Copyright O 1β73 American Telephone and Telegraph Company T H I BELL STBTEM TECHNICAL JODBNAI.

Vol. 62. No. 7, September, 1973 Printed in U.S.A.

Distribution of Σ.α„/η, α„ Randomly Equal to ± 1

By S. 0 . R I C E (Manuscript received February 6, 1973)

When Ol, α», · · · are independent random variables, each equal to ±1 with probability i , the sum Σ Γ «η /η is a random variable whose distribu­tion is dißcult to determine theoretically. This sum is of interest in the study of intersymbol interference in digital communication systems. Here the distribution of the sum is computed by numerical integration and the results tabulated. Asymptotic expressions are given for the tails of the distribution.

I . I N T R O D U C T I O N

The distr ibution of the r andom variable

I = Σ an/n, (1) • - 1

where Oi, Oj, · · · are independen t random variables equa l to + 1 or — 1 with probabil i ty J, is of some interes t in t he s tudy of intersymbo l interference in a digital communication system . Fo r example , t he sum of two independen t expressions of the form (1) occurs when the pulse t ra in Σ - - o»s in (< — nr)/(t — nr), o« randomly equa l to ± 1 , is sampled a t regularly spaced instants which are slightly ou t of s tep with the zeros of sin /. The theory of r andom variables of t Jφe (1) (in par­t icular with a„/3" in place of o„ /n ) has been studied by a numbe r of investigators . A survey of the field has been made recently by Hill and Blanco." Here we evaluate the distr ibution of χ numerically and give expressions for its behavior when χ is large . Questions of continuity and convergence are pu t aside.

Since the distribution is even abou t χ = 0, only values for a; έ 0 need be considered. F rom the characteristic fimction

f(u) = avg [exp (ixu)2 = Π [cos ( « / n ) ] (2) n—1

1097

1098 ΤΗΒ BBLL SYSTEM TECHNICAL JOUBNAL, 8BPTEMBBB 1973

we get an expression for the probabil i ty density p(x) of x:

p{x) = i Γ cos (xu)f(u)du, (3)

T._ u / ^ \ 1 1 /•" sin (xiu) . J . . . Prob > = 2 ~ ί io — « / ( u ) d u . (4)

The values of p(x) and Prob (a; > Xi) shown in Table I were obtained b y evaluat ing these integrals b y the trapezoidal r u l e ' * which works well for (3) and (4).

T h e aaymptot ic expressions (8) and (18) for p(x) follow from a saddle point analysis of (3). Both p(x) and Prob (χ > xi) decrease rapidly when χ (or Xi) > 3, the decrease being dominated by the factor exp C—exp(x — A)2 where A = 1.39 · · ·.

T h e rapid decrease of p(x) is interest ing because the divergence of Σ, l / n might lead one to expect t h a t p(x) would decrease slowly as χ—>Ό. Instead, p{x) actually decreases much faster t han a Gaussian probabil i ty density. I a m indebted to a referee for the observation t h a t the second, fourth, and sixth moments of p(x) are, respectively, irV6, 11TV180, and 233TV7560.

T h e reader m a y wonder why as m a n y as six decimal places are given in Table I. There are several reasons. One is t h a t the cost was low. Abou t 3 seconds were required b y a Honeywell 6000 Processor t o compute the values shown in Table I, and abou t 40 terms were required in each trapezoidal sum. This il lustrates the fact (apparent ly not well known) t h a t when integrals like (3) and (4) are to be evaluated numerically, the trapezoidal rule often performs bet ter t han most of the o ther conventional quadra tu re methods (bet ter t han Simpson's rule, for ins tance) . ' T h e six-figure accuracy is also used to gain an idea of the values of χ for which the asymptot ic expansion (18) for p(x) begins to be valid. This degree of accuracy also shows t h a t p(0) is equal t o 0.249 994 · · · and no t to | , as might be inferred from a four-figure tabulat ion.

