SPATIAL DISTRIBUTION OF STATIONS
TIME AND FREQUENCY DISTRIBUTION OF TRANSMISSIONS
EQUIPMENT AND PROPAGATION CHARACTERISTICS

oe

COMPUTE PROBABILITY
DISTRIBUTION OF
WANTED SIGNAL POWER

AVAILABLE AT A I
RANDOM RECEIVER COMPUTE PROBABILITY
S, dBW DISTRIBUTION OF
CO-CHANNEL INTERFERENCE
Ig , dBW I
ang COMPUTE PROBABILITY
DISTRIBUTION OF
SUM TO GET DISTRIBUTION NON-CO-CHANNEL
OF TOTAL INTERFERENCE, INTERFERENCE
LZ ike Io N Tg, dBW
1019 = 10! + 10+ 101°
af
ASSUME DISTRIBUTION OF
RADIO FREQUENCY NOISE
—-
N,dBW
a |
COMPUTE PROBABILITY DISTRIBUTION
OF SIGNAL-TO-INTERFERENCE RATIO
AT RECEIVER INPUT
S-I dBW
WW
OUTPUT TABLE OF PROBABILITY THAT
S/I <£Th

FOR SELECTED SET OF THRESHOLDS, Th

Figure 3. Block diagram of program for computing the probability of interference
in a congested environment.

14


4. RELATING CLASS A NOISE TO CONGESTED BAND SCENARIOS

The analytical solution for the probability of communications requires the
three parameters of Class A noise as input. Middleton (1979a,b) suggests that
these parameters must be determined by measurements and recommends measurement
procedures. However, it will be several years (at least) before an adequate set
of measurements is available. Another practical problem is that the probability
of interference often must be computed before deploying new systems. In this
case, it is impossible to measure the parameters of the Class A noise that will
exist after the systems are deployed. So the ability to compute Pa from a specifi-
cation of the characteristics and locations of interfering systems is necessary.
This section explores the possibility of making this calculation with the analyti-
cal model.

4.1 Interpretation of Factors in Class A Noise Formula

Equation (22) gives the probability density function of noise (plus interfer-
ence) for an undisciplined service for the same assumptions made by Middleton
(1972, 1976, 1979a). For M=~, it is

. © f.-u uTM .
(iq) = J [e a | Frjm(igim) » (34)
m=0
where U is the traffic intensity defined by equation (20). The term in the square
brackets is the probability that there will be exactly m simultaneous interfering
transmissions at any particular instant of time, and Film is the pdf of noise

given that there are exactly m interfering transmissions.
Now recopy the formula for Class A noise, (2):

w, (i) = t [e* elt exp(~i9,/7,) : (35)
m=0 n
where A is the traffic intensity perceived in the receiver at the output of the IF
filter. Notice the similarity of (34) and (35). If every transmission by an
interferer is observed by the wanted receiver, then A = U, and the factor in
square brackets in (35) is identical to the corresponding factor in (34). It is
interesting to consider whether the other factor in (35) can be identified as the
probability density function of interference power given that m transmitters are
transmitting for some distribution of transmitters and some propagation law.
Consider the term for m=0. The factor multiplying the square brackets is

35


“UOLINGLAPSLP BY} SUaPeOUg BdURLUeA OUBZUON “SSO| UOLSSLUSUeA. JO
AYLLLGELACA YNOYZLM pue YILM AaMod [eUBLS paALadad Jo UOLJOUNy AZLsUapP APLLLGeqoug “OL aunbLy

MaP ‘S
08- 06- 001- 0el- Ogi
T Y T if
- ~(s00
\ I
LU \ |
[- 1 ool oro
\ \
|
F \ | ae ovo
apo=% ——— \ (s)§
| apg= So —— \ | 020
eve \ |
Vy
- i) 820
: oro

31


b, /10
1 _y iv
10 =, mar * (47)

Dividing (46) by (47), eliminates in? and

by /10-b, /10

-—l0 —
B= 7" Bo/TO-by/TO\ (48)
a(-r0 )
Then
b /10 b,/10
7, = (1-8) 10° + Alo 1 . (49)

From Figure 13, by = -132 and b, = -74.6, so T = 7.28(107°) and i, = 8.67(107°).
Figure 14 compares the Class A noise pdf for these parameter values with the pdf of
noise and interference computed with the numerical model. The dashed line in
Figure 14 is copied from Figure 13 for easy comparison.

The Class A peak near -75 dBW is much higher and narrower than the peak com-
puted from the scenario (dashed line). If the conjecture in Section 4.1 is accepted
for the moment, this difference is probably due to the difference between the
geographic distribution of interferers assumed in the computation as given by (43)
and the distribution implied by the form of Class A noise, as given by (38). These
two different pdfs of interfering path lengths are shown in Figure 15. Curve B is
the path-length pdf given by (38) for this scenario, as discussed in Section 4.1.
Because there are almost no interferers closer than 4 km for Curve B, there is
virtually no probability of interference stronger than -65 dBW in Curve A of Figure
14. However, for Curve A of Figure 15, there is substantial probability of inter-
ferers located .1-4 km from the receiver. This explains the long tail above -65
dBW on the scenario pdf in Figure 14. The curve-fitting procedure forces both pdfs
to have the same area under the peaks on the right, so the narrower peak must be
higher.

At this point it is possible to test the accuracy of the interpretation of
Class A noise given in Section 4.1. Equation (34) (the computer model) was eval-
uated with the same parameters as in the scenario immediately above, except that
the pdf of interfering path lengths was Curve B of Figure 15 instead of Curve A;
that is, the pdf implied by the interpretation of the m=] curve was used. (In
addition, the standard deviation of transmitter power and the standard deviation of
transmission loss were both reduced to 1 dB to simulate the deterministic nature of

40


The distance between the wanted transmitter and the receiver is a random vari-
able, d. It is convenient to use a normalized distance,

dy = d/d. (9)

where d, is a reference distance related to the variance of the pdf of d. This pdf
is

= 2 2 f
fy(dy) =2 dy exp(-dy - va) 1,(2dy "oa) > dy >0 . (10)

In (10), "oa = d./d, > where 4d, is the distance between the receiver and the center of
Operations of the mobile transmitter. Middleton assumes that the mobile trans-
mitter performs a random walk around this center of operations. Naturally, "oa

may be zero. I(x) is the modified Bessel function of the first kind (Abramowitz
and Stegun, 1964).

Figure 1 shows the geometry of the transmitter, T; the receiver, R; and the
center of operations, C. Figure 2 illustrates the form of the path length pdf in
(10). In Curve A, dq, = 1 and "oa = 0. Curve B is for Toa = 0, but now the refer-
ence distance dy = 3. Comparison of curves A and B shows how dy can be used to
shape the path length pdf to fit the desired coverage area. Curve C in Figure 2 shows
the path length pdf for a center of operations 4 km from the desired receiver.

With these assumptions, the probability that the signal-to-noise ratio exceeds
the required signal-to-noise ratio, r, is (Middleton, 1979a)

. “AS an
P, = P(s/i, 2r) =1-e t mr Pit)» (11)
m=0
where
P(r) = { 2y exp(-y? - r?_ - mtn) Ij(2rgay) dy (12)
me oa” Thr (ayy)

Thus, (11) is an infinite series of definite integrals, which can be summed using
computer routines given by Middleton (1979a). (Although this is not an "analytical
solution" in this form, Middleton shows that (11) can be reduced to a triple infi-
nite series.)

Recapitulating, the input parameters required to compute Pa with Middleton's
(1979a) analytical solution, (11), are

(1) the Class A noise parameters that define the interference environment
(a) A, the overlap index,


99.9 - 7

9

80 [-

a

P(Y2R), %

7 ——=- Scenario Interference

Class A Noise

| | | | | | | |
50 =-40— 30 -20—-10 0 10 20 30 40
y,dB

Figure 20. Comparison of the probability of communications, Pas for the
scenario (dashed line) with Pg computed using the best-fit
Class A noise.

50


*(@ULL PLLOS) asou y sse19 IL4-3S9q
ULM (BULL paysep) OLueUadS puOdaS WOAy adUaVaJuaqUL-SN{d-aslou JO sypd Jo UOSLAeduiO) “6, auNnBLy

Map ‘x
09- 08- 001 - 02I- Obl- 09I-
2 i _ rt
“XN
z 100
= 200
= £00
(X)}
| - +00
asioN y ssD|ID ——
O|4DUaIgG
= c00

900

48


P(S-I 210), %

100

20

T T I

80}- 7
60}-— TT
401- ~
20;- 7
0 l |

0 5 10 15

M
Figure 23. Probability that the SNR exceeds 10 dB as a function of the

number of potential interferers, M. The points are taken from
Figure 22.

54


Berry, L. A. (1977), Probabilistic tradeoffs for efficient spectrum use with a "CB"
example, OT Report 77-117 (National Technical Information Service, Spring-
field, Virginia, 22161, PB 266669, $4.50).

Berry, L. A., and E. J. Haakinson (1978), Spectrum efficiency for multiple inde-
pendent spread-spectrum land mobile radio systems, NTIA Report 78-11 (NTIS
Access. No. PB 291539/AS).

Kleinrock, L. (1978). Queueing Systems: Volume I: Theory (John Wiley & Sons, New
York, N.Y.).

Longley, A. G. (1976), Location variability of transmission loss-land mobile radio
and broadcast systems, OT Report 76-87 (NTIS Access. No. PB 254472/AS.

Middleton, D. (1972), Statistical-physical models of urban radio-noise environ-
ments--Part I: Foundations, IEEE Trans. Electromagnetic Compatibility, EMC-
14, No. 2, May.

Middleton, D. (1974), Statistical-physical models of man-made and natural radio
noise, Part I: First-order probability models of the instantaneous amplitude,
OT Report 74-36 (NTIS, Springfield, VA 22161, Access. No. COM 75-10864/AS).

Middleton, D. (1976), Statistical-physical models of man-made and natural radio
noise, Part II: First-order models of the envelope and phase, OT Report 76-86
(National Technical Information Service, Access. No. PB 253949/AS).

Middleton, D. (1979a), Performance of telecommunications systems in the spectral-
use environment, Part V: Land-mobile and similar scenarios in Class A inter-
ference, NTIA-CR-79-4 (NTIS Access. No. PB 80-112097).

Middleton, D. (1979b), Procedures for determining the parameters of the first-order
canonical models of Class A and Class B electromagnetic interference [10],
TEEE Trans. Electromagnetic Compatibility, EMC-21, No. 3, 190-208.

