decibels (dB) is shown in Figure 1. 

Curves parametric in C/B are shown on the same figure to indicate 

the importance of this parameter. Note that when B = 1/τ the 

parameter C/B is the same as the information content, H. 

 It is apparent in Figure 1 that for a fixed power and noise 

more and more efficient communication is possible with an 

idealized system, as the bandwidth is increased. With no bandwidth limitation the signal-to-noise ratio per information bit, 

S/NoR, approaches a limit of ln2≈0.69 or -1.6 dB as noted by the 

horizontal slope of the curves for constant S. Although this 

limit is reached only for infinite bandwidth, it is within about 

1 dB when the packing ratio R/B = 0.5 b/s/Hz. In the power 

limited region where R is directly proportional in S coding, 

techniques provide a means of reducing power or increasing the 

data rate. Forney (1970) shows some of the potential gains 

achievable with various practical coding schemes. For example, 

the simplest possible sequential decoder working with rate 1/2 

codes, phase shift keying, and making hard decisions has a 

theoretical limit of S/NoR = 4.5 dB for essentially zero 

error probability. This can be compared with the -1.6 dB 

ideal limit. The theoretical limit of any sequential 

decoder on a white Gaussian channel accounts for 

3 dB of this loss. An additional 2 dB loss is due to the 
9
10

hard decisions and 1 dB to the choice of rate 1/2 rather than some 

lower rate. 

 Measured performance data obtained with one high speed 

sequential decoder are shown by Forney (1970) to be within 1 dB 

of the expected theoretical limit at Pe = 10-5 for the sequential 

decoded PSK system. 

 When the channel is bandwidth limited rather than power 

limited the channel capacity can be approached without coding by 

increasing the number of distinguishable levels, i.e., the 

packing ratio. 

 The curves in Figure 1 are replotted in Figure 2 to show 

idealized capacity as a function of signal power level and 

parametric in packing ratio. This plotting format is used 

repeatedly in Section 8 of this report to show the effect of 

modulation, channel distortion, and diversity reception on the 

signaling ratio achievable with some practical uncoded systems. 

 The bounds on error probability given by Equation 2-7 and 

other similar expressions involving constraints on signal design 

are often used as standards against which actual systems can be 

measured. In this report, however, the ideal (Shannon) limit on 

capacity is used to illustrate a common, though unachievable, 

upper bound on data rate. 

2.1 Practical Considerations 

 Shannon's channel capacity expression (eq. 2-6) 

provides an upper bound for error free transmission in a linear 

baseband channel. Radio channels require that this baseband be 

converted to a passband at radio frequencies. This involves 

modulation of a carrier so that the modulation envelope follows 

the baseband signal. Although a digital system may be considered 

wideband in terms of data rate, the radio signals are actually, 

narrowband relative to the carrier frequency in order to define 

this envelope. 

 Since radio channels are basically analog channels some form 

of modulation and demodulation (modem) is required to match 

digital baseband signals to the transmission medium. This 
11
12

baseband signal may be binary (2 level), coded binary, or m-ary 

(multiple level). Three basic techniques are used to convert the 

digital data into analog signals--amplitude modulation (AM), 

frequency modulation (FM) and phase modulation (PM). For digital 

systems they take the form of amplitude shift keying (ASK), 

frequency shift keying (FSK) and phase shift keying (PSK). 

Combinations of these are also used. 

 If the packing ratio for the baseband signal is R/B, then 

the packing ratio for double sideband amplitude modulation would 

be R/2B. For single sideband amplitude modulation, it stays the 

same. The packing ratios for FM and PM systems are more difficult 

to define because these are nonlinear modulation techniques. 

 Equation (2-6) gives the limitation on channel capacity 

imposed by white noise in a distortionless channel limited to an 

idealized bandwidth B. The expression also applies to certain 

other channels whose amplitude vs. frequency response may not be 

constant over B as shown by Gibby and Smith (1965). This, then, 

is a whole class of symbols whose amplitude spectra with linear 

phase meet the Nyquist criteria for signaling rates of 1/τ = 2B. 

In principle, however, it is physically impossible to generate 

such symbols because they have infinite tails in both directions 

and would require an infinite delay for an exact synthesis. The 

slowly decreasing tail is also extremely sensitive to timing 

errors and channel distortion. However, with sufficiently 

complicated equalization, it is possible to make interference 

very small. 

 Lucky and his colleagues (1968), describe one symbol with 

raised cosine characteristics which has been used extensively. 

The total spectrum occupies a bandwidth BT=kB where k is greater 

than 1 and is a function of the frequency rolloff. As k decreases, 

BT is less and response in the time domain becomes more oscillatory 

approaching the ideal response which has the form 

x

sin x . 

However, the problems of timing and channel equalization become 
13

more difficult. A typical value used is k=2, and the signaling 

rate in terms of the total bandwidth BT becomes 

= BT

τ

1

 (2-13) 

The data rate then is given by 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ≤ +

N B

B S R

o

T log 1

2 2 , (2-14) 

and the average received signal power is thus obtained from 

 S (raised cosine) = ( ) o

R B 2 T 1 BN 2 / −

 = ( ) o

R B

R B BN T

T

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + − 2

2 1 2 1

/

/ . (2-15) 

The signal power required for the idealized system was previously 

found to be (Equation 2-7), 

 S (idealized) = ( ) o

R B 2 1 BN / − . (2-16) 

In order to obtain identical data rates and occupied bandwidths 

with either system so that R/B = R/BT, the raised cosine system 

requires additional signal power by an amount in dB qiven by 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + = 2

2 1 10log ( )

( cos ) 10log

/

10

R B

S ideal

S raised ine

. (2-17) 

For R/B = 2,,4, and 8 bits/s/Hz, this amounts to 4 dB, 9.3 dB, 

and 21 dB, respectively. 

 Two other factors tend to limit the data rate. One is the 

variations in signal level (power fading) which are caused by the 

channel but do not distort the signaling element or cause intersymbol interference. In this case (defined as flat-flat channel 

in Section 3), it is necessary to find the average signal power by 

averaging over some fade distribution x. Various distributions 

have been used depending on the channel involved. 

 When a large number of multipath components exist without 

any major components as in some urban mobile links, the 

fading approximates a Rayleigh distribution. Other 
14

distributions are described in the literature. For example, see 

Nesenbergs (1967). 

 The other factor limiting data rate is multiplicative noise 

which is introduced by random distortions in the channel. 

Predictable amplitude and phase variations in channel can be 

corrected, at least in principle. Complete elimination, however, 

is difficult in practice and the remaining variations differ 

between similar links and in time on the same link in a random 

manner. 

 Under these conditions, the data rate has an upper bound 

given by 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

< +

DS N B

B SX R

o

T log 2 1

2 (2-18) 

where the symbol X reflects the non-distorting, flat-flat power 

fading of S and where DS is the noise due to random distortion in 

the channel. A similar expression is given by Sunde (1954) for 

wired systems where X = 1. 

 The distortion factor D is the result of unpredictable 

amplitude fluctuations and phase deviations in the channel. It is 

apparent that the multiplicative noise term DS places a limitation 

on data rate which unlike the additive noise term N B or the flatflat fading cannot be overcome by increasing the signal power. 

 The impact of these factors on the reliability or error rate 

of some uncoded systems in use today is illustrated in Figure 3. 

The error rates for several binary systems operating in additive 

white noise are bounded within the shaded curves. Numerous 

examples of specific curves of this type may be found in the 

literature. For three such examples see Schwartz, et al., (1966), 

Bello and Erhman (1969), and Frasco and Goldfein (1973). 

 Curves are plotted in Figure 3 for a non-fading channel, a 

Rayleigh fading channel, and a selective fading channel. The 

width of the curves bounds the performance between several 

modulation schemes including phase shifted keying (PSK) on the 

left and noncoherent frequency shift keying (FSK) on the right. 

