NTlA Report 82-95 

Digital System Performance 

Software Utilizing Noise 

Measurement Data 

A.D. Spaulding 

U.S. DEPARTMENT OF COMMERCE 

Malcolm Baldrigs. Secretary 

Bernard J. Wunder, Jr., Assistant Secretary 

for Communications and Information 

February 1982 

TABLE OF CONTENTS 

Page 

LIST OF FIGURES 

LIST OF TABLES 

ABSTRACT 

1. INTRODUCTION 

2. THE IMPULSIVE NOISE MODEL, TEST EXAMPLES 

3. SYSTEM PERFORMANCE CALCULATIONS 

4. CONCLUSIONS 

5. REFERENCES 

APPENDIX 

LIST OF FIGURES 

Page 

Figure 1 . Example of Class A ~q$e. 5 

Figure 2. Example of Class B Noise. 7 

Figure 3. Systems performance for NCFSK and DCPSK for the Class 

A noise example of Figure 1. 12 

Figure 4. System performance for NCFSK and DCiPSK for the Class B 

noise example of Figure 2. 13 

Figure 5. System performance for CPSK for the Class A noise 

example of Figure 1 . 19 

Figure 6. System performance for the class B noise example of 

Figure 2. 2 0 

LIST OF TABLES 

Tab1 e 1 . Class A and Class B Noise Data from Figures 1 and 2 8 

Table 2. ~rfc-l(2u) for Various a 15 

Table 3. CPSK System Performance for Class A Noise 18 

Table 4. CPSK System Performance for Class K Noise 18 

DIGITAL SYSTEM PERFORMANCE SOFTWARE UTILIZING NOISE MEASUREMENT DATA 

A. D. Spaulding* 

This report summarizes techniques that use the measured 

instantaneous envelope statistics of arbitrary noise or interference processes to calculate the degradation these processes 

cause to digital communication sys terns. Computer imp1 ementation of 

the techniques are also given. The computer algorithms are designed 

for the data obtained from a general purpose noise measurement 

device, termed the DM-4 (for "distribution meter-model number 4") 

recently developed by NTIAIITS. For ill ustration and for compari son with theoretical resul ts , two noise examples are employed, 

one for "narrowband" interference, and one for "broadband" interference. These examples are taken from the noise models recently 

devel oped by Mi ddl eton. 

Key Words: computer algorithms; digital system performance; nonGaussian noise; system performance software. 

1 . INTRODIJCTION 

Most currently used receiving systems are those which are optimum in 

Gaussian noise. Unfortunately, the actual interference environment is almost 

never Gaussian in character, but usually quite different, being impulsive in 

nature. By "impulsive" we mean only that there are significant probabilities 

of quite large instantaneous values of noise, which is a more general definition in that we can, and do, have both broadband (the usual definition, e.g., 

automotive ignition noise) and narrowband (e. g . , various combinations of 

interfering signals) "impulsive" processes. Recently there have been receiving 

systems designed to match this actual interference (e.g., Spaulding and 

Middl eton, 1977; Middleton, 1979). However, it is the purpose of this short 

report provide computer programs that will use noise measurements to calculate the performance of "normal" digital systems in arbitrary noise or interference (including, of course, Gaussian noise). By "normal" systems we mean 

those that are optimum in Gaussian noise, and, therefore, suboptimum in any 

other kind of noise or interference. The "normal" digital systems are "matched 

filter" or "correlation" systems. The common digital systems covered here are: 

*The author is with the Institute for Telecornmunication Sciences, National 

Telecommunications and Information Administration, U.S. Department of 

Commerce, Boulder, Colorado 80303. 
1. Binary non-coherent frequency shift keying (NCFSK) ; 7 

2. Binary differentially coherent phase shift keying (DCPSK) ; 

3. Binary coherent phase shift keying (antipodal or CPSK) ; 

4. Binary coherent frequency shift keying (othogonal or CFSK) ; and 

5. Binary coherent ON-OFF keying. 

In addition, the coherent signal detection system (Neyman-Person detection) is 

included. The performance of other systems, such as M level systems and minimal shift keying systems, can usually be obtained by appropriate extensions of 

the techniques summarized here. However, these extensions are not always 

straightforward. 

Recently, a general purpose noise measurement device, termed the DM-4 

(for "distribution meter-model number 4") was developed by NTIAIITS (Matheson, 

1980, DM-4 operation and maintenance manual, NTIA-TM 80-50), and the software 

presented here is designed to work specifically with the DM-4 measurements 

although no actual DM-4 measurements are used. For any received noise process, 

Z(t), we denote the probability density function (pdf) of the instantaneous 

amp1 i tude by pZ(z) . Denote the envelope of this received noise process by 

R(t) and pdf of the envelope by pR(r). The DM-4 measures the amplitude probability distribution (APD) of the envelope or Prob[R > RO], which we will 

denote by pR(r). Note that 

The DM-4 can also measure the average crossing rate characteristic of the 

received noise envelope, but these measurements are not considered here. While 

the DM-4 measures the APD in terms of the actual levels exceeded, referred to 

the input of the receiving system via calibration (e.g., dBm), we normally 

require the pR(r) or pZ(z) in normalized form so that the mean noise power is 

equal to 1. When we consider our desired signals, then the mean signal power 

is also the signal-to-noise ratio. For example, for the signal v?'!? Cos (oat), 

S is the signal power and also the signal-to-noise ratio. The DM-4 measures 

the APD at 31 cal i brated 1 eve1 s, i .e. , measures Prob[R > Ro] for 31 val ues of 

Ro. It uses a maximum sampling rate of 20 MHz, which means it can measure the 

output waveforms from systems of about 10 MHz bandwidth (IF) or less. The DM-4 

is designed to work with the detected logarithmic output of modern spectrum 
analyzers and EM1 meters, so that the 31 DM-4 levels are equally spaced in 

voltage, corresponding to 31 levels equally spaced in dB when referred to the 

receiving system input. For our purposes here we only need the calibrated 31 

levels (not necessarily equally spaced) and the Prob[R > Ro] for each of these 

levels as the input data to the system performance algorithms. 

For illustration and for comparison with theoretical results we make use 

of recently developed noise models. The models were developed for ITS by 

Middleton (1977, 1980) and Spaulding (1977). Two examples are selected, one for 

"narrowband" interference, termed Class A, and one for "broadband" interference, 

termed Class B. These examples of noise, from actual measurements (not DM-4 

however), are presented in the next section (Section 2) and are used then to 

simulate DM-4 "measurements." 

Section 3 presents the system performance algorithms, sample performance 

calculations, comparison with theoretical results when possible, and discussion 

of the algorithms. An Appendix then contains the actual computer software 

1 i stings in FORTRAN. 

2. THE IMPULSIVE NOISE MODEL, TEST EXAMPLES 

Recent work by Middleton has led to the development of a physical-statistical model for radio noise and interference. This model has been used to 

develop optimum detection algorithms for a wide range of communications problems (Spaul ding and Middl eton, 1977). It is this model which we wi 11 use here 

to simulate DM-4 "measurements." The Middleton model is the only one proposed 

to date in which the parameters of the model are determined explicitly by the 

underlying physical mechanisms (e.g . , source density , beam-patterns , propagation conditions, and emission waveforms). It is also the first model which 

treats narrowband interference processes (termed Class A), as well as the 

traditional broadband processes (Class B) . The model is a1 so canonical in 

nature in that the mathematical forms do not change with changing physical 

conditions. For a large number of comparisons of the model with measurements 

and for the details of the derivation of the model, see Middleton (1974, 1976, 

1977, 1978a, 1978b) and Spaulding (1977). We only summarize the results of the 

model which we need here. 

