Since th e re is little , if an y , engineering in te re st in th e s tre n g th of s tru c tu ra l m em bers of m axim um size, no in v e stig a tio n s w ere m a d e of sections of m axim um dim ensions.
Fig. 1— M om ent diagram for T ype A southern pine and D ouglas fir crossarm s per Specification AT-7075:
G raph 1—Resisting m om ents of arm s of nom inal dimensions, straig h t grained an d free from knots. (Fiber stress 5000 psi)
G raph 2— Resisting m om ents of arm s of minimum dimensions, having m axim um slant grain (1" in 8"), and containing knots of the m axim um sizes p e rm itte d (viz., sizes shown a t bottom of arm sketch). (Fiber stress 3250 psi)
G raph 3— Bending m om ents from a load of 50 pounds a t each pin position. j
S ection m odulus calcu latio n s w ere m a d e of each sh a p e of m in im u m a n d no m in al size, b o th w ith a n d w ith o u t k n o ts. T e sts h a v e show n t h a t, b e
R E L A T I V E B E N D I N G S T R E N G T H OF CR OS S A R M S 25
cause of th e d isto rtio n of th e grain t h a t occurs a ro u n d them , k n o ts are fully as injurious to th e stre n g th of s tru c tu ra l tim b ers as k n o t ho les.1 Therefore, m dealing w ith sections con tain in g k n o ts, it w as assum ed for th e purposes of th is s tu d y th a t th e k n o t exten d ed across th e section in th e sam e m anner as a hole h a v in g a d ia m e te r eq u a l to th e d ia m eter of th e k n o t. I t was also assum ed t h a t th e k n o t w as located in, or reasonably close to, th e m ost d am ag in g p o sitio n in th e a rm section.
I n th e calcu latio n s of th e section m odulus of all roofed arm sections, it w as necessary first to co m p u te th e m om ents of in e rtia of th e whole or p a rts of th e to p segm ents of such sections (viz. n om inal a n d m inim um sections
betw een pinholes, a n d nom inal a n d m inim um pinhole sections). A ccord
ingly, fo u r such co m p u tatio n s were m ade an d th e resu lts used in calculating th e section m oduli of all th e roofed sections investigated. T h e details of th e fo u r co m p u tatio n s a re show n in th e A ppendix. T o insure u niform ity in th e resu lts, th e degree of precision used in these com putations was con
sid e ra b ly g re a te r th a n is o rdinarily em ployed in dealing w ith tim b er p ro d u cts. A ll of th e w ork, how ever, was done on a com puting m achine, an d it w as ju s t a b o u t as easy to ca rry th e operations to eight decim al places (which w as th e ca p a c ity of th e m achine used) as to a lesser num ber. As a m a tte r of in te re st in th is connection, it was found b y a c tu a l tria l in C o m p u tatio n I th a t a b su rd resu lts w ould occur if fewer th a n five decim al places were used.
F o r convenience, all of th e section m odulus calculations were m ade in ta b u la r form . I n such form th e procedure em ployed w ould n o t be readily
1 Pg 6 D ep t C ircular 295, U. S. D ept, of Agriculture, “Basic Grading Rules and W ork
ing Stresses for S tructural Tim bers,” by J. A. Newlin and R. P. A. Johnson.
a p p a re n t. T herefore, a sam ple calc u latio n follows show ing th e m e th o d of
ponent p a rts m ultiplied by the squares of the distances of th eir own centers of g rav ity from the n eutral axis of th e compound section. T he section m odulus (S) of th is section is found,
R E L A T I V E B E N D IN G S T R E N G T H OF CROSS A R M S
Ta b l e 3 .— Section M odulus of Bolt Hole Sections
R E L A T I V E B E N D I N G S T R E N G T H OF CROSS A R M S 29
W ith respect to th e k n o t p ositions considered, it is a p p a re n t from a n exam in a tio n of th e th re e curves (Fig. 3) th a t k n o ts u p to approxim ately l \ " in d ia m e te r are m o st dam ag in g w hen located im m ediately below th e roofed p o rtio n of th e a rm ; a n d t h a t th e w o rst position for k n o ts over in d iam e te r is a t th e b o tto m of th e arm . H ow ever, since u n d er usu al loading
Fig. 3— Sections betw een pinholes. Section m odulus of crossarm sections containing knots of th e sizes shown on the base line and located in the positions indicated. The data
apply to sections of minimum size ( 3 ^ " x 4 ^ " ) .
