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## Material Information- Title:
- Application of sequential decision theory to voice communications
- Creator:
- Pettit, Ray Howard, 1933-
- Publication Date:
- 1964
- Copyright Date:
- 1964
- Language:
- English
- Physical Description:
- xvi, 159 leaves : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Error rates ( jstor )
Probabilities ( jstor ) Radar ( jstor ) Receivers ( jstor ) Reference systems ( jstor ) Signal detection ( jstor ) Signal to noise ratios ( jstor ) Signals ( jstor ) Statistics ( jstor ) Voice communications ( jstor ) Communication and traffic ( lcsh ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Mathematical statistics ( lcsh ) Voice ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: leaves 156-157.
- Additional Physical Form:
- Also available on World Wide Web
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000566141 ( AlephBibNum )
13640315 ( OCLC ) ACZ2568 ( NOTIS )
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APPLICATION OF SEQUENTIAL DECISION THEORY TO VOICE COMMUNICATIONS By RAY HOWARD PETTIT A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA June, 1964 ACKNOWLEDGMENT The author wishes to acknowledge his great debt to his advisor, Professor T. S. George, whose advice and encouragement made this dissertation possible. He wishes to thank Professor W. H. Chen for his assistance throughout the period of graduate study at the University of Florida, and the other members of his committee for their aid. The author also wishes to thank the Martin Company, Orlando, Florida, for providing the interesting work assignments which led him to return to graduate school and to pursue the research reported herein. A special acknowledgment is due the Lockheed-Georgia Company of the Lockheed Aircraft Corporation, where the author was employed during the course of this research. Assistance was provided by the Advanced Research Organization in forms too numerous to mention, most important of which was a research climate which facilitated the completion of this dissertation. TABLE OF CONTENTS Page ACKNOWLEDGMENT . . . . . . . ii LIST OF FIGURES. . . . . . . . . .. v KEY'TO SYMBOLS . . . . . . . . x ABSTRACT . . . . . . . . . . xiv Chapter I. INTRODUCTION . . . ... ..... 1 The Beginnings of Sequential Analysis. . 1 Description of the Sequential Test . . 2 Radar Application. ....... . 6 Data Application .... .... . . 12 Voice Systems. ...... . . ... 14 II. CONVENTIONAL AND SEQUENTIAL SYSTEMS WITHOUT IMPULSE NOISE CONSIDERATION. . . . 20 Introduction . . . . . . .. 20 QPPM, Double Threshold, Speech Statistics. 21 QPPM, Single Threshold, Conventional . 32 QPPM, Double Threshold, Largest of s . 35 QPPM, Largest of N . . . . . 40 Discussion of Results. . . . . 45 III. CONVENTIONAL AND SEQUENTIAL SYSTEMS WITH IMPULSE NOISE CONSIDERATION. . .. . 82 Introduction . . . . . 82 iii Chapter Page QPPM, Four Threshold, Speech Statistics . 83 QPPM, Four Threshold, Nearest to Reference. 90 QPPM, Single Threshold. . . . . . 96 QPPM, Largest of N. . . . . . . 96 QPPM, Double Threshold, Non-Sequential. . 97 QPPM, Single Threshold, Largest of N. .. 100 Discussion of Results . . . . . 103 IV. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH. . . . . . .. . . 146 Conclusions . . . . . . . . 146 Future Research . . .. . . . 147 APPENDICES. . . . . . . . . . . 149 A. Speech Probability Functions. . . . 150 B. Error Bound for Approximate Evaluation of the Integral in P8(cor) . . . . 154 BIBLIOGRAPHY. . . . . . . . . . . 156 BIOGRAPHICAL SKETCH . . . . .. . . 158 LIST OF FIGURES Figure Page 1. Functional Diagram, QPPM, Double Threshold, Speech Statistics System . . . . 23 2. Functional Diagram, QPPM, Single Threshold, Conventional System. .... ...... .. 34 3. Functional Diagram, QPPM, Double Threshold, Largest of s System. . . . . . . 37 4. Functional Diagram, QPPM, Largest of N System. 42 5. Probability of Error for Input Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, Without Impulse Noise.. . . . . . 47 6. Probability of Error for Double Threshold, Speech Statistics System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 5.9 Decibels . . . . . . . 50 7. Probability of Error for Double Threshold, Speech Statistics System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 9.0 Decibels . . . . . . . . 52 8. Probability of Error for Double Threshold, Speech Statistics System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 11.0 Decibels . .. . . . . . . . 54 9. Probability of Error for Double Threshold, Speech Statistics System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 12.1 Decibels . . . . . . . . . 56 10. Probability of Error for Double Threshold, Speech Statistics System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 13.0 Decibels . . . . . . . . . 58 11. Values of K1 and K2 for Double Threshold, Speech Statistics System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum Probability of Error . . . . 60 12. 9, andr s2 for Double Threshold, Speech Statistics System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum S Probability of Error. . .. .. . 63 13. Probability of Error for Single Threshold, Conventional System for K1 Between 0 and 1.0, and for Signal-to-Noise Ratios Between 5.9 and 16.0 Decibels. . . . . . . . . 65 14. Value of K1 for Single Threshold, Conventional System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum Probability of Error. . . . . . . . . . 67 15. Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 9.0 Decibels . 70 16. Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 11.0 Decibels. 72 17. Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 12.1 Decibels. 74 18. Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to-Noise Ratio 15.0 Decibels. 76 19. Values of K1 and K2 for Double Threshold, Largest of s System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum Probability of Error . ......... 78 Page Figure 2 20. K, s, and. sr for Double Threshold, Largest of s System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum Probability of Error *. . . . .. 80 21. Functional Diagram, QPPM, Four Threshold, Speech Statistics System, with Impulse Noise 85 22. Functional Diagram, QPPM, Four Threshold, Nearest to Reference System, with Impulse Noise. . . . . . . . . . 92 23. Functional Diagram, QPPM, Double Threshold, Non-Sequential System, with Impulse Noise. . 99 24. Functional Diagram, QPPM, Single Threshold, Largest of N, Non-Sequential System, with Impulse Noise. . . . ... . ... . 102 25. Probability of Error for Input Signal-to-Noise Ratios Between 9.0 and 16.0 Decibels, with Impulse Noise. . . . . . . ... 106 26. Probability of Error for Four Threshold, Speech Statistics System, with a and b at Optimum Settings, for Signal-to-Noise Ratio 12.1 Decibels. . . . . . . . . 109 27. Probability of Error for Four Threshold, Speech Statistics System, with a and c at Optimum Settings, for Signal-to-Noise Ratio 12.1 Decibels. . . . . . . . 111 28. Probability of Error for Four Threshold, Speech Statistics System, with a and d at Optimum Settings, for Signal-to-Noise Ratio 12.1 Decibels. . . . . . . . . 113 29. Values of K1, K2, K3 for Four Threshold, Speech Statistics System for Signal-to-Noise Ratios Between 9.0 and 16.0 Decibels, with Minimum Probability of Error . . . .. 116 30. Probability of Error for Four Threshold, Nearest to Reference System, with a and b at Optimum Settings, for Signal-to-Noise Ratio 12.1 Decibels. . . . . . .. . 118 vii Figure Page Figure 31. Probability of Error for Four Threshold, Nearest to Reference System, with a and c at Optimum Settings, for Signal-to-Noise Ratio 12.1 Decibels. . . . . . . . . 120 32. Probability of Error for Four Threshold, Nearest to Reference System, with a and d at Optimum Settings, for Signal-to-Noise Ratio 12.1 Decibels. . . . . . . . . 122 33. Values of K1, K2, K3 for Four Threshold, Nearest to Reference System for Signal-to- Noise Ratios Between 9.0 and 16.0 Decibels with Minimum Probability of Error. . . . 124 34. Probability of Error for Single Threshold, Impulse Noise System for K1.K2*K3 Between 0 and 2.0 and for Signal-to-Noise Ratios Between 9.0 and 16.0 Decibels. . .. . .. 