For the class A model, the expression for the pdf of the received noise 

signal, Z(t), is 
where 

and for the envelope, R(t), 

The Class A model has two parameters, A and r'. A is termed the impulsive 

index, and as A becomes larger (Q lo), the noise approaches Gaussian (still 

narrowband) and r' is the ratio of the energy in the Gaussian portion of the 

noise to the energy in the non-Gaussian component. In the above, the rms value 

of Z is equal to 1 , i . e., the process is a1 ready normal ized. 

For our sample DM-4 "measurement" of Class A noise, Figure 1 gives a 

measured Class A distribution and the appropriate model parameters are A=0.35 

and r0=0.5 X The first program, APDA, given in the Appendix, simply 

generates our "measurement" data. We compute pR(r) at 31 levels, starting 

with -59 dB (see Table 1) with the levels 3 dB apart. We further assume that 

the measurements after pR(r) = are zero. This gives us some zero "neasurements" that are likely from actual DM-4 measurements. The dynamic range 

covered, therefore, is 90 dB, from -59 to 31 dB. If we were actually measuring 

the APD of Figure 1, we would probably adjust the levels, so that the significant portion of the distribution (-50 to 20 dB, say) was more accurately 

covered. The above procedure (3 dB spacing), however, will be a better test of 

the system performance algorithms, but we want to keep in mind that accurracy 

can be improved by proper adjustment of the measurement levels. Once we have 

determined our 31 values of pR(r), we then assign these values to arbitrary 

(unnormalized) levels (3 dB apart) to check the normalization portion of the 

a1 gori thms, 
Denver 

- July 1,1972 - 

1520 - 1600 MDT 

200 kHz 

0 

- 6 kHz BW - 

o- Denotes Measured 

Points - 

- - 

- - 

- - 

- - 

A =0.35 

- - 

- 

- 60 

1 10 20 40 60 80 90 95 98 99 

Percent of Time Ordinate is Exceeded 

Figure 1. Exampl e of Class A noise. 
For the Class B model the pdf of the received instantaneous amplitude is: 

where, is a confluent hypergeometric function (Abramowitz and Stegun, 1964). 

The model has three parameters, a, Aa, and n. [A more detailed and complete 

model involving additional parameters has been developed, but (5) above is 

quite sufficient for our purposes.] The parameters a and Aa are intimately 

involved in the physical processes causing the interference. Again, definitions and details are contained in the references. The parameter n is a 

normalizing parameter. In the references, the normalization is n=l, which 

normalizes the process to the energy contained in the Gaussian positon of the 

noise. Here, we use a value of n which normalizes the process (z values) to 

the measured energy in the process. We cannot normalize to the energy computed 

from the model, since for (5), the second moment (or any moment) does not exist 

(i .e., is infinite). This is a typical problem with most such models for 

broadband impulsive noise. While the more complete model removes this problem, 

use of (5) will not 1 imit us here. The result corresponding to (5) for the 

APD is: 

On Figure 2, the parameter n was calculated with the assumption that pR(r) is 

zero for values of Ro > 40 dB. The program APDB, listed in the Appendix, is 

used to generate "measurement" from the APD of Figure 2. As before, a 90 dB 

dynamic range (here, -40 dB to 50 dB) is covered in 3-dB steps. A1 so, once the 

31 values of pR(r) are obtained, we assign arbitrary 3-dB step values for the 

corresponding Ro values to simulate actual measurements. The Class A example 

of Figure 1 is from Spaulding and Middleton (1977) and the Class B example of 

Figure 2 is from Evans and Griffiths (1974). Table 1 shows the outputs of APDA 

and APDB, and these then become our example "measurements." 
Percent of Time Ordinate Is Exceeded 

Figure 2. Example of Class B noise. 
Table 1. Class A and Class B Noise Data from Figures 1 and 2 

Class A Class B 

Prob [R > RO] 

- 

4 

Ro(dB) . -4.00000E+Oi 

-3~7000OE+Ol 

-3,400OOE+Oi 

-3.10000E+01 

-2.80000E+OI 

-2.5 000OE+Oi 

-2.20000E*01 

-1*90000E+01 

-1.6 00 OOE+Ol 

-1*30000E+OI 

-1~00000E+Ol 

-7.00000E+00 

-40OOOOOE+00 

-l~OOOOOE+OO 

2*00000E+00 

5- 0000OE+00 

8*00000E+00 

1~10000E*01 

1~40000E+Oi 

1~70000E+Ol 

2-00000E+Oi 

2~30000E+OP 

2*6DOOOE+OI 

2~90000E+Oi 

30 2000OE+Oi 

3~50000E+Oi 

3=80000E+Oi 

4*10000E+01 

4~400OOE+Ol 

4.70000E+Oi 

5*00000E+Oi 

Prob [R > RO] 

9.43528E-01 

8-914276-01 

7a98434E-01 

6m49084E-01 

4.51644E-01 

2-63070E-01 

1a44504E-01 

9-575056-02 

5,387636-02 

3-46832E-02 

2*25913E-02 

1*48055E-02 

9-73559E-03 

6-414O3E-03 

4m23040E-03 

2a79201E-03 

1a84343E-03 

1.21743E-03 

8- 04130E-04 

5*31191E-04 

3*50915E-04 

2-31830E-04 

1*53i61€-04 

1*01190E-04 

6.68539E-05 

6a41694E-05 

2a91821E-05 

Ia92803E-05 

la27383E-05 

0, 

Ow 
Finally, white Gaussian noise is a special case, and the performance of 

digital systems in white Gaussian noise has been treated in great detail. 

Here, we will occasionally refer to the well-known results of system performance in white Gaussian noise for comparison. The pdf for the instantaneous 

amplitude for Gaussian noise (mean noise power = 1 ) is: 

and for the corresponding envelope 

3. SYSTEM PERFORMANCE CALCULATIONS 

In this Section we want to present the results which are most advantageous 

for our use. We want to develop system performance algorithms which do not 

require particularly sophisticated numerical analysis techniques and which can 

be used on small scale computers. 

We start with the simplest. For arbitrary additive interference which is 

independent from an integration period (bit length) to the next and which has 

uniformly distributed phase, Montgomery (1 954) has shown that the probabil i ty 

of binary bit error, Pe, for NCFSK (non-coherent frequency shift keying) is 

given by: 

1 

Pe = Prob [noise envelope > rms signal 1 evel]. 