conditions k n o ts a t th e b o tto m of a n arm section are in com pression, and th u s w ould h av e less influence on stre n g th th a n th e y would h av e on th e ten sio n sid e,2 it w as felt th a t th e stre n g th value show n b y C urve 2 m a y be ignored; a n d t h a t th e values show n b y a sm ooth curve, com bining th e values
2 On Page 69 of U. S. D ept, of Agriculture Tech. Bui. 479, “ S trength and Related P roperties of Woods Grown in the U nited S tates” by L. J. M arkw ardt and T. R. C. W ilson, is the following statem ent: “K nots have approxim ately one-half as m uch effect on com pressive as on tensile stren g th .”
of C u rv e 3 u p to th e k n o t p o in t w ith th o se of C u rv e 1 for 2" a n d la rg e r k n o ts, w ould be th e p ra c tic a l m in im u m section m o d u li for roofed sections betw een pinholes. A ccordingly, such a sm o o th cu rv e w as c o n s tru c te d a n d is show n as C urve 2 in Fig. 4. T h e resu lts of C alcu latio n s 1 a n d 3 for no m
-0 1 2 3
K N O T D IA M E T E R -IN C H E S
Fig. 4— Sections between pinholes. Section m odulus of crossarm sections containing knots of the sizes shown on the base line and located in dam aging positions.
in a l a n d arm -en d m in im u m sections w ere also p lo tte d , a n d C urves 1 a n d 3 d ra w n for th o se sections.
Ro o f e d Pi n h o l e Se c t i o n s
Tw o calculations were m ade for th e pin h o le sectio n s: C a lc u latio n 4, in w hich th e k n o ts w ere assum ed to be lo c ate d a d ja c e n t to th e p in h o le in a
R E L A T I V E B E N D I N G S T R E N G T H OF C R O S S A R M S 31
vertical positio n ; an d C alculation 5, in w hich th e k n o ts were assum ed to be im m ed ia te ly below th e to p segm ent in a horizontal position. T h e results of these tw o calculations are show n in T ab le 2. I t h as heretofore been gen
e ra lly assum ed th a t in pinhole sections k n o ts less th a n in d ia m eter were m ore d am ag in g in a v e rtic a l p o sitio n th a n in a h o rizo n tal position. T he resu lts of C alcu latio n s 4 a n d 5, how ever, show th a t th e h o rizo n tal k n o ts im m ed ia te ly below th e to p segm ent are th e m ore dam aging. In ord er to com pare th e effect of k n o ts so located w ith th e effect of k n o ts a t th e extrem e
t o UJ Iu z to1 3D ÛO 2
ZO Hu u
t o
Fig. 5— Pinhole sections. Section modulus of crossarm sections containing knots of the sizes shown on the base line and located in damaging positions.
to p of th e section, th e following tw o com p u tatio n s assum ed 1" a n d 2" ho ri
z o n ta l k n o ts a t th e la tte r lo cation:
T ' {K n o t at Section Top:
A .0 2 8 7 5 ( 3 .0 9 3 7 5 ) ’ ! , 1
6
5 = 2 .9 6 3 1
2" K n o t at Section Top:
J
, 9 2 8 7 5 ( 2 .0 9 3 7 5 ^ =IP 6
5 = 1.3571
As th e section m odulus (S) valu es for sections co n tain in g 1" a n d 2" h o ri
R E L A T I V E B E N D I N G S T R E N G T H OF C R O S S A R M S 33 section m odulus associated w ith a sim ilarly located 1" knot would be 4.55.