II. TRAPEZOIDAL BULE CALCULATION

Prehminary computat ions showed t h a t | / ( u ) | < 1 0 - " when u > 15. Fur thermore , i t was found t h a t (3) and (4) could be evaluated t o within the desired accuracy by using a trapezoidal-rule spacing of Au = h 0.4. I n line with these values the trapezoidal sum was t runcated a t the 40th te rm (15/0.4 « 40).

INTERSYMBOL INTERFERENCE 1099

T A B L E I — V A L U E S OF p ( i ) AND P rob ( i > xi)

X or X i P(x) Prob ( i > i i ) X or Xi P(x) Prob (ζ > Χι)

0.0 0.249 994 0.500 000 2.0 0.125 000 0.056 599 0.2 0.249 970 0.450 003 2.2 0.091 768 0.034 949 0.4 0.249 802 0.400 021 2.4 0.061 647 0.019 683 0.6 0.249 073 0.350 118 2.6 0.037 148 0.009 912 0.8 0.246 778 0.300 494 2.8 0.019 592 0.004 357

1.0 0.241 222 0.251 623 3.0 0.008 777 0.001 623 1.2 0.230 408 0.204 357 3.2 0.003 222 0.000 494 1.4 0.212 852 0.159 912 3.4 0.000 927 0.000 118 1.6 0.188 3S3 0.119 683 3.6 0.000 198 0.000 021 1.8 0.158 232 0.084 949 3.8 0.000 029 0.000 003

2.0 0.125 000 0.056 599 4.0 0.000 003 0.000 000

T h e infinite product (2) for / ( u ) was computed by using

/(«) = ( n' [cos ( u / n ) ] ) exp Γ Σ In cos (« /n) 1, (5) \n-l / Ln-AT J

where iV is a large number such t h a t u/N « 1 for all values of « used in the computat ion (0 ^ μ á 16). T h e produc t I l f ~* in (5) was computed by s traightforward multiplication . T h e sum in (5) was computed by sett ing g{n) = In [cos ( u /n ) J in t he Euler-Maclaurin sum formula :

g(n) = £ g m + \g{N) - ^ ] g'^^N)

Β (.2k)! gi2k-u^N) + R,. (6)

Here t h e B 's denote Bernoulli 's numbers , B j = 1/6, Bi =- 1/30, Bt = 1/42 [we stopped a t Ρ» in our use of (6) ] , 9<*'(iV) denotes the value of (d/cU)^g(t) a.t t = N, and the remainder A* is the integral of

times the BernouUi "polynomial" of degree 2fc + 1 and period 1 (see pages 520-540 of Ref. 4) .

T h e integral in (6) can be evaluated to within the desired accuracy by sett ing u/t = y, dt = — udy/y*, expanding In (cos y) in powers of y with the help of

—In (cos y) = j t a n vdv = j V* , 2v* , 17t;' 3 + Ί 5 + 315 + ·

1100 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1973

r e s ruIN / g(i)dt = u ir ' In (cos y)dy

JN JO

2ΛΓ ^ 36ΛΓ» ^ 225JV« ^ 17640ÍV» ^

Expressions for the higher derivatives of g{N) in (6) can be obtained b y differentiating the series in (7) with respec t to N.

I n using (6) we s topped a t A; = 3 and neglected Rt. Three separate trapezoidal-rule evaluations of p(x) and P rob (χ > xi)

were made using eqs. (3) to (7) with h = 0.4, ΛΓ = 201 ; A = 0.38, Ν = 201; and A = 0.36, Ν = 301, respectively. Here A is t h e spacing used in the trapezoidal-rule evaluations of (3) and (4). T h e three sets of computed values differed only in the 7 th or 8 th decimal places, i.e., all agreed with the values shown in the table. The values 201 and 301 of Λ'̂ are so large t h a t te rms beyond fc = 3 in (6) and those shown in (7) are no t needed. T o check the computat ions, t he integrals of p(,x) a m d x ' p ( i ) from χ = 0 to i = «> (the upper limit used was actually X = 5) were computed b y the trapezoidal rule with a spacing of Δχ = 0.1. T h e trapezoidal values agreed with the known values, respectively i and TV12, to within 6 significant figures or be t te r .