Other systems whose performance falls within these bounds include 
15
16

coherent FSK, differentiated PSK, quadrature PSK, and minimum 

shift keying (MSK). The theoretical channel capacity limits for 

the ideal system and for a digital signal with segmented decoding 

are also shown on the figure. See Forney (1970). 

 The abscissa scale in Figure 3, is the critical parameter 

S/No, normalized by the data rate R to obtain 

o

b

o N

E

RN

S = (2-19) 

where Eb is the energy per data bit. It can be seen in Figure 3 

that the Rayleigh fading on a channel causes the error rate to 

vary in inverse proportion to the normalized signal-to-noise 

ratio Eb/No at large values. This is in contrast to nonfading 

signals where changes in Eb/No at large values produce exponential 

changes in error rate. The selective fading channel results in 

an error rate which cannot be reduced by increasing Eb/No for 

certain systems. The level at which this irreducible error rate 

occurs depends on the signaling rate and the channel characteristics. As this rate is increased causing more distortion or 

intersymbol interference, the errors increase and raise the 

irreducible level. This irreducible level can, however, be 

reduced by adaptive equalization prediction feedback or coding. 

3. RELATING CHANNEL CHARACTERISTICS TO 

SIGNALING CHARACTERISTICS 

 Long-distance information transfer requires some form of 

transmission medium and associated coupling mechanism or transducer. The transmission medium may be a cable, an optical fiber, 

an acoustic channel, or a radio channel. Ideally, whatever method 

is used the information transfer process should take place with no 

error. However, the outputs from real channels differ from the 

inputs because the transfer characteristics are not perfect and 

because of the omnipresent noise. 

 This report is concerned with terrestrial radio channels 

where the information transfer is accomplished by electromagnetic 

propagation through the atmosphere, at least on some portion of 
17

the path. Terrestrial radio channels tend to be dispersive by 

nature distorting multifrequency signaling wave forms in unusual 

and everchanging ways. The dispersive character of radio 

channels takes two independent forms: time dispersion and 

frequency dispersion. Time dispersion is manifested in the time 

domain by the spreading or smearing of pulses of short duration 

when they are propagated through the medium. Frequency dispersion 

is manifested in the frequency domain by the spreading or shifting 

of the frequency spectrum of narrow-band signals. Time and 

frequency dispersion may cause signals to fade in time, they may 

distort the symbols, or they may produce intersymbol interference. 

The following subsections describe the dispersive effects of the 

radio chaunel on the signal. The definitions used are essentially 

those given by Kennedy (1969). 

3.1 Time Dispersion 

 Time dispersive effects fall into two causitive classes, 

those due to the frequency dependence of atmospheric effects upon 

propagation paths and those due to mechanisms producing multiple 

propagation paths, i.e., multipath.

 In the first class, frequency dependence results from the 

complex refractive index of rain, water vapor and oxygen in the 

atmosphere. Broadband signals may be distorted when the amplitude 

of the channel transfer function varies with frequency or its 

phase varies nonlinearly with frequency. 

 In the second class, multipath results when the signals 

propagated along different path lengths arrive at their destination with different time delays. The resulting multipath spread 

causes frequency-selective fading which introduces signal 

distortion and, when severe, causes intersymbol interference. 

 Time-dispersion characteristics are determined by measuring 

the delay power spectrum (or impulse response) of the channel. A 

useful measure of the frequency selectivity is the coherence 

bandwidth. It is obtained by Fourier transform of the impulse 

response to give the frequency autocorrelation function. 
18

Coherence bandwidth is then defined as the frequency separation 

where the autocorrelation function falls below some specified 

value. Time dispersive effects tend to limit the maximum useable 

signaling rate (Bello and Nelin, 1964). 

3.2 Frequency Dispersion 

 Frequency dispersion is caused by time variations in the 

propagation path due to changes in the medium itself, terminal 

motion, or combinations of these. These dynamic changes in the 

channel can cause signals to fade and, when severe, cause timeselective fading. 

 Frequency-dispersion characteristics are determined by 

measuring the frequency power spectrum of the channel. A 

useful measure of the time selectivity is the coherence time. It 

is obtained by doing a Fourier transform of the frequency power 

spectrum to give the time autocorrelation function. Coherence 

time is then defined as the time separation where the autocorrelation function falls below some specified value. Frequency 

dispersion effects tend to limit the minimum useable signaling 

rate (Bello and Nelin, 1964). 

3.3 Channel Classifications 

 Time and frequency dispersion are independent measures of 

channel characteristics. A channel may be considered as nondisperslve, dispersive in time, dispersive in frequency or dispersive in time and frequency, depending on signal characteristics. 

Although all radio channels are dispersive to some extent, their 

distortive effect on transmission of signals depends on the 

magnitude and type of dispersion relative to the duration of signaling symbols and relative to signal bandwidth. If the symbol 

duration is long compared to the time dispersion, distortion 

is negligible. All frequency components of the signal fade 

together. This is called non-frequency selective fading or a 

time flat channel. As the symbol duration decreases, frequency 

components fade independently, the symbol is distorted and the 

channel becomes frequency selective. 
19

 When the bandwidth of the signal is wide compared to the frequency dispersion, the distortion is negligible. The amplitude 

and phase of a complete symbol or several symbols may still vary 

with time. This is called non-time selective fading or a frequency-flat channel. As the bandwidth of the signaling element 

decreases the amplitude and phase may vary over a single symbol 

causing distortion and the channel becomes tune selective. 

 A radio transmission channel with a given set of time and 

frequency dispersive characteristics may be called by different 

names depending on the signal format employed. When the signal is 

undistorted the channel is neither time selective nor frequency 

selective. This is called the time-flat and frequency-flat 

channel or just a flat-flat channel. Note that amplitude and 

phase of the symbols may vary slowly with time relative to the 

signaling speed. 

 Table 2 summarizes the effects of the channel on a signal, 

wherein signal distortion and intersymbol interference effects 

have been lumped together. 

4. RELATING CHANNEL CHARACTERISTICS 

TO DISPERSIVE MECHANISMS 

 Some of the various propagation mechanisms involved in a 

microwave radio channel are illustrated in Figure 4. Signals 

arrive at the receive terminal via direct, refracted and 

reflected paths where they combine to form a composite signal 

(Dougherty, 1968). The paths are different lengths and change 

with time. Along each path wideband signals are also perturbed 

by the dispersive nature of the medium as the complex index of 

refraction varies with frequency. They may also be perturbed by 

scattering from turbulent inhomogeneities in the atmosphere. 

Ideally, a system designer would prefer to use only one of these 

paths (the direct one in the situation depicted), but it is 

difficult to excite and resolve only one. 

 The propagation mechanisms illustrated in Figure 4 are 

represented by blocks in Figure 5. Although there is usually 
20
21
22
23

only one direct path (which may be diffracted) there may be 

additional refracted paths (usually only 1 or 2) and several 

reflected paths. The presence or absence of any path depends on 

link geometry, the operating environment, meteorological conditions and geographical location. Sometimes a particular propagating mechanism is desired and sometimes it exists when not 

wanted. For example, ionospheric refraction is used for long 

distance links in the HF band. 

 Interference from refracted paths may occur on microwave 

line-of-sight links at low grazing angles when layering in the 

atmosphere causes variations in the refractive index. 

 Reflected waves and additional refracted paths may be absent 

entirely on some links--a ground-to-satellite link, for example. 

Even the direct path can fade or not even exist. It may be 

blocked by an obstruction such as a building in the path of a 

mobile unit to a base station. 

 Reflected, refracted, diffracted, and direct paths may all be 

non-existent, e.g., transhorizon links use scatter mechanisms 

only--either ionospheric scatter or tropospheric scatter. In 

this case, the dispersive mechanisms are limited to medium 

dispersion and turbulent scatter. 