126 35. Value of Kl*K2*K3 for Single Threshold, Impulse Noise System for Signal-to-Noise Ratios Between 9.0 and 16.0 Decibels, and with Minimum Probability of Error . . . . 128 36. Probability of Error for Double Threshold, Impulse Noise System for Kl*K2*K3*K4 Between 0 and 1.0 and K1 K2 Between 1.0 and 3.6, and for Signal-to-Noise Ratio 9.0 Decibels . . 130 37. Probability of Error for Double Threshold, Impulse Noise System for K1*K2*K3*K4 Between 0 and 1.0 and K1'K2 Between 1.0 and 3.6, and for Signal-to-Noise Ratio 11.0 Decibels. . 132 38. Probability of Error for Double Threshold, Impulse Noise System for K1*Kp2*K*K4 Between 0 and 1.0, and K1.K2 Between 1.0 and 3.6, and for Signal-to-Noise Ratio 12.1 Decibels. . 134 viii Page 39. Probability of Error for Double Threshold, Impulse Noise System for K1*K2*K3*K4 Between 0 and 1.0 and K *K2 Between 1.0 and 3.6, and for Signal-to-Noise Ratio 13.0 Decibels. . 136 -40. Probability of Error for Double Threshold, Impulse Noise System for Kl*K2*K3*K4 Between O and 1.0, and K1*K2 Between 1.0 and 3.6, and for Signal-to-Noise Ratio 16.0 Decibels. . 138 41. Values of K1*K2 and K1-K2 K3*K4 for Double Threshold, Impulse Noise System for Signal- to-Noise Ratios Between 9.0 and 16.0 Decibels, with Minimum Probability of Error. .. . 140 42. Probability of Error for Single Threshold, Largest of N System for K1~K2 Between 0.3 and 3.5, and for Signal-to-Noise Ratios Between 9.0 and 13.0 Decibels. . . . . 143 43. Value of K1'K2 for Single Threshold, Largest of N System for Signal-to-Noise Ratios Between 9.0 and 16.0 Decibels, and with Minimum Probability of Error . ... . . . . 145 Figure Page KEY TO SYMBOLS Symb ol a a, a 1 A A b / B B c d erf(x) Ho H1 Io(x) K K1 K2 Description A threshold. Input signal-to-noise ratio. A preset value of a for the sequential test. Probability of false alarm in radar. An upper threshold number. Amplitude of signal pulse at the receiver. A threshold. Probability of false dismissal in radar. A lower threshold number. Amplitude of impulse at the receiver. A threshold. A threshold. x Error function, e Lu.. The hypothesis of noise alone. The hypothesis of signal and noise. Modified Bessel Function of first kind, zerot order. Average value for the number of slots for termination. Ratio b/A for systems of Chapter II, d/A for systems of Chapter III. Ratio a/b for systems of Chapter II, c/d for systems of Chapter III. Symbol K 3 K4 Am Am 11n n(t) N p Pl(R1) p2(R2) p (R3) P(K) P(s) P(err) Pl(cor) P2(cor) P (cor) P4(cor) P5(cor) Description Ratio b/c for systems of Chapter III. Ratio a/b for systems of Chapter III. Average number of impulses in N-1 noise slots. Probability ratio for m observations. Likelihood ratio at the mE stage. Number of impulses in N-1 noise slots. Input noise function. Number of pulse slots per speech sample frame. A priori probability that H is true. Probability density of noise envelope. Density of signal and noise envelope. Density of impulse and noise envelope. Probability of termination on the KI slot. Probability that the number of slots with envelopes in the "possible signal" zone is s, given that none are in the "signal" zone. Probability of error, l-P(cor). Probability of correct decision for QPPM, Double Threshold, Speech Statistics system. Probability of correct decision for QPPM, Single Threshold system. Probability of correct decision for QPPM, Double Threshold, Largest of s system. Probability of correct decision for QPPM, Largest of N system. Probability of correct decision for QPPM, Four Threshold, Speech Statistics system, with impulse noise. Symbol P6(cor) P7(cor) P8(cor) P (cor) P10(cor) PB(n) PB (n) Ps(j) pr(y) P (j/f) Q(x,y) R s s(t) 2 n s Description Probability of correct decision for QPPM, Four Threshold, Nearest to Reference system, with impulse noise. Probability of correct decision for QPPM, Single Threshold system, with impulse noise. Probability of correct decision for QPPM, Largest of N system, with impulse noise. Probability of correct decision for QPPM, Double Threshold, Non-Sequential system, with impulse noise. Probability of correct decision for QPPM, Single Threshold, Largest of N system, with impulse noise. Probability of n impulses in N-l noise slots. Probability of n impulses in j-1 noise slots. Probability of speech pulse in the ji slot. Probability of reference pulse in the Ib slot. Conditional probability that signal pulse in ji if reference is )t slot. Number of impulses closer than signal to reference. V2 yA+ XIL The "Q" function, fv.e'- r.(xv). -Jv, An envelope. Average number of slots with "possible signal" envelope. Input signal function. Average noise power at the IF. Variance of the number of slots with "possible signal" envelopes. xii Symbol wm(x,9) W1(x) W2(x1/x2 ,z) I Description Unknown parameter of a probability function. Joint probability density function of Xlx2,...xm when 9 is true. First probability density of the instantaneous speech amplitudes. Conditional density of speech amplitude x, when the value T seconds before was x2. xiii ABSTRACT Wald's Sequential Decision Theory is applied to voice communication systems of the quantized pulse position modu- lation type. When interference is narrow-band Gaussian noise, two sequential systems are considered. The "Double Threshold, Speech Statistics" system divides the speech sampled frame into N discrete slots. The three decision zones for each slot are: (1) signal is present, (2) signal is possibly present, and (3) signal is not present. If the end of the frame has produced no decision for the first zone, the receiver chooses one from the group representing the second decision zone. This choice is made on the basis of the one most likely to be the signal, as indicated by speech statistics. The "Double Threshold, Largest of S" system is the other sequential type. It is similar to the previously des- cribed technique, with the exception that when the choice must be made from the group representing the second decision zone, it is made on the basis of the largest one. When compared to the conventional, single threshold method, the sequential systems provide a reduction of about 1.5 decibels in required signal-to-noise ratio for a specified probability of error. Their performance is almost as good as a "Largest xiv of N" receiver, which considers all N slots and chooses the largest as signal. The sequential methods seem to represent a compromise between these two techniques, having some features of each. When interference is narrow-band Gaussian noise, with occasional impulse-like large amplitude pulse noise, the sequential systems have four thresholds. Thus the "signal is possibly present" and the "signal is not present" decision zones consist of two separate parts for each. The two sequential types are the speech statistics and the nearest to reference systems. The reference referred to is the decision zone representing signal present. Analysis is made on the basis of a mathematical model consisting of a Poisson distribution for the number of impulses in a frame, with an average number of one, and a modified Rayleigh distribution for the envelope of the impulse-like interference, with amplitude of the pulse three times the signal pulse. The error performance of the conventional single threshold and the "Largest of N" receivers is shown to be completely inadequate under these conditions. A conventional receiver modified to include an upper threshold for discrimi- nation against impulses gives a reasonable performance. The sequential methods, however, give an improvement of about 2.0 decibels over this. Roughly equivalent to this improve- ment is that given by a "Largest of N" system which is modified to include an upper threshold for impulse removal. xv The conclusion reached is that sequential techniques in the form described provide an improvement over conventional or modified conventional methods. The improvement is probably not large enough under most circumstances to warrant their use. However, specific situations may allow their beneficial application. It is felt that further study should be made of sequential techniques for other modulation methods in voice communications, possibly in other forms from those described here. xvi CHAPTER I INTRODUCTION The