T h e foregoing analysis for m inim um sections m ay be sum m arized as
(3) T h a t th e section m odulus associated w ith 1.875" to 2.25" k n o ts w ould be 3-2 5(L937^ l2 = 2.0334; a n d
6
(4) T h a t th e section m odulus valu es associated w ith 2§" a n d 3" k n o ts w ould be th e sam e as show n in th e C alculation 1 resu lts (T ab le 1).
Fig. 6— Brace bolt hole sections. Section m odulus of crossarm sections containing knots of the sizes shown on the base line and located in dam aging positions.
T h e resu lts of C alculation 6 (T able 3), a n d of th e foregoing analyses, to g e th e r w ith th e C alculation 1 resu lts for 2 \ " a n d 3" k n o ts, w ere p lo tte d in F ig. 6 for b o th m inim um a n d n o m in a l sections.
R E L A T I V E B E N D I N G S T R E N G T H OF CROSS A R M S 35
Re c t a n g u l a r Po l e Bo l t Ho l e Se c t i o n
T h e m ost d am aging position for k n o ts in th e pole b o lt hole section was assum ed to be a t th e to p of th e section. T h ey were so figured in
Calcula-0 1 2 3
K N O T D IA M E T E R -IN C H E S
Fig. 7— Pole bolt hole section. Section modulus of crossarm section containing knots of the sizes shown on the base line and located in damaging positions.
tio n 7, th e results of w hich are show n in T able 3 an d p lo tte d in Fig. 7 for b o th m inim um a n d nom inal arm s.
Re c t a n g u l a r Se c t i o n s w i t h o u t Bo l t Ho l e s
3 F or the purposes of specifying knot lim itations, crossarms under Specification AT-7075 are divided into a middle section (between brace bolt holes) and end sections (bevond brace bolt holes).
4 W here a brace bolt hole zone is less th a n four (4) inches from a pinhole zone these zones and the portion of the arm betw een them are considered as a single zone.
R E L A T I V E B E N D I N G S T R E N G T H OF C R O S S A R M S 37
follows, therefore, th a t th e average stre n g th of a n y lots of so u th e rn p in e o r
R E L A T I V E B E N D I N G S T R E N G T H OF CROS S A R M S 39
Fig. 8— Resisting moments and maximum bending moments for clear JW and W6
crossarms.
D IST A N C E FROM C E N T E R OF ARM - IN C H E S
include a new ty p e (“ W6” ) w ith 16 pin positions. I t was felt th a t, if th e ad d itio n a l pin holes in th e ty p e W6 did n o t u n d u ly w eaken th e arm , it could n o t only replace th e old ty p e “ J W ” arm w ith 8 pin positions b u t also be used in in stallatio n s w here g re a te r flexibility in wire spacings m ig h t be required.
<2 5 0
W6 ARM R E S IS T IN G MOMENTS
I n ord er to o b ta in a n e stim a te of th e stren g th relationship betw een th e two ty p e s, stre n g th te sts were m ade of 10 m atch ed arm s of each type. T h e te st arm s were m ade of air-seasoned, clear D ouglas fir. T h e dim ensions of the crossarm b lan k s were 3{" x 4 f " x 20'. I n selecting th e 10 blanks from w hich th e te s t arm s were m ade, only stra ig h t grained pieces free from
evidence of m a n u fa c tu rin g a n d o th e r defects w ere chosen. E a c h b la n k
R E L A T I V E B E N D I N G S T R E N G T H OF CROS S A R M S 41
2. I t is show n th a t th e critica l section of a crossarm is located a t th e pole pinholes. T h e p ra c tic a l v alue of th is observ atio n is th a t it em phasizes th e need for keeping th e pole pinhole sections an d th e p o rtio n of th e arm be
tw een th e m reasonably free from stre n g th reducing defects.