While Montgomery's result for NCFSK is in terms of the noise and signal envelopes 

at the input to an ideal discriminator, it has been shown (White, 1966) that the 

resul t is a1 so appl icabl e to most common FSK receivers (bandpass-fi 1 ter di scri - 

minator receivers and matched-f i 1 ter envelope detection receivers) . Using (lo), 

then, the performance for NCFSK can be obtained instantly from the APD measurements, once the APD has been normalized to its rrns level. For example, if the two 

NCFSK waveforms are given by 
Sl (t) = cos (w, t + g) , and 

s2(t) = LT cos (w2t + g), 

where 4 is the unknown (uniformly distributed) phase (i.e., incoherent signaling), 

m1 and me are the two frequencies, and S is the signal power, using (9) and (lo), 

performance in Gaussian noise is, therefore, 

For Class A noise, using (4) and (lo), 

The above, of course, is for the binary symmetric channel. That is, Sl (t) and 

S2(t) are equally probable. In short, the performance can be obtained by inspection from the normalized APD. If, for example, the probability that RO=l (0 dB) 

is exceeded is Po, then, for the signaling set given by (11 ), pe for a SNR of 

2(3 db) is P0/2, and so on, for aily SNR, Note the 3 dB "shift" for RCFSK. No 

algorithm is given in the Appendix for NCFSK, since all that is required is a 

normalized APDy and the normalization procedure is included in other system 

performance algorithms. Also, in any case, the APD measurement device, DM-4, 

would normally present the measurements in normalized form, although we do not 

make that assumption in this report in order to maintain as much generality as 

possible. 

In the bi-phase, DCPSK (differentially coherent phase shift keying) system, 

the receiver compares the phase 4 of a noisy signal with a reference phase F, to 

decide whether the corresponding pure signal relative phase $ was 0 or n ($ = 0, 

corresponding to the signal cos uot, is selected if I4-Fj<n/2, and $=T, 

corresponding to .m cos (mot), other~vise. ) The reference phase is obtained from 

the previously received signals; usually it is just the phase of the previous 

signal. Thus the analysis of this system is coniplicated by the fact that both 
@ and $ are affected by noise. This system also has adjacent symbol dependency, 

and, therefore, the occurrence of paired errors and other error groupings cannot 

be obtained easily, even with independent noise. Halton and Spaulding (1966) have 

given results for this system, including the occurrence of various error groupings 

However, it can be shown that for binary DCIPSK, the elemental probability of 

error, Pe, is the same as for NCFSK, with 3 dB less signal energy required. That 

is, for a given Pe, DCPSK requires 3 dB less SNR than does NCFSK for arbitrary 

additive interference that is independent from one bit time to the next. [For a 

geometrical derivation of this result see Arthurs and Dym (1962).] For example, 

therefore, for Gaussian noise for binary DCPSK; 

The performance of DCPSK can be obtained directly from the APD of the additive 

interference. If, for example, the probability that Ro = 1 (0 dB) is exceeded is 

Po, then for the above signaling set, Pe for a SNR of 1 (0 dB), is P0/2, and so on 

for any SNR. Figure 3 shows Pe versus SNR for the noise of figure 1 for both 

NCFSK and DCPSK, while Figure 4 shows Pe versus SNR for the Class B noise of 

Figure 2 for these two systems. Performance for Gaussian noise is also shown for 

reference. 

We next consider coherent binary systems. The performance of these systems 

can be obtained from the pdf of the additive interference envelope by means of the 

result : 

For the derivation of this result see Spaulding (1964), and for various other approaches which led to (15), see Arthurs and Dym (1962). For antipotal signaling 

(CPSK, coherent phase shift keying), the binary signal set is, 

s1 [t) = m cos (wgtl, and 

S2(t) = -m cos (wet), 
- I I 1 I I I I I I - 

+ 

- 

Class A Noise - 

- 

- - 

- - - - 

- - - 

- - 

- - 

- - 

- 

I 

- 

, 

- 

C 

P 

C 

1- 1 

- - 

+ 

- - 

C 

- 

- 

+ 

I- 

- 

- - - - d 

- - - - 

- + 

- 

- 

- ,-.. 

- 

- 

- 

- 

0 10 

SNR, S, dB 

Figure 3. Systems performance for NCFSK and DCPSK for the Class A noise 

exampl e of Figure 1 . 
SNR, S, dB 

- 

- 

- 

- 

d - 

- 

- 

- 

-1 - - i - 

F: 1 

d 

I 

tt - 

- 

r - 

C - 

L 

II 

!- 

i 

,L Gaussian 

i DCPSK-- -5 1 

Figure 4. System performance for NCFSK: and DCPSK for the Class B noise 

example of Figure 2. 

10 r - 

I 

,- 

i 

1- 

I 

II 

?- 

10-61 I L_I_-L_L_L I I 

- 

- 

A 

- 

-t 

I 

I I I 

- 30 - 20 -10 0 10 20 30 

1 1 40 
and in (15), K = 6. 

For coherent, orthogonal signaling, the signal set is, 

Sl (t) = V% cos (wet). and 

s2(t) = V% sin (oat) , 

and in (15), K =m. 

For ON-OFF coherent signaling, 

Sl(t) = cos (oat). and 

and in (15), K = my where we use the convention that the SNR is based on the 

average signal power of the two signals, Sl(t) and S2(t) This average is, of 

course, S/2, for a symmetric channel. 

The performance of a coherent Neyman-Pearson signal detection system can 

also be obtained via the integral in (15). We have two hypotheses: 

Ho: X(t) = Z(t) + S(t), and 

(19) 

H1: X(t) = Z(t). 

The received waveform, X(t), is composed of noise plus the completely known 

signal to be detected (Ho) , or it is composed of noise alone (HI ) . The NeymanPearson detector, which is optimum if Z(t) is Gaussian, decides between Ho and 

HI by presetting a probability of false alarm (deciding Ho, when H1 is true) a. 

Performance is then given by the probability of detection (deciding Ho when Ho 

is true), P,,, and the probability of a miss (deciding H1 when Ho is true), PM, 

and PD=l-P Performance for additive interference is given by (15), where M ' 

K = 43 - JZ erfc-'(~a), (20) 

for a desired signal of power S. The complimentary error function is given by 
Use of the K given in (20) in (15) gives the probability of a miss, pH. This 

is 

and PD = 1-PM, K given by (20). 

Table 2 below gives erfc-l(Za) for various probabilities of false alarm, 

a. 

Table 2. ~rfc-' (2a) for Various a 

For the above coherent systems in Gaussian noise, 

where K = 6 for antipodal signaling, K = &;/2 orthogonal signal ing, and 

K = for ON-OFF signaling. For the signal detection system in Gaussian 

noise, 

where K = 6 - fi erfc-l(Za) . 
Likewise, for Class A noise, we can obtain, 

CO 

Equation (25) gives the Pm when the K given by (20) is used. 

It now remains to develop efficient computer algorithms based on (15). 

The result (15) uses the pdf of the interference envelope, and the measurements 

are of the APD. Actually, the measurements at 31 levels of the APD also give 

an equally valid estimate of the pdf as well. Also if we attempt to modify 

(15) to a form that uses the APD directly, i .e., uses pR(r) rather than pR(r), 

we obtain computational complexities. For example, (15) can be transformed to: 

p e = 1bR(:) .rr 

-!- dr. 

m 

0 

Both (26) and (27) are improper integrals, and while this creates no problem 

analytically, very sophisticated numerical integration routines are required in 

order to obtain any accuracy for Pe, especially when pR(r) is given only in 

sampled data form. It turns out that it is much better to use (15) "directly" 

along with pR(r) estimated from the measured pR(r). 

The main algorithm presented in the Appendix is called SYSAPD. This 

program takes the 31 measured APD data points, normalizes the APD to its rms 

level, obtains the pdf, and then evaluates the integral (15) for the appropriate 

K. The program SYSAPD uses Gauss-Laguerre quadratures to evaluate (15) (Kopal, 

1961 ) . The Gauss-Laguerre quadrature formula is 
the points at which the integrand must be evaluated, y , and the corresponding j 

weight, H are obtained via the Laguerre polynominals. The program SYSAPD used j ' 

a fifteenth order quadrature [(n=15 in (28)l. The above means that the integral 

(15) is put in the form 

for evaluation. 