Beginnings of Sequential Analysis The voice communication systems discussed in this report have resulted from efforts to apply the ideas of sequential analysis to this field of study. It is there- fore appropriate to relate in this initial chapter some of the background information on sequential analysis, and to describe some earlier applications of this theory. Whether in radar, data, or pulse-coded voice communications the basic problem is the detection of a signal in noise. When the examination of the random process is to provide a positive or negative indication as to the presence of a signal the problem may be considered a test of statistical hypotheses. That is, based on the available information a choice is to be made between hypothesis H : signal is not present, and H1: signal is present. This may involve a statistical test of a mean, a signal-to-noise ratio, etc. Sequential analysis, including the sequential. probability ratio test, was devised by Abraham Wald1 for hypothesis testing. He and Wolfowitz2 showed that this test requires on the average fewer samples to terminate than do tests of fixed sample size, for equal probabilities of error. A distinguishing feature of the test is that it is conducted in stages, with the length not specified in advance, but determined by the progress of the test. Thus the sample size is a random variable. Although the length of the test is smaller on the average than that for any other test, it may be very large for a particular trial. Stein' showed that not only is the expected length of a sequential test finite, however, but all moments of the length are finite if the samples are independent. Thus the question was resolved as to whether sequential tests terminate. Because the length of a particular test may be longer than can be tolerated, many practical situations require that a test be terminated prematurely. At the point of truncation, some alternate criterion is adopted so that a choice of hypothesis may be made. A truncated test requires on the average a smaller number of samples, but the error performance is worse than for the untruncated sequential test. Truncation represents a compromise between the completely sequential test and the fixed-sample test. Description of the Sequential Test We let X. be the random variable representing a sample of a random process for which a statistical decision problem is formulated. Sample Xi is independent of any other sample Xj for i / j. The successive samples of X are denoted by X1 X2, ... The joint probability density function for the samples is considered known, except for some parameter 9 (or a set of parameters 5 in the general case), and is represented by Wm(X;9). The subscript indi- cates that the experiment is at the mt stage. Based on the sequential test a choice is to be made between two hypotheses: Ho, the value of 9 is g ; and H1, the value of 9 is 1. Therefore, if hypothesis H is true the joint probability density function of the m samples is W (X;G ); if hypothesis H1 is true the density is Wm(X;Q1). If the a priori probability that H is true is p, then the a priori probability that H1 is true is 1 p. More generally, of course, there could be more than two hypotheses from which to choose. Here we are interested in the binary case only. The likelihood ratio at the mt stage of the experi- ment is defined by (1 p)Wm(X;Ql) ,m (1.1) Am= Wm(X; ,) The carrying out of the sequential test requires this likelihood ratio to be computed at each stage of the experi- ment. Two threshold numbers AZ > 1 and B' < 1 are chosen. At the mt stage the decision is made to continue the test if B < Am AK for m = 1,2,...,n 1 (1.2) The test is terminated and hypothesis H accepted if at the n1 trial An< B (1.3) Similarly, hypothesis H1 is accepted at the nt trial and the, test terminated if An > A (1.4) Observe that if we set B = (1 p) B and A = (1 p) A p P the sequential test can be set up in terms of the conditional probabilities, independent of the a priori probabilities p and 1 p. This can be seen by considering the probability ratio W (X;,1) Am = m (1.5) m = o Note that Am is 1 Pm The test procedure is thus slightly modified. At the mtb stage the decision is made to continue the test if B< m< A for m = 1,2,...,n 1 (1.6) The test is terminated and hypothesis H accepted if at the ntb trial n B (1.7) Similarly, hypothesis H1 is accepted at the nt trial if Xn> A (1.8) The standout feature of the sequential test is the dividing of the decision zone into three parts (by means of the two thresholds). These are: (1) a zone of acceptance for H (2) a zone of acceptance for H1, and (3) a zone of indifference. The conventional fixed-number-of-samples test has associated with it only the two zones of acceptance and one threshold. As pointed out above, the sequential test provides a savings, on the average, in the required number of samples for given probabilities of error. The two types of error possible are the acceptance of Ho when H1 is true and the acceptance of H1 when H is true. Wald1 has showed that the thresholds A and B depend on these probabilities of error. The procedure followed in the conventional sequential test is to select acceptable error probabilities, thereby determining the thresholds. Of course, the smaller the error probabilities, the larger the average test length. It is emphasized at this point that a modified form of the conventional sequential test is utilized in the voice communications applications discussed in the body of this report. The several variations from the conventional will be explained in detail in the following chapters. Radar Application There has been much work in applying the sequential decision theory to radar systems. In the radar problem, the system decides on the presence or absence of a target in a particular region of space during a particular time period. Based on the signal received at the set, it may not be clear as to whether this is from the radar pulse returned from a target or whether it is noise alone. In this case, the radar continues to "look" until a decision can be made. Wald's theory, when considered from the point of view of detecting a radar signal in noise, offers a saving in the time required to make a decision within certain tolerable error limits as to the presence or absence of a target. Sequential detection of signals in noise falls into the general domain of Statistical Decision Theory. Van Meter and Middleton studied the general application to the reception problem of decision theory. Blasbalg,5 Fox, and Bussgang and Middleton7 have made valuable contributions with their works involving sequential signal detection. Much of the discussion here of the radar application will follow the results of Bussgang and Middleton.7 For radar, the parameter for which the sequential test is performed is the input signal-to-noise ratio a. Thus the two hypotheses being considered are H : the value of a is zero, and H : the value of a is greater than zero. Acceptance of Ho (rejection of H1) is a decision that no target is present. This is a "dismissal." Acceptance of H1 (rejection of Ho) is a decision that a target is present. This is an "alarm." As expected, the decision to accept H1 is more likely to be made as a is increased. A preset signal-to-noise ratio a is used to set up the sequential test. The value a is such that for a less than al, an acceptance of H is not too objectionable. The test involves selecting tolerable values for o< the probability of a false alarm, and for/ the probability of a false dismissal. The quantity/3 is determined by the probability of dismissal for a = a1. Sometimes oc and 3 are called the probabilities of error of the first and second kind, respectively. The "strength of the test" refers to these two probabilities of error and is denoted by (or ). The test is regarded as stronger as o< ,3 are made smaller. At this point we will consider the incoherent se- quential detection of the radar signal in noise. Following this will be a discussion of coherent sequential detection. For incoherent detection no phase information is available at the receiver. The random variable of interest is the envelope of a narrow-band Gaussian noise plus a sine wave at the center frequency of the noise band. The appropriate probability density function for this envelope R is shown by Rice8 to be 8 W(R;A ) = I -- for R > 0 (1.9) en 2n n 2 where a-n is the mean-square value of noise at the i-f filter and I (u) is the modified Bessel function of the first kind and of order zero. A is the peak amplitude of the sine wave. For the envelope of noise alone (A = 0), the density function is R R W(R;0) = -- 2 for R > 0 (1.10) C- 2 o- n n These are the Rayleigh distributions. The true signal-to- noise ratio a, which is the unknown parameter of these densities, is A a = o (1.11) -~c If, for convenience, we change the random variable from S to X, the densities become 2 2 W(x;a) = 2x e- a2 + x ) -I(2ax) x o (1.12) =0 x< . From this the probability ratio Am can be found, and the test carried out in terms of the threshold numbers A and B. An alternate procedure involves the consideration of the test in terms of the logarithm of the probability ratio. For this case, the zone of indifference is bounded by log A and log B. Under the usual condition of independent samples, the two error probabilities are Sb R. n-2 0 =e d .