Okumura, Y., E. Ohmori, T. Kawano, and K. Fukuda (1968), Field strength and its
variability in VHF and UHF land mobile radio service, Review of the Tokyo
Electrical Communication Laboratory, 16, No. 9-10.

Zehna, P. W. (1970), Probability Distributions and Statistics (Allyn and Bacon,
Inc., Boston).

58


999

99

95

80
R to
50

20 4
So 4
——- Scenario Interference
Class A Noise
0.1 +
| | al | | | | |
-70 =-60 = -50 -40 -30 = -20 -10 0 10 20
y, dB
Figure 17. Comparison of the probability of communications, Pa, for the

first scenario (dashed line) with P, computed using the best-

fit Class A noise.

46


Table 3. Probability that i (Out of Possible m) Transmitters Assigned
to a Channel are Transmitting

(U is the total channel utilization, U=(J2/60) m, where J is number of transmissions
per hour by a transmitter and & is average transmission length in minutes.)

U, Channel Utilization
i 0.25 0.5 1 2 3 4 5 10
0 .7788 -6065 -3679 -1353 0498 -0183 -0067 4.54(1 07)
1 .1947 -3033 -3679 +2707 +1494 .0733 -0337 - 0005
2 +0243 -0758 +1839 -2707 +2240 -1465 - 0842 -0023
3 .0020 -0126 -0613 +1804 -2240 -1954 1404 -0076
4 -0001 -0016 -0153 -0902 -1680 -1954 1755 -0189
5 -0000 -0002 -0031 -0361 - 1008 -1563 +1755 0378
6 - 0000 +0005 -0120 ~0504 -1042 -1462 -0631
7 -0001 - 0034 -0216 -0595 - 1044 -0901
8 -0000 -0009 -0081 -0298 -0653 -1126
9 - 0002 -0027 -0132 -0363 1251
10 -0000 -0008 -0053 -0181 1251
anf -0002 -0019 -0082 +1137
12 -0001 - 0006 - 0034 -0948
13 - 0000 -0002 -0013 -0729
14 -0001 +0005 .0521
15 -0000 -0002 -0347
16 -0000 -0217
7 0128
18 -0071
19 -0037
20 -0019
21 -0009
22 -0004
23 -0002
24 -0001
25 -0000

18


a

0

7

9
exp(-i 2 . (36)
n n

Using (3), (4), and (6), (36) can be reduced to

exp(-i,/n) (37)

a4

which is recognized as the pdf of an exponential distribution with mean n (Zehna,
1970, p. 142). Whenever it is assumed that the pdf of background noise, Frlo> is
given by (37), then the first terms of (34) and (35) are identical.

For m=1, the coefficient of the square brackets in (35) can be shown to be a pdf
of interference from one interferer. To do so, assume that the pdf of path lengths
from interferers to the wanted receiver is given by

wylr) = 2y pyarTM2¥! exp(-rpy/r?¥) (38)

where
A= g,/7, - (39)

Then, if the propagation law is given by equation (8), the interfering power from
one interferer is

te pyre . (40)

Applying the transformation in Table 2, the probability density of interfer-
ence from exactly one interferer is

9 _
. -d i _—
Fy Gl) = de An = 7 o7PC-m ifi,) > (41)

which is the coefficient of the square bracket for the m=] term in (35). Thus,
the m=1 terms of (34) and (35) are the same provided that the probability density
distribution of interferers is given by (38). This is the reason for choosing the
form of (38). Middleton (private communication, 1980) has shown that the strictly
canonical Class A noise model is exact when (38) is the pdf of source distance.

The spatial distribution of interferers is fixed by (38). If there are two
interferers, the pdf of interference power from each is given by (41), and inter-
ference power from two interferers is the sum of two random variables with this
pdf. Using the addition formula in Table 2, this pdf is

36


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

14.

16.

17.

20.

21.

22.

23.

A-1.

The probability density function of noise-plus-interference
for the first scenario in Section 4.2. The inset is the
interference portion of the pdf with an expanded vertical
scale. Most of the interference is due to one simultaneous
interferer... .... Se ee ee

Comparison of the scenario pdf (Curve B) of Figure 13 with
the best-fit Class A noise approximation (Curve A). The
scenario has higher probability of strong interference.
Curves C and D are fits for different values of igs see

Comparison of the pdf of interfering path lengths in the
scenario (Curve A) and the pdf of interfering path lengths
implicit in the m=1 term of Class A noise (Curve B), for the
best-fit curve of Figure 14. Curve B shows very small prob-
ability of paths shorter than 4 km. 2... 2 2 ee eee eee

Class A noise intensity pdfs for several sets of parameters.
The curves are attempts to fit the scenario interference
which is shown as the heavy dashed line... ...... we

Comparison of the probability of communications, Pag, for the
first scenario (dashed line) with Pa computed using the best-
fit Class Anoise........- rar

The probability density function of noise-plus-interference
for a scenario with interference only slightly stronger than
the background noise. ....... ars see

Comparison of pdfs of noise-plus-interference from second
scenario (dashed line) with best-fit Class A noise (solid
Tine) 2 ee ee ee ee ee .

Comparison of the probability of communications, Pg, for the
scenario (dashed line) with Pa computed using the best-fit
Class Anoise... 1... ee ee rrr)

Probability density function of noise-plus-interference for
different numbers of TR links assigned in a circle of radius
10 km. Each link has a traffic intensity U'=0.1.......

Probability that the signal-to-interference ratio exceeds a
threshold R, for the scenarios of Figure 21.....-+.+--

Probability that the SNR exceeds 10 dB as a function of the
number of potential interferers, M. The points are taken
from Figure 22. .... ee we we ee ee es

The pdf of interfering power, in watts, from exactly one
simultaneous interferer. Curve B is the pdf computed for the
scenario; Curve A is the canonical form of Class A noise.
Notice that the curves are on a log-log scale. ......--

The pdf of interfering power, in dBW, from exactly one simul-
taneous interferer. Curve B is the pdf computed for the
scenario; Curve A is the canonical form of Class A noise. . .

39

41

42

44

46

47

48

50

51

52

54

62

63


received signal power is recorded on a fixed path, it varies with time as atmo-
spheric conditions change. Signals over different paths of the same lengths vary
even more because of differences in terrain and man-made structures (see, for
example, Barsis, 1971; and Okumura, et al., 1968). This means that received signal
power is a probabilistic function of distance with a mean value that may vary with
distance like (8) and an approximately log-normal distribution at a fixed distance.

Figure 9 shows how realistic variability in transmission loss affects the
probability of communications. The mean value of received power is given by (8)
with y = 1. The received power in dBW at a fixed distance is assumed to have a
normal distribution with a standard deviation of 8 dB, a value appropriate for VHF
propagation in nonurban areas (Longley, 1976). The dashed lines in Figure 9 are
copied from Figure 8 to facilitate comparisons.

The most obvious effect of the variability in transmission loss is to smooth
out the plateau in Pa: The variability of transmission loss for any given path
length obviously broadens the distribution of received signal strength so that it
bridges the gap between the two bumps in Figure 4. This broadening is shown in
Figure 10 where the pdfs of received signal with transmission loss variability are
compared to the pdfs of received signal power without variability.

The important practical effect of transmission loss variability shown in Figure
9 is the lowering of the probability of communications for high transmitter power.
For example, the probability of communications when oa 7 4 and y = 25 dB is 99.7
percent without transmission loss variability, but is only 95.7 percent with vari-
ability. In other words, inclusion of transmission loss variability in the
calculation increases the probability of interference in this region by more than
an order of magnitude--from 0.3 percent to 4.3 percent. The messages lost were
transmitted over paths with transmission loss much greater than average.

Figure 11 compares Pa with and without transmission loss variability for the
more realistic value of y = 2. The plateau in Pa is not so prominent for this
case where there is no variability, so the inclusion of variability does not
change the curves so dramatically. There is still a decrease in Pa for reliabil-
ities higher than 90 percent. However, the curves are close enough together to
justify ignoring transmission loss variability (and using the faster Middleton
model) for estimates of Pa when reliabilities less than 95 percent are acceptable.

3.2 Effects of Variability in Effective Radiated Power

Often the probability of interference for a category of wanted transmitters
is of more interest than the probability of interference for a single transmitter.
In this case, the effective radiated power will vary from transmitter to transmitter

29



be combined with the pdf of distances (path lengths) using the formula in Table 2
to provide an unconditional pdf of transmission loss. The path length pdf can be
specified analytically, as in (10), or numerically. It is often generated off line
by using a real or postulated distribution of transmitter locations and tabulating
the resulting path lengths.

The signal, S, is the transmission loss subtracted from the transmitter power,
so the pdf of the signal available at a randomly selected receiver is computed
using the convolution integral given in Table 2.

A similar calculation produces the pdf for each category of interference.
Different categories are necessary when different pdf's are necessary to describe
any of the input parameters. For example, co-channel and adjacent channel inter-
ferers may have different path length pdf's because of circuit discipline (Berry,
1977), and local and skywave interferers have different transmission loss distri-
butions (Berry, 1978). There also may be an additional transformation (in the
receiver) for interference such as adjacent channel interference or intermodulation
interference (unpublished Technical Memorandum 79-14, "Probability of intermodula-
tion interference in an expanded CB service").

The result of the interference calculation to this point is the pdf of inter-
fering power at a typical receiver given that one randomly selected source in this
category is transmitting. It is possible that two, or more, sources are trans-
mitting simultaneously. If so, the two signals have identical pdf's (that is what
is meant by being in the same category). The pdf of the sum of the two powers must
be found using the rule in Table 2. It is assumed that interfering power from two
sources add incoherently. Similarly, the pdf for three or more interferers can be
computed; interference from different categories of interferers can be added
together; and the sum can be added to the background noise.

To perform these additions, the random variables must be transformed from dBW
to watts. If X is a random variable in dBW, with pdf fy(x), Tet ¥ = 10%/10,
that Y is the corresponding value in watts. The pdf of Y (using the transformation
rule in Table 2) is

so

10 lo
y

S10 &

fyly) = F,(10 logyg y) (14)

After adding the various contributions to the noise, the pdf of the sum, Z, can be
transformed back to dBW. Let V = 10 10949 Z. Then the pdf of V is

v/10)

fy(v) =p 10%/19 #,(10 ; (15)

15


0.35
0.0005

ic= 10710

006 |—

14 >
"

0.01}—

| |
-160 -140 -120 -100
Iy, dBW
Figure 4. Probability function for intensity of Class A noise in dBW for the
parameters shown on the figure. The bump on the left is the prob-

ability density of the gaussian noise; the bump on the right is the
nongaussian noise or interference.

