A mining research final report PCC 4679 04 

NOVEMBER 1981 

A FEASIBILITY STUDY FOR 

THE DETECTION OF WEAK 

ELECTROMAGNETIC 

SIGNAL/ BURSTS WITH 

HARD-LIMITED ARRAYS 

Contract 10318033 

University of Pittsburgh 

BUREAU OF MINES 

UNITED STATES DEPARTMENT OF THE INTERIOR 
The views and conclusions contained in this document are those of the authors 

and should not be interpreted as necessarily representing the official policies or 

recommendations of the Interior Department's Bureau of Mines or of the U.S. 

Government. 
- FCC 4679 04 . . - - - . -. - . - - - --. -.-- L. 7rt;c .8t :v!,tltle 5. Erporf Date 

A Feasibility Study for the Detection of Weak Electromagnetic Signal Bursts with Hard-Limited arrays. 

7. ~ufhor(s) 8. P~rIormtnl 0rp.nir.tlon icport No. 

M. Kanefsky 

- A.Q.K. Rajput 

II33 /' 

r:;in::ahon li.nr and ~dCrtrr 10. ProjestlI.rklW~rk Unlr No. 

University of Pittsburgh - - Department of Electrical Engineering 11. COO^,.^^ Or ~r.~t 10. 

Pittsburgh, PA 15260 ~ - -- - . - 

I Final Report I 

Wo channel polarity coincidence and polarity difference sratistics are 

analyzed. The signal, common to both channels, consists of sinusoidal bursts 

where the exact frequency of the signal is nearly known, but other parameters 

such as amplitude, phase, and pulse starting time are unknown. Thenoise 

inputs are dependent, narrow band, Harkov processes. 

It is shown that the performance depends not only on the signal uncertainties 

but on the precise shape of the cross-correlation functions between the noise inputs. By using two polairty difference statistics in addition to the polarity 

coincidence statistic, it is shown that the decrease in performance, as well 

as the cost of hard limiting due to correlated inputs, can be made small. 

!i. <.r>&:r.ht<:: '$ ; ,.: x=.es 

Array Detectors 

Correlated Noise Model 

Polarity Coincidence Correlation 

18. Availability Statrmenl 
FOREWORD 

This report was prepared by the University of Pittsburgh under USBM 

Contract number 50318033. This contract is administered under the Department of Electrical Engineering with Morton Kanefsky as the Principal Investigator. John Durkin is the contract admistrator for the Bureau of Mines. 

This final report is a summary of the work completed for this contract. 

during the period October 1980 to October 1981. This report was submitted 

by the authors on November 9. 1961. 
TABLE OF CONTENTS 

SUMMARY ............................ 

LIST OF FIGURES ........................ 

NOMENCLATURE .......................... 

1.0 INTRODUCTION ....................... 

1.1 General Background on the Detection of an E.M. 

Source Buried in the Ground .... 

1.2 Analog Array Detector ................ 

1.3 Polarity Coincidence Correlator (PCC) ........ 

1.4 Summary of Analysis ................. 

2.0 EVALUATION OF THE OUTPUT SNR FOR A PCC .......... 

2.1 The Computation of Q(k) ............... 

2.2 Stability Consideration and CPU Time Requirement ... 

2.2.1 Truncation and Stability ........... 

2.2.2 CPU Time Requirement ............. 

3.0 PERFORMANCE OF PCC AND COMPARISONS WITH UNCLIPPED 

DETECTOR ....................... 

3.1 Large Signals with Two Independent White Noise 

Inputs ..................... 

3.2 Small Signals with Correlated Noise Inputs ...... 

3.2.1 Inputs Positively Correlated ......... 

3.2.2 Inputs Negatively Correlated ......... 

4.0 MODELING OF TWO CORRELATED NOISE PROCESSES ........ 

4.1 Narrow Band Noise Processes ............. 

4.2 Wide Band Noise Processes .............. 
5.0 PERFORMANCE COMPARISON OF PCC WITH ANALOG DETECTORS 

FOR A WIDE RANGE OF CROSS-CORRELATION SHAPES ..... 

5.1 Positive Cross-Correlation .............. 

5.2 Negative Cross-Correlation .............. 

6.0 PCC DETECTION OF SINUSOIDAL BURSTS WITH UNCERTAINTIES 

IN SIGNAL PARAMETERS .............. 

6.1 Sampling Speed .................... 

6.2 Detection of Sinusoidal Bursts ............ 

6.2.1 Polarity Difference Statistics ....... 

6.2.2 Cost of Uncertainties in Signal Parameters . . 

6.2.3 Use of Microprocessors in Parallel ...... 

7.0 CONCLUSIONS AND SUGGESTIONS FOR FLlTURE RESEARCH ...... 

7.1 Conclusions .................... 

7.2 Suggestions for Future Research ........... 

APPENDIX A. .......................... 

APPENDIX 8. .......................... 

APPENDIX C ........................... 

APPENDIX D ........................... 

APPENDIX E ........................... 

APPENDIX F ........................... 

BIBLIOGRAPHY .......................... 
A prototype detector using electromagnetic signals for finding trapped 

miners has previously been built and tested. The signals used are generated 

from transmitters carried on the miner's belt and powered by his head lamp 

battery and the receiver is a human listener. This system has been found to 

be very efficient for the detection and location of trapped miners in most existing mines. However, for very deep mines, they are not adequate. Unfortunately, 

the signals employed (0.1 second bursts of a 1000 H, sinusoid repeated every 

second) while ideally suited for a human listener, is not useable for coherent 

detection schemes. 

This study investigates the use of a noncoherent detector based on the polarity coincidence statistic. In this study the problem of location was not 

considered, only the problem of detection. Since this two input statistic 

was always used previously for two noise inputs that are essentially uncorrelated, 

and since this is hardly the case with electromagnetic signals, a principle 

goal of this work was to analyse the proposed detector for varlous kinds of 

correlated noise inputs. Furthermore, this detector is normally used with a 

random signal. While the signal cannot be presumed to be known with precision 

it is at least partially known. Another principle goal was to determine 

whether other statistics using delayed versions of one of the inputs is 

practical. It is also necessary to determine the effect of the signal uncertainties. 

A general expression was derived for the detection parameter of the 

polarity coincidnece correlator, the inputs of which are assumed to consist 

of a comon signal plus correlated, stationarygaussian noises. An expression 
was obtained for the expectation of the product of four hard limited (clipped) 

gaussian inputs with arbitrary crosscorrelation, which is needed in order to 

evaluate the detection parameter of the PCC. A program to evaluate this expression 

has been implemented on the computer, evaluated and tested. This represents 

the most significant contribution of this study. 

A general model for the kinds of cross-correlation functions that would 

result when passing two heavily correlated noise processes through identical 

band-pass filters was developed. Based on this model, the performance of the 

PCC was evaluated and compared with the unclipped correlator and the sum and 

square detector. The sum and square detector is a near optimum detector. For 

large positive values of crass-correlation, the unclipped correlator outperforms 

the sum and square detector, and so may be optimum in this range. We observed 

that, for positive cross-correlations, while the performance of all three detectors fall off substantially with an increase in the correlation coefficient, 

the cost of clipping relative to the optimum increases only slightly, increasing 

from 2db to 3 db as the correlation increases form 0 to 0.5. For negative 

crosscorrelation, while the unclipped correlator performed as bef'ore, the 

performance of the FCC and the sum and square detector increased as the magnitude of the correlation coefficient increased. As before, the cost of clipping 

compared to the optimum increases slightly. The increase in performance of the 

PCC is comparable for negatively correlated noise to the decrease for positively 

correlated noise. 

The analysis is extended to two statistics, i.e., polarity coincidence 

and differences for 0.1 second duration sinusoidal bursts of nearly known 

frequency when the noise inputs are correlated. The decrease in performance 

relative to uncorrelated noise inputs is quite small. Indeed, for certain 

cross-correlation functions the performance increases. The increase in the 
cost of clipping with the correlation coefficient is also quite small. For 

the worst case, and perhaps the most likely case, where the cross-correlation 

function is proportional to the autocorrelation, i.e., R (T) = aR(r), the 

nln2 

degradation can be kept low using the polarity difSerence statistic. For a = 0.5, 

the degradation compared to the independent case is about ldb. The cost relative 

to the sum and square detector, for this case also increases by only by ldb. 

The detection parameter is proportional to the input signal-to-noise ratio. 

If we define the gain G as the ratio of the detection parameter to the input 

signal-to-noise ratio, it is determined that 

G is the gain for one pulse for either the standard PCC or the other statistics 

0 

considered where precise knowledge of the signal parameters are presumed.. It 

is precisely this gain that has been evaluated and compared with the unclipped 

detectors in the bulk of this report. D is the pulse sync error, y(Aw,~8) is 

the degradation due to uncertainties in the signal such as frequency (bw) and 

phase (EB), and T is the integration time is seconds. The values of the gain 

for a single pulse are undoubtedly less than the gain of the human observer; 

perhaps by a factor of 2 or more. However, the human observer cannot accumulate 

information from one pulse to the next. Thus for example, after 100 seconds (or 

100 pulses) the gain of the PCC statistic is improved by a factor of 10. 

The pulse sync error D is always less the 5 where M is the number of computation 

M 

intervals chosen for each 1 second period. Thus, if this period is divided into 

40 overlapping 0.1 second processing times, the degradation due to lack of pulse 

sync is at worst 718 which is acceptable. For the statistics considered in this 

study , Aw 

P cos(mnT- £0) - H Y(OW,E~) = 
where delays of mn are presumed (i.e. polarity coincidences and differences 

only) and pH is the correlation coefficient of the noise inputs. It is shown 

in this report that for m = + 1, the increase in G can be significantly larger 0 

than the decrease in y. However, for large values of m this is not the case. 