Consider first the Class A "measurement" data of Table 1. Table 3 gives 

Pe versus SNR for CPSK obtained from the program SYSAPD which uses (29). Note 

that in using the data (see program listing in the Appendix) arbitrary 3 dB 

levels are used. For Class A noise, (25) gives the "correct" theoretical 

performance. The program SYSCOR computes (25) and Table 3 also gives these 

results so that the approximation from the "measurements" can be compared with 

the "true" answer. Another program that is given in the Appendix is SYSGL. 

This program uses (29), but pR(r) is obtained from the Class A model mathematical expression (4) rather than from the corresponding "measurement" data. 

The Pe versus SNR for CPSK from this program is also given on Table 3. This 

shows the accuracy of the integration routine when these results are compared 

with the "true" results. It also indicates the accuracy of the normalization, 

pdf determination, and interpolation techniques used in SYSAPD. Finally, 

Figure 5 shows the results of Table 3 along with the standard performance in 

Gaussian noise (23) for further comparison. 

The above results are for the Class A example. For the Class B case, the 

simple Gauss-Laguerre quadrature used above does not give sufficient accuracy 

when the Class B "measurements" are used in the program SYSAPD or when the 

corresponding mathematical model for Class B noise is used with program SYSGL. 

[The result of using the Class B example in SYSAPD (or in SYSGL) is shown 

by the clashed curve on Figure 6.1 Because of this, a different integration 

routine must be used. This is given by program SYSWR, which used Weddle's 

Rule (Kopal, 1961) to perform the integrations. This integration routine uses 

(15) directly and, of course, is somewhat more sophisticated than the GaussLaguerre quadrature used previously, but it is still appropriate for small 

scale computers. For the Class B case, we have no "theoretical" results to 

use to check the accuracy of the integrations performed by SYSWP. 
Table 3. CPSK System Performance for Class A Noise 

Table 4. CSPK System Performance f~r Class B Noise 

t 1 

pe, SYSCOR 

1-60710E-01 

1.46402E-01 

1.4252RE-01 

1.40544E-01 

1,38309E-01 

1.352'0?€-02 

1.31073E-01 

1,25602E-01 

1.1835EE-01 

1-08950E-01 

9.67394E-02 

8-15E82E-02 

6.33627E-02 

4-33409E-02 

2.42702E-02 

9.99877E-03 

2- h9063E-03 

4.46870E-04 

4.05 12l.E-05 

1.39231E-06 

1.32516E-08 

1 

SNR (dB) 

-3eUCDDGE+G1 

-2r75CGtE+31 

-2.5 i003E+G1 

-2e2508CE+Cl 

-2e3EDiOE+Oi 

-1. 750 OCE+CI 

-1.5 EUOOE+CI 

-10 25GCOE+CI 

-1eOC006E+61 

-7.5@000E+CC 

-5.0 1003€+65 

-2.5 COOUE+@C 

5. 

2.5000fE+t!O 

5*00000E+OD 

7.5i!OGCE+OC 

I. OCOOCE+Cl 

I. ZSi3EGE+CI 

1e520f3PE+C1 

lr75003E+Gi 

2. O GG OCE+OI 

2.25000E+J1 

p, SYSGL 

1.447ICE-ji 

1r 437265091 

10 4241 6E-Ci 

I,GS6€9f-C1 

f. 38342E-Ci 

1.352441-01 

1. 31L2 5E-51 

1.25hE4E-Ci 

1. 19+585-21 

1.09; 14E-qi 

9*65926E-$2 

8.156f?lE-E2 

6,34648E-32 

40 3h341E-22 

2-43+UQE-02 

1.063E7E-92 

2.7P273E-93 

4,488765-34 

4e Cli C 2E-15 

1.39309E-26 

1.331 43E-38 

2, C 971 3:-11 

p,, SYSAPD 

1- 395llE-01 

1.38669E-01 

1.?755cE-01 

lm3605?E-01 

1.3425RE-01 

1.30951E-01 

1,26556E-01 

1.207F4E-01 

1-13917E-01 

1.04745E-01 

9-24013E-02 

7.69463E-02 

5.YZQ15E-02 

4.15365E-02 

2.21 76kE-02 

q-425R7E-03 

2-56 L02E-03 

4.70352E-04 

4,63972E-05 

2mn7974E-06 

5-?7832E-09 

0. I 
SNR, S, dB 

Figure 5. System performance for CPSK for the Class A noise example of 

Figure 1 . 
Figure 6. System performance for the Class B noise example of 

Figure 2. 
The accuracy was checked by using another integration routine, appropriate only 

for large computers, in which the desired accuracy can be specified in conjunction with the Class B mathematical model. Program SYSWP was found to give very 

good accuracy for all signal-to-noise ratios. Table 4 and Figure 6 show the 

results of the use of SYSWR with the "measurements" of Table 1. Two subroutines, 

both termed FUN13 are given. One for the "measurement" data and program SYSWR 

and one for the mathematical Class B model for use with SYSWR. 

In order to evaluate the integral (15) the pdf of the noise envelope is 

very easily obtained from the APD for Class A noise given in (4). However, obtaining the pdf for the envelope of Class B noise corresponding to the APD 

given by (6) is somewhat more involved. By differentiating (6) we eventually 

obtain the following, which has been put in a form suitable for numerical 

computation: 

The Appendix lists the appropriate programs and all the required subroutines used in the above example calculations. 

4. CONCLUS IONS 

This report has developed simple computer algorithms which use measurements of the APD of an interfering waveform (or a corresponding mathematical 

model ) to determine performance of various "normal " digital data systems. As 

can be seen from the examples above, we obtain very good estimates of system 

performance using SYSAPD and DM-4 Class A simulated noise measurements and by 

using SYSWR and DM-4 Class B simulated noise measurements. Of course, the 

Class A measurements can also be used with SYSWR. The algorithms developed are 

for binary digital systems (and the coherent Neyman-Person signal detection 

system), however, the performance of other systems (e. g. , M level systems, and 

minimal shift keying systems) can usual ly be obtained by appropriate extensions 

of the techniques developed here. 
5. REFERENCES 

Abramowi tz, M. , and I .A. Stegun (1 964), Handbook of mathematical Functions, 

National Bureau of Standards Applied Mathematics Series, No. 55, Section 

13, 503. (U.S. Government Printing Office, Washington, D.C. 20402). 

Arthurs, E., and H. Dym (1962), On the optimum detection of digital signals in 

the presence of white Gaussian noise: a geometrical interpretation and a 

study of three basic data transmission systems, IRE Trans. Commun. Sys. 

CS-10, No. 4, pp. 336-372. 

Evans, T.T., and A.S. Griffiths (1974), Design of a sanguine noise processor 

based upon world-wide extremely low frequency (ELF) recordings, IEEE 

Trans. Commun. COM-22, No. 4, pp. 528-539. 

Halton, J.H., and A.D. Spaulding (1966), Error rates in differentially coherent 

phase systems in non-Gaussian noise, IEEE Trans. Commun. Technol. COM-14, 

NO. 5, pp. 594-601. 

Kopal , Z. (1961 ), [lumerical Analysis, (John Wiley and Sons, New York, New York). 