[) o(e(-( R ] (1.13) o a 07 o a2. RRL e 2 J e3 /, 0_ o[Ir R(1.14) 0 In the above, oe and/3 are expressed in terms of the actual signal level thresholds a and b. Even though n, the final stage of the test, is a discrete random variable assuming integer values, it is convenient to consider it as a con- tinuous variable. Thus p(n/Hk;a) represents the conditional probability density function of n under the condition that Hk (k = 0.1) is accepted. The true value of the signal-to- noise ratio a is also required to specify this density. Simplification of the error probability expressions yields oO 0 0 62' r ( Y '/ H 0) z ,- '( 1 -1 5 ) (3= pd p 4){Q( r)a. r-),] + .[i-Q(^7^))J (1.16) where Q(x,y) is defined by Marcum9 as 00 v + X2 Q(x,3) = v.e I,(xv) v (1.17) Bussgang and Middleton' have studied this case ex- tensively. They show how the average number of samples required increases as the probabilities of error are lowered. The savings in average number of observations is paid for in the sample length's becoming random. The larger the variance of the sample length, the greater the savings in average length. As the variance gets smaller the savings is reduced and the sequential test approaches a non-sequential. Similarly, Bussgang and Middleton7 have thoroughly studied the sequential coherent detection case. Here, of course, it is assumed that the receiver has complete signal phase information. Statistically, the problem is the testing of the mean. Under H the true mean is zero; for H1 the mean is the value of the signal at the sampling time. The general technique for the test does not change. The probability ratio is constructed and tested against the two threshold numbers as before. The density functions for this case are Gaussian. Bussgang and Middleton' treat the general case of correlated samples, and give as an example the case with an exponential autocorrelation function. It is interesting to note the results of the compari- son between the sequential coherent and sequential non- conherent detectors. For signals smaller than the preset value al, the coherent detector is less likely to give a false dismissal than the non-coherent. In many cases this difference is very great. For signals larger than the preset value, the coherent detector is more likely to give a false dismissal. In the comparison of the average number of samples, it is found that in the weak signal case this average number depends on a for the coherent test and on a2 for the non- coherent. As an example of how this affects the average sample number, Bussgang and Middleton7 plotted al2 n(a) for coherent and al n(a) for non-coherent versus a/a. The peaks of the two curves are the same but occur at different values of a/a1. Since the case considered is for a1 very small, the coherent sequential detector therefore requires fewer average samples at this peak value. For the radar case, any truncation of the test re- quired would normally be set for some large number of samples. The result would be a test very close to the untruncated one. When a test is truncated the average number of samples required to terminate is reduced. How- ever the strength of the test ( o ,3 ) is reduced to (oc ,1). Complicated expressions relating c to c' and/3 to 13 have been derived, but they will not be repeated here. Data Application Data applications of the sequential theory are dif- ferent in several respects from the radar case. The most conspicuous difference is the use of a feedback channel for data. The receiver employs three decision zones as before: a zone of acceptance of Ho, a zone of acceptance of H1, and a zone of indifference. Here H represents the hypothesis that one of the binary data symbols was transmitted, while H1 represents the hypothesis that the other symbol was sent. The feedback channel is used whenever there is a reception falling in the zone of indifference. Whenever this situation occurs, the receiver notifies the transmitter over the feedback channel that it is not clear which symbol was sent. The transmitter repeats this symbol and the decision process starts over. The transmitter then either goes on to the next symbol or repeats the same symbol, depending on the decision at the receiver. With probability unity, the transmitter can eventually go on to the next symbol, although in specific instances, a long time may be required. When compared to the method of repetition a fixed number of times, it is seen that the sequential technique provides a savings which can be realized in increased data rate or reduced necessary signal power for a given probability of error. The receiver of the standard sequential system utilizes all past samples in deciding which signal is present and if another sample is required to give more accuracy. A modified form of this has received much emphasis in the literature. The technique referred to is the sequential test without memory, usually called null- zone detection. In null-zone detection, the same three choices of decision are available to the receiver. However, only the most recent sample is available on which to base the decision. For this situation there is a non-zero probability that the receiver would continue asking for a repeat of a particular signal, and thus never move on to the rest of the message. Consideration of truncation becomes very important here. Harris, Hauptschein, and Schwartzl0 have extensively studied the null-zone detection problem. Among other aspects of the problem they studied the manner in which the threshold levels should be adjusted in order to minimize communication costs. These costs depend on power, bandwidth, and transmission time. For non-truncation the uniform null (no adjustment) is optimum. For truncation, however, the levels should be adjusted for each sample so that the probability of the sample falling in the null zone is reduced over the previous sample. They show that the loss is small, however, if no adjustments are made. In these studies the feedback channel is usually considered error-free. This is reasonable since normally only a small amount of information compared to the channel capacity is transmitted over it. The probability of error can therefore be made very small. Boorstynll has also studied null-zone detection, concentrating his efforts on the case of extremely small average number of samples; i.e., between one and two. For the most part he deals with situations in which a repeat is required only a small fraction of the time, and only one repeat is allowed. He demonstrates that even for these small average sample sizes a significant improvement is obtained. Voice Systems With the above providing the background a description will now be given of the voice communication systems which have been devised from consideration of the sequential techniques. Just as there are differences between the radar and data sequential systems, so are the voice systems dif- ferent from them in several respects. Search of the literature shows no previous successful efforts along these lines. The problem is of a somewhat different nature from the radar or