Thus, this work suggests that three statistics should be used, the standard 

PCC and 2 polarity difference statistics. 

Overall, the analysis is quite encouraging and strongly suggests that 

the proposed detector be implemented and tested. 
LIST OF FIGURES 

Figure No. 

1. The geometry of horizontal loop buried at 

a depth of h .................. 

Block diagram of optimum two channel receiver . . 

Block diagram of polarity coincidence correlator. 

Architecture of polarity comparison detector . . 

Dependence of relative error on truncation ... 

Approximation of R(k) .............. 

Dependence of CPU time on truncation ..... 

Performance comparison of PCC and unclipped 

correlator .................. 

Performance comparison of PCC and unclipped 

correlator for large input signals ...... 

Cost of clipping versus input SNR ........ 

Performance comparison of PCC with unclipped 

detectors for positively correlated inputs . . 

Cost of clipping for positively correlated 

inputs .................... 

Performance comparison of PCC with unclipped 

detectors for negatively correlated noise 

inputs .................... 

Cost of clipping for negatively correlated 

inputs .................... 

Autocorrelation function for narrow band noise 

processes .................. 

Cross-correlation functions ........... 

Performance comparison of PCC with correlator 

and sum and square detectors for positively 

correlated noise inputs using general model . 
Figure No. 

26-34. Performance comparison of PCC with correlator 

and sum and square detectors for negatively 

correlated noise inputs using general model . 

Performance of PCC versus sampling speed .... 

Performance comparison of PCC with no shifts; 

2 180' shift for wide range of crosscorrelation ................. 

Performance of PCC versus various shifts in 

exact signal and aligned pulse ....... 

Improvement in performance versus various 

shifts when signal has 0%, t5%, 210% error . 

Timing diagram and block diagram of parallel 

microprocessor ............... 

Comparison of two channel detector with one 

channel envelope detector with error in 

signal frequency .............. 
NOMENCLATURE 

Symbol 

C.. 

13 

Definition 

Hypothesis that signal is absent 

Hypothesis that signal is present 

Expectation 

Detection parameter 

Variance of the signal 

Variance of the noise 

Noise inputs 

Number of observations 

Signal input 

Correlation coefficient 

~[sgn xl (t) sgn x2(t) sgn xl(t+k.r) sgn x2(t+k?)l 

Autocorrelation of noise 

Cross-correlation between n and n 1 2 

Probability density 

Represents correlation matrix 

Elements of P-I 

Magnitude of cross-correlation 

Determines the combination of odd and even crosscorrelation function 

Determines the combination of narrow band and wide band 

autocorrelation function 

Input signal-to-noise ratio 

Correlator 

Sum and square 
PCC 

N,N(u) 

T 

B 

s 

S (w) 

7 

Polarity coincidence correlation 

Noise spectrum 

Integration time 

Bandwidth of the signal 

Test statistics 

Signal Spectrum 

Delay between inputs 
1.0 INTRODUCTION 

1.1 General Background on the Detection of an E.M. 

Source Buried in the Ground 

The feasibility of finding trapped miners by detecting electromagnetic signals generated from transmitters carried on the miner's 

belt and powered by his head lamp has been investigated for roughly 

10 years. Wait ('I* determinedin 1971 that the location of a horizontal loop buried in a homogeneous earth can be determined from the 

horizontal and vertical magnetic field components at the earth surface. The geometry of this problem is shown in Figure 1. The 

horizontal magnetic component achieves a null and the vertical component a maxima directly above the loop. Based on these ideas, a 

detector was developed and a prototype built and tested. While 

some location errors resulted from an inhomogeneous earth and nonhorizontal earth interfaces, detection was achieved in almost all 

cases. For mines deeper than 1000 feet, of which there are few 

presently, detection itself becomes a problem. 

The signals employed by these systems are 0.1 second long bursts 

of 1000 Hz sinusoids repeated every second. This signal is ideally 

suited for the constraints of the problem which are (a) long life (12 

hours or so) for the transmitter, (b) low background noise power 

levels, and (c) signals that are most easily detected in noise by 

human listeners. For very deep mines, the human observer is an 

'Parenthetical references placed superior to the line of the 

text refer to the bibliography. 
Figure 1. The geometry of horizontal loop buried at a deph of h. 
inadequate detector and some kind of signal processing based on data 

accumulated over many seconds or minutes is required. It is not clear 

whether the processor should make the detection decision or merely 

preprocess and display the data in a form that a human observer, with 

his remarkably adaptive brain, can use for detection. 

Coherent detection techniques, which presume accurate knowledge of 

the signal, are the most powerful. Unfortunately the signals used in 

the present system do not lend themselves to coherent reception. For 

one thing the frequency is much too high, relative to the skin depth of 

the earth, to presume accurate knowledge of the signal at the earth's 

surface. Also signal bursts, rather than continuous signals, complicate 

coherent detection. For these reasons most of the signal processing 

proposals and studies suggest very low frequency, less than 10 Hz, 

continuous signals. (3-5) While the level of the background noise is 

quite high, in this low frequency range, it is highly correlated in 

time and frequency and optimal coherent techniques could be very effective. There are a number of problems, however, with this approach. 

Even though most of the hardware complexity would involve the receiver, 

the transmitter would require precise timing references and be more expensive. Sophisticated techniques for reducing the heavy background 

noise and remaining phase locked with the signal rely on stationarity 

assumptions that may not always be valid. Perhaps most important, the 

signals employed cannot be heard by human listeners. This means that 

in the event of equipment failure, or for that matter a failure of the 

stationarity assumptions, a simple back-up of the human observer is not 

possible even for relatively shallow mines. 
It is certainly important to study the feasibility of incoherent 

detection techniques using the same signals of the present prototypes. 

It should be remembered that the performance of any detector which accumulates or integrates data over some period of time (T) improves as 

T increases. Also the integration time of a coherent detector is 

limited by the uncertainty in the received signal. Beyondthis limitation, integration must be incoherent in any event. Given the possibility of a sufficiently long decision time, say many minutes, a clever 

incoherent technique can be relatively efficient. 

1.2 Analog Array Detectors 

Optimum receivers for detecting the presence of a random signal 

common to two or more receivers with additive Gaussian noise have been 

discussed by Bryn and others. (6'7) Such an analysis is carried out in 

Appendix A where the noise inputs are presumed to have an arbitrary 

cross correlation function (e. they are not independent). It is 

determined in this Appendix that the optimum receiver is shown in 

Figure 2 where the + sign is employed when the signals are in phase and 

the - sign when they are 180' out of phase. Since the optimum filters 

depend on the signal-to-noise ratio, which is unknown, one usually implements the locally optimum detector (optimum as SNR + 0) which has 

the same structure with the filters replaced by 
Filter 

Square 

+-?=-q~

Filter 

Figure 2. Block Diagram of Optimum Two-Channel Receiver. 
Finally, since in our case the signal spectrum is qui e narrow, 

the filters can be replaced with narrowband filters cente d at 1,000 

Hz. It is seen in Appendix A that the structure of the op I imum 

detector, not the performance, is in fact independent of the degree 

of correlation between the noise inputs. In Appendix B, the output 

signal-to-noise ratio for this optimum detector is evaluat d under the 

assumption of an arbitrary space/time noise correlation. t is 

assumed that the signals are either identical to both inpu I 

actly 180' out of phase. The output signal-to-noise ratio 

where T is the signal duration, T the delay between inputs1 and 

where pll(.) is the time autocorrelation function of each hoise input and p (-) 1s the correlation function between the noise inputs. 12 

We see that while the performance increases with signal duration, 

the extent of the improvement depends, in a rather complicated way, 

on the space/time noise correlation. It must be remembered that this 

receiver is said to be optimum under the assumption that the signal 

I 

is random. However, for the locally optimum detector, the statistics 

of the signal need not be known. It is worth considering htw this 

receiver compkres with a coherent detector. In Appendix C the overly 

simplifying assumption that the cross-correlation between The noise 
inputs is identically zero is made. This receiver is then compared with 

a one-channel ideal envelope detector (essentially equivalent to coherent detection) which accumulates the results of each pulse incoherently. It is found that the two-channel detector performs better 

particularly if the signal frequency is not known very accurately. Of 

course, if the inputs from two channels are summed prior to envelope 

detection and the signal frequency is known with precision, it will 

perform a little better than the non-coherent detector. This discussion 

is included as part of the justification for the use of a two-channel 

non-coherent detector. It must be shown, however, that when the noise 

inputs are heavily correlated, the degradation is not severe. 

1.3 Polarity Coincidence Correlation (PCC) 

Polarity coincidence correlation is a well documented technique 

for detecting a random signal imbedded in additive noise. (7-13) A 

detector which operates on the polarity coincidence of both channels 

is called a polafity coincidence correlator (PCC). The two-channel 

PCC, shown in Figure 3, "clips" both inputs and compares their 

polarities. If the polarities of two hard limited incoming signals 

tend to be the same, the decision that the signal is present is made. 

We now make the point that for many two-channel acoustic applications the data is hard limited and the detection is made with a 

polarity coincidence correlator (8'10-12115) or for signals 180" out 

of phase a polarity difference correlator. It is known that when 

the inputs are uncorrelated, the cost of this data reduction is quite 
yard limiter 

Polarity coincidence output 

Figure 3. Block' Diagram of Polarity Coincidence Correlator. 
small (about 0.5 db). (16'17) There are two major advantages to compensate for this cost. The most obvious advantage is that the hardware 

complexity, both the data acquisition and the detector implementation, 

is greatly reduced. This makes the possibility of a small hand 

carried receiver quite feasible. A second advantage is that the hand 

limited receiver has certain nonparametric qualities, and indeed can 

outperform the so-called optimal receiver when the noise is impulsive 

rather than gaussian, (17-19) when the input samples are independent. 