21


In (19),
U=vT , (20)

where v is the average number of transmitter turn-ons per second, and T is the
average length of a transmission.
Substituting (19) into (18) yields

- -u ut 21
P(Iy st) = } e Gp P(Iystlmon) - (21)
0 !

P(Iy < t) is of course the probability distribution of Iy. Taking the derivative
with respect to Iy results in the pdf:

M -u ym
f,(Iy) = neo ear Fy jm(Tylm) : (22)

When 0 transmitters are on, the noise consists entirely of the background or
ambient noise. The pdf of this noise, Frlo° must be supplied as input. When one or
more transmitters are on, there is noise plus interference, and the pdf is computed
using the process described above culminated by equation (15). Thus, equation (22)
is computable and is the pdf of interference-plus-noise for given traffic intensity,
U, and number of potential interfering transmitters, M. Theoretically, M can be
infinite. Table 3 shows values of the coefficients of the pdf in (22) for various
traffic intensities, U. In land mobile radio channels, U is usually less than one.

The pdf of the signal-to-interference ratio can now be computed for the random
variable S-I using the convolution integral shown in Table 2. The probability of
noninterference is the probability that S-I exceeds the required value R and is
given by (13).

Recapitulating, the input allowed in the numerical model is

(1) for the wanted signal,

(a) the pdf of radiated power of the wanted transmitters,

(b) the conditional pdf of transmission loss, given wanted
path length,

(c) the pdf of wanted path length;

(2) for each category of interferer,

(a) the pdf of radiated power,

(b) the conditional pdf of transmission loss, given inter-
fering path length,

V7


99.9

99

95

80

°
Py %

90

20

Ol a”

olL

|
-40 -20 20 40

y, dB

Figure 9. Probability of satisfactory reception, P_, with and without (dashed
line) location variability of transmissiOn loss. Variability of
transmission loss smooths out the plateau in Par The propagation
factor y = 1.

30


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(7k <r) uu

s(heyz yu pu

(x)%y uayy «(uk Mh) u pu = q pue «(le ly) xew = ¥ SA UO L[BUOLZLPUOD SL X JJ :S[LCUOLZLPUOD

a pue «(z/'x «lxyxew = y uaya ‘a < Lx pueg< !x gr sap (AYA #(Az)%y [A] i = (z)44
a

v
‘quepuadapu, ave , pue x dr “Ap (A SAz)AXy [A] f = (2)45 vaya SA/X = Z SI
a

Y
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q

y
‘quapuedapul aue , pue X J] “Xp & «x yAXy dal = (z)75 uayd SAX = Z JI
q

“xp (x-z)Ay (x)Xy } = (2)44 *quapuadapuL
q

aue , puke X fT *Xp (X-Z «xy AXy ¥ = (2)45 uayy s(taez oyu Lu = g pue «(Nez ly) xeu =V‘A+tX=Z41
q

PST
“fae )%s T= (A Sa + xe = A dE SaeLMoLqued uy

*(A)Y go ansgeasaap aya st (A),4Y a4aum *] (A) ul ((A)U)%s = (A)AZ *(x)6 = AST
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|, “A ¥aALB *X 40 spd LeuoLaLpUCD AUR SL (A]xyAlXy

x

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4

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Salqelue, wopuey Jo suolzLsodwo) Auejuaewe,y *Z alqel

12


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payeino_eo aq 0} adUadayuazUL MOLL SznduL asayl

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Y ‘SdddajuaquL JO saluobazeo LeuaAes aq ued auayL

*yauaguaqzuL Jo Auobazed sLyy
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-ady **6'8) UOLJOUNJ vaysuesz JaALaday + LeuoL3dQ

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ALL LGeqoud ayz OJUL pauLquiod ave AZLsuazUL IL
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56



necessary to obtain a tractable result. Each restriction or approximation was
carefully incorporated into the notation, so that there would be no confusion between
the general beginnings and the somewhat specialized results. Although this practice
ensured clarity, it produced a cumbersome notation. This report deals only with the
final solution for one class of noise, so a more simplified notation is used with
the restrictions treated as given. Each symbol will be defined when it is intro-
duced, and Table 1 defines’ all symbols and shows their relationship to Middleton's
notation.

2.1 Middleton's Analytical Solution for Land Mobile Radio
Communications
Middleton's formal solution for Pao the probability that the SNR exceeds the
required threshold, is (Middleton, 1979a)

P, = P(x>r def fz Wg (xz|@,) wy (z 187) dz dx , q)
x, 0
where
x is the signal-to-noise ratio,
r is the required SNR threshold,
Wg(s|@g) is the probability density function (pdf) of the signal, given the
the input parameters, 855 that influence the signal, and
wy (2/67) is the pdf of interference given the input parameters, 6; that
influence the interference.
To actually evaluate (1), the pdf's must be specified in computable form, and the
indicated double integration must be performed.
Middleton (1979a) assumes that the interference environment is represented by
a Class A noise model. The probability density function (pdf) of the noise power,
measured after the IF filter in the receiver, is

AS Am
Lat y

wi) =e" are exp(-ig /i,) (2)
Ih'n ! nm on

where in is the total (instantaneous) received noise power in watts, and

- AQ +T) (3)
Sn FAP *
In (3),
Te nfi, (4)
is the ratio between the mean gaussian noise, n, and the mean nongaussian noise,
i: (Throughout this report a bar over a variable will denote the mean of the

4


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27


20

Figure A-2. The pdf of interfering power, in dBW, from exactly one
simultaneous interferer. Curve B is the pdf computed
for the scenario; Curve A is the canonical form of
Class A noise.

63



Form OT-29
° U.S. DEPARTMENT OF COMMER
tore oF FcR oF TELECOMMUM CATIONS

BIBLIOGRAPHIC DATA SHEET

T. PUBLICATION OR REPORT NO] 2, Gov't Accession No. |3. Recipient's Accession No.

NTIA Report 80-41

4, TITLE AND SUBTITLE 5, Publication Date

COMPARISONS OF ANALYTICAL AND NUMERICAL CALCULATIONS OF | June 1980

COMMUNICATIONS PROBABILITY 6. Performing Organization Code
910.01

9. Project/Task/Work Unit No.

7, AUTHOR(S)

Leslie A. Berry

8, PERFORMING ORGANIZATION NAME AND ADDRESS
U.S. Department of Commerce 9107136

National Telecommunications & Information Admin. 10. Contract/Grant Now
Institute for Telecommunication Sciences
Boulder, Colorado 80303

11, Sponsoring Organization Name and Address

12, Type of Report and Period Covered

U.S. Department of Commerce
National Telecommunications & Information Admin.
Washington, DC 20005

14, SUPPLEMENTARY NOTES

15. ABSTRACT (A 200-word or less factual summa: of ignil in i
BST zy of most significant information. ; i
bibliography ot literature survey, mention it here.) ignificant information. If document includes a significant

Middleton's analytical formula for the probability of communications ina
Jand mobile environment and Berry's computer program for probabilistic calculations
of interference are shown to produce identical answers for identical inputs. The
assumptions, input, and output of the two models are compared. Middleton's assump-
tion that interfering sources transmit randomly and independently is of particular
interest for mobile radio services. The effects of the greater variety of input
assumptions possible with the computer method are illustrated numerically.

To use Middleton's formula to compute the probability of interference for a
detailed scenario, the parameters of Class A noise must be determined from the
scenario descriptors. Methods for finding these parameters are ex lored.

To illustrate possible applications, the number of statistically identical |
links that can operate with specified reliability in a given area is computed with
the numerical model.

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

Class A noise; communications probability; mobile radio; probabilistic EMC;
probability of interference

17. AVAILABILITY STATEMENT 18. Security Class (This report) 20. Number of pages
UNLIMITED. i i
a Unclassified 68
19, Security Class (This pa, 21, Price:
([) FoR OFFICIAL DISTRIBUTION. TM a nse
Unclassified

U.S. GOVERNMENT PRINTING OFFICE: 1980-0-678-890/442 9716-P73




this scale. Middleton plots Pa over a limited range of the independent variable.
This range is extended in this report by using the program listed in Middleton's
report to compute additional values.

Figure 6 shows a comparison of the output of the two methods. for a propagation
parameter y = 2 and for two values of the noncentral displacement, "oar The two
methods give essentially identical results, except for very small Poe

The shape of the curves can be understood by referring to Figure 4, which is
the pdf of Class A noise for the same case as in Figure 6. Remember that the bump
on the left in Figure 4 is the continuous background noise, while the bump on the
right is the intermittent interference present about 30 percent of the time. Assume
that the observed transmitter 1 km away has controllable power, that you are measuring
the signal-to-noise ratio at the receiver, and that the required signal-to-noise
ratio is 20 dB. Then in Figure 6, y = received signal +80 dB.

Begin with a low radiated power so that the received signal is less than -140
dBW. Then interference occurs unless the noise is less than -160 dBW. Figure 4
shows that this is very improbable, so P,® 0. Now increase the power 10 dB so the
received signal is -130 dBW. Then the required S-I is achieved when the noise is
-150 dBW. The probability of this occurrence is the area under the curve in Figure
4 to the left of -150 dBW, a small but nonzero probability. As the power is
increased more and more, moving right on Figure 4, the area under the curve left of
the threshold increases, and Pa increases until the received signal power is -105
dBW. At this point, S-I = 20 dB if the noise is -125 dBW. As the power is increased
10 or 15 dB above this level, Pa remains almost constant, because there is virtually
no area under the curve in this region. The received signal is powerful enough that
it exceeds the background noise nearly all the time. However, Ps is about 0.7
rather than one, because the signal is too weak to compete with the interference
whenever the interference is present.

As the power is increased still further, so that the signal is 20 dB greater
than the noise at -110 dBW, Pa begins to increase again. The signal is now strong
enough to beat the interference part of the time. Pa continues to increase with
increasing transmitter power, until the received signal is about -65 dBW. At this
power, S-I is greater than 20 dBW nearly all the time, and Pa = 1. Even when the
interference is on, the signal is more than 20 dB stronger.