Middleton, D. (1 974), Statistical-physical models of man-made radio noise, Part 

I : First order probabi 1 i ty models of the instantaneous amp1 i tude, Office 

of Telecommunications Report OT 74-36 (NTIA Access. No. COM75-10864PAS). 

Middleton, D. (1976), Statistical -physical models of man-made and natural radio 

noise, Part 11: First-order probability models of thn envelope and phase, 

Office of Telecommunications Report OT 76-86 (NTIS Access. No. PB253-949). 

Mi ddleton, D. (1 977), Statistical -physical models of electromagnetic interference, IEEE Trans EMC EMC-19, No. 3, pp. 106-127. 

Mi ddl eton, D. (1 978a), Stati sti cal -physical model s of man-made and natural 

radio noise, Part 111: First-order probability models of the instantaneous amplitude of Class B interference, National Telecommunications and 

Information Administration Contractor Report 78-1 (NTIS Access. No. 

PB284862lAS). 

Middleton, D. (1978b), Statistical -physical models of man-made and natural 

radio noise, Part IV: Determination of the first-order parameters of 

Class A and Class B interference, National Telecommunications and Information Administration Contractor Report 78-2 (NTIS Access. No. ~B288467). 

Middl eton, D. (1 979), Canonical non-Gaussian noise models : Their imp1 ications 

for measurements and for prediction of receiver performance, IEEE Trans. 

EMC EMC-21, No. 3, pp, 209-220. 
Middlet;on, D. (1980), A tutorial review of the new EM1 models and their effects 

on receiver performance, National Telec~ommunications and Information 

Administration Contractor Report 80-7 (NTIS Access. No. PB80-211188). 

Montgomery, G. F. (1 954), Comparison on amp1 i tude and angle modulation for narrowband communications of bi nary-coded messages in fl uctuati on noise , 

Proc. IRE - 42, pp. 447-454. 

Spaul ding, A. D. (1 964), Determination of error rates for narrow-band communi - 

cation of binary-coded messages in atmospheric radio noise, Proc. IEEE 

- 52, No. 2, pp. 220-221 . Spaulding, A.D. (1 977), Stochastic model ing of the electromagnetic interference 

environment, Conference Record - International Conference on Communications, 

ICC '77, Chicago, 42.2-114-123 (IEEE Catalog No. 77CH 1209-6C SCB). 

Spaulding, A.D., and D. Middleton (1977), Optimum reception in an impulsive 

interference environment - Part I: Coherent detection; Part 11: Incoherent reception, IEEE Trans. Commun. COM-25, pp. 910-934. -- 

White, H.E. (1966), Atmospheric noise FSK error probabilities for an envelope - 

detector receiver, IEEE Trans. Commun. Technol. COM-14, No. 2, pp. 288- 

289. 
APPENDIX 

Program Listings 

In this Appendix we simply list the computer programs (in Fortran 4) used 

for the sample calculations given in the report and required for similar calculations. The programs are essentially self-explanatory via the comment statements, but some further explanation may be helpful. 

The first two programs, APDA and APDB, compute the APD "measurement data" 

from the Middleton Class A and Class B models. The output is given in Table 1. 

The program APDB requires Subroutine CONHYP and FUNCTION GAMMA which are also 

gi ven . 

The next set of programs are for the Class A example. The program SYSCOR 

computes the theoretical performance and requires FUNCTION CERF. The program 

SYSGL computes performance via Gauss-Laguerre integration but obtains the 

required pdf of the envelope values directly from the Class A model and requires 

SUBROUTINE FUN. Finally, one of the main results is program SYSAPD which 

computes performance from measured APD val ues. Sections of this program normal ize 

the APD and estimate the pdf from the measured APD. These routines are useful 

in their own right for various purposes. The outputs of these programs are 

contained in Table 3. 

The final set of programs are for the Class B example, but can also be used 

for any noise example. The program SYSWR computes performance via Weddle's 

Rule integration. Two FUN1 subroutines are given for use with SYSWR. One calculates the envelope pdf from the mathematical model given by (30) and the 

other one calculates the envelope pdf from the measured APD. The outputs of 

SYSWR are given in Table 4. 

In some of the programs IRAY and SYSTEMC are used. This is to suppress an 

exponent underflow error message for the particular computer used (CYBER 170/750) 

and is not, in general, required. 
PROGRAM APDA( INPUT,ClUTPUT) 

PR'OGRAM USED TO OBTAIN APD VALllES FROM CLASS A WODEC* 

EQUATION 4 OF TEXT* 

DTVENSION IQAY(6) 

DATA IRAY/-1,-1~-1~0,-1,-1/ 

CALL SYSTEYC(115,IRAY) 

PRINT 6 

FOQMAT( 1H1) 

A~0.35 

CAY=0.5E-3 

DO 40 1=1931 

RDBs-59*+3**( 1-10 

P=lO.**(RDB/20* 

SM.0. $ FJn1. 

DO 20 JJ=1,2h 

J=JJ-1 

IF(J*NE.Q) FJ=FJ*J 

SIGSQ=(J/A+GAM)/(l.tGAMI 

T=((h**J)FJ)*FXP(-R*R/SIGSQ) 

SM=SYtT 

CONTINUE 

P=SM*EXP(-A) 

IF(PatTo1.OE-6) GO TO 10 

GO Tg 15 

P.O. 

PRINT ~JRDBPP 

CONTIrJUE 

FORYAT( lOXt2(1PFlZe5,3X) 

END 
PROGRAM APDB( INPUT,OUT PUT) 

PROGRAM USED TO OBTAIN APD VALUES FROM CLASS B MODELEQUATION S 3F TEXT. 

DIMENSION IKAY (6) 

DATA IRAY/-19-1 ,-I,O,-I,-lt 

CALL SYSTEMC4 3.15 r IRAY) 

PRINT 6 

FORHAT (IHI) 

AA=1* 

4LPHA=l. 2 

OMEG4=0.00079433 

DO 40 I=1,31 

RD8=-40m+3,+I 1-1-1 

IF(ROB*GT.45. GO TO 10 

R=10er*#RD9/20e) 

ZN=R*R/OHEGA 

SM=O* E FN=I* $ SMI=O. 

00 20 N=1*25 

FN=FM+N 

CALL CONHYP(Le-N+ALPHA/2.,2-,ZN,S,IOVFLW) 

T= (4 (-AA)*+N) /FN)*GAMY A (1-+N*RLPHA/2.) "S 

IF( IOVFLHmNE. I) GO TO 14 

SMl=SMItT 

GO TO 20 

SM=SM+T 

CONTINE 

FP=O* 

IF IZNmLT*675,) FP=EXP(-ZN) 

P=FP-ZN*(FP*SM+SMIl 

GO TO 15 

P=Om 

PRINT 7, RDB,P 

CONTINUE 

FORMAT (lOX, 2( IPEiZe5,3X) 

EN3 
SUBROUTINE CONHYPtA ,ST X,S? IOVFCW) 

CoooemCOYPUTES IFI(A,3,X) FOS REAL A*B,X 

CoooaaIF X GREATER THAN 741. AN OVERFLOW WILL OCCUR* SEE 

Co.oo,COMMENTS BELOW0 

S=ie $ Y=i. 