Nonparametric detectors have test statistics under the hypothesis 

which are invariant over some class of noise process, and are therefore based on the assumption that the statistical properties of the 

noise are not completely known. ('O) For fast sampling rate, however, 

the PCC is usually analyzed using the assumption ofgaussian statistics. 

If the target, for independent noise, is not directly in line with the 

center of the array, the delays from the target to each receiver are 

sufficiently different to enable the location (bearing angle) of the 

target to be determined. Unfortunately, for electromagnetic signals 

with receivers reasonably close, the noise inputs at each receiver are 

heavily correlated and signal delays may be too small to measure. 

While the small delays mean that location by some kind of triangulation may be difficult, they have a benefit in that the signals at each 

receiver are nearly in phase if the horizontal coordinates of the 

transmitter is in the vicinity of those of the receivers. 

If a delay of one half period (.5 m sec) is introduced in one 

channel, an equivalent detector determines polarity differences. 

Other statistics which compare coincidence of one pulse with those of 

anothermaybe more desirable. In fact a receiver that can implement 

many different statistics should be very useful. A receiver which has 

this flexibility and which overcomes the lack of pulse synchronization 

is shown in Figure 4. Each microprocessor, operating in parallel, 

assumes a particular pulse sync and accumulates data (1200 binary 

pulses per input) over 0.1 seconds. During much of the remaining 0.9 

seconds, before the next pulse, a series of statistics are computed. 

These include polarity coincidences between simultaneous samples, and 

samples displaced by multiples of 1 m sec (or 1000 Hz cycles). Also, 

polarity differences betkieen samples displaced by odd multiples of 0.5 

ml sec are computed. Other statistics such as polarity coincidences 

with the data from a previously stored pulse can also be computed. 

These statistics can be accumulated and after some normalization displayed as a line on a CRT. The accumulation continues until either a 

target is visually observed, by a bright spot or line on the display, 

or the computations are stopped. 

It is conceivable that, after detection, the inputs, perhaps 

sampled at a much faster rate, could be used for location via triangulation. In this event it should be noticed that the best statistic 

and pulse synchronization have been determined by the brightest spot. 

? .4 Summary of Analysis 

The detection parameter for weak input signals and noise is discussed in Chapter 2 and given by (D-13) in Appendix D as, 
where 

represent the correlation coefficient under the alternative and hypothesis 

respectively, R (k) is the noise cross-correlation function, as 2 2 

n n 12 "n 

is the input signal-to-noise ratio, N refers to the number of sample accumulated by the test static, and 

~(k) = E~ ISgn xl(t) sgn x2(t) Sgn xl(t+kr) Sgn x2(t+kr)l . (1-4) 

Equation (1-3) is asymptotically valid as the input signal-to-noise ratio 

2 2 approaches zero. It canbe extended to include finite values of as /an 

provided the signal is gaussian as well as the noise. For non-gaussian 

signals, only the asymptotic evaluation is possible. 

The technique for evaluating the expected value of the product of 

four hard limited functions (Equation 1-4) is discussed in Chapter 2 

and carried out in Appendix D. This evaluation is one of the main 

contributions of this research, without which the only possible way of 

evaluating the PCC would have been extensive simulation. The solution 

of this problem is carried out by a computer algorithm. The memory 

required, computation time, and stability considerations are evaluated. 
In order to test the programs, the PCC is analyzed in Chapter 3 

for a first order Markov signal common to two dependent noise inputs, 

whose cross-correlation is proportional to the autocorrelation. 

A general model of cross-correlation functions for narrow band 

and wide band noise process is developed in Chapter 4. Using this 

model, the PCC is evaluated for a wide variety of cross-correlation 

shapes in Chapter 5. There is shown to be a significant difference 

in the performance for even or odd cross-correlation functions. 

The results are extended in Chapter 6 in a number of important 

ways. The signal is assumed to be an electromagnetic burst. Signals 

of this sort have been considered for the detection of trapped miners. 

The exact knowledge of this frequency is also not available because of 

variations in the transmitter. The frequency, in a two-channel 

receiver, can be assumed to be the same, but because of the different 

locations of the receivers, the amplitude and phase of the signals can 

be different in both channels. These uncertainties in the signal are 

the primary reason for considering a noncoherent PCC rather than a 

coherent receiver. Nevertheless, because the frequency is nearly 

known.other statistics are possible. For example, by introducing a 

delay in one channel which causes a 180' phase shift and employing a 

polarity difference correlator, it is possible to get improved performance. The performance depends not only on theamplitudesand phase 

difference in the channels and more critically on the frequency uncertainty, but on the precise shape of the cross-correlation function. 
In order to carry out such an analysis, the kinds of crosscorrelation functions that would result when passing two heavily 

correlated noise processes through identical band pass filters are 

used. The PCC is first evaluated for identical input signals for a 

wide range of cross-correlation shapes, and is comparedwith the unclipped correlator and sum and square detectors. The sum and square 

detector is locally optimum when the inputs are independent. For 

dependent inputs, however, the unclipped correlator can, under certain 

conditions, outperform the sum and square detector. Next the cost due 

to differences in the amplitudes and phase of the signal as well as 

the uncertainty in both the frequency and pulse locations of the signal 

is analyzed. Finally, other statistics based on the fact that the 

frequency is nearly known are evaluated. 

The main results of this analysis are: 

1. The cost of clipping (i.e., the comparison between the PCC 

and analog devices) is quite modest even for significant 

amounts of cross correlation. 

2. The degradation in performance due to the correlation between 

the inputs is not severe, particularly if one employs two 

polarity difference statistics in addition to polarity coincidence correlation. Statistics other than these three do 

not seem worthwhile. 

3. The degradation in performance due to signal uncertainies can 

be made quite small. 
These results convince the author that the use of the proposed 

modified PCC is quite feasible for the detection of the electromagnetic signals used by trapped miners. 
2.0 EVALUATION OF THE OUTPUT SNR FOR A PCC 

A detector is often evaluated by computing the output signal-tonoise ratio, 

EK(S) - EH(S) 

D = 

d VarK (S) 

where S is the test statistic, H refers to the hypothesis of noise only 

and K refers to the alternative of an additive signal common to both 

inputs. If one is only interested in a small input signal-to-noise 

ratio, this equation can be greatly simplified by replacing the variance 

of the test statistic under the alternative with the variance under the 

hypothesis. Assuming independent noise inputs, independent samples, and 

small signal power, the output signal-to-noise ratio becomes for 

gaussian statistics f8,9) 

2 where oS /on2 is the input signal-to-noise ratio and N refers to the number of samples accumulated by the test statistics. For the unclipped 

2 2 correlator, the output signal-to-noise ration is (oS /o m(8'9) and 

n 

the cost of clipping is n/2 or 2 db. 

If one wishes to change some of the assumptions, particularly that 

of independent noise inputs and independent samples, it is necessary to 

examine the test statistic more closely. 
where sgn x(t) = 1 if x(t) > 0 and -1 if x(t) < 0. T is the sampling 

interval, and where xl(t) and x2(t) refer to the two inputs. From Van 

- 1 

Vleck ('l) is is known that E [sgn xl(t) sgn x2(t)] = (2/~) sin p where 

p is the correlation coefficient between two inputs. If pH and pK 

represent the correlation coefficient under the hypothesis H and alternative K respectively, the output signal to noise ratio from (D-13) in 

Appendix D is 

- 1 2/71 /fT [sin-' pK - sin pH] D = 

PC'= N 

[I-{ (2/n)sin-l pK12 + 2 Z [l-1/N] [QK(k) -((2/rr)sin-' pK)2]]1/2 

k=l 

(2-4) 

where 

Since xi (t) and xi (t+kr) become independent as k increases, it follows 

that 

~,(k) + E~ [sgn xl(t) sgn x2(t) 1 EK [sgn xl(t+kr) sgn x2(t+k7) 1 

kHence the terms in the summation of equation 4 are decreasing to zero 

and since this summation can be truncated, the detection index 

ultimately depends on fi. 
If the common noise inputs are assumed to be independent, p =O. H 

For small signal power, the variance is evaluated under the assumption 

of the hypothesis and 

2 T sin [pK] 

- 

N 

where Q' (k) 3 E [SF xl(t) SF sl(t+k't)l EH[sgn x2(t) 'gn x2(t-kT)1 H H 

2 2 

- Os /Os Rn [k~l 

where P~ - l+US 

22' /on P,(~I = 7 

On 

and R (k~) is the autocorrelation of the noise. 

n 

For independent samples [pn(k)=O], equation (2-7) reduces to 

equation (2-2). Equation (2-7) has been evaluated for dependent 

samples,") where the cost of clipping has been found to decrease with 

an increase in sampling speed. 

For either dependent noise inputs or large signal power, we must 

compute Q(k) which involves 16 integrals. 

2.1 The Computation of Q(k) 

The problem is to compute ~Ir(x)) - where 

r = n sgn xi(ti) 

i=l 
and 5 is zero mean gaussian or 

-1/2 4 4 

p(5) = (27~)-~ [det PI exp {-(1/2) z z q..x.x.). 

13 1 3 (2-8) i=j j=1 

1 The coefficients q.. are the elements of P-l, and both P and P- are - 13 

symmetric [q.. = q..]. Since r(x) is either +1 or -1 depending on 13 31 

which of the 16 "orthants" x lies in, it follows that 

- 

8 orthants 8 orthants 

where r = +1 where r = -1 

By a simple change of variables, each of the 16 integrals can be written 

where W(.) is identical top(.) with appropriate changes in the signs 

of the q.. for ifj, and v is a dummy variable. 13 

From (E-19) of Appendix E, equation (2-10) can be written as: 

where 

j-s i-k-m 2i+j+r-2 kc rC S m C C i+l j+k+lIr(r+s+m+l i+'+r-k-5-m+l 

'12 34 23 '13 24 14 r r 2 )r( 2 1 

9 = (j-s)! (i-k-m)! k! s! m! r! 