So when the mean of the gaussian noise is much less than the mean of the inter-
mittent interference, Pa will be small when y is small; increase to a plateau near
1-e7* as y increases; remain nearly constant for a while; and then increase again to
a value near one. This is exactly the behavior of the curves in Figure 6.

24



COMPARISONS OF ANALYTICAL AND NUMERICAL CALCULATIONS OF
COMMUNICATIONS PROBABILITY

Leslie A. Berry*

Middleton's analytical formula for the probability of communica-
tions in a land mobile environment and Berry's computer program for
probabilistic calculations of interference are shown to produce identi-
cal answers for identical inputs. The assumptions, input, and output
of the two models are compared. Middleton's assumption that inter-
fering sources transmit randomly and independently is of particular
interest for mobile radio services. The effects of the greater variety
of input assumptions possible with the computer method are illustrated
numerically.

To use Middleton's formula to compute the probability of inter-
ference for a detailed scenario, the parameters of Class A noise must
be determined from the scenario descriptors. Methods for finding
these parameters are explored.

To illustrate possible applications, the number of statistically
identical links that can operate with specified reliability in a given
area is computed with the numerical model.

Key words: Class A noise; communications probability; mobile radio;
probabilistic EMC; probability of interference

1. INTRODUCTION

Two models have recently been developed for computing the probability of com-
munications in a congested spectral-use environment. Middleton has derived an
analytical formula--an infinite series of integrals--for the probability that the
signal-to-noise ratio (SNR) exceeds a specified threshold (Middleton, 1979a).

He calls this probability the probability of noninterference (PNI) and denotes its
value by Pa Berry had developed a computer program to compute this same probabil-
ity (Berry, 1977). Successful frequency management depends in part on knowledge
of the probability of interference, so these models are potentially valuable tools
for frequency managers. The purpose of this report is to make both models immedi -
ately accessible to frequency managers. The modeling assumptions, required input
data, and output are compared to help the user decide when each method is appro-
priate and capable.

“The author is with the U.S. Department of Commerce, National Telecommunications
and Information Administration, Institute for Telecommunication Sciences, Boulder,
Colorado 80303.


Table 1.

Table 2.
Table 3.

Table 4.

LIST OF TABLES

Notation Used in This Report Related to Notation Used in
Middleton (1979a). . 2... 2... Se ee

Elementary Compositions of Random Variables. .....

Probability that i (Out of Possible m) Transmitters Assigned
to a Channel are Transmitting. ... . ee ee eee ee

Comparison of the Available Analytical Solution for P(s/i > r)

(Middleton, eet and the Available Computer Program for the
Same Probability (Berry, 1977) ....... eee ee

vi

Page

5
12

18


for calculating the probability of communications that exploits this capability.
Because the integration is done numerically, the pdf which best describes the
parameter variation can be used as input. The pdf need not even be analytic--a
table of measured values can be used as the input pdf. This program has been
described in detail by Berry (1977, 1978). Its structure will be briefly outlined
in this section.

In Berry's model, all power is expressed in dBW rather than in watts. Although
either set of units can be used, most measured data on transmission loss and noise
are given in dB, and the pdf's of these measured data are approximately normally
distributed in dB.

The formal solution for Pa evaluated by the computer program is

P, = P(S-I > R) = if foiy (x) dx, (13)

where
S is the received signal strength in dBW,
I is the received interference in dBW,
R is the required SNR in dBW, and
Fy_y(x) is the pdf of X = S-I.

Thus (13) is the equivalent of (1) for powers given in dBW. The procedure is to
find fo_y(x) numerically.

The overall plan in the program is to compute the pdf of the wanted signal,
then compute the pdf's of each "kind" of interference. The powers from the dif-
ferent sources of interference are added together with the ambient noise to get the
pdf of the total noise plus interference. This sum is combined with the signal pdf
to obtain the probability distribution of the signal-to-noise ratio. The probabil-
ity of achieving a specified signal-to-noise ratio is then computed. Figure 3 shows
the major steps in this calculation.

To compute the distribution of the desired signal at a random receiver, the
pdf of wanted-transmitter effective radiated power (erp) must be known. In prac-
tice, this has included the effects of antenna gain and antenna height gain, but
these factors could be specified separately and used to compute the pdf of erp.

In most terrestrial problems, the transmission loss for a fixed path length is
not a constant but varies with time and the specific details of the propagation
path. Thus, transmission loss is a conditional random variable with a statistical
distribution for each distance. In practice, the mean of the distribution is
specified as a function of distance, often with a formula like (8), and the varia-
tion around this mean is specified. The conditional pdf of transmission loss can

13


only the available analytical solution (Middleton, 1979a) with the available com-
puter program (Berry, 1978).

In the next section of this report, the analytical solution derived by Middle-
ton (1979a) and the numerical evaluation are compared. The differences between
the approaches and the input required and allowed are explored. Numerical results
for identical input are compared and found to agree. This agreement lends some
credence to both methods. It is unlikely that agreement between two such different
approaches is merely fortuitous.

To get an analytical solution, Middleton (1979a) idealized some features of
the problem and restricted the scenario to certain specialized cases. The computer
Program can accept a greater range and variety of scenarios as input. So, in
Section 3, the computer program is used to calculate the changes inthe probabil-
ity of interference for different scenarios. These differences illuminate the
effects of Middleton's assumptions. :

In Middleton's (1979a) formula for PNI, the noise and interference environ-
ment is described by the three parameters of Class A noise.' In a practical
application, one is more likely to know the powers, locations, and traffic patterns
of interfering transmitters. So to use the formulas, these known parameters of
the scenario must be related to the Class A noise parameters. Physical interpreta-
tion of the terms in the series for Class A noise and reference to the formal
solution provide some insight, but no satisfactory analytical solution for strictly
canonical Class A noise parameters as functions of scenario descriptors could be
found. (Such a direct calculation is possible with a newly developed, quasi-canoni-
cal Class A model [Middleton, private communication, 1980].) It is apparently
necessary to compute Class A noise parameters numerically using a program similar
to Berry's. Section 4 of this report describes the search for a connection between
a scenario and Class A noise and illustrates the hybrid calculation that is appar-
ently necessary in a practical application.

2. COMPARISONS OF THE TWO MODELS

Middleton's (1979a) formula for the probability that the signal-to-noise ratio
will be satisfactory is an extension of his development of canonical noise models
(e.g., Middleton, 1972, 1976). To develop these models, he began with quite general
situations and fundamental equations, then added restrictions and approximations as

1A similar formula for Class B noise will soon be published (private communication
from D. Middleton, 1980).


(c) the pdf of path lengths,
(d) the traffic intensity, U, of interferers in this category,
(e) the receiver transfer function, if any;

(3) the pdf of ambient noise; and

(4) the required signal-to-noise ratio.

Any pdf may be given analytically, in which case the defining parameters (e.g.
Mean, variance) must be given, or the pdf may be defined by an input table of mea-
sured values.

2.2.1 Specialization of the Numerical Model for Comparisons

To compare output from the numerical model with calculations made using equa-
tion (11), two modifications are necessary. First, the calculation of the pdf of
noise plus interference from input descriptors of the power, locations, etc. of the
interferers, (22), is replaced by a subroutine to calculate the pdf of Class A
noise, (2).

Second, the calculation of wanted signal is simplified because the analytical
solution, (11), assumes that there is no variability in radiated power and the
transmission loss is a deterministic (rather than probabilistic) function of path
length. Converting the signal in watts (equation (8)) to dBW,

S = 10 logyq Py} - 20 ¥ logy_d dBW. (23)

But d is a random variable, so S is a random variable whose pdf is found using the
transformation rule in Table 2. It is

£,(S) = d(10'8-S)/20%) (ig lB-S)/20r) (24)
where B = 10 10919 Py and Fy (x) is given by (10).

2.2.2 The pdf of Class A Noise Given in dBW

The numerical model treats power in dBW, so the pdf of Class A noise, equa-
tion (2), must be transformed as shown in Table 2. This can be done mechanically
by the computer, but it is instructive to perform the transformation analytically.
Let ly = 10 10949 in: Then the pdf of ly is

-A @ AM Sn Gn Oly

f(y) =e Y =D exp(Dl, -=e °)
I ! j
N mo TM in N iy

: (25)

19


NTIA-Report 80-41

Comparisons of Analytical
and Numerical Calculations
of Communications
Probability

L. A. Berry

oy

%
Stares OF

U.S. DEPARTMENT OF COMMERCE
Philip M. Klutznick, Secretary

Henry Geller, Assistant Secretary
for Communications and Information

June 1980


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42


Table 1. Notation Used in This Report Related to Notation Used

in Middleton (1979a)
(continued)

This Report Middleton
Symbol Units Notes Symbol
Ps * P(s/i, > 1) = P(S-Iy > R) Ps
Pa * An integral, see equation (12) Pn
Py watts Signal power received at 1 km; defines los
power of transmitter
r * Required ("threshold") signal-to-noise X
ratio
"oa * Poa = da/dy "oa
R dB R= 10 10919 r ---
s watts Wanted signal power in receiver I,
Ss dBW $= 10 10949 s ---
o km = d/v2 5
T s Average length of an emission from an ---
interference source
Tp Ss Average length of an interfering emission Ty
measured after the IF filter, Tp <T
U * Traffic intensity, U=vT >A ---
y * XQ/Xq = Py/(i,r) XO/X



Table 1. Notation Used in This Report Related to Notation Used

in Middleton (1979a)

This Report Middleton
Symbol Units Notes Symbol
A * A= Var Ay
d km Distance from transmitter to receiver ny
ds km Distance from receiver to transmitter's r
center of operations

dy * Normalized distance, dy = d/d, vy
dq, km A reference distance in path length pdf tS
D * D=0.1 log, 10 ---
e&, volts Limiting sensitivity voltage of receiver eff)
Sn * Gq = ACL + T)/(m + AL) (207 4)”

* = n/i '
T T ni, T A
Y * Exponent in propagation law, decay rate Y
i, watts Total noise power, i, ent i, Iy
in watts Mean of i,, i, = 7,(1 + 1) Ty
Iy dBW ly = 10 10949 iy ---
i, watts Nongaussian noise, “interference” ---
I, dBW I, = 10 10919 i, oon
ig watts Mean of i, Qn
n watts Gaussian noise ---
nT watts Mean of n %
N dBW N= 10 10949 n ---
v sl Average number of emissions by inter- ---

fering sources, per second

YR sl Average number of interfering emissions v5

measured after IF filter, vp <v



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fal) = Tea (43)

The effective radiated power of interfering transmitters has a mean value of -40 dBW
and is log normally distributed with a standard deviation of 3 dB. The transmission
loss is normally distributed in decibels with a mean of -40 -40 log d (equivalent to
y=2) and a standard deviation of 5 dB.