3. O nly b y b rea k in g te sts can th e actual bending stre n g th of crossarm s be determ in ed . T h e relative bending stren g th s, how ever, of tw o o r m ore arm s of different ty p e s or q u a lity m a y be estim a ted w ith sufficient accuracy b y m eans of th e m o m en t diagram , regardless of th e fiber stress used in its con stru ctio n .
4. I f th e fiber stress fac to r em ployed is dependable, th e m om ent d iag ram m a y b e used to e stim a te th e m inim um fac to r of safety th a t would o btain for a n arm of a n y ty p e or a n y assum ed q u ality . I n th is connection, it is believed th a t th e stre n g th of Bell S ystem crossarm s is well above th e m in i
m um required to su p p o rt th e loads o rd in arily carried.
5. T h e section m odulus curves of Figs. 4, 5, 6 an d 7 will sim plify th e con
stru c tio n of m o m en t diagram s for arm s of th e sam e sizes shown in th e figures b u t differing w ith respect to ty p e an d qu ality .
T h e uses listed lead to th e general conclusion th a t th e crossarm m om ent d iag ram is a co nvenient an d reasonably reliable engineering tool.
A P P E N D I X
Computation I . M oment of Inertia of Top Segment of M i n i m u m (3y&" x
4 £2") Section between Pinholes:
T h e m o m e n t of in e rtia ( I T ) of a segm ent (T) w ith respect to an axis th ro u g h its ce n te r of g ra v ity a n d p ara llel to its base m a y be found b y th e form ula
I T = I b b - A x2
w here I bb is th e m o m e n t of in e rtia of th e segm ent a b o u t th e axis B B , A th e area of th e segm ent a n d £ th e d istan c e betw een th e tw o axes. T h e values I B b , A a n d a; are given b y :
T h e significance of r a n d a in these form ulae, a n d of th e o th e r sym bols used in th e co m p u tatio n s th a t follow will b e clear from a glance a t F ig. 9.
D = 4.09375" Sin a - 1/2 b - 0.39843750
r
b = 3.1875" a = 23° 2 8 ' 49.93"
§ b = 1.59375" a = 0.40981266 ra d ia n s
r = 4" 2 a = 46° 5 7 ' 39.86"
r2 = 16.000000 Sin3 a = 0.063252925
(1 /2 b)2 = 2.540039 Sin 2 a = 0.73089017
p2 = 13.459961 Cos a = 0.91719548
p = 3.668782" Sin a Cos a = 0.36544507
d = p + (D - r) = 3.7625"
A = 0.7099 sq. ins. [Area of T b y F o rm u la (2)]
X = 3.8018" [By F o rm u la (3)]
g B x f p = 0.1330"
I bb = 10.2654 [By F o rm u la (1)]
A.x2 = 10.2601
I T = 0.0053
(N o te: W hile th e resu lts of th is a n d th e following co m p u ta tio n s are show n to four decim al places, th e a c tu a l w ork w as done b y m ach in e a n d c a rrie d to eig h t decim al places as m e n tio n e d in th e te x t.)
Since th e w idth of th e section in th is c o m p u ta tio n a n d th e ra d iu s of its roof is th e sam e as for th e m inim um 3 j^ " x 4" section a t th e end of th e a rm , th e to p segm ents of th e tw o are iden tical, a n d th e on ly v a lu e t h a t will differ
R E L A T I V E B E X D I N G S T R E N G T H OF C R O S S A R M S 43
will b e th e d ep th (d) of th e rec tan g u la r p o rtio n of th e section, w hich for th e sm aller w ill be p + (D — /•), or
3.6688 + (4 - 4) = 3.6688"
Computation I I . M oment of Inertia of Top Segment of N om inal ( 3 j" x 4y s") Section between Pinholes:
As th is c o m p u tatio n was m ade in exactly th e sam e m an n er as C o m p u ta
tio n I, only th e resu lts are here show n:
d = 3.8593"
g = 0.1317"
A = 0.7168 sq. ins I T = 0.0053
Computation I I I . M oment of Inertia of Top Segment of M i n i m u m (3ys" x
4 ^ " ) Pinhole Section:
I t will be n o te d in Fig. 10 th a t th e to p segm ent is divided into four p a rts : th e sm all segm ent (2\ ) a t th e to p of th e pinhole, th e rec tan g u la r p o rtio n
d, «
ft
" — ^ 1N © 1
9
D y D c,
d
\
I zR -r
\
R— b,
-Fig. 10—Crossarm pinhole section.