III. DISCUSSION OF TABLE I

Table I shows t h a t p(x) remains nearly equal t o p(0) = 0.249994 for 0 S X á 1, passes through p(2) = 0.125000 (is i t exactly i ? ) , and then decreases rapidly . T h e question as to whether ρ (2) is exactly J remains unanswered, b u t p(0) = 0.249994 does no t seem to be an erroneous calculation of J . For if pj(w) is the probabil i ty density of u = Σ ί " O n / n , t hen

p(x) = i p , ( x - 1) -f- i p , ( x -I- 1).

Sett ing X equal to 0 and 2 and combining the results give a result I owe to J . E . Mazo,

P(2) = ip (0 ) + ip , (3 ) > i p ( 0 ) .

Fur thermore , replacing ip»(3) b y p(4) — ip i (5 ) gives

P(2) = ip (0 ) + p(4) - ip , (5 )

0.125000 = 0.124997 + 0.000003 - i p , (5 )

which is satisfied by the tabula ted values when p2(5) is assumed to be negligibly small.

and integrat ing termwiae:

INTERSYMBOL INTERFERENCE 1101

00

exp £-xy + φ(ν)3, ψ(ν) = [cosh (2//n)]. (11) η - 1

When y is large we can show t h a t <p(y) = y In y-y +Ay+ ^1α2 + r(y),

A = 27 - In (T /4) = 1.39599 · · · , ^ '

where r(y) has roughly the same magni tude as exp (—2^). We first outline the derivation of the expression (12) for φ(ν), and

then apply (12) to obtain the asymptot ic expression (8) for p(x). The derivation of (12) is based upon the Euler-Maclaurin sum

formula (6) with .V = 1 and g{l) = In [cosh ( y / i ) ] . T h e integral in t h e sum formula is

j ' In [cosh iy/t)2dt

— y j 1Q (cosh v)dv

= — In (cosh y) + yQxi y) t anh y — y (In v) sech ' vdv

= ln2-y + y In y-yj^ (In w) sech ' vdv + Olye-^' In j / ] , (13)

IV. ASYMPTOTIC EXPRESSIONS FOR LARGE X

We shall show t h a t t h e rapid decrease of p(a.) for a: > 3 is described by

P(-r) ~ (2/o/ir)ie-'O, y, = exp [.r - 27 + In ( ir /4)] = exp [a; - 1.39599 · · · ] .

where 7 denotes Euler 's constant , 0.577215 • • · . In tegrat ing (8) gives P rob (χ > Χι) ~ erfc (j/i) ~ p(xi)/yo, (9)

where yo is computed from the second of equations (8) with a-i in place of .r. Bounds for the distribution involving exponential functions of

have been obtained by L. A. Shepp in unpublished work. T o obtain (8) we rewrite the integral (3) for p(.r) as

P W = ¿ / _ " e ' " / ( M ) d w . (10)

As is often the case for such integrals , the asymptot ic value of p(a;) is given by t h e contribution of a saddle point , Wo = iyo, lying far ou t on the positive imaginary u-axis (the pa th of integration being deformed so as t o pass through t he saddle point) . Fo r u = iy t h e in tegrand in (10) becomes

1102 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1973

j ' ( I n υ) sech ' vdv = - y + \n ( i r /4 ) (14)

which can be obtained by (i) replacing In ν in (14) by (it) differ­ent iat ing the known value (formula 3.527-3, page 352 of Ref. 5) of the resul t ing integral with respect to μ, and (Hi) set t ing μ = 1. T h e derivative of g(t) with respect to t, ff"'(Oi in the sum formula (6) is

g^'Kt) = - yt-* t a n h (y/t) = - yt-*{l - 2e-*y" + 2e-*»i' ) ,

where the exponential te rms become negligible when y becomes large and ί = 1. I n general, for Ζ = 0, 1, 2, · · · , ?("+'>(!) is equal t o — (21+ l)\y plus negligible terms. Therefore, the right side of the sum formula (6) is the integral plus

^(y-\n2)+^y + ^y+---+yRi (15)

plus te rms which are negligible when y is large. T h e sum of the co­efficients of y in (15) is known to be equal to y (page 529 of Ref. 4) . Hence (15) is equal to yy ~ i In 2. Addition of (13) and (15) and use of (14) gives the expression (12) for ^(y) .