(2-12) 

b and C.. - * q../q where the sign depends on which of the 16 orthants 

1 - 13 11' 

of the integral we are evaluating. 
Observation reveals the Symmetry in Equation (2-9), which reduces the 

16 integrals to 8. It is clear by examining equation (2-12), that 

the J, (i,j,r,k,s,m) terms for each of these 8 integrals differ only in 

their signs. Thus we can write 

where I)' corresponds to the J, where all the C.. are the normalized 

11 

negative values of P-I denoted by i.. and y can be determined to be 

11 

+k Using the fact that = (-1) , this can be simplified to 

Y = 0 if either i, (x + s + r), or (k + j) is even, { 8 if i, m + s+ r, and k + j are all odd. (2-15) 

A A Changing variables (x = m + s + r and y = k + j) and recognizing that 

n+l n-1 r2) = (2)! for n an odd integer, equation (2-13) becomes 

- - 11 i=O j=O r=O y=j s=O x=r+s 

odd odd 0 dd 
where 

2 i+j+r i-1 

u(i,j,r,~,~,~) = (j-s) ! (i+j+s+r-x-y) ! (y-j) !s! (x-5-1) !r! ' 

Computing E{I(~) 1 by equation (2- 16) is computationally simpler and 

requires significantly less CPU time than computing each integral I 

via equation (2-11). 

2.2 Stability Considerations and CPU Time Requirement 

2.2.1 Truncation and Stability 

The infinite summations of equation (2-16) have to be truncated 

to a finite interval T, forcing a compromise between CPU time and 

truncation error. The truncation error is mre severe the more the inputs are correlated. For a strong, additive, Markov signal that is 

common to two independent, white noise inputs, the relative truncation 

error of equation (2-16) is plotted in Figure 5 for k=l (the worst case) 

and an input signal-to-noise ration of 0.5 as a function of the truncation T. The overall error in computing the detection parameter of 

equation (2-4) is considerably less. Since the - i summation in equation 

(2-16) is evaluated only for odd values, it is not surprising that 

significant decreases in truncation error occur for odd values of T. 

For T=5, the relative error for k=l is about 1%, the overall error in 

the evaluation of equation (2-4) is an order of magnitude better. 
Figure 5. Dependence of relative error on truncation 
There is no need to consider a truncation T greater than 9. In fact, the 

observations reveal that for most cases, equation (2-16) will converge 

very fast. However, for small signal-to-noise ratios and heavily correlated noise inputs of the order of 0.6 and more, and for small values of 

k (1 s k 5 3), equation (2-16) does not converge as the truncation is 

increased. In this unstable region (which is evident from the eigen 

values of the P-I matrix and hence can be isolated) computation approximations may or may not be possible. 

The computations can be made stable with rapid convergence if the 

diagonal elements of the P matrix are all increased sufficiently. If 

the results of this computation are compared with the true results for 

those values where convergence is no problem, approximation can often be 

made. Such a comparison is shown in Figure 6 for a = 0.6, where the 

approximations for the three points where convergence did not take place 

are self evident. 

While there may well be problems for which our solution fails because of lack of convergence, we anticipate that for a wide class of 

problems convergence will take place or can be approximated. 

2.2.2 CPU Time Requirement 

The CPU time required to evaluate Equation (2-16) is plotted in 

Figure 7 for various truncation values. Since this time is essentially 

independent of k, this plot (appropriately scaled) also represents the 

overall CPU time required to compute the detection parameter Dpcc. We 

observe 15-fold an increase in CPU time when T was increased from 5 to 9. 
Figure 6. Approximation of R(k) 
Figure 7. Dependence of CPU time on truncation 
The evaluation of the detection parameter of equation (2-4) requires the computation of Q(k) N times. Therefore, if it takes n 

seconds to compute Q(k) via equation (2-16), the total CPU time will be 

nN seconds. This time can be greatly reduced, however, by using the 

following two procedures. 

For different values of k, the only part of equation (2-16) that 

changes is the B(i,j,r,y,s,x) terms. Therefore, we can calculate all 

of the a(i,j,r,y,s,x) terms and store them in a one-dimensional array. 

The time to compure Q(k) is greatly reduced by this time. This procedure does increase the core storage requirement as show in Table 1. 

Truncation T 

TABLE 1 

5 7 9 11 

Core Storage Required for 

the a terms (words) 1098 5120 17,375 47,880 

For the Markov process, and typically in general, the correlation 

between input samples decreases as k increases, and, for k large enough, 

Q(k) approaches a limit. Therefore, PCU time can be further reduced by 

finding k=I in equation (2-4) beyond which Q(k) remains nearly 

-1 [(2/n) sin pk]. Thus the summation of equation (2-4) is partitioned 

into: 

I N 

Z (1-k/N) [Q(k) - [(2/n) sin-'pK] + Z (1-k/N) [(Z/n) sin-' pK) 

k= 1 k=I+l 

I 2 - [(2/n) sin-' = Z (1-k/N) (Q(k)-[(2/n) sin-' ] 

k=l 

(2-18) 
and we need to compute Q(k) only I times. 

For most cases, the value of I = 30 in equation (2-18) was found 

to be adequate and the total CPU time to compute the detection parameter in equation (2-4) by using T = 5 on the DEC-10 computer, in 

Fortran-10 language, was 30 seconds. However, if the input data in any 

channel is to be shifted, for more accurate results, the I in equation 

(2-18) should be increased by that amount. 
3.0 PERFORMANCE OF PCC AND CONPARISONS WITH 

UNCLIPPED DETECTOR 

3.1 Large Signals with Two Independent White 

Noise Inputs 

The output signal-to-noise ratio of the PCC has been evaluated 

under the assumption of white independent noise and a common Markov 

signal. The correlation coefficients between signal samples kr seconds 

apart are assumed to be e-k'4. Since a gaussian Markov process has an 

autocorrelation function of the form o e 

s -2aB1r1, where B is the signal 

bandwidth, this corresponds to a sampling rate of BnB. This is sufficiently fast in that the output signal-to-noise ratio cannot be increased by faster sampling.(lO) The covariance matrix P used to evaluate 

equation (2-16) is 

2 

2 2 2 where o = oS + on and pK = 

Os 

X us2 + 0 2 

n 

The detection parameter D of equation (2-4), or output signal- PCC 

to-noise ratio, is ploted in Figure 8 as a function of the input 

signal-to-noise ratio. Shown in the same figure is the detection 
- 

4-00 o:ss a:15 s:z' o'.sz 

INPUT SN!.~' 

0.4s 0:66 9'.64 

Figure 8. Performance comparison of PCC and unclipped Correlator 
parameter based on the erroneous assumption of equation (2-7) or small 

input signal-to-noise ratios, as well as the performance for the unclipped correlator, which is 

2 N k where Var(S) = 1+2R [l + 2 E (1 - rj) . e-k'2] + 2R, 

k= 1 

2 2 and R = o /a input signal-to-noise ratio. 

s n 

It can be seen that the assumption of small input signal-to-noise 

ratios appears valid over a wide range. Actually the error in the 

variance calculation is compensating the error in the numerator. 

All three curves are continued in Figure 9 for large input signalto-noise ratios. ThedifferencebetweenthePCCperformance andthe correlator, or the cost of clipping, is plotted in decibels in Figure 10. 

For large input signal-to-noise ratios the cost of clipping decreases. 

3.2 Small Signals with Correlated Noise Inputs 

3.2.1 Inputs Positively Correlated 

In this example, the signal is assumed to be weak so that the 

variance of the test statistic can be evaluated under the hypothesis. 

Thus, equation (2-4) and (2-5) are closely approximated by 
Dv~c ban.< en erroneous 

... YVtlDn 

00 0:40 0:~ 1:20 1:60 ~:00 ~:40 Z:BB 5:28 

INPUT SNR 

Figure 9. Performance comparison of PCC and unclipped 

Correlator for large input signals 
INPUT SIGNAL-TO-NOISE RATIO 

Figure 10. Cost of clipping versus input SNR 
2/n (sin-' pK - sin - 1 

D = 

PCC 

S1) 

k [I-{(2/n) sin-' pH>' + 2 Z (1- iS){Q(k)-((2/n) sin" pH)2}]1/2 

k=l 

where Q(k) = EH {sgn nl(t) sgn n2(t) sgn nl(t+k?) sgn n2(t+k?)} . (3-5) 

2 We will assume that pH = E[nl(t)n2(t)]/on = a where a ranges from 0 to 

1. More specifically we will assume that the noise inputs are rapidily 

varying Markov processes [R (kr) = o 2e-k/4] and the crosscorrelation 

n n 

function between them is Rn ikr) = o 2ae-k/4. The analysis in no way 

12 n 

depends on these assumptions, rather they are made as an example. pK 

2 2 2 2 in equation (3-4) becomes (us /on +a) / (1 + Us /on 1. 