R h w ith a w id th of bx an d a d e p th of d h a n d tw o p o rtio n s d esignated Tc.
T h e pu rp o se of th is co m p u tatio n is to determ ine th e m om ent of in ertia of one of th e Tc p o rtio n s w ith respect to its g rav ity axis p arallel to its base.
T h e m om ent of in e rtia of th e tw o Tc p o rtio n s a b o u t th e axis B B m a y be
found b y d ed u c tin g th e m om ents of in e rtia of T i a n d R x a b o u t th is axis from
R E L A T I V E B E N D I N G S T R E N G T H OF CR OS S A R M S 45
As prev io u sly show n, 2I TcBb = 4.1751 2 Tc Z“ = 4.1738
2 I T c = 0.0013 I T c = 0.0007 D - r = 0.09375"
z = 3.7680"
3.8618"
d = 3.7625"
g for T c = 0.0993"
T h e resu lts of th is co m p u tatio n ap p ly also to th e m i n i m u m 3 x 4 " p in hole section a t th e ends of th e arm . T he d ep th (d) of th e rec tan g u la r p o r
tio n of th e end pinhole sections will be th e sam e as a t th e extrem e ends of th e arm , viz. 3.6688".
Computation I V . M oment of Inertia of T op Segment of N om inal ( J |" x 4y s") Pinhole Section:
Since th is co m p u tatio n was m ade in th e sam e m aim er as C o m p u tatio n I I I , only th e resu lts are h ere show n:
d = 3.8593"
g p 0.1019"
T c — 0.1630 sq. ins.
I T c = 0.0008
b y s . o . RICE {Concluded f rom J u ly 1944 issue)
P A R T I I I
S T A T IS T IC A L P R O P E R T IE S O F R A N D O M N O IS E C U R R E N T S 3 .0 In t r o d u c t i o n
I n th is section we use th e re p re se n ta tio n s of th e noise c u rre n ts giv en in section 2.8 to derive som e sta tis tic a l p ro p e rtie s of l{ t) . T h e first six sec
tions are concerned w ith th e p ro b a b ility d is trib u tio n of I ( t ) a n d of its zeros a n d m axim a. Sections 3.7 a n d 3.8 are concerned w ith th e s ta tis tic a l p ro p erties of th e envelope of I (t) . F lu c tu a tio n s of in te g rals in v o lv in g f i t ) are discussed in section 3.9. T h e p ro b a b ility d is trib u tio n of a sine w ave plus a noise c u rre n t is given in 3.10 a n d in 3.11 a n a lte r n a tiv e m e th o d of deriving th e resu lts of P a r t I I I is m en tio n ed . P rof. U h len b e ck h as p o in te d o u t th a t m uch of th e m a te ria l in th is P a r t is closely co n n ected w ith th e th e o ry of M arkoff processes. Also S. C h a n d ra se k h a r h as w ritte n a review of a class of p h y sical problem s w hich is re la te d , in a g en eral w ay , to th e p re se n t su b je c t.2"
3 .1 T h e D i s t r i b u t i o n o e t h e N o i s e C u r r e n t23
I n section 1.4 it h as been show n t h a t th e d is trib u tio n of a sh o t effect c u rre n t app ro ach es a n orm al law as th e expected n u m b e r of e v e n ts p e r second, v, increases w ith o u t lim it.