IOVFLW=O 

KUNDEF=O 

IF(AoGToOoIG3 TO 101 

K=- A 

ENA=-K-I 

VA= A-ENA 

IFIVAoEQolo*3R*VAoEQoO 0) GO TO 110 

101 IF(BoGToOoIG3 TO 130 

J=- 8 

ENB=- J-I 

VB=B-ENR 

IFIVBoEQ.1moOR-VBaEQoOo 11209 130 

110 KUNDEF=I 

GO TO I01 

120 IF(KUNDEF*EQ. I)PRINTIOOOTA~R 

IF(KUNDEFeNEoilPRINTiOOi,B 

RETURN 

130 IF(KUNDEFoEQ* 1)GO TO 10 

5 IF IXoGEolOOo) GO TO 60 

6 IF(X*GEoIOo) GO TO 10 

NN=IOO 

GO 70 15 

10 NN=300 

15 IFIKUNDEFeEQo 1) NN=-At 1 

DO 20 N=l,NY 

O=N'( (B+N-1s 0) **20 1 

Y=(A+N-IoOO*(Y/O) 

Y= Yr (B+N-I 00) 

Y=Y*X 

IF(SoEQo (S+Y 1) GO T 0 50 

S=S+Y 

20 CONTINUE 

50 RETURN 

Cam -a APPROXIMATES IF1 (A,BTX) FOR REAL A92 ,X BY USING THE 

Coo l em ASYMPTOTIC E%PANSIONo SEE PAGE 1073, INTRODUCTION 

C-• l om TO STATIST ICd L COMMUNICATIONS THEORY I MIDOLETONC*o* -*IF XoGE.675- AN OVERFLOH WILL OCCUR FROM EXPO 

CooeaoTO AVOID THIS, THE VAQIABLE IOVFLH IS SET TO i AND 

Coo*--THE FUNCTION VALUE IS CRLCULATEC WITHOUT THE EXP(X1 FACTOR. 

Co*. *-SO THAT THE VALUE RETUqNEO IS S/EXP(X) 

60 NN=20 

00 100 N=lrNN 

Y=Y* (8-AtN-lo I *(N-A) 

Y=YJ (N*X) 

IF(SoEQo1S+Y) )GO TO 150 

S=S WY 

100 CONTINUE 

150 S=S*(GPMMA(8)/GAMHA(A))*(X*+(P-8)) 

IF(XoLTo675o)GO TO 190 

IOVFLW=I 

GO TO 200 

190 S=S*EXP(X) 
200 RETUSN 

100 0 FO?MAT I//TiXT' CANNOT EVALUATE EXPRESSION SINCE BOTH+r 

I+ A BNO 8 AiiE NEGATIVE INTEGERS OR ZERO* A=*,F10.2** B=+T 

2F10129//) 

1001 FORMATl//rlX,+ BAD VALUE FOR e GIVES INFINITE RESULT FOR S", 

P *, B=*~F10.29//) 

END 

FUNCTION GAMMA(Y) 

C RETURYS TI'E GAMMA FUNCTIOY F'YR REAL ARGUMENT. 

C NOTE. THE GAMMA FUNCTION IS NOT DEFIYED FOR A NEGATIVE INTEGER OR ZER 

C INPUT 

C X = THE REAL ARGUYEYT* 

C OliTPIJT • 

C GAMYAIX) = THF GAYYA FUNCTI3fl 7F ARGUMENT XO 

75 FQRMAT(66H GAYMA FUVCTION OF 4 NEGATIVE INTEGER* OR OF ZERO, IS NO 

IT DEFINED*) 

5 IF(X) 10980915 

10 N=-X 

EN=-Y-1 

V=X-EN 

IF(VeEQe1.)80,20 

15 N=X 

EN =Y 

VrX-EN 

20 GAV~A~l.+V*(~422784337+V*(~4118402518+V*(eO81578?1878+V* 

I.( e07423790761+V*(-~0002109074673+V*(~010973495~4+V*~~~0024~674790

2+V*~o9015397b8105-Vs(.00034427~Z04~-V*.00006?710~71~7~~~~~~

IF(EY-2.) 37925930 

25 2FTURN 

30 N=N-l 

Drl 35 I=29N 

FIST 

35 GAMM4=GAMMAt(FI+V) 

RETURY 

37 N=?*-EN 

DO 40 I=19N 

FI=?-T: 

40 GAMYA=GbMYA /(FI+V) 

RETUqW 

80 PRIUT 75 

CALL EXIT 

END 
PROGRhM SYSCOR( INPUTJOUTPUT) 

THIS PROGRAM COYPIJTES THE PROBASILITY OF EQROR FOP BINARY 

CPSKt CFSY, 4ND COHERENT ON-DFF IN CLASS A~NOISEp THAT 1st 

MID0LETON"S CLASS A MODEL* 

HFRE, THE E9FC FUNCTION IS TESYED CERF, TO r3YPASS THE 

SYSTEYS INTERNAL ESFC ROUTINE. 

FOR CPSY, AK=SQRT( S) 

FOR CFSKt AK=SQRT(S/2r) 

FOR ON-OFF, AK=SQRT(S/4*) 

Ax9.35 

GArJIs0.5E-3 

PRIVT 6 

FflRMAT(lH1) 

nfl 40 J~lt29 

SOQ*-30.+2*5*(J-1) 

S=IO***(SOB/lO~) 

AK=SQQT(S) 

SUY=Oe $ FKsl* 

DO 20 KK=ltZS 

K =KY-1 

IF(Y*qE*O) FK=FY*K 

SIGSQ*( K/A+GAM) /(1 etC4M) 

SIG=SQRT(SIGSQ) 

T=((A**K)/FK)*CERF(AK/SIG) 

SUM=SUM+T 

COYTINUE 

PEnEXP(-A)*SUM/Z* 

IF(PE*LT*l*E-9) C4LL EXIT 

PRINT 89 SDBIPE 

CONTINUE 

FORYATllOXy 2(1PE12*5,3X) 1 

END 

FUNCTION CERF (XI 

C SEE APPRflXIMATIONS FOR DIGITAL COYPUTERS 

C RY C* HASTINGS, PRINCETON U* PRESS, 1955, 

C PAGE 169. ALSO IN ARRAM3WITZ AND STEGUN* 

E=1*0/( 1*3+0.3275911sX) 

S~(((((0~940646070*E)-1~2R78224533*E+1~259695130~*€~0~25212866

1+0*225836846)*E 

XSQ=Y**Z 

ExPFX=O*O 

IF(XSQ*LT*709*0)EXPFX~EXP(-XSQ) 

CERF=S*EXPFX*1.1?9379167 

RE TURN 

END 
S(lBR'3UTIYE FUq( X,P,Y) 

THIS FUNCTION ROUTINE IS FDR CLASS 4 NOISE F39 USE 

WITH PROGRAM SYSGL. 

A=0,35 

GAYs0.5E-3 

v= X+P 

SM=O* 8 FJ.1. 