data problems. With speech there is not enough time to allow for many samples before making a decision, as in the radar case. But, instead, a decision must be made during every speech sample interval. Similarly, no provision can be made for a feed- back communications path, as in the data case, for the purpose of requesting a retransmission. (Of course, in duplex communications the listener can simply request that a word or sentence be repeated, but here a particular sample of the voice wave is being referred to in the preceding statement.) All of the discussion here will be concerned with systems of the quantized pulse position modulation type (QPPM); i.e., information regarding the speech amplitude at the sampling instant is denoted by transmitting the pulse in a particular one of the N discrete pulse position slots in the speech sample interval. The pulse will be detected incoherently. It is believed that techniques similar to those described here can also be applied to other types of modulation, but this will be left for future study. Two cases will be discussed. First, the interference will be considered to consist of a narrow-band Gaussian noise. A double threshold QPPM sequential system is des- cribed in Chapter II which is effective against this type of interference. Second, the interference is a narrow-band Gaussian noise and an impulsive noise. In Chapter III is described the four-threshold level QPPM sequential system which effectively combats this kind of interference. In each case, the decision zones are of the three usual types: signal present, signal absent, and the zone of indifference. The receiver must examine the N pulse position slots sequentially during each speech sample interval, beginning with the first slot and continuing through the Nt slot or through the termination slot if termination occurs before the end of the frame. During the time a slot is being examined, only one sample is available from which hypothesis H or hypothesis H1 is decided if possible. At this stage, the receiver is acting like the null-zone detector of the data case (the sequential test without memory). If hypothesis H1 is accepted for the slot under examination (thedecision is made that this slot contains the signal) the test is terminated until the next speech frame begins. Acceptance of H for a slot does not termi- nate the test. This is a difference from the previous cases discussed. The receiver has memory of those slots for which the sample falls into the zone of indifference. Use is made of this information for those frames in which no slot sample calls for acceptance of H1. That is, at the end of the NT slot, if no slot has been chosen as definitely having the signal, a choice will then be made from only those slots which were not rejected by choice of hypothesis H The test is therefore truncated after N slots. There are alternative criteria available for use in choosing the signal slot from the several "possible signal" slots. Use may be made of speech statistics for this choice, for example. Samples of the speech wave represent a Markov process. The probability density function for a sample is conditional upon the amplitudes of the preceding samples. Therefore, for example, based on the position of the pulse in the previous frame, that one of the "possible signal" slots could be chosen which represented the most probable slot. The mathematical model used for the speech marginal and conditional probability density functions is 12 based on the experimental results of Davenport.12 A more complete and detailed discussion of this model is given in. Appendix A. For the double threshold level situation, an alternate criterion is to make the choice from the "possible signal" slots on the basis of the one with the largest amplitude pulse. For the four threshold level situation, a possible criterion is to make the choice on the basis of the slot whose pulse amplitude is nearest to the range of amplitudes representing acceptance of H1. Other criteria are possible, but these are the ones considered in this study. For all of these there is a non-zero probability that all slots in a frame represent acceptance of H When this occurs the receiver chooses the center slot (the Nt) and proceeds to the next frame. Of primary interest is how these systems compare to conventional QPPM systems. The two standard QPPM methods are the single threshold, incoherent detection type, and the largest of N type, without a threshold. In the first, the receiver selects as the signal slot the first slot whose pulse amplitude exceeds the threshold. In the second, the receiver selects as the signal slot that one whose pulse amplitude is the largest. For comparison purposes there are many criteria possible, such as: (1) output signal-to-noise ratio, (2) bandwidth required, (3) average cost in the Bayes sense, (4) complexity, and (5) probability of correctly choosing the proper slot. For mathematical tractability, the latter one is used in this study. In addition to the comparisons reported here, para- metric studies of the systems are also given. For various input signal-to-noise ratios, the threshold settings are allowed to vary and the effect noted on the probability of correct choice. The difference between an adaptive and a non-adaptive system can also be seen. The sense in which the systems are adaptive is that the threshold settings are self-adjusting with changes in the noise level so as to be optimum. Other quantities of interest are the mean and the variance of the number of "possible signal" slots from which a choice is to be made if no "signal" slot is available. These have a bearing on the complexity of that part of the system which performs this task. As the thresholds are varied, the report describes the variation of these quantities with particular input signal-to-noise ratios. Also of interest is the mean of the number of slots before termination of the test. With this chapter providing the introduction, the remainder of this report will concern itself with more details of the system descriptions and with a complete discussion of the results. CHAPTER II CONVENTIONAL AND SEQUENTIAL SYSTEMS WITHOUT IMPULSE NOISE CONSIDERATION Introduction In this section is a discussion of four QPPM systems, with the interference taken to be narrow-band Gaussian noise. Two of the systems are conventional types while the other two have been devised from a consideration of sequential principles. It should be noted that there have been made several modifications of the sequential techniques as used in radar and data for these voice applications. The radar receiver can afford a very large number of samples before choosing a hypothesis. The voice receiver, however, must make a decision no later than the end of each speech sampling frame. Therefore, for voice we are dealing with a moderate number of samples and the maximum number allowed is fixed. Minimization of the average number of samples is not the primary consideration for voice. Instead of fixing the values of the two types of error, and thereby fixing the thresholds so as to minimize the average number of samples, we now vary the thresholds to obtain a minimum probability of error. Of course, a test with a truncation point does not have fixed thresholds for minimization of average samples. The thresholds instead should move closer together in some manner, not yet clear, as the test proceeds. Thus, fixing the errors for fixed thresholds is not appropriate. The data application allows a feedback channel for requesting retransmission of a questionable reception. For the same reason that the voice receiver must make a decision no later than the end of each frame, no feedback channel can be used with it. Another difference which will be made clear in the system descriptions is that the test terminates only with the acceptance of the signal hypothesis. Acceptance of the noise alone hypothesis does not stop the test. QPPM, Double Threshold, Speech Statistics A functional diagram of this system is given in Figure 1. The signal and narrow-band Gaussian noise are input to an IF filter and then envelope detected. The optimum decision thresholds will be shown later to be adaptive ones in the sense that