 The medium couplers depicted in Figure 5 are the transmit 

and receive antenna systems. Directive antennas may be employed 

to reduce the multipath dispersion by discriminating against 

reflected components. Narrow beamwidths tend to reduce the time 

spread due to scattering mechanisms. Multiple coupling antenna 

configurations are often used to provide space diversity to 

improve performance on fading channels. 

 Table 3 lists the gross characteristics for some microwave 

channels. The estimates given are only approximate and were 

obtained from a variety of sources. Pertinent sources are listed 

later in Section 10 in Table 9. The values listed here for the 

delay spread of the dispersive medium and turbulent scatter 

mechanisms are rms values of the delay profile. The delay indicated for refracted and reflected components is the maximum delay 
24

Table 3 

Gross Channel Characteristics for Microwave Channels 

Channel Characteristic

Delay 

Spread Time Variability

 

Medium Dispersion 

• Clear Sky <0.1 ns Diurnal & Seasonal 

• Heavy rainfall <0.2 ns Minutes to hours 

 

Turbulent Scatter 

• Atmosphere (non-ionized) <1.0 ns Seconds 

• Troposcatter <1.0 μs Seconds to Minutes 

• Ionospheric <100 ns Seconds 

 

Refraction 

• Fixed Terminals <10 ns Minutes to hours 

• Moving Terminals <10 ns Seconds to minutes 

 

Reflection 

• LOS Links 

 Grid to Grid <10 ns Hours 

 Air to Grid Link <10 ns Minutes 

 Air to space <10 ns Minutes 

• Mobile links 

 Urban <5 μs Seconds 

 Suburban <0.5 μs Seconds 
25

relative to a direct path. The time variability indicates 

typical values for the periods of time over which the channels' 

characteristic parameters can no longer be considered stationary. 

 Diffractive effects have been neglected here although they 

can have considerable impact on the observed channel response 

because the strength of various components can change due to 

diffraction. In the mobile environment the signals at shorter 

delays may be enhanced at lower frequencies due to diffraction 

around obstructions. See Turin, et al., (1972). 

5. RELATING PROPAGATION MECHANISMS TO CHANNEL 

RESPONSE OF SPECIFIC LINKS 

 This section indicates some qualitative response functions 

obtained on certain specific links and quantitative response 

functions for mobile links. Here, emphasis is on the impulse 

response rather than on the frequency transfer function of the 

channel. Either would suffice since one is the Fourier transform 

of the other. Generally, the impulse response is easier to 

relate to the physical characteristics of the channel. The 

square of the absolute value of the response is sometimes called 

the delay power spectrum or multipath profile. It is a measure 

of the received signal power distribution in the time domain. 

 An instantaneous impulse response is obtained when the 

channel is excited by an impulse whose duration is much less than 

the dispersive character to be measured. Since the response 

usually changes slowly with time and space, an average of many 

instantaneous responses may be used to describe the actual 

response as a statistical representation of the channel over 

short periods and distances. The time variability of the response 

over longer periods and distances defines the rapidity of fading 

due to the dynamics of the path. 

 Some delay power spectra averaged over short periods are 

illustrated qualitatively in Figure 6 for certain representative 

links. The variability with type of link is clearly indicated. 

Although the detailed characteristics of a given type of link 

will vary, the general characteristics tends to remain the same 
26
27

This variation depends on many factors including range, geographic 

location, carrier frequency, meteorological conditions, link 

geometry, terminal motion, antenna height, beamwidth, and 

pointing angle. 

 For any specific link, the general character of the response 

depends on the propagation mechanisms involved. Table 4 indicates 

this relationship for representative links. 

 There are numerous methods used to describe a delay 

profile. A gross description is obtained from approximating the 

total width including all discrete components falling above the 

background noise threshold. Some typical values are given on the 

responses in Figure 6. 

 A more quantitative single number representative of the time 

delay is the rms spread, Sd, given by the square root of the 

second moment. For example, see Cox (1972a). 

( )

1/ 2

1 1

2

⎥

⎦

⎤ ⎢

⎣

⎡ = ∑ − ∑ = =

m

k

k

m

k

Sd τ k τ o Pk P (5-1) 

where the mean path delay is given by the first moment with 

respect to the first arrival delay, τa, i.e., 

∑ ∑ = =

= −

m

k

R a

m

k

o k PR P

1 1

τ τ τ (5-2) 

The summations from k=l to m include an arbitrary delay before 

the minimum arrival delay τa where the profile amplitude is 

insignificant and a sample at m corresponding to the last delay 

of significant amplitude. Other measures of width used by Cox 

(1972a) include the delay between the outer -3 dB or -10 dB 

points. 

 A fairly complete discussion and presentation of descriptors 

is given by Nielson (1975) for profiles obtained over mobile 

links. In addition to those given above, these include the 
28
29

number of resolvable components, distribution of delays between 

resolvable components, and the rms-to-average ratio of the 

profile amplitude. Nielson also indicates that the delay spread 

parameter, Sd, does not adequately describe profiles containing 

discrete, closely spaced components. 

 Mathematical functions which approximate the shape of a 

given profile are often found in the literature. See Gans 

(1972). Examples include Gaussian, exponential, rectangular, and 

discrete-valued functions. Representing different channels in 

this manner permits the subsequent mathematical manipulations 

(Fourier transforms, for example) to become more tractable. The 

Gaussian function is often used to describe the received process 

when multiple paths cannot be distinguished physically. This is 

often the case when scattering mechanisms are involved and the 

number of scatterers is fairly large. 

 Hartman (1974) has collected together many of the theories 

and techniques used in solving multipath problems including the 

classification of multipath and its effect on systems. He 

indicates, for example, how to treat a two-discrete-paths model 

when the statistics enter the problem only through the probability 

that a particular delay will occur. 

5.1 Time Dispersion in Mobile Links 

 Land-mobile links are commonly used for short-range communications distances up to about 30 km. Portions of the VHF and UHF 

frequency bands have been allocated for this purpose. 

 The multipath characteristics for mobile links operating in 

urban and suburban environments have been measured by various 

groups. One readily measured characteristic is the amplitude 

variation of the envelope of a signal transmitted over a mobile 

link as the terminal position is changed. Several workers have 

made such recordings at frequencies ranging from 50 MHz to 

12 GHz. A mathematical theory and review is given by Clark 

(1968). Over short distances the fading envelope tends to be 

Rayleigh distributed indicating the resultant field consists of a 
30

number of reflected waves with random amplitudes and angles of 

arrival and uniformly distributed phases. When there is a 

significant direct path signal, the envelope tends to be Rice 

distributed. 

 At longer distances, the fading tends to approach a lognormal distribution because of the terrain and the effects of 

other obstacles. These characteristics are important for evaluating the performance of systems when the signaling rate 

corresponds to a time-flat, frequency-flat channel. For selective 

channels, the time and frequency spread are of interest. These 

have also been measured using different techniques and at 

different frequencies. The general scheme for measuring delay 

spread is to transmit short pulses and observe the echoes. 

Pseudo-random (PN) codes are also used. When keyed at high 

rates, such codes can be cross correlated with an identical code 

at the receiver, yielding short impulse functions. The width of 

the pulse in either case determines the system resolution 

capability. Table 5 summarizes the characteristics of systems 

used by various workers. 

 The reflected and scattered signals arriving at the measuring 

terminal will be frequency (Doppler) shifted by an amount depending on terminal velocity and the angle of arrival relative to the 

terminal's velocity vector. These frequency shifts can be 

measured with the PN code equipment when quadrature coherent 

detection is used and stable frequency references are available. 

See Cox (1973). 