The interfering transmitters transmit at random times, with traffic intensity
U=0.25. The background noise power is normally distributed in decibels with a mean
of -132 dBW and a standard deviation of 5 dB.

The series in (22) converged in six terms. Figure 13 shows the pdf of noise
plus interference for this case. The dashed line shows the interference plus noise
when only one interferer is transmitting. Clearly, there is more than one inter-
ferer for only a small fraction of the time, because the total interference is only
slightly more than that for one interferer. Mathematically, the coefficients of
the higher-order terms for U = 0.25, given in Table 3, are about an order of magni-
tude smaller than the coefficient for m=1.

The curve in Figure 13 looks generally like the one for Class A noise in
Figure 4. To use formula (11), the Class A noise parameters A, I, and i, must be
known. Assuming that all interfering transmissions are perceptible in the receiver,
A=U.

One way to determine T and iy is based on the partial identification at the end
of Section 4.1--that (34) and (35) can be approximately equated term by term.
Substituting (3) into (30) for m=0 and m=1, yields

q
o

. -—
1y(0) = b, = 10 ros ol, Ti) : (44)
and

my

- = Jar
11) by = 10 Toa; 9(F, har) : (45)
Recall that Ty(m) is the location of the maximum of the pdf for the specified value
of m, which can be read from the curve in Figure 13. So there are only two unknowns
in (44) and (45). These equations can be written:

b /10

10° zy T

"1 TF ° (46)

and

38


In the thought experiment above, the distance between the receiver and trans-
mitter is fixed. It is a random variable for the calculations that led to Figure 6
so there is a range of signal levels rather than a fixed value, to compare to the
noise. This results in some smoothing of the plateau effect because, when the mean
of the received signal is between the bumps in Figure 4, the tails of the signal
distribution extend into both bumps. To illustrate, Figure 7 shows the distribution
of the received signal for both cases shown in Figure 6. The distribution of the
received signal for oa = 0 is over 30 dB wide, easily bridging the 15 dB interval
between the bumps in Figure 4. Thus, the plateau in the curve for r
6 is not very pronounced.

oa” 0 in Figure

The signal distribution of the received signal for a center of operations dis-
placed 4 km from the receiver (ro, = 4) is much narrower. This is because the ratio
of farthest-to nearest-probable transmitter location is not so large for this situ-
ation (see Fig. 2). The effective range of the signal distribution for oa = 4 is
about 15 dB (Fig. 7), which fits neatly in the gap between the bumps in Figure 4.
Therefore, the plateau in Pa is quite noticeable in the curve for "oa = 4 in Figure
6.

For free-space propagation (y = 1), received signal strength does not vary so
much with path length, so the pdf of the signal is narrower in Figure 7. Asa
result, the plateau in Pa is more pronounced in Figure 8, which compares the analyt-
ical and numerical methods for y = 1.

Without comparing curves for all of the parameter combinations used by Mid-
dleton, it is clear that the two methods produce the same probability of communica-
tions for the same input parameters.

3. EFFECTS OF DIFFERENT WANTED SIGNAL INPUT ASSUMPTIONS ON P,

Comparison of the summaries at the end of Sections 2.1 and 2.2 shows that the
available computer program allows more freedom in the specification of input param-
eters than does Middleton's analytical solution, equation (11). Middleton's formal
solution, (1), is less specialized but would require a (as yet unwritten) computer
program for evaluation. One set of input parameters determines the pdf of the
wanted signal. Differences in the probability of communications, Pa resulting from
greater flexibility in this set of input parameters are explored in this section.
The even greater differences in ways of specifying the interference and noise
environment are examined in Section 4.

3.1 The Effects of Variability in Transmission Loss
The analytical solution (11) assumes that transmission loss is a deterministic
function of path length given by equation (8) (see footnote 2, page 7). However, if

26


these quantities in the derivation of Class A noise.) The right (interference)
bump exactly matches. that of Class A noise in Curve A of Figure 14.

Another way to determine T and i, is suggested by Appendix A-I of Middleton's
report (Middleton, 1979a). The mathematical details of this procedure are in the
Appendix of this report; the result for the scenario described at the beginning of
this section is

s -2
i, = <e5> py (d.4.,) Ay (50)

eis a “limiting sensitivity voltage," and <> means average;
Py is a normalized power of an interfering source;

is the shortest distance to an interferer; and
d_ is the maximum distance to an interferer.

In the present scenario, qd, = 0.1, qi = 10. I believe that if Py = Py (Table 1)
then <eg> = 1, so i, = A (107") = 2.5(1075). Then r = n/7 = 10783 +2/7, = 2.52(107°).
Curve C of Figure 14 shows the Class A noise pdf for this set of parameters. The
interference bump is the same size and shape as for Curve A, but is much further to
the right--centered at -40 dBW.

Perhaps the product <e> Py has been incorrectly interpreted. As a test, the
pdf was computed for <eo> Py = 107®, so that iy = 2.5(10"7). Curve D of Figure 14
is the Class A noise pdf for this set of parameters. The only change is that the
interference bump is now centered over -60 dBW.

By now it should be clear that as long as A=0.25 and I<<1, the interference
bump (nongaussian noise) will have the same size and shape. Changing i, only
translates the bump along the abscissa.

In the comparisons above, A=U, and the fits are unsatisfactory. Figure 16
shows several heuristic attempts to achieve a better fit by changing A. Curves
(1), (2), and (3) were attempts to get more probability of interference at levels
above -65 dBW and to reduce the right peak of Class A noise to about the same
height as that of the scenario noise. The result is a poor fit to the left of -75
dBW, without achieving the required probability to the right of -60 dBW.

The parameters for curves 4 and 5 keep the maximum of the right peak at the
correct location and bring its height down nearer the height of the scenario inter-
ference. These curves do not have sufficient probability of interference above -65
dBW.

Curves (1) through (5) in Figure 16 suffer from the defect A#U, which is the
one parameter known from the scenario description. This results in insufficient

43


T T
x
99.9;— / x
/
Xx
99 Numerical Model loa of |
X X Analytical Model Xx |
/ x
95 |- , / 4
x x
7
“x /
80 |- xx v* 4
py» ae X—X— XX foa=4
a2/0 LY x
Lo
50 x / a
/ x
/* /
20 y* / +
5 x
JL Class A Noise
/ Tg = 10" w
Ie x A =035 4
x I = 0.0005
x x Path Length pdf: dy = 1km
ore Propagation: Y =2 4
l l | |
-40 -20 0 20 40

y, dB

Figure 6. Probability, P,, that the signal-to-noise ratio exceeds the vequired
threshold as a° function of Middleton's parameter y = Xo lx,

Notice the plateau in Pa near Pa =e-A, The propagation Qetor ya2.

25



Figure A-1.

10 T T

The pdf of interfering power, in watts, from exactly one simultaneous
interferer. Curve B is the pdf computed for the scenario; Curve A is
the canonical form of Class A noise. Notice that the curves are on a
log-log scale.

62


1.1 General Comparison of Analytical Solutions and Numerical Methods

In this report, an analytical solution is an equation of the desired quantity
and an algebraic combination of specific mathematical functions. An analytic
solution does not include indicated derivatives or integrations, nor purely sym-
bolic functions whose form is yet to be determined.

Analytical solutions are distinguished from formal solutions in which the
expression for the desired quantity may contain symbolic operators (such as inte-
grals or derivatives) and general functions whose form has not yet been specified.
A formal solution is essentially a plan for deriving an analytical solution or for
getting numerical results. To get the results, one must specify the form of the
general functions and carry out the indicated operations analytically (if possible)
or numerically.

All other things being equal, analytical solutions are preferable to computer
programs for at least two reasons. First, the analytical solution provides a
direct, explicit relationship between the input variables and the desired answer.
If the solution is simple enough, the effect of varying different parameters of
the problem can be perceived just by studying the relationship. The behavior of
the answer for limiting values of the parameters can be derived. Second, if the
solution is complicated enough that numerical evaluation is required, the programming
is usually straightforward, requires only a medium-to-small computer, and program
execution uses relatively little computer time.

Unfortunately, analytical solutions have not yet been derived for many practi-
cal engineering problems, including some for which formal solutions exist. In
these cases numerical evaluation of the formal solution can provide needed results
immediately. Computer programs can also evaluate formal solutions for general
cases that will never yield to analytical solution. The disadvantages of the
numerical approach are the requirement for greater computer power, longer running
time, and lack of direct insight into the behavior of the solution because the
relationships are implicit rather than explicit.

1.2 Overview of This Report

These general differences between analytical solutions and numerical evalua-
tions of formal solutions are illustrated by the comparisons in this report. Both
the computer program and the analytical solution are based on quite general formal
solutions for the probability of interference in a congested environment. In
their most general forms, these two formal solutions are essentially similar and of
little immediate use to frequency managers, who do not have time to program com-
puters nor to carry out difficult derivations. Therefore, this report compares


where D = (1og,10)/10. This is the pdf of interference, given a specific number of

interferers.
The unconditional pdf of interference is computed as follows. In general, the
probability that the interference is below a given value t is

P(Iy <t)= P(LIy < t and 0 on] or [ly < t and 1 on] or...
---or [Iy <t and Mon]) , (16)
where M is the total number of potential interferers. In (16), the phrase "m on"

means "exactly m transmitters are transmitting."
Because the events in square brackets are mutually exclusive,

M
P(Iy < t) y P(Iy <tandmon) , (17)
m=0

nT

M
} P(Iy < t|mon) P(mon) . (18)
m=0

The probability, P(m on), depends on the average traffic intensity of each
link and on the operating practices in the service. In a disciplined land-mobile
service, a user will usually wait until the channel seems to be clear before trans-
mitting. This means that P(m) is very small form > 2. P(m) for m> 2 would be
zero if every operator had perfect information about others' use of the channel.
However, there is a possibility that an operator will not hear another transmitter
even though it is transmitting and will then emit some interference. Thus, com-
puting P(m on) is quite difficult in a disciplined service.