Dn 20 ~~11~26 

J=JJ-1 

IF(JeVE.0) FJ=FJ*J 

SIGSQ=(J/A+GAM)/(l.+GAM) 

T=(2,*V/SIGSQ)*((A**J)/FJ)*€XD(-V*V/SIGSO) 

?M=$Y+T 

70 CONTINUE 

Y=SM*FXP(X)*ACOS(P/V)*EXP(-A) 

QFTUQN 

E NO 
PROGRAM SYSAPD( INPUT9OUTPUT) 

C THIS PROGRAM MAKES DIRECT USE OF THE YEASURED AQD DATA931 LEVELS, 

C AND ESTIMATES THE PDF OF THE ENVELOPE FOR USE IN THF GFNERAL 

C COHCRENT SYSTEM PERFORIIANCE 4LGORITYMa THE PRO0 OF RIT EQROR 

C FOR BIWARY CPSKtCFSKtAYD COHERENT Oq-OFF IS ESTIMATED* THE 

C SIGNAL POWER IS GIVEN RY St AND THE NOISE IS 

C NORMALIZED SO THAT THE MEAN YOISE POWER IS UNITY* THEN 

C THE SIGNAL-TO-NOISE RATIO IS AC5O GIVEN BY Sa SIGNAL-TO-NOISE 

C RATIOS FRnY 400R TO -3008 ASE COWERED IN 2e5DR STEPS* THE 

c PROGRAY USES GAUSS-LAQUESSE QUADRATURES TO EVALATF THF INTEGRALS, 

C FOR CPSK, AK=SQRT(S) 

C FOR CFSK, AK=SQRT( S/2* 1 

C FOR OFF-ON, AK*SORT( S/4* 

C NOTE, BEFORE THIS PROGQAM IS EFFECTIVE, THE EQTIRE 4PD MUST 

C BE COVERED BY THE 31 (OR LESS) LEVELS* THAT IS, p SYqULD 

C RANGE FROM ABOUT 0.95 OR HIGHER OQWY TO ABOUT 1eOE-5 OR YMALCESa 

OIYENSION CL(31)tSL(31)~P(31)~CXI31)9SX(31)~SP(31) 

DIMENSION Z(15)tH(15) 

DATA CL/1a~4*,7a,10a~13a~16a~19a~22~~25a~38a~31a~34av37~

140e~43a~46~~49r~52*~Fi5~~58et61a*b4a~67et70a~73at7ha~79a

282e,95*988*,91./ 

DATA P/ a99929 a99659 e9930, e98609 a9724,.9460**8964, .8@85~ a66969 

la49469a3519ta29979 a29499 a2944v~29349a29169.2879,a2807~ a26699 

2~2415~a1981~a1343~~0h372~a01670~a002116~~000I047t~0000P10

3a0~*0~a09 a0/ 

C CL IS THE 4PD LEVELS IY 09 4ND p IS THE PROB48ILITIEF AT 

C THESE LEVELS. 

OATA Z/0a09330781~0a49269174~1a21559541~2a7_b994952~3e667622?2~ 

15*42533663,7a565916239 lOa12022857~ 13 a130282489 lhe654407719 

220a776478Q0~25a62389423~31a40751W17,38a53068331~48e02608557/ 

DATA H/e21823489,*342210189a263027589 a126425829 a40206865E-19 

1a85639778E-2~a12124361E-2~a11167439E-3,~64599268E-5~~2?2631b9F-6~ 

2~42274304E-8~~39218973F~13~a14565153E-1Z~a1~8302~1E~15~ 

3al6005949E-19/ 

C M AND Z ARE AS GIVEN IY EQa 28 OF TEXT* 

CX(1)=1*5*CL~l)-0*5*CL~2) 

SX(1)~10,**(CX(1)/20~) 

DO 20 1~2~31 

CX(I)=(CL(I-1)+CL(I) )/2* 

SX~I)~10***~CX(I)/20a) 

20 CONTINUE 

C COMPUTE ?MS 

SUM*SX(l)*SX(lI*(l.-P(1)) 

00 30 Ja2931 

T=SX(J)*SX(J)*(P( J-l)-p(J)) 

SUM=SUM+T 

30 CONTINUE 

RMS~(SUM+lO***(CL(31)/20*)sp(31))**0a5 

C NORHALTZE TO RMS LEVEL AND COMPUTE PDF 

CL(l)~CL(l)-20~*ALOGlO(QMS) 

SLYl)=lOa**(CL(l) /20*) 

SX(l)=SX(l)/RMS 

CX(1)*20a*ACOGlO(SX(1)) 

SP(l)=(l*-P(l))/SL(l) 
DO 40 K~2931 

CL(K)aCL(K)-ZO**ALOGlO(RMS) 

SL(K)=lO,**(CL(K)/20.) 

SX(K)=SX(K) /RMS 

CX(K)=2O.*ALOGlO(SX(K)) 

SP(K)=(P(Y-1)-P(K))/(SL(K)-SL(K-1)) 

CONTINUE 

PRINT 6 

FORflAT(lH1) 

00 80 N~1929 

SOB=-304+2.5*tN-1) 

S=lO***(SDS/10,) 

AK-SORTcS) 

SUMzO* $ MM=2 

DO 70 L=1,15 

V=Z(L)+AY 

IF(VeLT*SX(l)*OPeVeGT*SX(31)) 15~16 

DO 17 MaMHp31 

IF(V*LEoSXtY)*AYDoVeGTeSX(M-1)) GO TO 18 

COYTINUE 

IF(SP(M-l)*EQ.O*oOR.SP(M)*EQ*O*) GO TO 15 

YDB~(20,*ALOGlO(V)-CX~M-l) )/(CX(M)-CX(M-1)) 

YDR~YD8*(20~*ALOG10( SP(M) )-2O,*ALOGlO( SP(M-1) 1) 

YDB* Y08+20~*AL~G10( SP(M-1) 1 

YalOe**(YOB/ZO* 1 

MM-M 

GO TO 19 

Y=Oo 

T=Y*EXP(Z(L )*ACOS(AK/V) 

SUY*SUM+T*H(L 

CONTINUE 

PE~SUY/3.141592054 

PRINT 89 SDBpPE 

IF(PEeLTo1eE-9) CALL EXIT 

CONTINUE 

FORMAT( lOX,Z(lPElZe5,3X) 

E NO 
'JROGRAY SYSWR (INDUT,OUTPUT) 

THIS PROGRAY USES MEDDLES RULE FOR THE 

INTEGRATIONS REQUIRED Iq THE DETERYIVATION OF 

SYSTEY PE2FORMANCEe 

DIFFEqEVT FUN1 SUSROUTINES AQE USED F3R 

DTFFERENT UOISE M3DEL5 AND/OR NOISE MEASUREYENTSe 

QIMENSION Z(7) 

COMM7N/QQQ/AA, ALPHAYOMEGAY AK 

PRINT 5 

FORMAT(lY1) 

AA=1*0 

ALPH9=1.? 

OYEGA=3*00379433 

AK=l. 

CALL FUYl(l.,ZZ) 

DO 60 J=1,15 

?DR=-33.+5e*(J-l) 

S=lfi**+( S93/19* 1 

AK=S3*3*5 

CUY=C* 

07 53 I=1,15 

AKD9=23**4L'lG10(4X 1 

4DS=A<3'3+4**(1-1) 

BDR=AK95+4**1 

A=lOe**(AD3/20e) 

Bp10***(9D5/20e) 

!lX=(B-A)/6* 

DO 40 Y=1,7 

X=A+(Y-l)*3X 

CALL FUY?(X,I(K)) 

CONTIYUE 

SS=Oe3*DX*(Z(1)+5e*Z(2)+Z(3)+6**i!(4)+Z(5)+5.*Z(6)+Z(7)) 

IF(SSsEQo9,) GO TO 55 

SU"ISUY+SS 

CONTINUE 

PE=SUY/3*141592654 

PRINT f31 YDB, PE 

IF(PF*LT.l*E-?) CALL EXIT 

CONTIN'JE 

FORMAT(~~X,F~*~,ZXJ~PE~~~~) 

END 
SUBPOUTIYE FUN1 (X9Y 

C THIS SUBRCIUTIYE IS FOR USE WITH PR3GRAM SYSWK. 