 Different workers used different methods to analyze their 

results. Young and Lacy (1950) made tests at 450 MHz in the 

lower part of Manhattan in New York City. Pulse profiles are 

presented as sequences of motion-picture frames along several 

streets. Their statistical analysis shows the data in a form of 

probability that minor paths will appear within specified intervals of relative delay. The maximum time delay spread is about 5 

microseconds with an approximately exponential distribution. 
31
32

 Turin and his colleagues (1972) report multipath spread 

measurements in the East Bay region of the San Francisco Bay area 

and in San Francisco itself. Observations were made at three 

frequencies from about 500 MHz to 3 GHz. On some links, line-ofsight paths were unobstructed. Turin et al., concluded that, in 

their experiment, the multipath spread depended almost entirely 

on the local environment of the receiver. The results obtained 

at the three different frequencies were essentially the same. 

The total delay was always less than 7 μs and usually 5 μs for 

high rise portions of the city. Smaller towns and suburban areas 

showed spreads of 3 μs to 4 μs. 

 Local multipath statistics for a suburban area are described 

by Cox (1972a and 1972b) based on measurements along several 

business and residential streets in suburban New Jersey. He 

presents data in the form of average delay profiles, cumulative 

distributions of signal amplitude at fixed delays and frequency 

shift spectra at fixed delays. Typical delay spreads for the 

suburban area are 0.25 μs although in extreme cases significant 

components were observed with excess delays from 5 μs to 7 μs and 

the rms delay spread was approximately 2 μs. 

 Multipath profiles in urban areas of New York City have also 

been measured by Cox (1973) using similar analysis, measurement 

equipment, and presentation of the data. In some locations 

significant amplitudes were observed with excess delays of 9 μs 

to 10 μs. The rms delay spreads for these areas were 2 μs to 

2.5 μs. Cox concludes from his results that it appears reasonable 

to model the urban radio channel as a Gaussian quasi-wide-sense 

stationary uncorrelated scatter channel (Bello & Nelin, 1963), 

at least within a bandwidth of 10 MHz and for intervals up to 

30 meters along the street. 

 Nielson (1975) measured various mobile channel characteristics at 1340 MHz along several paths in areas around San Francisco. Nielson classifies different paths by degree of urbanization into five terrain categories: dense urban, moderate urban, 

limited urban, suburban, and rural. He finds however, that this 
33

classification does not reflect large differences in the impulse 

responses observed. The total time delay spread varied from 

5 μs to 6 μs in the urban areas to less than the equipment 

resolution capabilities (50 ns) for direct line-of-sight paths. 

The spread over intermediate terrain categories fell between 

these extremes but was essentially undifferentiable. 

 More recently (July 1977) the equipment used by ITS and 

described by Linfield et al. (1976) was employed to measure the 

impulse response at 8.6 GHz on mobile links in Boulder, Colorado. 

The delayed power spectra (including quadrature phase 

components) are being recorded on magnetic tape for subsequent 

analysis on the laboratory's analyzer and computer. Although 

this analysis has not been completed, it is of interest here to 

show some of the responses obtained because of the high resolution 

of the equipment and the higher carrier frequency used. 

 Figure 7 is a series of oscilloscope photographs of the 

instantaneous envelope of the response measured along a 

suburban street in an area of one-story houses near the OT/ITS 

laboratory. The line-of-sight path to the transmitter is 

partially obscured by buildings and trees. The three pictures 

were taken with the receiving terminal stopped at three points 

about 3 meters apart. The total delay varies from about 50 ns 

to 100 ns over this distance. 

 Figure 8 illustrates oscilloscope photographs taken one 

block south of the previous site. Both photographs were taken 

at the same location while the receiver terminal was stationary. 

Trees with foliage were blocking the line-of-sight path to the 

transmitter. The upper photograph contains only a few responses 

whereas the lower photograph shows several hundred responses 

averaged by the storage oscilloscope over a one-minute period. 

Ten sweeps were obtained each second. The maximum spread observed 

is about 120 ns. Fast changes in amplitude were attributed to 

the movement of the trees and their foliage which changes the 

attenuation on the paths at this frequency. 
34
35
36

 The photographs shown in Figure 9 were, like those in Figure 

7, taken at three points 3 meters apart but in an area of fairly 

isolated multi-storied apartment buildings. The earliest arriving 

signal in the upper photograph is severely attenuated over a 

reflected component arriving 200 ns later. A third component of 

significant amplitude also occurs 320 ns after the first component. The multipath structure appears entirely different when 

the reception point is changed by only a few meters as shown by 

the middle and lower photographs. 

 For many locations, including the one-story housing area, 

signals were attenuated below the receiver noise threshold. Even 

small structures such as a highway sign could completely block 

out the direct signal. At these higher frequencies diffraction 

effects around buildings and signs are considerably less. There 

appear to be other differences attributed to variations in 

attenuation caused by movement of small objects including foliage 

on trees. 

 Other authors have reviewed available data (See Jakes (1975) 

and Clark (1968), for example) and they have concluded that an rms 

delay spread of 0.25 μs is typical for a suburban area and 2.5 μs 

is typical for an urban area. These values are probably 

reasonable estimates as an upper and lower bound for almost any 

area where a significant number of reflections exist and for 

frequencies below about 3 GHz. At higher frequencies data are 

extremely limited but it appears that the delay profile may be 

considerably different. 

 Table 6 summarizes the total delay spreads measured in 

different environments by the various groups. 

 Table 7 indicates some analysis results from Cox for New 

Jersey and New York, and from Nielson for San Francisco. The rms 

spreads shown for San Francisco are averages from several hundred 

records. Specific values varied approximately ± 1 μs from these 

averages. No coherent bandwidths are given by Nielson. However, 

Dr. R.A. Whiteman (private communication) of the Electromagnetic 

Compatibility Analysis Center, Annapolis, Maryland, has computed 
37
38
39
40

the rms delay spread for one profile given by Nielson which 

represented an average of 100 profiles in an urban area. His 

results for an rms delay spread of Sd = 2.25 μs yielded a 

coherent bandwidth of 34.2 kHz for a correlation coefficient of 

0.9. 

5.2 Frequency Dispersion on Mobile Links. 

 As a mobile terminal moves with a constant velocity a 

continuous wave signal transmitted at a carrier frequency will be 

received with randomly varying amplitude and phase. The maximum 

effective bandwidth of this received signal is bounded by a 

maximum frequency shift of ±V/λ where V is the terminal velocity 

and λ is the wavelength of the carrier frequency. At 1 GHz, for 

example, a vehicle traveling at 56 km/s (35 miles per hour), the 

bandwidth occupied by the cw signal is ± 50 Hz. This frequency 

spread or Doppler shift can be normalized in units of cycles per 

meter of travel by dividing by the terminal velocity. Frequency 

spreads have been measured and evaluated by Cox (1972a & b, 1973) 

for various points on the delay curve. The doppler spectra at 

different delays are quite different although none exceeded 

± l/λ cycles/meter. At 910 MHz where the measurements were made 

l/λ = 3 cycles/meter. By observing the shift spectra at different 

delays, Cox was able to deduce the direction of arrival of 

components producing dominant features on the power impulse 

response. 

 Aircraft and other moving reflectors in the path can cause 

increases in observed shifts, and significant variations in 

observed multipath profiles even when the transmit and receive 

terminals are stationary. 

 In some instances, Nielson (1975) observed significant 

variations when a dominant path became obstructed. Such variations could present serious communication problems for certain 

wideband systems; for example, modems which resolve individual 

paths by synchronizing on the strongest path. Loss of synchronization would then occur if this path disappears or is switched off 

momentarily by an obstruction. 
41

6. RELATING LINK RESPONSE TO COHERENT BANDWIDTH 

 Coherence bandwidth is a function of the time dispersion. 

It defines the separation between two frequencies, which under 

dynamic conditions will fade essentially in unison. Signals 

whose bandwidths exceed the coherence bandwidth will be distorted 

because the phase and amplitude of different frequency components 

fade independently. 

