Technical Report TR-63-3-BF A Short Table of a Multiple Alternative Error Integral. Albert H. Nuttall U 1~ April 30, "

Transcription

1 Communication Sciences Laboratory Data Systems Division Systems, nc. SLitton 221 Crescent Street 54, assachusetts SWaltham Technical Report TR-63-3-BF A Short Table of a ultiple Alternative Error ntegral by Albert H. Nuttall U 1~ April 30, " This technical report covers work performed on the Acoustic Signal Processing Study with the Office of Naval Research on Contract N(. Nonr 3320(00). "Reproduction in whole or in part is permrtted for any purpose of the United States Government." BEST AVALABLE COPY

2 ABSTRACT n this report are tabulated the values of P(a, b) = S (x),-l (ax+b) dx fora= 1.0(0. 1)1.4, b= 0(0.25)5, and=2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 32, 64, 128, 256, 512, 4(x) is the normalized Gaussian probability density function and O(x) is the normal cumulative probability. The table was prepared by calculating P(a, b) with an accuracy of approximately , and rounding off to five places. Therefore an occasional error of one unit in the fifth place occurs. i! -ii-

3 TABLE OF CONTENTS Page No. 1. NTRODUCTON 2. SPECAL CASES 4 3. TABLE 5 -iii-

4 1 1. NTRODUCTON The probability P(a, b) 4(x) W *'1 (ax+b) dx arises in at least two different modes of multiple alternative communication. ý(x) is the normalized Gaussian probability density function, 0 (x) = (2nw) /2 exp(-x 2/2), and O(x) is the normal cumulative probability, x 0 *(X) = 5 (y) dy) n the first, one of equiprobable equal energy equi-correlated signals or the null signal is transmitted. At the receiver, the largest of the sampled output s of phase-coherent matched filters is compared with a threshold. For the null signal transmitted, the probability that the threshold is exceeded is called the false detection probability, PF' and is given by** P(x, r) = 1 -P t77, r where X is the (common) correlation coefficient of the signal set and r is a normalized threshold. A. H. Nuttall, "Error Probabilities for Equi-Correlated -ary Signals Under Phase-Coherent and Phase-ncoherent Reception", RE Trans. on nfo. Th., Vol. T-8, No. 4, pp ; July (See eq. (73).) Also, Technical Report TR-61-1-BF, Litton Systems, nc., Waltham, ass.; June 15, (See Section 5. ) [ ** A very small table (240 values) of P (x, rx) is available as Appendix D of TR-61--BF, through the use of eq. (5.-3A). The present table complements and greatly supplements the previous tabulation. --

5 * The second mode* of multiple alternative communication is phase- coherent, through (fast) Rayleigh fading. One of equiprobable equal energy S orthogonal signals is transmitted; at the receiver, the outputs of radiometer- type ** filters are sampled and a decision made in favor of the largest filter output. The probability of correct decision is approximately given by P(a, b) through appropriate identification of a and b. Sn general, the probability P(a. b) furnishes the answer to the following statistical problem (which is not tied to any particular application). Consider a set {xkl of independent Gaussian random variables with = m; xk = 0, k> 2 and Sa 2 2 2an (x 1 ) a 1 ;a (xk)=a2, k> 2. Then the probability that the variable x is larger than all the other variables is Pr (x 1 > x 2,... x) Xl = S dxl pl(xl)5.''' dx"'..dxp2(x2)...p2(x) F (xl 2' x 2 Wo exp2e dx a dx 2 p 2r21 A. H. Nuttall, "Linear Signal Processing Theory and easurements", Technical Report TR-63-2-BF, Litton Systems, nc., Waltham, ass; April 30, (See Section 5.) ** R. Price and P. E. Green, Jr., "Signal Processing in Radar Astronomy- Communication by Fluctuating ultipath edia", Technical Report No. 234, Lincoln Laboratory, ass. nst. of Tech.; October 6,

6 dx += )e- x + d2 [~.20 -/-, i Also, for completeness, we note that the probability that one of the other variables (not x 1 ) is largest is Pr(x2 > x, x3... x = Pr(x3 > x, x2, x4...,x 3 P 3 1,

7 2. SPECAL CASES Urbano. a = 1 SFor (1, b) reduces to an integral already tabulated by For = 2, P 2 (a, b) can be readily integrated to yield P2 (,b)=* b V'1i+a 2 1* and 4. The results are P 2 (a, 0)=- For b = 0, P(a, 0) may be integrated in closed form for a 2, 3, P 3 (a, 0)= -- sin 1! 1 1 1/ a 2 / SP 4 (a, 0)= 13 sin- n 2 (l a2) 4(2 ir SP(, 0) 1/ And for a = 1, b = 0, we have, for all, These special cases allow for numerous checks on the computations, which are tabulated in the following section. -4 R. H. Urbano, "Analysis and Tabulation of the -Positions Experiment ntegral and Related Error Function ntegrals", AFCRC-TR , Electronics Research Directorate, Air Force Cambridge Research Center, Cambridge, ass.; April, **Nuttall, "Error Probabilities, etc. ", TR-61--BF, loc. cit., eqs. (5. 36)-(5.48).

8 a1.0 b b

9 a= 1.0 b b ' S S S~-6-

10 it l a =1.0 b b

11 a =1.0 b a=1.1 b O (. -8-

12 a b b

13 a=1.1 b b "

14 a=1,1 j b b w

15 a =.2 b ' b

16 a = 1.2 b b

17 {a j a=l1.2 b S b S~-14-

18 a=1.2 b a1.3 h S S

19 a =.3 b b

20 a =1.3 b b ( ! S~-17-

21 a =.3 b b

22 a 1.4 b b