-1 - 1 Using Taylor's expansion it is seen that sin pK = sin a 

+ os2/on2 . Thus equation (1-4) becomes 

The covariance matrix for this example is 
The detection parameter Dpcc, or output signal-to-noise ratio, 

is plotted in Figure 11 as a function of the amount of positive crosscorrelation. Shown in the same figure are the performance of the unclipped correlator and the sum and square detector, whose detection parameters can be derived from equation (D-20) and equation (D-25) of 

Appendix D, respectively as follows: 

and 

where a can be positive or negative. 

c = correlator 

ss = sum and square 

As expected, the performance of all three detectors fall off as 

the noise correlation increases. The cost of clipping in decibels is 

shown in Figure 12. It can be observed that while the performance of 

both the PCC and the sum and square detector fall off sharply with an 

increase in correlation, the cost of clipping does not change substantially, increasing from 2db to 3db for the assumed sampling rate 

and u=+0.5. The performance of the unclipped correlator does not fall 

off as sharply and, for a sufficiently high correlation coefficient, it 

outperforms the "sum and square" detector. 

3.2.2 Inputs Negatively Correlated 

Because the results are quite different for negative crosscorrelations, the evaluations of all three detectors is discussed 

separately. 

All three detection parameter curves are repeated in Figure 13 

for negative a. The cost of clipping is shown in Figure 14. In this 

case the performance of the PCC and the "sum and square" detectors increase by roughly the same amount (the cost of clipping is between 2db 

and 3db for a=-0.5). The increase in performance is essentially the 

same amount for negatively correlated noise as the decrease for positively correlated noise. The unclipped correlator performs the same 

for negative as for positive cross correlations and is, therefore, less 

attractive for negatively correlated noise inputs. 
Figure 13. Perfornance comparison of PCC Figure 14. Cost of clipping 

with unclipped detectors for 

negatively correlated inputs 
4.0 MODELING OF WO CORRELATED NOISE PROCESSES 

4.1 Narrow Band Noise Processes 

Consider two correlated noise processes ni(t) that have been passed 

through identical narrow band filters whose transfer functions are both 

H(f). 

ni(t1 H(O \-fii(t) 

The output spectra are given by 

and 

where S, (f) and S, (f) are real and even. 

"1 "2 

If it is assumed that they are constant in the vicinity of the filter, 

then 

and 
Substituting the low pass equivalent 

H(f) = 1/2 HLP(f - f,) + 1/2 HLP(f + f,) 

Equation (4-3) can be changed to 

m 

I~~~(f)1~ cos 2rfot cos 2nft df 2 -m 

For a first order low equivalent 

~(t) = ~(0) e-'ltl cos 2nfot . (4-6) 

(f) does not have to be real and even but its inverse must be 

real. Let 

where Seven(f), Sodd(f) are both real (and denoted Se(t) and So(f) 

for convenience). The inverse transform of Se(f) is real and even and 

the inverse transform of j So(f) is real and odd. 

By definition 

m 2 

Rnln2(tl = I /H(f)I Sfi fi (O st - j sin*) df, 

12 

- m (4-9) 

Substituting (4-7) in (4-9) 
m m 

2 I 2 = IH(~) 1 \(f) coswt df + J H(Q / ~~(0 sin* df . -m -m 

Using the low pass equivalents and assuming that Se(f) and So(f) 

are nearly constant in the vicinity of f and -fo, one obtains 0 

m 

2. 

Rn1"2 

(t) i aj IH~~(~) I SlnR cos 2niot df 

-m 

2. + a j IH~~(Q / slnn cos infot df 

-m 

where 

- 

= [aRLp(t) + bii(t)] cos 2nfot 

. I 2. 

R(t) = .IHLP(f) 1 sinwt df . 

Equation (4-11) in general has no closed form solution. In order 

to simplify the model, we assume instead, 

In fact the inverse of the above expression is not equavalent to an 

imaginary constant in the vicinity of fo. The shape is nontheless 

similar to a numerical computation of equation 14-11). Therefore, the 

simplified model becomes. 

Thus a is related to the magnitude of the correlation and y(0 < y < 1) - - 

determines the proportion of even and odd terms. When y = 1, Rn1n2 (k) 

is proportional to R(k) and the proportionality constant becomes the 
correlation coefficient. When y = 0, R (k) is an odd function whose 

"ln2 

shape has similar characteristics to R(k). The magnitude of the crosscorrelation is proportional to a, but a can no longer be considered the 

cross-correlation coefficient. For y between 0 and 1, a wide variety of 

shapes are possible, all of which have characteristics resembling R(k). 

In Figure 15 the auto correlation R(k) has been plotted, which is 

an even function. In Figure 16 (a,b,c), a is set equal to 1 and y is 

set equal to 0, 0.5, 1 respectively. The resulting curves give some idea 

of the possible shapes. For these curve 5 was set equal to 4, fo = 1000 

Hz, the filter bandwidth = 500 Hz, and the sampling rate 12000 Hz. We 

shall see that a sampling rate of 12xfo is quite reasonable. 

4.2 Wide Band Noise Processes 

The autocorrelation function in equation (4-6) is extended from 

the low-pass Markov process to a combination of a low-pass Markov process 

and a band-pass process by defining 

The parameter a. is allowed to vary from 0 to 1 and for ao=O equation 

(4-14) is the same as equation (4-6) and the cross-correlation model in 

equation (4-13) is the narrow band noise process. For y and a. between 

0 and 1, a wide variety of shapes are possible, all of which have charactistics resembling R(k). 

In figures 16(d,e,f,g,h,i), u is set to 1 and a. is set to (0.5, 

1) while y is set to equal 0, 1/2, and 1. In general the resulting 6 
Figure 15. Autocorrelation function for AO=O 

Figure 16. (a) Cross-correlation function 

54 
(b) Cross-correlation function 

(c) Cross-correlation function 

Figure 16 (cont'd.) 
(d) Cross-correlation function 

(e) Cross-correlation function 

Figure 16 (cont'd.) 
" 

(g) Cross-correlation function 

Figure 16 (cont'd.) 
(h) Cross-correlation function 

(i) Cross-correlation function 

Figure 16 (cont'd.) 
curves and the 4 curves in the preceding section give some idea of the 

shapes possible. In Figure 16(c,f,i), where y = 1 we see the effect a 

0 

has on the autocorrelation R(k), since for a=y=l, both R(k) and R (k) 

"l"2 

are the same. In Figures 16(a,d,g), where y= 0, the cross-correlation 

is constrained to be an odd function. In Figures 16(b,e,h), where 

y = 1/2, the cross-correlation is nearly one-sided. 
5.0 PERFORMANCE COMPARISON OF PCC WITH ANALOG DETECTORS 

FOR A WIDE RANGE OF CROSS-CORRELATION SHAPES 

5.1 Positive Cross-correlation 

The performance of the PCC is evaluated based on the developed 

general model for cross-correlation functions in equation (4-13) with 

R(k) equation (4-14) or the the shapes in Figure 16(a) through Figure 

16(i), and for common input signals. The covariance matrix to calculate Q(k) in equation (2-4) is given by 

where ROE = Rn (0) 

12 

The detection parameter D or output signal-to-noise ratio is pcc' 

P(k) = 

plotted in Figure 17(a) through Figure 25(a) for three values of 

y(0, 0.5, l), and three values of a (0, 0.5, l), and for positive a. 0 

Shown in the same figures are the performance of the unclipped correlator 

and sum and square detector. For odd cross-correlation functions (y = O), 

the performance of all detectors are nearly independent of a or the 

- 

R(O) ROE 

ROE R(O) 

(5-1) 

R(k) (k) R(O) ROE 

magnitude of Rn (k). For even cross-correlation functions (y = l), 

12 

ROE 
1 

mpcc - polarity coincidence correlator 

Figure 17 (a). Performance comparison for (b) Cost of clipping for positiv

positive correlation correlation 
- 

qrr S.,O o:n dr om o:r o..~o- 0:- 

SPACE CROSS-CtWV3ELATION 

Figure 18 (a). Performance comparison (b) Cost of clipping 


Figure 21 (b). Cost of clipping Figure 21 (a). Performance comp


8 

00 'd 

SF 

0.4 

Y It 

#2 

o r'o 40 c'o r.0 r'o wo .or* 'ws A & 

the performance of the detectors decrease for an increase in a. The 

correlator outperforms the PCC and sum and square detectors for a sufficiently large a. The decrease in performance of the PCC is about 

-2db for a = 0.5. 

The cost of clipping compared to the correlator and sum and square 

detectors are plotted in Figure 17(b) through Figure 25(b), which remains 

almost constant compared to the correlator but drops off with compared 

to sum and square detector for y=O. For y = 1, on the other hand, the 

cost compared to the sum and square detector increases from 2db to 3db 

with moderately correlated inputs (a < 0.5) and compared to the corre- - 

lator detector increases from ldb to above 3db. The sum and square 

detector is locally optimun when the inputs are independent. For dependent inputs, however, the unclipped correlator can under certain conditions, outperform the sum and square detector. 

5.2 Negative Cross-correlation 

Since the performance of all three detectors is quite different, 

in most cases, for negative a, this case has been considered separately. 

All three detection parameter curves are repeated in Figure 26(a) 

through Figure 34(a) for y = 0, 0.5, 1 respectively and for negative 

cross-correlation. It can be observed that for y = 0 the performance 

remains the same as for a positive a. The cost of clipping remains constant for y = 0.5, but increases mostly for y = 1 and for higher correlation. The cost comparison curves for negative correlation are shown 

in Figure 26(b) through Figure 34(b). 



Figure 29 (a) Performance comparison (b) Cost of clipping 





For even cross-correlation functions (y = 1). the performance of 

the PCC decreases for positive values of a(-2db for a = +0.5) and increases a like amount for negative value of a(+2db for a = -0.5). For 

cross-correlation function even and odd components (y = 0.5) the performance lies somewhere in between the even and odd case. 