 The coherence bandwidth, Bc, of a radio channel is considerably different from the bandwidth of a conventional filter or the 

ideal filter discussed in Section 2. Signaling power is severely 

attenuated at frequencies outside the ideal filter bandwidth, B, 

or the practical filter bandwidth, BT, given in Section 2. A 

radio channel, however, can transmit power at frequencies which 

far exceed the coherent bandwidth. Signals whose bandwidth 

begins to exceed B or BT are quickly degraded while signals whose 

bandwidths begin to exceed B are only gradually degraded. 

Conventional filter bandwidths are usually defined by the -3dB 

points, i.e., the frequency separation between points on the 

response curve at one-half the peak power. Coherence bandwidth 

on the other hand is defined as the frequency separation where 

the correlation between the two frequencies falls to some specified value. Different correlation values have been specified in 

the literature ranging from 0.9 to 0.37 depending on the author's 

view, on the analytical approach, and on the particular system 

under consideration. 

 The single correlation coefficient used to define correlation 

bandwidth is obtained from the frequency autocorrelation 

function. This function may be measured directly (Gallager, 

1964) or may be obtained by Fourier transform of the impulse 

response. 

 The exponential distribution of time delays was found by 

Clarke (1968) and Gans (1972) to be a reasonable fit to impulse 

response data obtained by Young and Lacy (1950) for mobile links 

operating at a frequency of 450 MHz in New York City. The 

exponential distribution function is also a good approximation to 
42

later measurements at higher frequencies by Cox (1972a & b) and 

Turin, et al., (1972) for suburban as well as urban areas. 

 The frequency autocorrelation function for the exponential 

distribution is given by Gans (1972) as 

( )

1/ 2

2 1 2

1

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ ∆ =

d π fS ρ (6-1) 

where Sd is the rms delay spread from Equation (5-1) and ∆f 

is the frequency separation. 

 A curve of the autocorrelation function ρ versus ∆f is shown 

in Figure 10 for the exponential delay profile. Using this curve 

it is possible to define a coherent bandwidth, Bc, for any 

selected value of ρ. For example, when ρ = 0.9, then 

d

c S

B

4π

1 = (6-2) 

and when the rms delay spread Sd=l microsecond, the coherence 

bandwidth Bc = 80 kHz. Other values for Bc as a function of Sd

and parametric in ρ are plotted in Figure 11 for the exponential 

distribution. 

 Different values of Bc are obtained for the same p as the 

assumed shape of the delay profile changes. Gans (1972) compared 

the autocorrelation function for the exponential, Maxwellian and 

two valued profiles. For values of ρ greater than 0.75 his 

results indicate that the coherence bandwidth is only slightly 

dependent upon the shape of the profile. He, therefore, approximates the coherent bandwidth as Bc = 1/8 Sd since this corresponds 

to specific combiner degradation in diversity systems using a 

pilot tone for determining weighting factors. 

 Jakes (1975) indicates the relationship between rms delay 

spread Sd and the coherence bandwidth for the Gaussian distribution is given by 

d

c S

B

π

2 = (6-3) 
43
44
45

when the correlation coefficient falls to 1/e, i.e., ρ = 0.37. 

This is shown as a dashed line on Figure 11 and it is apparent 

that the coherent bandwidth for the exponential and Gaussian 

distributions are approximately the same for ρ = 1/e. 

 Bailey and Lindenlaub (1968) have theoretically compared the 

performance of a binary DPSK system using both Gaussian and 

rectangular delay profiles and using different symbol shapes. 

Their results show that there is a larger difference between 

achievable error rates when different signals are used in the 

same channel than when different channels are used for the same 

signal. This is also shown by Monsen (1973) discussing adaptive 

equalization techniques on a troposcatter channel. Monsen's 

computations indicated that the average probability of error was 

completely insensitive to the choice of shape of the delay power 

impulse response for a fixed ratio of Sd to symbol width. 

 The results presented in Section 7 assume an exponential 

distribution for the channel impulse response with the coherence 

bandwidth defined at the ρ = 1/e points as given in Figure 11. 

7. RELATING COHERENT BANDWIDTH 

TO SIGNALING RATE 

 Bello and Nelin (1963) calculated error probabilities for a 

time dispersive channel and Gaussian noise for FSK and PSK systems 

which utilize diversity combining. They assume that the time-flat 

frequency-selective channel has a Gaussian impulse response and 

that the signaling element is rectangular. Bailey and Lindenlaub 

(1968) have extended these results for the PSK system to include a 

rectangular impulse response for the channel and a raised cosine 

signaling element. 

 One example of the performance obtained for the binary PSK 

system using differentially coherent detection is shown on 

Figure 12. The value for Bc indicated on the selective fading 

curves is the correlation bandwidth for ρ = 1/e. The irreducible 

error rate caused by the frequency selective fading is indicated 
46
47

for three different signaling rates in terms of Bc. When this 

irreducible error rate is reached, it cannot be changed by 

increasing signal power. 

 The irreducible error rates versus l/τBc and parametric in 

order of diversity, L, for differential PSK systems and noncoherent FSK systems are plotted in Figure 13, assuming a rectangular signaling element is used. The curves shown assume that the 

signaling rate is such that intersymbol interference from 

immediately adjacent symbols occurs. 

 At a fixed error rate it is possible using points from the 

curves in Figure 13 and the coherence bandwidth from the 

ρ = 0.37 curve in Figure 11 to relate the signaling rate to rms 

delay spread, Sd. Such curves are plotted in Figure 14 for both 

systems and parametric in L for an error rate Pe = 10-5. 

8. RELATING SIGNALING RATE TO SIGNAL POWER 

 In Section 7 and Figure 14, the signaling rate for two 

binary systems (PSK and FSK) operating at different levels of 

diversity (L=l, 2, and 4) was related to the rms delay spread, 

Sd, over a range from 0.1 μs to 10 μs. It was noted earlier that 

the data rate 

R log2 m

1

τ = , (8-1) 

so that for cosine signaling with a binary system where m=2, the 

signaling rate is equal to the data rate, i.e., R = l/τ bits/s. 

For cosine signaling with frequency roll-off of k=2, the signaling 

rate l/τ = BT and the packing ratio is R/BT = 1. Note that the 

ordinate is the data rate in bits/per second which for these 

binary systems corresponds to the signaling rate in symbols/s. 

 In Figure 15, the effect of the level of diversity, L, on the 

irreducible data rate is indicated for the case where the error 

rate Pe = 10-5 and the rms delay spread Sd = 1 μs. As the L 

increases, the rate at lower signaling levels tends to approach 

the no-fading curves as expected. Ultimately, a transition point 

is reached when the rate cannot be increased by increasing the 

signal power.
48
49
50
51

 In Figure 16, the level of diversity and the rms delay spread 

are held constant and the error rate allowed to vary. 

 For the curves plotted in Figure 17, the error rate is constant (Pe = 10-5) and the parameters L and Sd are varied. The 

curves are for the DPSK system. Figure 18 shows a similar set of 

curves for the non-coherent FSK system. The FSK system is less 

sensitive to the effects of frequency selective fading than DPSK, 

although DPSK is better than FSK by 3 dB for a flat-flat channel. 

 The maximum signaling rate for either system increases for 

both types of modulation as the order of diversity increases. The 

type of diversity used has a definite impact. In Figures 17 and 

18 the rate increase is due to space diversity requiring the use 

of separate antenna configurations, separate receivers and postdetection combining. The use of such redundant channels is 

denoted as "explicit" diversity. The use of more than four 

explicit diversity levels is generally considered unpractical due 

to system complexity and economic consideration. 

 An alternative scheme utilizes the "implicit" diversity 

inherent in the channel itself. Implicit diversity may be interpreted physically as a time diversity produced by resolving 

multipath components in the dispersive channel. Multipath 

diversity has been used in the RAKE system to reduce fading 

effects in a flat-flat channel as described by Price and Green 

(1958). RAKE uses a spread spectrum carrier and a tapped delay 

line receiver that continuously measures the multipath structure 

and coherently combines the energy from various paths. 