they change with changing noise level. This system utilizes two thresholds, a and b, with the b larger. During each of the N pulse slots, the receiver is called upon to attempt a choice of hypothesis. If the envelope during the slot is greater than b, the signal hypothesis H1 is chosen. Thus the receiver has selected a Figure 1 Functional Diagram, QPPM, Double Threshold, Speech Statistics System DECISIONS ADAPTIVE IF ABOVE b CHOOSE H s F n) ENVELOPE THRESHOLDS IF BELOE a, CHOOSE H H0 DETECTOR a, b IF BETWEEN a AND b, _ a<-b CAN NOT DECIDE NOTE TIME OF BLANK STORE SIGNAL REMOVE FROM STORAGE HI OCCURENCE. FOR SLOT AS A THOSE SLOTS FOR THIS IS SIGNAL REMAINDER REFERENCE FOR WHICH SIGNAL WAS SLOT OF FRAME FOLLOWING FRAME QUESTIONABLE H THIS IS A NOISE AT END OF FRAME, IF SLOT. REMOVE FROM ALL DECISIONS WERE H0, CONSIDERATION AS CHOOSE CENTER SLOT POSSIBLE SIGNAL SLOT AS SIGNAL SLOT. AT END OF FRAME, IF NOT ABLE TO CHOOSE HI BEFORE, ? NOTE AND STORE THE CHOOSE THE ONE OF THE SLOT BEING CHECKED STORED SLOTS WHICH IS NEAREST TO THE PREVIOUS FRAME'S SIGNAL-SLOT slot as the signal slot. The remainder of that frame may now be blanked out and possibly used for the reception of non-real-time data, for example. The receiver stores this slot as a reference for the following frame to use if necessary. All previous slots for the frame which were in the'"possible signal" category are removed from storage at this time. If the envelope is below a, the noise alone hypothesis H is chosen. This does not halt the test for the frame, but o removes the tested slot from consideration as a "possible signal" slot. At the end of the frame, if all slots have Ho associated with them, the receiver selects the center slot as the desired one. Whenever the envelope is between a and b for a slot, the receiver stores this as a "possible signal" slot, and proceeds to the next slot. If some following slot has the signal hypothesis accepted, this information is removed from storage. If, however, at the end of the frame there has been no selection of a signal slot, this slot is compared with any other stored "possible signal" slots. By using speech statistics the slot most likely to be the signal slot is then chosen. This is the slot nearest to the position occupied by the signal of the preceding frame, as stored in the receiver. The speech probability expressions required are developed in Appendix A. The first is the a priori probability that the signal is in the jb slot. This is Ps (J) 0.3 3 e + e +o.i [erf 24 -)} -erM2) (2.1) where X z er (x) =- e du (2.2) The upper sign is to be used for 1 j N and the lower sign for 2 + 1 < j < N The second required expression is the probability that the speech sample's amplitude corresponds to the jt position, given that the previous position was thetmb. This is given by AJ C -e.:!L .-i-A J-e NL-+ e + ('--- 2 e7 7 ..-- ( =m ((2.3) The desired expression for the probability of correct decision during a frame is composed of three parts: Pl(cor) = P1 (correct, signal in "signal" zone) + P1 (correct, signal in "possible signal" zone) + P1 (correct, signal in "noise alone" zone) (2.4) The probability of a correct decision, given that the signal is in the jtb slot and that its envelope exceeds b, is the probability that the j-1 preceding slots with noise alone do not exceed b. The probability densities of the envelopes of narrow-band Gaussian noise alone and sine wave plus noise are Rayleigh and Modified Rayleigh, respectively, as.discussed in Chapter I. Therefore, P1 (correct, signal in "signal" zone) N ba J=l where 00 XZ 2+ Qo/3, )= x-. 1o (cx)-x (2.6) The second part of the expression is more complicated. Given that the signal envelope is in the "possible signal" region and that none of the "noise alone" envelopes exceed b, we can develop the probability of a correct decision with the reference slot being the jb. We have to consider only z/ and double the resulting expression, since there is an equal contribution for) > N due to the symmetry of the speech first probability density. Under these condi- tions, the probability of correct decision is the probability that none of the noise slots closer to the reference than the signal slot is, are in the "possible signal" zone. For the special case where a noise slot which is "possible signal" is the same distance from the reference as the signal slot is, the receiver chooses the one nearer to the center of the frame. The range of values for j, the signal slot, must be separated into three parts: jj 9 j + 1 4 j ~ 21 , and 21 + 1 j < N The result is Pi (correct, signal in "possible signal" zone) =Q( )- b" N-I the probability of correct decision is the probability that P1 (correct, signal in "noise alone" zone) (.- L -e / 1I For the case of signal in the "noise alone" region, the probability of correct decision is the probability that all of the noise envelopes are below a and that the signal slot is the center one. Therefore, P, (correct, signal in "noise alone" zone) S Q iif eLj"" (2.8) Combining the three contributing parts gives the final result )N bL .J- P,(o- ( ',)* P(r i [i' Ie 2 j=1 an N-Pr ) r r .2 2 r -a i Ps -7 TO-' S a 2. i. S ,-a I -)., e- - ff"*[*-Q )I- e (2.9) The expression derived above for the probability of a correct decision has considered only two frames at any one time. Higher order speech statistics were not available to allow more than the preceding frame to be used for reference purposes. This would seem to have only a small effect and therefore could be safely neglected. Furthermore, the fact that the reference position is taken to be correct each time is acceptable since, for the practical situation, the probability of error for a particular frame is small. Another quantity of some interest is the average number of slots tested before termination occurs. This is K = K- P(K) (2.10) K=I where P(K) is the probability of termination on the Kb slot. The three cases to be considered are: signal slot is the Ki,,signal slot is before the Kb, and signal slot is after the Ki. A termination occurs on the Kt slot for the first case if the signal envelope exceeds b and none of the envelopes of the preceding K-1 slots exceed b. For the second case, the signal envelope is below b, as well as the envelopes of the first K-2 noise slots. The Kt slot con- taining noise alone does exceed b. The last case requires the first K-l slots to have noise envelopes below b, with the Kb above b. Special consideration must be given to the termination on the last slot. This can occur from the envelope of the last slot, whether signal or noise, exceed- ing b and the envelopes of the other slots being below b. It also can occur due to the test truncation requirement when there are no envelopes exceeding b during the entire frame. Consideration of these facts gives the result zi bK -K-I A +. 62[ PQ ( 2, a, S j= bI N 30 j= K+( + e- - I ^ SI- Q ^ (2.11) Other quantities of interest are the mean and the variance of the number of slots with envelopes in the "possible signal" zone, when no slot's envelope exceeds b during the entire frame. These are given by N s= ZS. P(S) (2.12) 5=1 and a:= Ts -()T () (5) (2.13) 5=1 where P(S) is the probability that the number will be s, given that the test did not terminate due to an envelope being above b. The two cases to be considered are that the signal slot makes up one of the s, and that it does not make up one of the s slots. It is thus seen that (-' [, )].[er, -c I -P- e L sL--e Ql2 AVs S b s Ni-I-s (2.14) Therefore, a2- Z. - a b a2 N-I- -') a -7()-' e e-H.