We can see that the shape of the autocorrelation R(k), as 

determined by a has a modest affect on the performance of the PCC 0 

that is nearly independent of y. The shape of the cross-correlation 

(k), as determined by y on the other hand, has a significant effect. 
6.0 PCC DETECTION OF SINUSOIDAL BURSTS WITH 

UNCERTAINTIES IN SIGNAL PARAMETERS 

6.1 Sampling Speed 

In Figure 35, the detection parameter of the PCC is evaluated for 

sampling speeds in the range of 1200 sampleslsec to 24000 samples/sec. 

While this curve is for the y = 1, a = +0.5 case, it does not vary much 

with these parameters. The band width of the noise is of the order of 

2,000 Hz but the nonlinearity used to obtain polarities spreads the 

spectrum further. For this reason we observe a substantial improvement 

in performance as the sampling rate is increased to about 8000 samples/ 

sec. But after that, the improvement slows down and increasing the 

sampling rate from 12000 to 24000 results in a gain of only 0.4db. Therefore, sampling rates of 12000 samples/sec are assmed throughout. 

6.2 Detection of Sinusoidal Bursts 

Electromagnetic signals consisting of sinusoidal bursts have been 

considered for the detection of trapped miners. The signal is assumed 

to be 0.1 second long bursts of a 1000 Hz sinusoid repeated every second. 

For this non-gaussian signal the asymptotic analysis of PCC is carried 

out under various assumptions, as the input signal-to-noise ratio 

decreases to zero. 
Figure 35. Performance of PCC versus sampling speed 
6.2.1 Polarity Difference Statistics 

Assuming that the signal is a sinusoidal burst of nearly known 

frequency, it is possible for the PCC detector to perform other statistics in addition to polarity coincidences. By adding a suitable delay 

in one channel the signal will be nearly 180' out of phase rather than 

in phase. For these signals, computing polarity differences is more 

appropriate. Computing polarity differences when the signals are out 

of phase and the noise inputs have a positive correlation is identical 

to computing polarity coincidence when the signals are in phase and the 

noise inputs have a negative correlation. Therefore, when the signals 

are out of phase polarity difference correlation can be evaluated using 

equation (2-4) by changing the sign of pH. Using this technique, the 

polarity coincidence statistic is compared with polarity differences in 

Figure 36 (a,b,c) for y = 0, 0.5, 1. The shift in either of channels 

is considered or in other words a shift of _+180e. From Figure 36(a), 

(y = 0) we can see that by adding a suitable delay an improvement can be 

achieved for both negative and positive cross-correlations. But for 

y = 0.5 and y = 1 in Figure 36 (b,c) we can see that possible improvement is for positive cross-correlation only with -180' shift. Ingeneral, 

we can say that for negative cross-correlation, polarity coincidences 

will outperform polarity differences whereas for positive crosscorrelation, the opposite is true. To perform this analysis, it has to 

be kept in mind that pH in equation (2-4) and ROE in matrix (5-1) have 

to be redefined. 
GAMA = 0.00 

.so -b.so 4.40 -0.20 4.00 6.20 O'.W o'.w O'.W 

SPACE CROSS-CORRELATION 

Figure 36 (a) Performance comparison of PCC with no shift, 

+180° shift for a wide range of crosscorrelation 

GAMA = 1 .OO 

Figure 36 (c) 
It is of course possible to consider other phase shifts such as 

m, where form even polarity coincidences are performed and for m odd 

polarity differences are performed. For the case y = 1, and a = +0.5 

the statistic D is plotted for various values of m in Figure 37. It 

PCC 

seems that the improvement of 1.3db for m = 1 can be increased somewhat 

with a larger m. However this analysis does not take into consideration 

the possible error in the frequency. 

6.2.2 Cost of Uncertainties in Signal Parameters 

Let the difference or delay between the beginning of the 0.1 second 

pulse and the start of the processing be denoted by D. That is, D = 0 

corresponds to perfect synchronization and D = 1 corresponds to missing 

the pulse completely. The numerator of equation (2-4) can be written as, 

-1 - 1 - 1 (1 - D) sin (pK) + sin pH - sin pH 

where 

- 1 

= (1 - D) [sin-'pK - sin pH] , 

and Sl(t). S2(t) are the signals in the two channels. If the entire 1 

second interval (or time between pulses) is divided into M overlapping 

5 0.1 second processing times, then it follows that D c E. Thus if 
w d.m w.m Jo.w w.w ib.w &o.w b.w dc.w &.w *.w ib. 

Wlrr 

Figure 37. Performance of PCC versus various shifts 
M = 40, in the worst case the degradation is only 0.58db. 

Let 

S,(t) = A(l + AA) Sin {(w +A w)t + 8) 

S2(t) = A(l + AA + EA) Sin {(w + Aw)(t-T) + 8 + €8) (6-3) 

where PA, Aw, and 8 are unknown but the same for both channels and EA, 

€8 are unknown differences in Sl(t) and S2(t). It is assumed that 

m 

T = - where m = 0,1,2, ... which means that we are going to consider 

W 

polarity coincidences and differences only. In computation of equation 

(6-2), to a second order approximation, EA has almost no effect and we 

get, 

Aw 

0 2 0 2. Cos (m-- - s/n w ce) + OH 

PK - 2+1 

's Ion 

where o is the average signal power in the two channels. It is seen 

S 

that ~8 willnot significantly degrade the performance unless it is large. 

The results of Figure 37 are extended in Figure 38 for €8 = 0 and 

Af = 0, + -5%, + 10% of the assumed frequency. This figure shows the 

degradation caused by an error in the frequency. We observe that a real 

improvement is achieved for m = 1, but further shifts do not help unless 

the frequency is known with some precision. 

We conclude that three statistics should be performed in addition 

to the polarity coincidence; polarity difference with a 180' delay in 

each channel should also be performed. 
VIF : Variation in frequency , 

Figure 38. Improvement in performance versus various 

shifts when signal has 0%, +5%, 210% error 
On combining Equations 6-1, 6-4, and 2-4 we obtain 

p 2 

Dpcc= 

2 2 1 2 -1 2 '1- ink + 2: 1-Q - (; sin P,) I (6-5) k= 1 

- as2 

On2 cos(mn % - €0) + pH 

where pk= 

1 + as 

2 

02- 

n 

5 and where D(the loss in pulse 'sync) is bounded by D 2~ where M is the number 

of processing intervals. 

For very small input signal-to-noise ratios, the denominator can be 

evaluated under the Hypothesis of noise only, and the numerator can be approximated by the lSt two terms of a Taylor's expansion 

us2 

Incorporating these changes, and recognizing that N = 1,200 T(seconds) for 

our presumed sampling rate, we obtain fl 2 

Since D 2 5 , the gain G (defined as D /os2/un2) is therefore bounded by 

M PCC 
where 2 & 34.64 

GO= 

- 

n 14~ 

and the bracketed term represents the cost due to signal mcertainties. 

The value of Go (between 10 and 20) is probably less than the gain of the 

human observer; perhaps by a factor of 2 or so. However, the human observer 

cannot accumulate information from one (pulse to the next (i.e. no term. 

Thus, for example, after 100 seconds (or 100 pulses), the gain of the PCC is 

improved by a factor of 10. 
6.2.3 Use of Microprocessors in Parallel 

We can effectively process the output of the PCC using microprocessor technology. For 40 overlapping intervals to be processed and 

assuming 0.1 seconds to read in the data and initial processing, and an 

additional 0.1 second to complete all three statistics,* we can process 

the data using only 8 microprocessors in parallel. The idea is shown 

in Figure 39(a) as the timing diagram of 8 microprocessors, (25-27) and 

the block diagram of an array processor in Figure 39(b). In the timing 

diagram the second microprocessor will start processing after 1/40 

second and by the time the eighth microprocessor receives data, the 

first microprocessor has finished. Therefore, one processing cycle is 

complete after 0.2 seconds. Assuming 12000 samples/sec, only 300 words 

(each word corresponding to 8 quantized samples) are processed by an 

individual microprocessor which is quite reasonable for any available 

microprocessor in the market. 

*The actual time depends on the microprocessor choice and the 

software package. 

93 
- (0.1 sec)Initial Processing Time j p0.1 sec) Time to complete a11 Statistics 

l-p 

m 11 111 

I 

I\ 

0.2 sec 0.8 sec 0.2 sec 

SR -Shift Register 

Fig. 39(a) Timing Magram 

P= Processor 

Fig. 39(b) Block Diagram of an Array Processor. 
7.0 CONCLUSIONS AND SUGGESTIONS FOR FUrURE RESEARCH 

7.1 Conclusions 

A general expression was derived from the detection parameter 

of the polarity coincidence correlator, the inputs of which are assumed 

to consist of a common signal plus correlated, stationary gaussian 

noises. An expression was obtained for the expectation of the product 

of four hard limited (clipped) gaussian inputs with arbitrary crosscorrelation, which is needed in order to evaluate the detection parameter of the PCC. A program to evaluate this expression has been 

implemented on the computer, evaluated and tested. This represents 

the most significant contribution of this study. 

A general model for the kinds of cross-correlation functions that 

would result when passing two heavily correlated noise processes through 

identical band-pass filters were developed. The model was extended from 

band-pass to a combination of band-pass and low-pass processes. Based 

on this model, the performance of the PCC was evaluated and compared 

with the unclipped correlator and the sum and square detector. The sum 

and square detector is optimum for independent noise inputs and for 

negative magnitudes of cross-correlation. For positive and large magnitudes of cross-correlation, the unclipped correlator outperforms the sum 

and square detector, and so may be optimum in this range. We observed 

that, for positive cross-correlations, while the performance of all three 

detectors fall off substantially with an increase in the correlation coefficient, the cost of clipping relative to the optimum increases only 
slightly, increasing from 2db to 3 db as the correlation increases from 

0 to 0.5. For negative cross-correlation, while the unclipped correlator 

performed as before, the performance of the PCC and the sum and square 

detector increased as the magnitude of the correlation coefficient increased. As before, the cost of clipping compared to the optimum 

increases slightly. The increase in performance of the PCC is comparable 

for negatively correlated noise to the decrease for positively correlated 

noise. 