 More recently Monsen (1971, 1973, 1974) described a multipath 

diversity system which employs a receiver having adaptive 

equalization with decision feedback. The system does not require 

a spread spectrum carrier for resolving the multipath components 

as RAKE does. Monsen shows that the dispersive effects of the 

channel can actually improve performance if a proper structure is 

employed to combine the available implicit diversity paths. A 

simulation of troposcatter channels indicates that high signaling 

rates could be realized with this scheme. The rates which could 
52
53
54
55

be realized are three times a rate equal to the inverse of the 

rms delay spread. Using both explicit and implicit diversity, 

the system performance, in terms of error rate and signal power 

required, approximates the performance attained in the case of no 

fading. The adaptive equalization technique is apparently only 

applicable to channels with slow time variations relative to the 

delay spread (e.g., a troposcatter channel). Whether this or 

similar techniques could be used in the mobile channel, where the 

frequency dispersion is much greater, is a matter of conjecture. 

 The signaling rates achievable for some rms delay spreads 

considered typical for mobile systems operating in urban and 

suburban environments are illustrated in Figure 19. As noted 

previously, the delay spreads of 0.25 μs and 2.5 μs are probably 

reasonable estimates for upper and lower bounds in almost any area 

where a significant number of reflections exist. 

9. RELATING SIGNALING RATE TO DATA RATE 

 The previous section showed how the signaling rate (or 

binary system data rate) for a given modulation scheme depends on 

a) the delay spread, b) the acceptable error rate, and c) the 

level of diversity. A question still to be answered is how much 

can the data rate be increased for a fixed signaling rate by 

sending more bits per symbol assuming the bandwidth is limited 

and the transmitter power is not limited. 

 Each symbol in a stream of symbols can be made to convey 

more than one bit of information by letting one symbol characteristic assume several distinguishable states or by varying more 

than one characteristic. 

 An example of the effect on one systems data rate is indicated in Figure 20 for an m-ary PSK system assuming Pe = 10-5, L=1 

and Sd = 1 μs. The solid line corresponds to a PSK system with 

R/B=2 and the dashed line to R/B=8. It is apparent that increasing the maximum data rate by a factor o£ four over coherent 

bandwidth limited channels for this system requires a substantial 

increase in received power to maintain the same error rate. 
56
57
58

It was noted previously in Section 2 that, for this reason, there 

is not much point in letting R/B be greater than approximately 10. 

 Maximizing the packing ratio within these limits is 

obviously one way to increase data rate as long as the signaling 

rate is such that no selective fading occurs. Such a signaling 

rate means that the occupied bandwidth does not exceed KBc. If 

the allocated bandwidth Ba is greater than KBc, then one way the 

data rate can be increased by occupying more bandwidth while not 

increasing the signaling rate is to utilize additional transmitting frequencies. The number of adjacent channels that can be 

used depends on the bandwidth occupied by the signal in each and 

on the filter design required to separate the channels. 

 If there are n frequencies transmitted one at a time with m 

types of symbols per frequency, then the total information 

conveyed is log nm bits/symbol. If each channel is KBc wide 

then the total bandwidth Ba required is nKBc (assuming ideal 

filtering) and therefore 

n

nm R Ba

log / = (9-1) 

which reaches a maximum when n=3. 

 Values for n=Baτ vs. Rτ = log mn have been plotted in 

Figure 21. Here it is assumed that the n frequencies are separated by l/τ and ideal filters are used. The actual data rate 

achieved in a practical system using non-idealized filters would 

be less depending on the response of the filters employed. Note 

that information can be transferred when m=l as long as n is 

greater than 1. 

 As an example, assume that a 50 kHz bandwidth has been 

allocated for a mobile link in a suburban environment where 

Sd = 0.25 μs. The maximum undistorted signaling rate from 

Figure 19 is about 10 k symbols/s for a DPSK system with no 

diversity (L=l). Therefore 

5

10

50 = = kk

n (9-2) 
59
60

 Using Figure 21 the maximum rates achieved for an m-ary PSK, 

n-ary FSK system can be determined as a function of m and n. 

Results are tabulated in Table 8 for n=5. Note, however in 

Table 8, that about 10 times as much power is required to maintain 

the same error rate when m increases from 4 to 64 with a corresponding increase in data rate from 43 b/s to 83 b/s. 

10. ESTIMATES OF THE MAXIMUM SIGNALING RATE 

FOR VARIOUS TYPES OF LINKS 

 Previous sections have shown how signaling rates may be 

limited by dispersive mechanisms in the channel. Increasing the 

signaling rate beyond a certain limit merely increases the number 

of errors observed. These errors cannot be overcome by increasing 

signal power. 

 However, at an acceptable undistorted signaling rate, the 

data rate can be increased almost indefinitely if sufficient 

power is available. This is accomplished by increasing the 

number of information bits conveyed by each symbol. 

 Estimates of the maximum signaling rate in symbols/s, have 

been derived for certain systems operating over channels where the 

rms delay spread varies from 100 ns to 10 μs. 

 The actual spread observed depends on a particular type of 

link. In Figure 22, the estimated maximum signaling rates for 

various types of binary links are illustrated. The data used to 

develop Figure 22 were obtained from a variety of sources which 

are given in Table 9. 

 These results do not include systems employing wideband 

carriers, e.g., spread spectrum systems. Even when multipath 

conditions exist a wideband carrier with bandwidths 100 to 1000 

times the coherent bandwidth may, with a properly implemented 

receiver, resolve individual propagation paths. Since these 

individual paths are extremely broadband, when the receive 

process isolates one path, the coherence bandwidth limitation 

no longer exists. 
61
62
63

Table 9 

Data Sources for Figure 22 

Channel Source Reference

EHF & SHF Medium Dispersion Ma and Dougherty (1976) 

Roche, et al. (1970) 

 

SHF Atmospheric Turbulence Bello et al. (1973a) 

SHF Atmospheric Multipath Dougherty and Hartman (1977) 

UHF-SHF Troposcatter with 

 Adaptive Equalization 

Monsen (1971, 1973, & 1974) 

 

UHF-SHF Ionospheric 

 Scintillation 

Nesenbergs and Ott (1974, 

 Private Communication) 

 

UHF-SHF Troposcatter Schartz, et al. (1966) 

Bello (1969a, 1969b) 

 

UHF Aeronautical Frasco and Goldfien (1973) 

Sutton, et al. (1973) 

Schnieder (1973) 

 

UHF Land Mobile 

 Urban and Suburgan 

This Report (Fig. 19, L=1) 

 

VHF Ionoscatter Schwartz, et al. (1966) 

HF Skywaves Schwartz, et al. (1966) 
64

11. COMPARING WIDEBAND TACTICAL NETS WITH 

FM (30-76 MHz) SYSTEMS 

 It would be useful to compare the results presented here 

with results obtained from actual wideband systems operating at 

high data rates comparable with those which appear realizable. 

Unfortunately, such data are severely lacking for mobile links. 

There are, however, some data on the use of VHF-FM radios in 

Army Tactical Data Systems where a number of sets share a common 

frequency in the 30 MHz to 76 MHz band with a 50 kHz channel 

spacing. A summary is given by Tucker (1972). 

 The operational characteristics of typical FM tackical radio 

equipment are as follows* 

Equipment 

Descriptor 

Transmitter 

Power 

(watts) 

Frequency 

Rated 

(kHz) 

Deviation 

Used 

(kHz) 

Necessary 

Bandwidth 

(kHz) 

AN/PRC-25 2 to 3 ±8 ±5 16 

AN/VRC-12 35 +8 +5 16 

A center-fed whip antenna is normally used with no diversity. 