C IT '44<ES USE CIF YEASUQED APD DATA. SEE PRnGQAP' 

C SYSAPD FD4 iU9THU9 DETAILS. 

QTMFNSIOY CL(71),SL(31),P(31),CX(31)9SX(31)~SP(31) 

COPM?Y/Q/AK 

2ATA CL/1.94.17.r10.,13.,16.919~922.925.9?3.~3I.93+.93?.~ 

140.,43.946.,44.95?.,55.,58.,31.9h4.967.~70.973*975.979.9 

282.~95.988.991./ 

OATA P/.9435,.?914,.7384,.6491~.4516~.263I~.1445~.O8~7j+*O~387~ 

1.0344A,~02?59,.014R1~~0'3~734~~005414~~004230~~00279?r~OQ18

2.001217~.~008041,.0005312~ci~003503~.OO023l8~.000~532~~~

3.0003h~85,~00004427~.0C00291R~~00001928~~00001274~0~~Oo/ 

C CL IS THF 4p3 LEVELS IN DB AND P IS THE PQqBARILITIES AT 

C THESE LEVELS. 

CX(l)=1.5*CL(1)-0.5*CL(?) 

SX(1)=1O.**(CX(1)/2o,) 

Dn 23 I=2,31 

CX(I)=(CL(I-l)+CL(I))/2. 

SX(I)=13.**(CX(I)/20.) 

70 C~VTIYUF 

C CPM2UTE ?'IS 

SUY=Sx(l)*Sx(l)*(l.-P(1)) 

DO 30 J=2,31 

T=SX(J)*SX(J)*(P(J-l)-P(J)) 

SUi.l=SIJYtT 

30 CCIYTIUUi 

YMS=(SUY+l3.**(CL(71)/2O~)*P(31))**Oe5 

C YOR~4LIZE TO YYS LEVEL AN0 COMPUTE PDF 

CL(l)=CL(1)-20~*4L~G10(Q~lS) 

SL(1)=13.**(CL11)/20.) 

SX(l)=SX(lI/RYS 

CX(1)=20.*ALOGlO(SY(l)) 

SP(1)=(1.-?(1)1/SL(l) 

on 40 ~=7+31 

CL(K)=CLtK)-20.*4LOGlO(RMS) 

SL(K)=lO.*:k(CL(K)/20.) 

SX(K)=SY(<) /ZYS 

CY(K)=70.*ALPGlO(SX(K) 

SP(Y)=(P(<-1)-P(K))/(SL(K)-SL(K-1)) 

40 CnNTINUF 

MY=M 

FVT'IY FUV2 

v=x 

IF(V.LT.SY(1)~OR.V~GT.SX(31)) 15915 

15 Dn 17 Y=~A 

IF(V.LE.SX(M).AND.V.GT.SX(Y-1)) G3 TO 18 

17 CnnlTIVUE 

18 IF(SP(Y-l).EQ.O..OS.SP(M).EQ.O.) GO TO 15 

YD8=(?3.*4L7Gl9(V)-CX(M-l))/(CX(Y)-CX(F4-1) 1 

YD3=YD3*(??~*ALOGlO(SP(M))-2O.*ALflGlO(SP(M-I))) 

YDP= Y3E+77.*4L9GlO(SP(fl-l)) 

PDF=13.**(YDB/?0a 

Gfl T'II l? 

15 PDF=O. 

14 Y=PDF*ACqS(AK/V) 

2 FTURN 

FYD 
SIIBRCIUTINF FUN1 (X, Y 1 

TPIS F'JNCTIOY QPUTINF IS F'l3 CLASS 9 NOISE FOP USE 

WITH P?OGQAY SYSd?. 

CUMY~~Y/QQO/AA,ALPkiA,OYEGA,AY 

ENTRY FUN2 

v=x 

ZY=V*V/GM€SA 

SY=O. B FU=le 4 SY1 so. 

DO 70 Yq=1,26 

N=NN-1 

IF(Y.UE.g) FY=FY*V 

CALL C?VHYP(l.-N*hLPHA/2.t2epZN,S,IOVFLW) 

CALL C~NHYP(le-N*ALPHA/2.~3.~ZN~SSpIr)VFLW) 

SSS=(ZY/?.)*(l.+~*ALPHA/2.)*SS 

T=(((-4A)**N)/FN)*GAYM4(1.+U*ALPYA/2.)*(S-SSS) 

IF(I'7VFLd.UE.l) G3 TO 14 

SMl=SMl+T 

GO TO 30 

14 SY=S".r+T 

70 CONTIhllJE 

FP=O. 

IF(ZNeLT.675.) FP=EXP(-ZN) 

?DF=(2 e*V/IYEGA)*(FP*SM+SYl) 

Y=PDF*ACqS(AK/V) 

QFTLlKY 

EYI) 

FORM NTIA-29 U.S. DEPARTMENT OF COMMERCE 

(4-80) NAT'L. TELECOMMUNICATIONS AND INFORMATION ADMINISTRATION 

BIBLIOGRAPHIC DATA SHEET 

1 NTIA Report 82-95 1 I 1 

1. PUBLICATION NO. 

P RFOR N 0 GANIZATIONNA LAND A I 'hat, on2 Ye9 ecommuni cay1 ons YRfAormation ~dmi nix1 

2. Gov't Accession No. 

Digital system performance software utlizing noise 

measurement data 

A. D. Spaulding 

Institute for Telecommunication Sciences 

325 Broadway 

3. Recipient's Accession No. 

5. Publicat~on Date 

February 1982 

6. Performing Organization Code 

NITA/ ITS 

9. Project/Task/Work Unit No. 

10. Contract/Grant No. 

15. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant b~bliography or literature 

survey, mention it here.) 

Boul der, CO~ orado 80303 

11. Sponsoring Organization Name and Address 

Department of the Army 

U.S. Army Communications Electronics Engineering 

Installation Agency 

Ft . Huachuca , Arizona 8561 3 

This report summarizes techniques that use the measured instantaneous 

envelope statistics of arbitrary noise or interference processes to calculate the 

degradation these processes cause to digital conimunication systems. Computer 

implementation of the techniques are also given. The computer algorithms are 

designed for the data obtained from a general purpose noise measurement device, 

termed the DM-4 (for "di stri bution meter-model number 4") recently devel oped by 

NTIAIITS. For illustration and for comparison with theoretical results, two 

noise examples are employed, one for "narrowband" interference, and one for 

"broadband" interference. These examples are taken from the noise models recently 

devel oped by Mi ddl eton. 

12. Type of Report and Period Covered 

Technical Report 

13. 

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

14. SUPPLEMENTARY NOTES 

computer algorithms; digital system performance; non-Gaussian noise; 

system performance software. 

17. AVAILABILITY STATEMENT 1 18. Security Class. (This report) 1 20. Number of pages 

a UNLIMITED. 

FOR OFFICIAL DISTRIBUTION 

19. ~ecurity~lass. (This page) 21. Price, 

1 Unclassified 1 1 

U.S. GOVERNMENT PRINTING OFFICE: 1982-577-8181149 Region No. 8