The analysis is extended to two statistics, i.e., polarity coincidence and differences for 0.1 second duration sinusoidal bursts of 

nearly known frequency when the noise inputs are correlated. The decrease 

in performance relative to uncorrelated noise inputs is quite small. Indeed, for certain cross-correlation functions the performance increases. 

The increase in the cost of clipping with the correlation coefficient is 

also quite small. For the worst case, and perhaps the most likely case, 

where the cross-correlation function is proportional to the autocorrelation, i.e., (T) = aR(r), the degradation can be kept low using the 

polarity difference statistic. For a = 0.5, the degradation compared to 

the independent case is about ldb. The cost relative to the sum and 

square detector, for this case also increases by only ldb. The degradation caused by uncertainties of the signal parameters seem quite modest. 

A scheme to implement polarity coincidence and difference detectors, 

using microprocessor technology, is presented for the above problem. 
7.2 Suggestions for Future Research 

The analytical form of the expectation of the product of four 

clipped gaussian signals with correlated inputs, as calculated in 

Appendix E, diverges in certain regions. In this dissertation a scheme 

is presented to project those unstable points to a stable region. However, there is a need to develop a solution which converges in all 

regions. To possibly achieve that, the following four ways should be 

investigated. 

a) A determination of the joint pdf of the outputs of four 

clippers whose inputs are jointly gaussian and correlated could be 

attempted. This joint pdf is expected to be the combination of delta 

functions, because every output can only take two values. Having this 

pdf in a closed analytical form can lead to an easier and possibly 

more accurate computation of the expectation, evaluated in Appendix E. 

b) Using a transformation of cartesian coordinates into 4- 

dimensional spherical coordinates, the individual integrals in equation 

(E-5) of Appendix E might be formulated in a more stable form. 

c) Another way to facilitate the computation of the above integrals originates from the fact that the correlation matrix between 

four gaussian inputs is symmetrical. Hence, one could find a suitable 

transformation to diagonalize this matrix, immediately transforming 

the 4-D integrals into product of four 1-D integrals. 

d) has generalized Van Vleck's approach to calculate 

the bivariate expected value of two random variables where these 

random variables are arbitrary functions of gaussian random variables. 
While Gupta (291 has shown that multivariate normal expectation cannot be 

analytically expressed in'closed form, a formulation based on Rice's 

approach may lead to a more stable numerical computation of this expectation. 

Another subject for further research is related to implementation. 

The polarity coincidence correlator is simple in its hardware implementation. Since the PCC has been analyzed for a wide range of crosscorrelation shapes and the results are, or can be made, very encouraging, 

its implementation can be the subject of further research. 

Chapter 6 of this dissertation gives some idea about the arrangement of microprocessors. However, there is a need to develop this 

hardware structure and software carefully based on particular microprocessors. There are a few significant aspects of the implementation 

that we have not explored. We have not explored the manner in which either 

the supervisor microprocessor or the parallel processors accunulate the 

statistics from each signal pulse for display on a CRT. While we have 

determined that dividing each 1-second segment into 40 intervals seems 

reasonable, the number of parallel processors needed will undoubtedly be 

significantly less. This number will depend on the particular microprocessor 

and the efficiency of the software. 

We are confident, however, that if the receiver suggested in this 

report were implemented and tested, it would prove to be a valuable detector. 
APPENDIX A 

OPTIhiUM TWO-CHANNEL DETECTOR 

Consider the problem of detecting a random, gaussian signal that 

is common to two channels containing stationary gaussian noise. If 

the product of the integration time (T) and the signal bandwidth (0) is 

large, both signals can be accurately represented by the Fourier Series 

where the coefficients are statistically independent [see reference 3 

for example]. Thus we can write 

and 

where j = 1 or 2 and is the channel index. The assumptions that the 

signal is either common to both channels or 180° out of phase, that 

each noise input has the same power spectrum N(f), and that the coefficients are independent, enables us to write, for each frequency i, 

the following spatial correlation matrices for the signal only (P) and 

noise only (Q) cases. 
where S(i) is the value of the signal spectrum at frequency iwo and 

where 

- T 

c. c. (1 ) R (u) cos m wou du 

Pi = 11 12 . T 12 

T (A-5) 

fm (1- ;) R,(u) cos moou du 

1.1 iz 

where Rn(u) is the autocorrelation of the noise (or Fourier Transform 

of N(f)) and R12(-) is the cross correlation between the two noise 

inputs. Under the assumption TB>>l, this is closely approximated by 
where N12(f) is the Fourier Transform of R12(~). 

If z is a vector representing the four Fourier Series coefficients -1 

of the detector inputs (ai, bi for each channel) at each frequency i, 

then 

f z/si a1 and noise) where A is the likelihood ratio, f[zlnoee only) , and 

The optimum detector [any statistic nonotonically related to the 

likelihood ratio] forms the statistic 

After performing the algebra implied by equations A-9, A-8, A-3, and 

A-4, it can be shown that 

BT 

D =z 'i 2 

opt i=l 2 2 s. 1 [(ai12ai2)2 + (bil+bi2) I, 

(l+Pi) Ni (1+2 - I Ni (l+pi) (A- 10) 

where the + sign corresponds to signals exactly in phase and the - 

sign corresponds to signals 180° out of phase, and ail, bil. ai2. bi2 

are the variables of zi. 

We now wish to show that this statistic can be implemented by 

the following receiver: 
xl(t) = z Gail cos i u o t + JTbil sin i u t 

0 

x2(t) = Z eai2 cos i wot + qbi2 sin i uot . 

If H(w) is the transfer function of the filter, the output of the 

summer (difference) can be written as 

sin i wot + B ) 1 (A- 11) 

where Bi is the phase of H(iwo). The square law device and integrator 

calculates the energy in this signal or 

This is the same statistic as that of equation (A-10) provided 

(A- 13) 

Thus we have found the optimum detector if we can implement the filter 

indicated by equation (A-13). 
APPENDIX B 

DETECTION PARAMETER (OUTPUT SIGh'AL-TO-NOISE RATIO) 

OF THE TWO CHANNEL DETECTOR THAT IS LOCALLY 

OPTIMUhI FOR GAUSSIAN NARROW BAND IXPUTS 

'i For very small input signal-to-noise ratios - , the optimum filte~ 

Ni 

(Equation A-13) can be replaced with 

2 IH(iwo) 1 = s1 

2 (B-1) Ni (]+Pi) 

The resultant detector is called "locally optimum." For very narroK 

band signals, this amounts to just a narrow band filter that just passes 

the signal. If n. (t) are the noise in each channel just after the 

1 

filter, the test statistic can be written as 

where Ho represents the hypothesis of noise only and 

'I' 

s = I12 s(t) +nl(t) + n2(t-r)12 dt, (B- 3) 

H1 0 

where H1 is the alternative of an additive common signal. The output 

signal-to-noise ratio (D ) is given by opt 

After some lengthy but straight-forward calculations, it is determined 

that 
and 

2 k 

varH (s) = 4 on T J (1- [pll(k) + pZ2(k)+ ~~~(k-~)f~~~(k+~)1~ dk, 

0 0 

2. where on is the variance of the noise, T is the integration time, 

P12(T) is the normalized cross correlation function between n (t) and 1 

n2(t), and P (T) is the autocorrelation function of nl(t) or n2(t). 11 

Substituting into equation (B-4) we get 

where 

q(k, TI = [pll(k) + PZ2(k) + p12(k-?) 2 p12 (k+~)]~ . (B-9) 

Let us now make the very unrealistic assumption that the noise 

inputs are uncorrelated or p (T) = 0 for all T. Assuming further 12 

that pll(k) = pZ2(k) = ~(k), equation (B-7) becomes 

m 

8 T I un2p2(k)dk, for BT >>l 

,m 
and in the frequency domain the above expression becomes 

where B is the filter bandwidth. If the signal is a sinewave with 

amplitude A, then equation (B-6) becomes 

A2 

E~l 

(S) - EH (S) = 4 7j- T. 

0 

Substituting again into equation (9-4), 

(B- 10) 

Finally, if there are N bursts of 0.1 second duration signals, then 

T = 0.1 N and 
APPENDIX C 

DETECTION PARAMETER OF AN IDEAL ENVELOPE DETECTOR AND 

COMPARISON WITH THE TWO CHANNEL LOCALLY 

OPTIMUM DETECTOR 

In this Appendix we evaluate the ideal envelope detector under 

the assumption of independent noise inputs and compare it with the two 

channel results of Appendix B. The output of an envelope detector is 

given by 

T T 2 S = ~(t) cos wct dtj2 + [I z(t) sin wct dt] , (C-1) 

0 0 

where under the hypotheses Ho, z(t) = N(t) and under the alternative 

H1,z(t) = s(t) + N(t). It is easily seen that 

It follows that 

T T 

EH (S) = J J Rn(tl-t,) - cos wc(tl-t2) dtldt, (C-2) 

0 

- 

0 0 

and 

T T 

E (S) - E,, (S) = I J s(tl)s(t2) cos wc(t,-t,) dtldtZ . (C-3) 

H1 0 0 0 

We let s(t) = A sin [(wc + Aw) t+81 where Aw is the unccrtaint!. 

in the signal frequency. Equation (C-3) can he evaluated as 
A 2 cos [(2wc + Aw) T+ ] - cos 9 

cos (AwT+B) - cos$ 2 

(S) - E (S) = 7 I + 

Ho 2wc + Aw Aw 

i 

2 Sin [2wc +Awl T+9] - sin 9 

A I sin(AwT+9) - sin 9 

2 

+ - (C-4) 4 (2wc + Aw) Aw 

The (2wc + Aw) terms are much smaller than the other terms. A good 

approximation involves ignoring these terms, and equation (C-4) can be 

manipulated into 

2 E (5) - PH (S) = -4 A''' sinc =A ft 

H1 0 

It can be determined that for gaussian noise, 

The output signal-to-noise ratio D can be determined by dividing ENV 

equation (C-5) by the square root of equation (C-6) or 

L 

- A~T sinc nAfT 

D~~~ - 

Jl+sinc2 2nfcT 

For T = .1 sec, fc = 1000 Hz, this becomes 

A2 = - 2 nAf 

D~~~ 2~1~ sinc (T) . 
Figure C-1. Died/Dopt vs. Delta F 
Finally for N pulses, 

- - - A~ 2 nAf 

D~~~ 20No fl sinc (rn) . 