In a completely undisciplined service, it is easier to model P(m on) because
transmission times are random and independent. Under quite general conditions then,
the number of transmitters on at any instant of time is poisson distributed.
Examples of the mathematical assumptions that are required are given by Zehna (1970,
p- 135) and Kleinrock (1975, p. 60). In the present application, the assumptions
Mean that transmitter turn-ons are independent (no courtesy) and that the number of
potential interferers is large compared to the average number transmitting at any
given instant. These are the assumptions made by Middleton (1972, 1974, 1976,
1979a). Under these conditions (Kleinrock, 1975),

P(m on) = ve eu, (19)

16


service is R=10 dB. Figure 23 shows the probability that this is achieved as a
function of M, the number of links in the area. If the reliability that the average
link is to achieve is specified, then the number of links that can operate with this
reliability can be read from the curve. For example, if the desired reliability is
75%, the number of links is about M=5. If the desired reliability is 90%, the
number of links that can operate satisfactorily is only about M=2.

The analytical solution for P,, equation (11), cannot be used alone for determin
ing the number of links that can operate satisfactorily in a given area because the
source distribution and density do not appear in the formula. However, if the
source distribution is one for which the canonical Class A noise model is a good
approximation, then a procedure similar to that used by Middleton (1978a, Section
5.4) could be used. An acceptable source distribution is Curve B of Figure 15
(equation 38) or a distribution similar to it. Note that the uniform source density
distribution used in Section 4.2 does not approximate (38).

The answer to this section's question that was found with the numerical model
must be evaluated in the light of the assumption made about operating procedure.

For direct comparison with Middleton's model, it was assumed that transmitter turn-
ons were independent--that is, that there was no circuit discipline or courtesy.
This assumption, which is basic to Class A noise (Middleton, 1972, 1976), is perhaps
a good approximation for the Citizens Band radio service, but is probably not valid
for any other service. However, the assumption of independent turn-ons is not
necessary to the numerical model. All that is necessary to make a more realistic
calculation of the assignment capacity with the numerical model is to replace equa-
tion (19) for P(m on) with a more realistic formula. The method followed above will
then produce the desired result.

6. CONCLUSIONS

The conclusions in this section apply only to the available analytical solu-
tion, (11) (Middleton, 1979a), and the available computer program (Berry, 1977).
These are the tools which a frequency manager can use to get immediate quantitative
results. The two models produce identical output for identical input, as shown in
Section 2.3.

Table 4 summarizes the essential assumptions for both models, the required and
optional input, and the normal output.

It is clear from the input list in Table 4 that the numerical model has more
flexibility than the present analytical solution, equation (11), but is computa-
tionally slower. The analytical solution was derived from a formal solution with
flexibility comparable to the numerical model, but the formal solution has not been
Programmed, and its programming would be nontrivial. Indeed, such a program would
be very simiTar to the already-available program for the numerical model.

53


ae sford<d<d.. (A-3)
m s
For the specific scenario of interest, qi, = 10 and d, = 0.1.

Now that d is random, (a 2) must be 1 modified so * that the average of er)2¥ =
iat is taken. Let v = vat . Then the pdf of v is found using the transformation

in Table 2 of the report. It is

fy(d) =

-3/2
flv) =—*——_ , vycvew, s (A-4)
a(do - qd)
~ a4 = yf .
where Yo = q, and Ym = Ys ° Then the expected value of v is
Vin 1 : Vin
E(v) = f vfj(v) dv = |? “| : (A-5)
Yo 2(d?-d2) Yo
So
-2
E(v) = (dod,)°° (A-6)

This is the value Middleton derives in a slightly different way in his private com-
munication. Inserting this value and the other values determined above into (A-2)
yields

2% = 2 2 2 2 q2 -
<Bo> = (<eo> 9¢/ (4m) )2 py / (dg qn) : (A-7)
The parameters e& and g, are not given in the scenario, but for now take
g,= 40 (A-8)
so that
2a 2 2 q2 .
<B2> = <e?> 2p,/(d2 d2) (A-9)

This is the result used in (50) of the report. It is clear from the discussion
following (50) that the assumption in (A-8) is not critical. It is also clear from
this discussion that direct calculation of Class A parameters does not yield a
satisfactory fit to the scenario interference.

The fundamental reason that the Class A parameters that fit scenario interfer-
ence cannot be calculated directly is that the canonical shape of the Class A pdf is

strikingly different from the shape of the pdf that naturally arises from the scenario

in Section 4.2. The pdfs of interference, given that exactly one interferer is

60


because of different antenna types, different antenna heights above terrain, and
different designs or maintenance of transmitters. The numerical model described
in Section 2.2 accepts a distribution of transmitter power as input, but the
analytical solution described in Section 2.1 does not.

Calculations are not necessary to estimate the effect of this difference
because variability of radiated power simply increases the variability of received
signal power just as variability of transmission loss does. A broader distribution
of received signals would further smooth out the plateau in Pa and decrease Pa for
large values of y.

3.3 Effect of Different Desired Path Length pdfs

To use the analytical solution (11), the probability density function of
wanted path lengths must be given by (10). There are two adjustable parameters in
(10), so many actual distributions of wanted path lengths can probably be approxi-
mated by it. The numerical model accepts any pdf of path lengths as input so it
can be used to test the effect of different pdfs on the probability of communica-
tions.

The simplest model assumes that wanted transmitters are poisson-distributed
in a circle around the wanted receiver. This means that the probability that any
small area contains the wanted transmitter is proportional only to the size of the
area. Then the pdf of path lengths is

fad) = Gd (33)

where d is the radius of the circle.

Figure 12 compares Pa for the parameters given in Figure 6 with Pa for the
same parameters except that (10) is replaced by (33) with q,=1- (This corresponds
to d, = 1] in (10).) The probability of communications is higher for the calcula-
tion using (33) because no paths are longer than 1 km. There are some paths
_ longer than 1 km when (11) is used. The signal over these longer paths is less
likely to be strong enough to overcome the noise.

The only point of this comparison is that the probability of communications
depends on the form of the pdf of path lengths. The available analytical solution,
(11), is restricted to one form of path length pdf--that given by (10). The com-
puter program can use whatever path length pdf best fits the operational situation.

33


T T
99.9 - 4
99 F
9
80 4
Fa %
50 4
7 Class A Noise
ta Tg =107 w
20 A=0.35 _
Y T = 0.0005
y=2
Ae 4
IF 4
OF 4
| | | l
-40 -20 0 20 40
y, dB
Figure 12. Probability of communications, P_, for two different distributions

of distance between the transmit@er and receiver. The distance pdf
used to compute curve A is given by (33) in the text; the distance
pdf for curve B is given by (10).

34


probability of interference stronger than -110 dBW. As before, it is unlikely that
a better fit can be achieved because there is no way to spread out the nongaussian
bump in Class A noise.

Figure 20 shows the probability of communications for both models of noise in
Figure 19. Again, the calculation using Class A noise overestimates the probability
of communications for a given transmitter power. For 99% reliability, the required
power is underestimated by 18 dB; for 90% reliability, it is underestimated by 8
dB.

Recall that the point of this section was to see if Class A noise parameters
can be determined from calculations of the noise-plus-interference for an arbitrary
scenario. The possibility of doing this approximately has been illustrated for two
cases. It is possible to define scenarios that produce much closer fits. For
example, if, in the scenario at the beginning of this section, the pdf of path
lengths is changed from (43) to (38)--that is, from Curve A to Curve B in Figure
15, then the calculated pdf of noise plus interference exactly matches the Class A
noise pdf of Figure 14. On the other hand, using the insights gained from these
examples, it would be possible to define scenarios that result in interference pdfs
much different from the Class A pdf.

5. THE NUMBER OF CO-CHANNEL LINKS THAT CAN OPERATE IN A REGION

How many co-channel transmitter-receiver (TR) links can achieve a specified
reliability in a given geographic region? This question is basic to spectrum
Management because its answer defines the assignment capacity of a frequency band
and area. The answer is useful only if the transmitters are confined to a specified
area or if the geographic density is given. (It is clear without making any calcu-
lations that an infinite number of TR links can operate in an infinite area.)

This question can be answered using the computer program described in Section
2.2. Assume a scenario in which all links are statistically similar to the one
described at the beginning of Section 4.2, except for the traffic intensity, U.
Instead assume that the average traffic intensity for each individual TR link is U'.
Then the total traffic intensity in the region is

U=MU' ,

where M is the number of links in the region.

Figure 21 shows the pdf of noise and interference for U' = 0.1 and different
values of M. Figure 22 shows the probability that the signal-to-noise ratio exceeds
Ras a function of R. Figure 22 can be replotted to show the same probability as a
function of M, for fixed R. Suppose that the required signal-to-noise ratio for a

49


Iy(0) #10 logy 7,0 = Ty +10 logg? . (31)

which for the parameters of Figure 4 is -133 dBW.
Form>OandA<<T<1,

Ty(m) = 10 logy m Tp/A . (32)

Now i, is the mean interference power averaged over time, and, roughly speaking,

A is the fraction of time the nongaussian interference is present? (see Table 1).
So T/A is the mean nongaussian interference power averaged over only the time that
it is present. The maximum of the m=1 term is at this value in dBW. Notice that
the maximum of the nongaussian interference bump in Figure 4 is near -95 dBW, the
value Iy(1). This suggests that the higher order terms have only a second order
effect on the value of the interference in this case.

Figure 5 shows plots of fy for two other sets of values of the parameters of
Class A noise. As expected from the above analysis, the distance between the two
maxima is determined by the ratio of the two means, T, in dBW. If Tf > 0.1, the two
bumps merge. The absolute location is determined by Ty: The ratio of the area of
the interference bump to the area of the noise bump is eA - 1. If A << 1, this
ratio is roughly equal to A--the fraction of time the interference is on.

2.3 Numerical Comparisons of the Two Models

Middleton (1979a) shows plots of the probability of noninterference, Poe
P(s/i >), for various values of the input parameters. Pa was calculated for
identical input with the computer program described in Section 2.2 using the spe-
cializations in 2.2.1. Specifically,

(1) the pdf of wanted path lengths was given by (10),

(2) the propagation law was deterministic and was given by (8), and

(3) interference had a Class A pdf, given by (25).

Examination of the computer program in Appendix A-IV of Middleton's report
(1979a) shows that he converts y = RIK to dB by taking 20 times the 109) 9 of it.
The ratio itself is dimensionless, but the elementary factors in the ratio are in
watts. Therefore, it seems more conventional to define dB = 10 1094 9: This latter

definition is used in this report, and Middleton's results have been replotted to

3More accurately, the fraction of time that interference is present is
1 - P(O0 on) = 1 - eA (see (19)). But when A << 1, eA] - A, so 1 - P(O on) SA.