--et] - []"'1 S-S 3 N-1- S s)- -'. IJ N- t (2.15) [*- (, .)J [ i- e- J J and a2 b s-I N-S 1, )J. ( azAS- 1 L- N-s dsz b- Q^-s He'^ e'^ ]3 [ l- e 3"' zA -b . ((2.16) QPPM, Single Threshold, Conventional The functional diagram for this system is given in Figure 2. The output of the envelope detector is compared to a single threshold b in this case. The slot whose envelope is the first to exceed b is selected as the signal slot. That is, for each slot, either the signal hypothesis H1 or the noise hypothesis Ho is selected. As soon as a slot has H1 associated with it, the test is stopped and the remainder of the frame could be blanked out. Should Ho be selected for a slot, the receiver proceeds to examine the next one. As before, if all slots in the frame lead to choices of Ho, the receiver chooses the center slot as the signal position. Figure 2 Functional Diagram, QPPM, Single Threshold, Conventional System s(t)+n(t) ENVELOPE ADAPTIVE IF DETECTOR THRESHOLD b V DECISIONS IF ABOVE b, CHOOSE HI IF BELOW b, CHOOSE HO BLANK OUT REMAINING SLOTS IN FRAME ---e- HI -- H0 NOTE TIME OF OCCURENCE WITHIN FRAME OF THE SAMPLE. THIS IS SIGNAL SLOT H- H0 GO TO NEXT SLOT FOR REPEAT OF DECISION PROCESS IF LAST SLOT STILL LEADS TO Ho, CHOOSE AS SIGNAL SLOT THE CENTER ONE The probability of correct decision within the frame is then seen to be the probability that signal envelope is above b and all preceding noise envelopes are below b, or that all envelopes are below b and signal is the center slot. Therefore, 3 j=i + )** (2.17) QPPM, Double Threshold, Largest of s This system is shown in Figure 3. As can be seen, it is very similar to the Speech Statistics system of Figure 1. In fact, they are identical up to the point where a hypothesis has been chosen. When H1 is chosen for a slot, that slot is selected as the signal slot. The remainder of the frame is blanked, and all slots stored as "possible signal" slots are removed from storage. Note that it is not required that the signal slot be stored as a reference, because a new criterion is used for resolving the "possible signal" slots. Choice of H for a slot removes that slot from any consideration as signal, except when all slots are Ho. In this case the center slot is used as signal. One additional storage requirement when a slot's envelope is in Figure 3 Functional Diagram, QPPM, Double Threshold, Largest of s System s(t)+n() ENVELOPE DETECTOR ADAPTIVE THRESHOLDS a, b a_ b NOTE TIME OF BLANK OCCURENCE. FOR THIS IS SIGNAL REMAINDER SLOT OF FRAME NOTE AND STORE THE SLOT BEING CHECKED. STORE PULSE AMPLITUDE DURING THIS SLOT HI Ho DECISIONS IF ABOVE b, CHOOSE HI IF BELOW a, CHOOSE HO IF BETWEEN a AND b, CAN NOT DECIDE -~ H --*.** H 0 REMOVE FROM STORAGE THOSE SLOTS FOR WHICH SIGNAL WAS QUESTIONABLE AT END OF FRAME, IF ALL DECISIONS WERE H0, CHOOSE CENTER SLOT AS SIGNAL SLOT. AT END OF FRAME, IF NOT ABLE TO CHOOSE HI BEFORE, CHOOSE THE ONE OF THE STORED SLOTS WITH LARGEST AMPLITUDE ?i THIS IS A NOISE SLOT. REMOVE FROM CONSIDERATION AS POSSIBLE SIGNAL SLOT *-- the "possible signal" zone is that the envelope amplitude, as well as the slot position, is stored. At the end of the frame, if there has been no selection of H1, the receiver selects as the signal slot the one of the s stored whose envelope is largest. The probability of correct decision for this case is also made up of three parts. P (cor) = P3 (correct, signal in "signal" zone) + P3 (correct, signal in "possible signal" zone) + P3 (correct, signal in "noise alone" zone) (2.18) The first and last of these are identical to the first and last terms of Pl(cor). The second probability is the probability that j of the noise envelopes, in the "possible signal" zone, are less than the signal envelope, also in the "possible signal" region, while the remaining noise envelopes are in the "noise alone" region. That is, P3 (correct, signal in "possible signal" zone) R i a [-^~ j ( 2 1 9 ) J=O a. a o Now, ,C) e J d = I--- (2.20) o 6 K a -^ J *?}'d^ = -^ -"' aa R I~ca,i~ e 2 " By the binomial expansion ,e Z R j -e J - e<7~ J 2. a (II) It follows that P3 (correct, signal in "possible signal" zone) N-o a2 a R+A2 j=0 Cl e" IN-rJ i e-- Rs+A J *e- 'e e" '` O R If now we let -+-1'R be replaced by x, the integral becomes 2 A2. -2 A x 101 41 0 n j dOx V*- Az A .--- ( 0; e a~7r7a e A. e ; 0- Ia)I bHa,37 - and (2.21) (2.22) N-1 j=0 e )20 r.(-i) *g (2.23) (2.24) Rt -A ;rat e " (J} \nl f-7 OL &I YI) wnt+ C __x Ab i 4, The final expression is N bL 'j-. '(CID r) ( p5(j) [I given in equations 2.11, 2.15, and 2.16, respectively. j=0o t=o QPPM, Largest of N Z C-"' (2.25) ThThe receiver stores the N envelopes received during system frame. At the end of thare identity selects as thderived previougnal slot the given in equations 2.11, 2.15, and 2.16, respectively. QPPM, Largest of N This system is functionally described in Figure 4. The receiver stores the N envelopes received during a frame. At the end of the frame it selects as the signal slot the one whose envelope is largest. A correct decision is made by the receiver whenever all of the noise envelopes are below the signal envelope. This is represented by Figure 4 Functional Diagram, QPPM, Largest of N System s()+n(t) ENVELOPE __ STORE AMPLITUDE FOR W DETECTOR EACH OF N SLOTS CHOOSE AS THE SIGNAL SLOT THE LARGEST OF THE N P or) = 00 R SP, (R J- [<(F, J.". A a 2. 00 R +A CR ~ <^ - - o (-, AR R e N-I R li dK 00 2 2 = e " '0(7A) 11- '4-I e dt R1 R *A := .-.e" I. O co R (+i) +AZ o S R Z'~~~ 0 h l(AR R ) d/ K (AR~) (N K) KN- <=0o (2.26) K 2 .- 2 e2- dR Replacing Rif-K by x, the integral becomes oO .x .- A A 00 0 \(K I A +/----0% -4; * leC K+-1 (4 K / X 2 AL e .7*To -dX K Az K+l Thus, we have N-I P+ r)= -o. K=o K A2A (K+1/-2i (2.28) K(-) (K.) K 71 f- )O) (2.27) I. -r Discussion of Results The primary result of this chapter is depicted in Figure 5. This allows a direct comparison to be made for the systems discussed. The criterion of comparison is minimum probability of error for a given signal-to-noise input ratio. Thus, the curves in this figure are based on optimum threshold settings for the given noise conditions. It is interesting to note that the sequential system based on speech statistics is approximately 1.5 decibels better than the conventional, single threshold system, throughout the range of interest of signal-to-noise ratios. The sequential system based on a largest of s selection is about 1.0 decibel better than the conventional. Slightly better than these is the system based on selecting the largest amplitude out of N possible choices. This is a more complex system, however, and therefore has its advantage offset to some degree. It will be shown that the average value of s, with optimum thresholds, is about 1 or 2. Therefore, it would be relatively simple to select the most probable signal slot out of s under these conditions. Thus, the sequential systems offer a compromise between the simplicity of the conventional technique and the higher theoretical performance of the largest of N system. The differences noted above would appear to be insignificant under most circumstances. However, there are, foreseeably, FIGURE 5 Probability of Error for Input Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, Without Impulse Noise 100 P] (err), N = 16 --- P2(err), N = 16 --- P3(err), N = 16 --- P4(err), N = 16 0- P (err); N = 8, 0, P2(err); N =8,~ -s O0 P3(err); N =8, \\\ \\ \\ 10 - 10 \ \ 4 6 .8 10 12 14 A2, DECIBELS 2,-2 2o-n2 situations in which use of the sequential systems would be advantageous. The calculations for the curves of Figure 5 were based on a value of 16 for N. To obtain an indication of what changes might result for other values, single points were obtained for each system for values of 8 and 32 for N. The relative changes between systems in the probabilities of error are about the same amount. As a result, the conclusions reached above are unchanged. The results of a comprehensive parametric study for the speech statistics system are given in Figures 6, 7, 8, 9, and 10. These are for signal-to-noise ratios of 5.9, 9.0, 11.0, 12.1, and 13.0 decibels, respectively. Within the range of values shown, the effect on the probability of error is seen for any combination of threshold settings. The optimum thresholds can be obtained from these curves. As can be seen, the optimum thresholds are not particularly critical, in the sense that variation in the vicinity of the optimum point does not increase the probability of error by a large amount. The variation of the optimum thresholds with signal, to-noise ratio is given in Figure 11. For the practical range of interest K2 has the approximately constant value of 0.7, with K1 slightly greater. For a reasonably narrow range of signal-to-noise ratios, K1 and K2 could be fixed FIGURE 6 Probability of Error for Double Threshold, Speech Sta- tistics System for K2 Between 0 and 1.0, and for Signal- to Noise Ratio 5.9 Decibels SK1 =0.6__ 0.7 I- 0.9 1.0 A2 -- 5.9 db 2an2 I 1 0.4 K2 0.6 2 1 0.2 I 0.8 1.0 FIGURE 7 Probability of Error for Double Threshold, Speech Sta- tistics System for K2 Between 0 and 1.0, and for Signal- to-Noise Ratio 9.0 Decibels 0 0.2 0.4 0.6 0.8 FIGURE 8 Probability of Error for Double Threshold, Speech Sta- tistics System for K2 Between 0 and 1.0, and for Signal- to-Noise Ratio 11.0 Decibels 0.7 -A= 11 db 20\2 0.2 0 FIGURE 9 Probability of Error for Double Threshold, Speech Sta- tistics System for K2 Between 0 and 1.0, and for Signal- to-Noise Ratio 12.1 Decibels 0.2 0.4 0.6 0.8 FIGURE 10 Probability of Error for Double Threshold, Speech Sta- tistics System for K2 Between 0 and 1.0, and for Signal- to-Noise Ratio 13.0 Decibels 0.2 0.4 0.6 0.8 FIGURE 11 Values of K1 and K2 for Double Threshold, Speech Sta- tistics System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum Probability of Error 60 1.4 1.2 1.0- K I KI 0.8- K2 0.6- 0.4- 0.2- 0 4 6 8 10 12 14 A2 DECIBELS 2 n 2 beforehand. The degradation would be slight from the variable threshold situation. The other quantities of interest for the speech statistics system are C, s, and a-2 under optimum threshold conditions. This information is given in Figure 12. For a particular signal-to-noise ratio, the difference between the length of a frame and the value of K gives the number of slots which perhaps could be used in some other manner; for example, in the transmission of non-real time data. The value of 5, along with s2, gives an indication of the complexity of that part of the receiver which chooses the one of the s possible slots as the signal slot. As 5 gets larger, the complexity increases. As is seen in Figure 12, s is between 1 and 2 for the range of interest for the optimum settings, with small variance. The indicated complexity is not great. The parametric study of the conventional system is depicted in Figure 13. Here, of course, only one threshold is involved. The effect on probability of error is seen for a change in the threshold, for a wide range of signal- to-noise ratios. The variation of the optimum threshold with signal- to-noise ratio is given in Figure 14. It is very close to the variation of K1 shown in Figure 11 for the speech statistics system. FIGURE 12 K, s, and a2 for Double Threshold, Speech Statistics System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum Probability of Error 10- -2.0 8- -1.6 b 6 -1.2 4- -0.8 2- -0.4 0 I 0 i 0 4 6 8 10 12 14 A2 .A DECIBELS 2a,| FIGURE 13 Probability of Error for Single Threshold, Conventional System for K1 Between 0 and 1.0, and for Signal-to- Noise Ratios Between 5.9 and 16.0 Decibels 100 65 10 12 0\411.5K1 db0 12.1 db 10-2 \13 db/ 16 db 10-3 10---0.20.j 060.. 0 0.2 0.4 K1 0.6 0.8 1 0 FIGURE 14 Value of K1 for Single Threshold, Conventional System for Signal-to-Noise Ratios Between 4.0 and 14.0 Deci- bels, with Minimum Probability of Error 67 --I- I 4 6 8 10 12 14 A2, DECIBELS 2 2 n Similarly, a parametric study of the largest of s system was carried out, and the results plotted in Figures 15, 16, 17, and 18 for signal-to-noise ratios of 9.0, 11.0, 12.1, and 13.0 decibels, respectively. As before, the effect on probability of error of change in thresholds is easily seen. It is to be noted that the curves dip for values of 0 and about 0.7 for K2. It is to be expected that probability of error would decrease as the upper threshold increases and the lower threshold approaches zero, since the largest of s system approaches the largest of N system. However, the system performance is almost as good if the optimum threshold is taken as the one for the second dip in the curve. This allows the value of s to be small, with a corresponding decrease in complexity. The variation of these threshold settings is shown in Figure 19. Again, these could be fixed beforehand for a reasonably narrow range of signal-to-noise ratios, with only small degradation in error performance. For this system also, in Figure 20, the variations of K, s, and -2 are given. For the practical range of signal-to-noise ratios, the values are not significantly different from the speech statistics system. This completes the report in this chapter of the study of the two sequential systems, the conventional system, and the largest of N system, for the case of no FIGURE 15 Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to- Noise Ratio 9.0 Decibels 1.0 0 0.2 0.4 0.6 0.8 FIGURE 16 Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to- Noise Ratio 11.0 Decibels 100- 10-1 10-2, 10-3 L. 0 0.2 0.4 0.6 0.8 2 --= 11 db 2 2 1.0 FIGURE 17 Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to- Noise Ratio 12.1 Decibels 1.0 0 0.2 0.4 0.6 0.8 I FIGURE 18 Probability of Error for Double Threshold, Largest of s System for K2 Between 0 and 1.0, and for Signal-to- Noise Ratio 13.0 Decibels 76 00 K1 =0.3 1.0 0.4 / 0.9 1.0 ~ 0.8 0.9 0.5 9 0.8 co0.7 -2 0.6 ~ 0. 2 -13db 2 22 -3 I I I i I 0 0.2 0.4 0.6 0.8 1.0 Ko FIGURE 19 Values of K1 and K2 for Double Threshold, Largest of s System for Signal-to-Noise Ratios Between 4.0 and 14.O Decibels, with Minimum Probability of Error 4 6 8 10 12 2 DECIBELS 2o-n FIGURE 20 - 2 K, s, and s2s for Double Threshold, Largest of s System for Signal-to-Noise Ratios Between 4.0 and 14.0 Decibels, with Minimum Probability of Error 80 2 $2.4 2- 0 -2.0 8 -1.6 CN b 6 -1.2 S 4 -- 0.8 2- -0.4 -2 0- O ---s- ------ 4 6 8 10 12 14 A2 DECIBELS 2o-n2 impulse noise. Included have been fairly comprehensive studies of the error performance of each system under various conditions. The 1.5 decibel improvement over the conventional, for the speech statistics system, would appear to be small under most circumstances. However, there are possible situations where this would be justi- fiably enough. The sequential systems provide other choices, other than the conventional and largest of N. They apparently offer a compromise between the simplicity of the former and the error performance of the latter. CHAPTER III CONVENTIONAL AND SEQUENTIAL SYSTEMS WITH IMPULSE NOISE CONSIDERATION Introduction The QPPM voice systems discussed in this chapter are considered to be under the influence of narrow-band Gaussian noise and an impulse-like noise. The impulse noise consists of pulses of amplitude B, greater than the signal pulse amplitude A. It arises from sources external to the receiver. The noise pulse carrier frequency is the same as the signal carrier. Of the N-l noise slots in a frame, n will be considered to have impulses of the type described. The remainder have only narrow-band Gaussian noise. The probability distribution of n will be taken to be Poisson for purposes of this discussion. That is, -X ,X n=v o,i, z...= N-I (3.1) where X is the average number of impulses per frame. This is felt to be representative of certain special communication situations. Descriptions of the systems considered and the derivations of the expressions for the probability of correct decision are given in the following. QPPM, Four Threshold, Speech Statistics This system is functionally described in Figure 21. The similarities between Figure 1 and Figure 21 should be noted. In this case, the output of the envelope detector is compared to the adaptive thresholds a, b, c, and d. Herb, a is less than b, b less than c, and c less than d. If the envelope is between b and c, the receiver chooses the signal hypothesis H1. If the envelope is below a, or above d, the noise hypothesis H is selected. If between a and b, or c and d, neither hypothesis is chosen at this point. The remainder of the receiver processing is identical to the QPPM, Double Threshold, Speech Statistics system of Chapter II, shown in Figure 1. Note that the difference is the additional noise zone and the additional zone of indifference. The probability of correct decision is again a sum of three contributing terms. That is, P5(cor) = P5 (correct, signal in "signal" zone) + P5 (correct, signal in "possible signal" zone) + P5 (correct, signal in "noise alone" zone) (3.2) If the signal, in the ji slot, is in the "signal" zone, a correct decision will be made if none of the preceding j-1 noise slots has an envelope in the "signal" zone. Here we are interested in the number of noise impulses in the j-1 |