If we compare the two channel detector, equation (B-11) for a 

filter bandwidth (B) of 30 Hz with this one channel envelope detector, 

we get 

(C- 10) 

This equation is plotted in Figure (C-1). For greater uncertainties 

it is considerably worse. 

Of course this analysis is very preliminary because of the unrealistic assumption that the noise inputs are independent. If we 

compare two-channel detector with two outputs of I.E.D. then (C-10) 

becomes 

D~~~ -= 2 nAf 

D 1.7 sinc - 

opt 

10 

In theory, we can extend coherent analysis beyond one pulse. But 

from Figure -1, for coherent detection, beyond one pulse of duration 0.1 sec, we should know the frequency to within 0.01%. 
APPENDIX D 

CALCULATION OF DETECTION PAMETERS 

I. Polarity Coincidence Correlator (PCC) 

The detection parameter for the PCC is 

iqith hypothesis 

all t E (0,T) 

n(t), S(t) are noise process and signal, and 

where 

the xi(t) refer to two inputs and N is the number of obversations. 

The variance of the test statistic is 
N 

= N Var {h[x(t)]} + 2 (N-k) {E[h[x(t)] h[x(t+kr)]] 

k=l 

From Van Vleck (I4) it is known that E [Sgn xl(t) Sgn x2(t) ] = 

2 . -1 (2/n) sin p, where p is the correlation coefficient. Using this 

result, 

4 

Var {h[i(t)]} = - - l/n2 (sin-lp) 4 (D-5) 

and 

1 1 E {h[i(t)] h[i(t+k)]} = - 4 + l/n sin-lp + ;r E [Sgn xI(t) 

After substituting the results of (D-5) and (D-6) in (D-4) we obtain, 
N 

N Var [S] = [l - (2/n sin -' k PI' + (1- E) {~[~gn xl(t) Sgn x2(t) 

k=l 

Sgn xI (t+k) Sgn x2(t+k)] - (2/n sin-' p)' I] (D-7) 

We now calculate the shift in the mean of the statistic or 

where a = - (1/2) (x1,x2) [PI 

and correlation matrix 

After changing the integral of (D-8) into polar coordinants and using 

the identity, 

= (~-r~)-~/~ tn/2 + sin-lr) , 1-r sin 26 

0 

and after some simplifications, we get 

Rx (0) N E [S] = Ti[n/2 + sin -1 I 2 

K I 
Similarly, 

Rn (0) 

N -1 1 2 EH[S] = ;[n/2 + sin 

Rn(0) I 

Utilizirig results of (D-6), (D-) , and (D-12) in (D-1) we get, 

dN(2/n) [sin-' pK - sin - 1 

D = %I 

P Cc N 

k 7 1/2 

[I-{ (21~) sin-' pK12+ 2 1 (1- [Q(k) - {(ZIT) sin-' pKli]l 

k= 1 

(D-13) 

(0) 

where, - PK - R 

x1x2(0) 

Anevaluation of (D-14) is presented in Appendix E. 

11. Detection Parameter for the Unclipped Correlator (Dc) 

The test statistic for the correlator is 
where x.(t) refers to two zero mean gaussian inputs under the hypothesis. 1 

For the purpose of the evaluation of the detection parameter Dc, we will 

assume a general shape of the autocoreelation and cross-correlation. 

Thus in this case, 

EXIS) - EH(S) = N RS[o) (D- 16) 

where R (0) is the signal power S 

NN N 

= E~[II xi(ti) x2(ti) x (t.1 x (t - EH[l xl(ti) x2(ti)l 2 

11 2j i j j (D- 17) 

The first part of the above expression for zero mean gaussian random 

variable can be expanded as follows: 

Using the above expression in (D-16) and after some simplification, 

we can write 
where R(k) and Rx (k) are the auto-correlation and cross-correlation 

12 

of inputs x (t) and x,(t) respectively. Using (n-16) and (D-19) the 1 & 

detection parameter for the correlator can be written as 

111. Detection Parameter for the Sum and Square Detector (DSS) 

The test statistic of the sum and square detector is defined as 

The shift in mean becomes 

also 

Expanding.equation (D-23) and using the following equivalents 
and EH[S] = 2N[Rx (0) + R(0)1 , 12 

the variance becomes 

2 2 Var,(S) = E[S ] - E [S] 

H H 

. [2R2 (k) + 0.5 (R 2 

(k)+Rxx (-k) + 2(Rx (k) 

X1 X2 12 12 

Therefore, 

where the variance is given in (D-24) 
APPENDIX E 

THE EXPECTATIOI2 OF THE PRODUCT OF FOUR CLIPPED 

GAUSSIAN RANDOM VARIABLES 

We use the following notation 

Variables Random Variable 

(samples of Stochastic Process) 

1 

---------------- xl (t) 

X2 

---------------- x2 (t) 

X3 ---------------- xl(t + k7) 

X4 

---------------- x2(t + kT) 

where the x. are zero mean gaussian. 1 

The general correlation matrix C is given as 

where R..(kr) and R..(k'r) are the autocorrelation and cross11 11 

correlation between x. and x. (i,j = 1,2) . 1 I 

The joint gaussian p.d.f of four variable can be written as 
where the coefficients q.. are the elements of C-I. Ke are to compute 13 

where Sgn x(t) is 1 if x(t) > 0 and -1 if x(t) < 0. 

mmmm 

which can be written as a sum of 16 integrals in equation (E-5) 

found on the folloi~ing page. 

Introducing a suitable change in variables, each of the 16 

ingegrals can be written as 

mmmm 

where the coefficients w.. are identical to q.. but with appropriate 

1 I 11 ' 

changes in signs which are due to the change of variables. 

In this Appendix, the solution of one of the 16 integrals will be 

presented, where the remaining 15 can be solved with a similar result 

4 

and the summation of the 16 results is the expectation of Il Sgn xi . i=l 

Let (E-6) be written as 

mmmm 

where 

Integrating with respect to x first, 1 

0 

where A represents all of the constant terms. 

Using the identities, 

(E-9) can be written as 

where 

Rearranging for x2 
0 

B represents the terms and constants which are not involved in the 

integral with respect to x2. Using the binomial expansion, 

and integrating 

Again defining some E which represents the terms not involved in 

evaluation of the remaining integrals, we can write 

where c = -(1/2) {w~~x~~ + W 44~4~) + W34X3X4 
i-k 

i-k (W13X3+W14x4) = (w~~x~)~ (Ul4x4) i-k-m Ii.7 

m=o 

The term involved in the remaining integrations is 

I I r+j-s+i-k-m e - (1/2) ({w~~x~~+v~~x~~I)~x 3 dx 4 (E- 17) 

X4 

Putting (E-15) and (E-18) together we have the final results. 

APPENDIX F 

A Flow-Diagram for the FORTRhl program that computes the 

expectation of the product of four clipped, jointly gaussian 

functions is given. Another general Flow-Diagram follows, which 

uses this expectation. Subroutine to compute the Detection Parameter for the Polarity Coincidence Correlator and compares results 

with Analog Detectors. 
Flow Diagram. 

IT, C, DET :I 

NORMALIZE C '7 

1 YES 
YES 

m 

J=J+1 

I=I+1 

OUT= SlJM/((m)2fi ~(1317) 

I 

RETURN 
FORTRAN Implementation. 

SUM 

T 

OUT 

List of Principal Variables 

Program Symbol Definition 

IT Truncation of Infinite Sumation. 

C Inverse of Correlation Matrix P. 

DET Determinant of Matrix P. 

I,J,IR Index of Infinite Surmnations. 

K,L,M Index of Finite Smnmations. 

IODD Checks Whether I or (K+J) or (M+L+IR) 

is Odd. If any of them is Even then 

that Calculation is Skiped. 

Partial Result 

Value of the Product of Elements of Upper Triangle 

of C Matrix, while Elements are raised to the 

Powers as mentioned in Appendix E. 

Expectation of the Product of Four Clipped 

Gaussian Functions whose Correlation Matrix is P. 
Flow Diagram. 0 tart, 

I Define Parameters I 

Compute (COMMON) 

Coeffecients to be used 

in Expectation Subroutine 

Parameters 

Alpha,AO,Gama to Define 

Compute Elements of 

Matrix P 

Which are Independent 

9 Sum -0 

r Initialize: k=O I 

Compute Elements of 
Compute Determinant 1-1 

1 Compute P + Call EXPECT 

Update 2mmation 

1-1 *[ (xi)] 

YES 

r 

Compute and Store 

Detection Parameter 

For PCC 

Compute and Store 

Other Detection Parameters 

For Correrator & Sum & Square 

Change Variable Parameters 

, 

Alpha, A0 , Gama 
I YES 

Print and/or 

Plot Results 
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