Tests were conducted by Salava (1970) using a FT-524/VRC-12 radio 

set having a 3 kHz bandpass characteristic for voice band and 

17 kHz bandpass characteristic for the wideband mode. Data on 

error rate vs. signal-to-noise ratio were obtained at 300, 600, 

and 1200 bits/s over ranges from 2 miles to 22 miles∗

. The 

average bit error rate varied inversely as the ratio. This 

relationship would be expected for a Rayleigh fading channel or 

additive impulsive noise. At the maximum rate of 1200 bits/s an 

irreducible error plateau was not apparent at least within the 

minimum error rates presented (Pe≈l0-6). The multipath disturbances which were experienced were attributed by Salava to low 

flying aircraft in the vicinity of the test area and possibly 

passing vehicles. 

 Similar tests were performed by Ebisu and Leen (1971) at 

120 b/s using FSK modulation over non-line-of-sight paths where

 

∗ Private communication from Electromagnetic Compatibility Analysis 

Center, Annapolis, Maryland.
65

the intervening terrain varied from relatively flat to hilly. 

Analysis of the errors attributable to fading was difficult to 

quantify because other error-causing factors (lightning and 

ignition noise) were also present. At these rates, however, one 

would expect the fading to be flat, i.e., non-frequency selective. 

 Tests at higher data rates of 19 k bits/s were performed by 

Mikulski (1969) of Motorola Inc. using the same AN/VRC-12 equipment. An irreducible error rate at approximately 2xl0-4 was 

attained but this was attributed primarily to the response of the 

radio set operating in the wideband (17 kHz) mode. This error 

rate was deemed acceptable for digitized voice transmissions. 

During periods of signal distortion due to multipath Mikulski 

experienced difficulties in maintaining bit synchronization. A 

movement of just a few feet caused the error rate to change from a 

low value to 5 x 101

. The multipath spread is unknown. This 

effect may be attributed to obstructions momentarily appearing in 

the path which suppress the dominant path as described in Section 

3.3. Multipath effects caused by aircraft were observed at 

ranges exceeding 16 miles (26 km). Multipath from other sources 

appeared to be a significant source of errors at short ranges for 

the 19 k bit/s system. 

 Note that a 19 k bit/s irreducible rate falls between an rms 

delay spread of 0.1 μs to 1 μs (Figure 18) for the single 

diversity systems and assuming Pe = 105

. 

 Bussgang and his colleagues (1976) have considered these 

results in developing design criteria for a VHF tactical simulator 

for land mobile links. Considering Milulski's results, Bussgang 

notes that at 19 k bits/s, the multipath spread could become a 

substantial portion of the signaling element duration assuming the 

signaling rate equals the data rate, then a 10% spread would be 

5.3 μs at 19 k bits/s. 

 Conventional FM carrier modulation techniques waste bandwidth. The packing ratio is reduced by an amount which depends on 

the modulation index used. Packing ratios are typically on the 

order of 0.1 bits/s/Hz. See Bello (1969b). The RF bandwidth 
66

occupied by the signal must at the same time be less than the 

coherent bandwidth of the channel to eliminate selective fading. 

Even in a flat fading channel most errors occur when the system is 

below FM threshold. Thus the use of digital modems on 

conventional FM carriers appears to be an inefficient way to 

transmit data. 

 The use of spread spectrum techniques also reduces the 

packing ratio (R/B < 0.01 bits/s/Hz) and suffers the same limitation of RF bandwidth where conventional reception is used as 

noted by Bello (1969b). However, if a RAKE type of receiver 

structure is used, the RF bandwidth no longer needs to be confined 

to be less than the correlation bandwidth. 

 When a system is dealing with many users rather than one, 

as in the land mobile service, the spread spectrum technique 

appears to have expanded capabilities over conventional modulation 

and spectrum accessing techniques. The number of simultaneous 

users in a cellular system using spread spectrum techniques far 

exceeds the number possible in an FM system. In addition to the 

high simultaneous usage rate the spectral efficiency is 

substantially improved. See Eckert and Kelly (1977). Spread 

spectrum carrier modulation appears to be useful in many other 

multiple access applications. 

 With more conventional systems when sufficient power is 

available, maximum data rates are achieved using a combination 

modulation scheme of multiple PSK and transmitting at different 

frequencies to increase the packing ratio as described in Section 

9. 

 As an example, assume that a standard channel bandwidth of 

50 kHz has been allocated for the FM carrier system where the 

packing ratio is typically 0.1 b/s/Hz for a modulation index of 

3. The maximum expected rate in the allocated bandwidth is 

therefore 5.0 k b/s. However, if the link is dispersive and 

frequency selective at this rate, then a lower signaling rate 

must be used to avoid a high error rate caused by distortion. 

Thus the actual data rate may be considerably less than 5 k 
67

bits/s. Although no actual data are available, 1 k bits/s seems 

reasonable for suburban areas. 

 This can be compared with the data rates achievable with an 

m-PSK, n-FSK system designed to utilize fully the allocated 

bandwidth. For n=5 and m=8 the data rate is found to be 53 k b/s 

which is an order of magnitude better. 

12. SUMMARY AND CONCLUSIONS 

 Communication system designers are faced with an everincreasing demand for higher and higher data rates over radio 

channels. Systems utilizing such channels are adversely affected 

by frequency selective fading which perturbs the signal in 

unusual and ever-changing ways. At high rates this so-called 

"multiplicative noise" can be the primary source of errors rather 

than the omnipresent additive noise. This is because the unpredictable distortion, both amplitude and phase, and intersymbol 

interference caused by the channel ultimately limits the number 

of distinguishable characteristics of the signaling elements or 

symbols. Distortion thereby limits the rate at which a sequence 

of signals can be transmitted with an acceptable number of errors. 

In binary coded systems where only two symbol characteristics are 

changed (the minimum number required for transferring information), this signaling rate equals the data rate. 

 Even in an idealized system there are two physical limitations to the capacity of a communications channel—bandwidth 

and noise. These determine respectively the signaling rate, or 

number of symbols transmitted per unit time, and the information 

content, or number of information bits transmitted per symbol. 

Together, this rate and content determine information rate on a 

time basis. 

 Shannon's ideal capacity formula was used in Section 2 to 

define interrelationships between data rate, bandwidth, noise, 

and signal power for an idealized system. This relationship was 

plotted in a form different from the one commonly used. The 

curves for the idealized system show that the capacity is 
68

directly proportional to signaling power for a constant noise 

power spectral density and for a constant packing ratio--the data 

rate per unit bandwidth. Similar curves for certain practical 

systems operating over selective radio channels are given in 

Section 8, which show that the signaling rate or binary data 

rate is not directly proportional to signal power when signal 

distortion occurs in the channel. In fact, when distortion does 

occur, those rates are independent of the signal power. Most of 

the report discusses the causes for this distortion and its 

effect on the rates achievable with uncoded binary systems. 

 In summary, it is shown that the distortion (and intersymbol 

interference) is caused by the occurrence of multiple transmission 

paths between the transmitter and receiver terminals. This 

multipath exists because of scattering, reflecting and refracting 

mechanisms in the propagation medium. At high signaling rates, 

the multipath causes frequency-selective fading which distorts 

digitized signals. Even when individual paths are resolved, 

some distortion occurs due to medium dispersion and atmospheric 

turbulence. These individual paths however are extremely 

broadband. 

 The maximum undistorted signaling rate which can be achieved 

depends on the impulse response of the channel. The impulse 

response varies according to a number of factors including: 

• operating frequency 

• type of link and its geometry 

• geographical location 

• meteorological conditions 

• antenna directivity and pointing angle. 