22


T
999 - 4
99
95
80
8
e
Al
150
2
a
20 ~~ a
~
~.
TM~.
ob \
M=1 —-——
| MeS |
Mz IO ceseeeee
M=20 —-—
ue 4
l l l !
-20 0 20 40 60
R,dB

Figure 22. Probability that the signal-to-interference ratio exceeds a
threshold R, for the scenarios of Figure 21.

52


transmitting, will be used to demonstrate the difference. This will be sufficiently
accurate, because, as Figure 13 shows, interference from more than one interferer at
a time has only a second-order effect on the pdfs for A=U=0.25.

The pdf of interference from one interferer for the scenario is proportional to
Fv) in (A-4) and is plotted as curve B in Figure A-1 for the scenario value E(v) =
1.

The corresponding canonical form for Class A noise for m=1 is g e79¥, where gz
1/E(v) when << 1, as it is in this case. This function is plotted as Curve A
in Figure A-1. Notice that the shape of Curve B is unlike that of Curve A.

The difference in shapes is even more dramatic when we convert the variable v
from power to dBW. Figure A-2 shows the pdfs after this transformation. Again,
Curve B is that for the scenario and Curve A comes from the canonical form of Class
A interference. Curve A has the familiar shape of the nongaussian interference bump
(see, for example, Figure 14 of the report).

However, the shape of Curve B is somewhat different from that of the scenario
interference in Figure 14. This is because the scenario leading to Figure 14
included location variability in the propagation and had a statistical distribution
of transmitter power, but the scenario in this Appendix did not. If there is no
propagation variability, an interferer at the maximum range of 10 km produces exactly
1/(10") W, or -40 dBW interference. There are no interferers at greater ranges, so
there is zero probability of lower levels of interference, and the pdf (Curve B of
Figure A-2) has an abrupt jump at -40 dBW.

In the scenario leading to Figure 14, the power received from a fixed distance
(like 10 km) has a statistical distribution, so that an interferer at a distance of
10 km might produce less than -40 dBW interference. Therefore there is no abrupt
jump in the scenario interference in Figure 14--the left end of the interference
bump has the same smooth shape as the distribution of transmission loss. If there
had been no variability in transmission loss in the scenario in Section 4.2, the
disparity between scenario noise and Class A noise in Figure 14 would have been even
greater.

The disparity in the shapes of Curves A and B in Figure A-2 is the reason that
direct calculation of Class A parameters from the scenario results in a worse fit
than the approximate method described in Section 4.2 of the report.

61


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

10.

LIST OF FIGURES

Geometry of wanted path for Middleton's model. The trans-
mitter performs a random walk around its center of operations,
C, which is da km from the receiver. The random variable d
is the distance from the transmitter to the receiver. .....

Probability density function of the wanted path length, equa-
tion (10). The center of operations of the mobile transmitter
is da km from the receiver. . 2... ee ee ee ee ee eee

Block diagram of program for computing the probability of
interference in a congested environment ............

Probability function for intensity of Class A noise in dBW for
the parameters shown on the figure. The bump on the left is

the probability density of the gaussian noise; the bump on the
right is the nongaussian noise or interference. ........

Probability density function for Class A noise in dBW for the
parameters shown on the figure. Comparison with Figure 4
shows that increasing T moves the bumps closer together;
increasing A changes the relative sizes of the bumps. .....

Probability, Pa, that the signal-to-noise ratio exceeds the
required threshold as a function of Middleton's parameter
y = X/x., dB. Notice the plateau in Pa near Pg = e7*. The
propagat?on factor y=2. 2. 6 2 ee ee ee ee

Probability density function of received signal power S for
different values of the propagation parameter and for different
path length pdfs. Notice that the distributions are narrower
for y=1 than for y=2 and are narrower for d,=4 than for

amet ee ee ee

Probability, Pa, that the signal-to-noise ratio exceeds the
required threshold as a function of Middleton's parameter,
ye Xo/Xq> dB. The propagation parameter y=] .........

Probability of satisfactory reception, Pa, with and without
(dashed line) location variability of transmission loss.
Variability of transmission loss smooths out the plateau in Pg.
The propagation factor y=]. 2... 1. ee ee ee ee ee ee

Probability density function of received signal power with and
without variability of transmission loss. Nonzero variance
broadens the distribution . 2... ee ee eee ee ee eee

Probability of satisfactory reception, Pa, with and without
(dashed line) transmission loss variability for propagation
Parameter y=2.....-.--4.2.220202088 rs

Probability of communications, Pg, for two different distri bu-
tions of distance between the transmitter and receiver. The
distance pdf used to compute curve A is given by (33) in the
text; the distance pdf for curve B is given by (10)......

iv

10

14

21

23

25

27

28

30

31

32

34


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[- 9000 00
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GZO=Nn
100

39




Transmitter

Receiver

Figure 1. Geometry of wanted path for Middleton's model. The transmitter per-
forms a random walk around its center of operations, C, which is da km
from the receiver. The random variable d is the distance from the
transmitter to the receiver.


“L'0=,N APpsuaquy

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S4aqUNU JUaLaJJLP AOJ BOUaIaJ4aZUL-SN{d-aslou Jo UOLZoUNJ AZLSUaP AZLLLGeqoud “LZ aunbiy

map ‘Ny
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100
200
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(NI)}
500
900

100

800

51


There is an intriguing similarity between the series for the pdf of Class A
noise (2) and the pdf of background noise and interference from an infinite number
of interference sources which transmit independently. Under appropriate assump-
tions, the series have the same sum, the first terms of the two series are identi-
cal, and all other corresponding terms contain an identical factor. These conditions
are not sufficient for a term-by-term identification, but the identification is
useful for determining approximate Class A parameters from a scenario, as shown in
Section 4.2.

The Class A noise parameters required by equation (11) can be approximately
determined from calculations of signal and interference for a scenario. How well
the approximation fits the scenario noise depends on how closely the scenario
conforms to that implied by the interpretation discussed in the preceding paragraph.
For the example in Section 4.2, the approximation overestimated the probability of
communications.

A basic assumption in the derivation of the Class A noise pdf is that inter-
fering sources transmit randomly. In land-mobile radio terms, this assumption means
that there is no circuit discipline--no operator courtesy. The numerical model does
not require this assumption. Any degree of circuit discipline can be modeled by
specifying the probability that m sources are transmitting for m=0, 1, 2, ... .

The numerical model can be used to compute the number of links that can operate
with specified reliability in a given area. The analytical solution, (11), can be
used for this calculation for only those scenarios whose interference source distri-
butions are approximated by (38).

7. ACKNOWLEDGMENTS

Dr. David Cohen of NTIA/Annapolis stimulated this study and contributed com-
ments and information during it. Dr. David Middleton introduced me to his valuable
work and explained some of the difficult points to me. His rigorous review of the
first draft of this report resulted in substantial revision. Dr. A. D. Spaulding of
NTIA/ITS provided computer programs which evaluate the analytical formula for S/N
probability.

8. REFERENCES
Abramowitz, M., and I. A. Stegun (1964), Handbook of Mathematical Functions,

National Bureau of Standards AMS 55 (U.S. Government Printing Office, Washing-
ton, D.C., 20402).

Barsis, A. P. (1971), Radio wave propagation over irregular terrain in the 76-to-
9200-MHz frequency range, IEEE Trans. Veh. Tech., VT-20, No. 2, 41-62.

57


(b) ip the mean of the total noise, and
(c) IT, the ratio of the mean of the gaussian noise to mean of
the nongaussian noise;
(2) the path length or geographic parameters for the wanted signal
(a) ‘oa = d/d)> where qd, is the distance between the desired
receiver and the center of operations for the trans-
mitter, and
(b) qo» a reference distance used to fit the pdf to the desired
coverage area;
(3) the attenuation rate parameter, y;
(4) the system parameters
(a) p,» the transmitter power and
(b) r, the required signal-to-noise ratio.
(Actually, the ratio p,/r is all that is needed.)

2.2 Numerical Model for the Probability of Communications

In many congested spectrum-use environments, there is significant variability
in almost every factor influencing communications success. Transmitters may be
mobile or their locations may be unknown, either because data are not available or
because the systems have not yet been deployed. Antenna heights may depend on
availability of favorable locations or on desired coverage of individual systems.
Transmitter powers may vary by specification or because of aging and different
maintenance procedures. Transmission loss varies with details of the transmission
path and often with time because of weather or other geophysical events. Message
traffic or channel usage may vary with the requirements of the users. Therefore,
any realistic estimate of communications success must be probabilistic. It should
be based on consideration of statistical distribution of all significant factors.

The first order description of variation of a random process is provided by the
probability density function (pdf). When several random processes interact to
produce an effect, that effect is itself a random process with a pdf. If the pdf's
of each of the contributing random variables and the nature of their interactions
are known, rules such as those shown in Table 2 provide procedures for finding the
pdf of the effect.

Analytical closed-form solutions for the desired pdf's are possible for only
a few special forms of contributing pdf's because most of the rules involve integra-
tion of products. However, numerical integration is an accurate and efficient
procedure on electronic computers. Berry (1977) has developed a computer program

nN


TABLE OF CONTENTS

LIST OF FIGURES . 2 1. 1 ee ee ee ee ee

LIST OF TABLES. 2 2 1 we ee ee ee ee

ABSTRACT. 2 2 6 6 ee ee ee ee ee

1. INTRODUCTION «1 ww ww ee ee ee ee

1.1 General Comparison of Analytical Solutions and Numerical Methods. .

1.2 Overview of This Report... 1. eee ee ee te ee ee

2. COMPARISONS OF THE TWO MODELS... 2. eee ee ee eee ee ee ee
2.1 Middleton's Analytical Solution for Land Mobile Radio

Communications. ©. 2. 6 6 1 ee ee ee

2.2 Numerical Model for the Probability of Communications .......

2.2.1 Specialization of the Numerical Model for Comparisons. . . .

2.2.2 The pdf of Class A Noise Given in dBW... . 2... 2. ee ee

2.3 Numerical Comparisons of the Two Models... 2.1... ee ee eee
3. EFFECTS OF DIFFERENT WANTED SIGNAL INPUT ASSUMPTIONS ON Pa see eee

3.1 The Effects of Variability in Transmission Loss .........s

3.2 Effects of Variability in Effective Radiated Power. ........
3.3 Effect of Different Desired Path Length pdfs... ......---
4, RELATING CLASS A NOISE TO CONGESTED BAND SCENARIOS... ..... 2 ee
4.1 Interpretation of Factors in Class A Noise Formula. ........
4.2 Class A Parameters Using Scenario Simulation. ..... 2.2.24
5. THE NUMBER OF CO-CHANNEL LINKS THAT CAN OPERATE INA REGION. ......
6. CONCLUSIONS . 2. 1 we ee ee
7. ACKNOWLEDGMENTS 2 1 6 1 ee ee ee ee
8. REFERENCES. 2. 1 1 1 ee ee ee ee ee

APPENDIX: DIRECT CALCULATION OF CLASS A PARAMETERS FROM SCENARIO
DESCRIPTORS. ©. 2 1 eee ee ee ee ee ee


99.9}— a

l | | |
=40 =2 0 20 40

y, dB

Figure 11. Probability of satisfactory reception, P_, with and without (dashed
line) transmission loss variability for fropagation parameter y = 2.