Next we use the expression for <f>(,y), with the small term r(y) neglected, to obtain t h e asymptot ic form of p(x). T h e saddle point of interest occurs atuo = iya where yo is the zero of the derivative of the exponent in (11). T h e exponent is —xy + ^»(y), and yo is t he zero of

-X + ip'{y) =- X + \xiy ->r A.

T h u s 2/0 = exp {x — A). This y o is t he same as the y o appearing in the asymptot ic expression for p{x) s ta ted in (8). The exponent itself has the value —xyo + ¥>(i/o) = — j /o + i In 2 a t j / o . B y making use of <p"{y) = 1/y and tUe higher derivatives of φί^), the exponent can be expanded in a Taylor series about j /o . F rom this expansion i t follows t h a t near Wo the integrand in the integral (10) for p{x) can b e writ ten as

exp -2'" + 2 ^ 2 + ί/οΣ^(5^^ (16)

where ζ = u - Uo . Sett ing (16) in (10), changing the variable of integration from ω to τ = z / j /o = (w - MO)/2/O, and assuming t h a t p{x) is given asymptotically (as χ and j /o tend to «>) b y the contribution of

where we have integrated by par t s twice. T h e last integral in (13) has the value

INTERSY.MBOL INTERFERENCE 1103

p(x) ~ (2r)~'2lyoe~'™y exp dr. (17)

Here the nominal pa th of integration is the real τ-axis. The classical saddle point asymptot ic expansion obtained from (17) is

P(a;) ~ (yo/rye-<"> 1 , J 2 3 _ ^ 24j/„ 1152i/g ^ (18)

The coefficients of the powers of l/yo in the series can be determined by a general procedure described in Appendix D, page 1999, of Ref. 6.

T h e asymptot ic expression s ta ted in (8) is the leading term in (18). An idea of the accuracy of the asymptot ic expressions can be obtained by considering the case χ = 3. For i = 3, j/o is 4.973 and Table I gives the "exac t" value p(3) = 0.008777. The asymptot ic value of p(3) obtained from (8) is 0.008710, the first tΛvo terms in (18) give 0.008783, and the first three terms give 0.008776. Table I gives the "exac t " value Prob (χ > 3) = 0.001623 and eq. (9), namely Prob (.τ > 3) ~e r f c (i/i) = erfc (2.230), gives 0.001612.

REFERENCES

1. Hill, F. S., Jr., and Blanco, M. Α., "llandom Geometric Series and Intersymbol Interference," accepted for publication in IEEE Trans. Information Theory.

2. Rice, S. O., "Efficient Evaluation of Integrals of Analytic Functions by the Trapezoidal Rule," B.S.T.J., 5S, No. 5 (May-June 1973), pp. 707-722.

3. Kendall, D. G., "A Summation Formula .associated With Finite Trigonometric Integrals," (juart. J. Math. (Oxford Ser.), 13 (1942), pp. 172-184.

4. Knopp, Κ., Theory and Application of Infinite Series, London: Blackie and Son, 1928.

5. Gradshteyn, I. S., and Ryzhik, I. W., Tables of Integrals, Series, and Products, New York and London: Academic Press, 1965.

6. Rice, S. O., "Uniform Asymptotic Expansions for Saddle Point Integrals— Application to a Probability Distribution Occurring in Noise Theory," B.S.T.J., 47, No. 9 (November 1968), pp. 1971-2013.

the saddle point a t uo give

(ir)"