 Given the impulse response or its statistical representation, 

then the maximum achievable signaling rate depends on: 

• user's reliability requirements or error rate 

• type and level of diversity 
69

• signal design 

• modulation scheme 

 The maximum undistorted signaling rates have been given for 

a variety of links and for their responses. If this maximum rate 

is exceeded, then the information content of the signaling 

elements cannot be increased by conventional means. If this 

maximum signaling rate is not exceeded, then the informationcarrying capacity of the channel can be increased by increasing 

the information content of each signaling element. This, however, 

may require complex implementation schemes and an excessive 

amount of power. Even for an idealized system, Shannon's 

capacity equation shows that it is extremely impractical to send 

more than 10 bits per cycle of available (allocated) bandwidth 

because of the power required. Since the available bandwidth of 

channels is limited to about 10% of the carrier frequency for 

technological reasons (Section 1), this means that the maximum 

achievable data rate for systems using the electromagnetic 

spectrum is about equal to the carrier frequency. Some typical 

rates in use today are compared with this limit in Figure 23. 

These are gross estimates and can vary widely depending on path 

length, transmitter power, antenna gain, reliability and system 

design. 

 The technologically limited region shown in Figure 23 

assumes the bandwidth is 10% of the carrier frequency with an 

upper bound of 10 bits/s per unit of bandwidth and a lower bound 

of 1 bit/s per unit of bandwidth. The actual rates for typical 

systems fall well below these bounds because allocated bandwidths 

are not necessarily 10% of the carrier frequency or because of 

dispersive mechanisms in the channel. Also, maximizing the 

number of bits per hertz of bandwidth has not been an urgent 

consideration because bandwidth has been readily available 

particularly in the higher spectrum bands. 

 The principal conclusions reached from the results of this 

study and literature research program are 
70
71

1. Dispersive mechanisms in a radio propagation channel 

can, at certain signaling rates, cause selective 

fading which distorts the signal and limits the 

maximum and minimum signaling rate. These limits 

cannot be overcome by increasing transmitter power. 

The maximum rate is limited by time dispersive 

mechanisms which cause frequency selective fading. 

2. Where signaling rates are reduced so frequency 

selective fading does not limit performance, then 

the data rate can be increased by maximizing the 

number of information bits transmitted by each 

symbol. 

3. Where channels are bandlimited by allocation or for 

technological reasons, the data rate can be increased 

by increasing the packing ratio. The maximum data 

rate per hertz of bandwidth should probably not 

exceed 10 bits/s/Hz due to the excessive power 

required. 

4. The impact of frequency selectivity on a channel is 

a function of the response of the channel, the signal 

design, and the rate of change of that signal. Channel 

responses depend upon the link being used and 

the propagation conditions for that link. 

5. The response in the time domain, i.e., the impulse 

response of a channel is a measure of the transmission 

capabilities for that channel. Statistical 

representations of the responses of several 

channels for a given type of link can be used to 

determine system performance over that link. 

6. The measured response for mobile links indicate 

maximum signaling rates may vary from 103

 symbols per 

second to 106 symbols per second depending on the 

environment, the modulation scheme, level of 

diversity and the acceptable error rate. Some 

frequency effects may also occur due to varia- 
72

 tions in the reflection coefficients with frequency 

and due to the attenuation of trees and foliage 

at high frequencies. 

7. The maximum achievable data rate over a channel 

with a fixed signaling rate depends on the allocated 

bandwidth and transmitter power. For a mobile 

link with a 50 kHz allocated bandwidth and 

using quadrature PSK in conjunction with 

multiple frequency shift keying, the maximum 

data rate was estimated to be 43 k bits/s in 

suburban areas, and 7.6 k bits/s in urban areas. 

Estimates for military tactical nets using FM 

carrier modulation were about 1 k bits/s for the 

urban area. 

8. Since the signals arriving via individual paths in a 

multipath channel are extremely broadband, it 

appears feasible to structure a system which combines 

each path coherently and thereby takes advantage 

of this broadband feature. This adaptive 

equalization process greatly increases transmission 

capability but requires that the path characteristics 

change relatively slowly compared to other system 

response times. 

9. Preliminary impulse response measurements at 

8.6 GHz using high resolution equipment indicate a 

considerable difference in the delay profiles 

from those obtained at lower frequencies. This 

is attributed to a lessening of the effects of 

diffraction around obstructing objects, the 

variation in attenuation caused by smaller objects 

such as leaves on trees, and differences in the 

reflection coefficient of reflecting objects. 
73

13. A RECOMMENDATION 

 The signaling and data rates for various links and channel 

responses estimated in this report are based on certain assumptions and a limited amount of data. One's confidence in these 

estimates would be greatly enhanced if comparisons could be made 

with actual performance data. Unfortunately, such data are 

severely lacking. Several workers have measured responses on 

radio channels and others have measured the error rates of 

specific systems over radio channels. Seldom, however, have both 

response and performance been measured simultaneously and at rates 

where selective fading occurs, in spite of the fact that both 

measurement techniques are well established. 

 For some types of links in the future, some people expect a 

more stringent channelization of spectrum bandwidth as filter 

technology advances. See Cuccia (1976). This coupled with the 

increasing demand for higher data rates, implies a need for 

signal designs and modulation techniques that maximize the 

number of bits that can be transmitted on a per-unit-of-bandwidth 

basis. For other types of links, less channelization may be 

better. Multiple access systems for example. Spread spectrum 

system technology could provide substantial improvement over 

conventional (FM) modulation and accessing techniques for the land 

mobile services. See Eckhert & Kelly (1977). 

 Designers of such systems will need a comprehensive data 

base to characterize the channel and its effect on various 

signals. As time goes on, it becomes more and more difficult to 

obtain this data base because it requires broadband measurements 

over clear channels. Already portions of the spectrum are 

overcrowded with users making it difficult to make broadband 

impulse measurements and wide band performance measurements 

without encountering man-made interference. This is particularly 

true in the mobile area where already the lower bands are filled 

with users in every city. Nielson (1975), for example, was 

unable to measure the impulse response in the 400 MHz land mobile 

band because of the intense interference which existed in the 

San Francisco Bay Area. 
74

 Because a firm data base is so essential to the design of 

advanced systems making maximum utilization of the limited 

spectrum, it is recommended that measurement programs be 

instituted to characterize various channels (at frequencies and 

locales that are still clear) as soon as possible. This program 

should include both additive noise and response data (either in 

the frequency domain or the time domain) over a variety of links 

and channels. Impulse response measurements should use the 

highest practical resolution. 

 Continuous recordings of the response and the noise should, 

after processing, be capable of providing appropriate measures of 

the time and frequency dispersiveness of the channel and a 

statistical representation of the noise. 

 At the same time, the error performance of a system should 

be measured at various keying rates and using a convenient 

modulation scheme such as quadrature PSK so that performance 

predictions can be verified. 

 Throughout this report, the additive noise has been assumed 

to be white with a constant noise power spectral density. This 

of course is often not the case and particularly not true for the 

mobile environment where the primary error-causing additive noise 

is more likely to be man-made (either ignition or power-line 

noise) with impulsive properties or interference from adjacent 

channels. Measurement programs which are designed to characterize 

the channel should include additive-noise measurements commensurate with the response and performance measurements, and, an 

attempt should be made to resolve the error-causing effects. 

 An accurate wideband characterization of the multiplicative 

and additive noise in the radio channel is essential to the 

future development of advanced communication systems. Statistical 

models based on experimental data are also useful for simulation 

experiments in order to avoid costly hardware tests of ad hoc 

systems. An example of this latter application is given by Suzuki 

(1977) who utilized experimental data to establish a statistical 

model for the urban radio propagation medium. 
75

14. ACKNOWLEDGMENTS 

 The author wishes to thank D.D. Crombie and J.A. Hull of 

OT/ITS for suggesting the study; J.A. Zoellner and R.A. Whiteman 

of ECAC for their interest and support; many associates at ITS 

and particularly W.J. Hartman and R.W. Hubbard for helpful 

discussions; L.E. Pratt and C.H. Minister for the impulse response 

photographs in Section 5; J.M. Harman for drafting the figures; 

E.L. McCoy for typing; Evelyn M. Gray for content review; and 

H.T. Dougherty, G.A. Hufford, and M.C. Thompson for technical 

review. 
76

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