32


where in is the mean of total noise power, iy? in watts. (Remember that D =
(1og,10)/10. )

An approximation appropriate for land-mobile radio channels results from
assuming that the mean of the background (gaussian) noise is much less than the
mean of the co-channel interference (nongaussian noise) and that the traffic
intensity (channel occupancy) is less than one. That is, assume that

T<<A<1 . (26)
In (3), then,

Ig © 1/T and

Gy Alm ifm#O . (27)
Equation (6) shows that
i= ig (28)
when T << 1, so that
afi, * Fi, a yleun. (29)
n

To the approximation in (26), the first term of the series in (25) is a function of
only the mean of the gaussian noise, n, and thus, it is the pdf of the background
noise in dBW.

Figure 4 shows a plot of (Ty) for A = 0.35, T = 0.0005, and i, = 197! watts,
or I, = -100 dBW. These values are "in the ball park" for low-band VHF land-mobile
radio. The gaussian and nongaussian noise are clearly separated in Figure 4. The
bump on the left is the probability density of the background noise, and bump on
the right is the probability density of the interference--the nongaussian noise.
Analysis of the details of these bumps presents a clear physical picture.

First, the maximum of each term in (25) can be found as a function of ly by
equating the derivative to 0. The result is

Iy(m) = 10 logig(7,/g,) > (30)

where Ty(m) denotes the value of Iy at which the mth term is a maximum. For m=0 and
the approximation in (27), the maximum is at




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Figure 8. Probability. P,, that the signal-to-noise ratio exceeds the
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ye XQ/Xq> dB. The propagation parameter y=1.

28


=i
. 2424 n
Frj2(i,l2) =? ine : (42)

which cannot be reduced to the coefficient of the square bracket for m=2 in equa-
tion (35). Therefore, (34) and (35) cannot be equated term-by-term even under the
most favorable assumptions.

Indeed, the fact that two infinite series have the same sum is never sufficient
reason to expect the corresponding terms to be equal. However, the identification
of the first two terms provides useful physical insight when A=U is small, because
then the first two terms are a good approximation for the pdf except for the tail
on the right end. In fact, for an example in Section 4.2 below, the interpretation
makes (34) an excellent approximation for (35) over the entire interesting range.

Middleton (1974, 1976) made some mathematical approximations in the derivation
of equation (2). A mathematical approximation to a solution for a given physical
situation is equivalent to some modification of the physical situation. It is
usually not possible to determine directly what the modification is, but identifica-
tion of the approximate solution with a solution for a different physical situation
throws some light on the effect of the approximation. This is the value of the
partial identification above. Even though Class A noise does not arise from such
a situation, the pdf is the same as if the interference was produced by equal-
amplitude transmitters distributed according to (38).

4.2 Class A Parameters Using Scenario Simulation

The numerical model described in Section 2.2 computes the pdf of interference
from a number of interferers as an intermediate result, using (22). If this pdf
can be approximated by Class A noise for some set of parameters, and if these
parameters can be determined, then the probability of noninterference, Pa can be
computed using Middleton's formula (11) instead of the slower numerical procedure.
Of course some detail is lost, as shown in Section 3, but the results are accurate
enough for some purposes.

As an example, the pdf of interference was computed using the numerical model
for the following input assumptions:

The interfering transmitters are randomly located in a ring with inner radius
of 0.1 km and outer radius 10 km with the victim receiver at the center of the
circle. The probability that any small area contains an interfering transmitter
is proportional to the size of the area. This implies that the pdf of path lengths
from interferers to victim is

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variable.) The total noise, in (which includes both "noise" and "interference"),
is the sum of background "noise," n, which is gaussian in nature, and structured
or impulsive "interference," ig: That is

in zigtn, (5)
and
i, ig ta i, +P) : (6)
The “overlap index," A, is
A= vplp > (7)

where VR is the average number of interfering emissions per second measured in the
receiver after the intermediate frequency (IF) filter, and Tp is the average length
in seconds of those measured emissions. In the derivation of (2), Middleton
assumes that emissions from interfering sources begin at random times and are inde-
pendent.

Middleton assumes that the parameters of the Class A noise environment have
been determined by measurement and has described methods for making the measurements
(Middleton, 1979b). Adequate measurement programs will take considerable time and
money, and some frequency planning problems include systems that are not yet
built. It will sometimes be necessary to estimate the Class A noise parameters
from a description of the systems and their deployment. This problem is discussed
in more detail in Section 4. In this section, it is assumed that the parameters
A, T, and i, are known.

The propagation law assumed by Middleton results in a received signal,

Py

s = a watts, (8)
where Py represents the effective radiated power of the transmitter by giving the
received power at a distance of 1 km, and d is the distance between transmitter
and receiver in kilometers. The signal attenuation rate, or transmission loss
rate, is given by y, which, in practice, takes on values from 0.5 to 3 or more. For
VHF and UHF propagation between low antennas over normal ground, y 2. Notice
that propagation is assumed to be a deterministic function of distance with no
provision for location or time variability.”

2Propagation variability is included in the formal solution (Middleton, 1979a) and
will be es in an analytical solution soon (private communication, D. Middle-
ton; 1980).


APPENDIX: DIRECT CALCULATION OF CLASS A PARAMETERS
FROM SCENARIO DESCRIPTORS

In Section 4.2, Class A noise parameters are determined for a scenario by
fitting the pdf of interference computed numerically. In a private communication,
Middleton has suggested that the parameters can be computed directly from scenario
descriptors. In this appendix, this direct calculation is attempted, and the
failure of the attempt is explained.

The scenario is the same as that given at the beginning of Section 4.2, which
led to the interference pdf shown in Figure 13. For this case, A=U=0.25, and the
Mean of the background noise (-132 dBW) can be taken for the mean of the gaussian
noise. The mean of the interference must be computed. Middleton (1979a) gives
these relationships (converted to the notation of this report):

i, = A<BS>/2 > (A-1)
and
<Bo> = <eg><ug><AZ><|Apr(8) |?> <gg/ (40 ca)eY> > (A-2)
where
By is the basic envelope of the interference measured in the
receiver after the IF filter;
e& is the limiting sensitivity voltage of the receiver;
uy is the normalized interfering signal waveform;
Ay is the peak amplitude of the interfering signal waveform;
Apr (8) is the combined antenna patterns of the receiver and trans-
mitter;
9, is a constant when the distribution of interferers does not
depend on azimuth around the receiver; and
cAsd is the distance of an interferer from the receiver (c is the

speed of light).

As usual, the notation <> indicates an average. (Middleton (1979a) inadvertently
omitted the average over the factor on the right in A-2.)

For the scenario in Section 4.2, Apt (8) = 1 and y = 2. Clearly, <ug><A5>/2 =
Py is a power associated with the interferers (who all have the same power). Con-
sistent with earlier usage, it is assumed that these quantities are measured 1 km
from the transmitter so that Py = Py> the effective radiated power.

For the distribution of interferers in the scenario, d has the probability den-
sity function (pdf)

59


area under the right bump in all the curves; that is, the probability of interfer-
ence from gaussian noise is not large enough.

Study of Figures 14 and 16 makes it hard to imagine any set of parameters that
would make Class A noise fit the pdf of Figure 13. The basic problem is that the
Class A nongaussian interference bump has about the same shape for all reasonable
values of the parameters, when a different shape, with a long tail to the right, is
needed. There is no parameter in the Class A noise formula that can be used to
spread the pdf of nongaussian interference. Perhaps this should have been antici-
pated, because the Class A parameters involve only the means of the two kinds of
noise and not their variances.

It should be noted that (2) is an approximation to the pdf of the envelope of
Class A noise (Middleton, 1976). The more exact form includes terms that extend
the tail to the right. However, no analytical solution for Pa exists for this
form; so its consideration at this time is fruitless.

Of more practical importance than the pdf of noise is the probability of
communications for a given scenario. Figure 17 compares Poe the probability that
the signal-to-noise ratio exceeds the required threshold for noise curves A and B
in Figure 14. In both cases, the pdf of distance of the wanted transmitter from
the receiver is given by Curve A of Figure 2. Transmission loss is given by (8)
with y=2; it has no location variability. The dashed line in Figure 17 is Pa for
the scenario interference; the solid line is P, for the Class A noise. The two
curves are fairly close together for Pa less than 70%--that is, until the interfer-
ence becomes important and the difference between the two bumps on the right in
Figure 14 becomes important. In this region, the calculation using the Class A
noise approximation consistently overestimates the probability of communications.
Looked at another way, if a reliability Pa = 99% is required, the Class A approxi-
mation underestimates the required transmitter power (represented by y) by 14 dB.
If a reliability of 90% is required, the approximation underestimates the required
transmitter power by 5 dB.

Figure 18 shows noise plus interference for a scenario with less separation
between noise and interference. The geographic distribution of interferers is the
same as for Figure 13. The traffic intensity U=0.5 and the reference power is Py
= -85 dBW. The dashed line in Figure 18 is the pdf of interference from a single
interferer. Using the locations of the two peaks, the parameters for a Class A
noise model can be found using (48) and (49). They are Tf = 0.137 and iy = 5.31
(10713), Figure 19 compares the Class A noise for these parameters with the noise
from the scenario. As in the previous case, the nongaussian Class A noise has a
higher peak than the interference from the scenario, and the scenario has a higher

45


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