I n line w ith th e sp irit of this P a r t, P a r t I I I , we sh all use th e re p re s e n ta tio n
A'
I ( t ) = x (a n cos wn t + bn sin Cx)n t) (2.8-1)
71= 1
to show th a t I{t) is d is trib u te d according to a n o rm a l law . T h is is o b ta in e d a t once w hen th e p ro ced u re o u tlin ed in sectio n 2.8 is follow ed. Since a„
a n d bn are d is trib u te d no rm ally , so are an cos bcnt a n d bn sin cont w hen I is reg a rd e d as fixed. l i t ) is th u s th e su m of 2N in d e p e n d e n t n o rm al v a ria te s a n d con seq u en tly is itself d is trib u te d norm ally.
22 Stochastic Problem s in Physics and A stronom y, Rev. of Mod. P hys., Vol. 15, pp 1-89 (1943).
23 An interesting discussion of this subject by V. D. L andon an d K. A. N o rto n is given in the I.R .E . Proc., 30 (Sept. 1942) pp. 425-429.
46
M A T H E M A T I C A L A N A L Y S I S OF R A N D O M N O I S E 47
T h e average v alu e of I ( t ) as given b y (2.8-1) is zero since d n = b n — 0:
T(¡0 = 0 (3.1-1)
T h e m ean sq u a re v alu e of I( l) is
N ^_______________ ___
P ( t ) =
E («»
cos2u n t + b \ sin2 con i)7 1 = 1
N
E
m f r d A f (3.1-2)Jo
f
M f ) i f = m sI n w ritin g clown (3.1-2) we h av e m ade use of th e fa c t th a t all th e a’s a n d b’s are in d e p en d e n t a n d co nsequently th e average cf a n y cross p ro d u c t is zero.
W e h av e also m ade use of
a n = b l = w ( f n)Af, f n = nAf, u n = 2irf n
w hich were given in 2.8. f ( r ) is th e correlation fu n ctio n of 1(f) and is re la te d to w (f) by
ypr = f ( r ) = [ w { f ) cos 2-irfr d f (2.1—6) Jo
as is explained in section 2.1. In this p a r t we shall w rite th e arg u m e n t of ip(r) as a su b sc rip t in order to save space.
Since we know th a t I (t ) is norm al a n d since we also know t h a t its average is zero a n d its m ean square value is f o , we m ay w rite dow n its p ro b a b ility d en sity function a t once. T hus, the p ro b ab ility of I ( f ) being in th e range I , I + d l is
3:^: : : ' (3-‘- 3>
T h is is th e p ro b a b ility ol finding th e c u rre n t betw een I a n d I + d l a t a tim e selected a t random . A nother w ay of saying th e sam e thin g is to s ta te t h a t (3.1-3) is th e tra c tio n of tim e the c u rre n t spends in th e range 2 ,7 + d l.
I n m a n y cases it is m ore co nvenient to use the rep rese n tatio n (2.8-6) 7V
l i t ) = E °n cos ( u n t — <pn), c n = 2 w ( f n) A f (2.8-6)
in w hich ip i, • • • ipn are in d e p en d e n t ran d o m phase angles. In order to deduce th e norm al d istrib u tio n from this rep rese n tatio n we first observe
th a t (2.8-6) expresses I{t) as th e su m of a large n u m b e r of in d e p e n d e n t r a n dom v aria b les
X n = Cn COS ( 0)nt — <pn)
a n d hence th a t as N —> <» I ( t ) becom es d is trib u te d according to a n o rm a l law . I n order to m ake th e lim itin g process definite we first choose N a n d A / suctr th a t N A f = F w here
w here e is som e a rb itra rily chosen sm all p o sitiv e q u a n tity . W e now le t N —» oo a n d A / —■> 0 in such a w ay t h a t N A f rem a in s e q u a l to F. T h e n
B = I X i I3 + • • • + |i X N I3 = 2 ( 2 w ( f n ) L f ) 3' 2 I CO S (cO n t — <pn) |3 1
where th e b ars denote averages w ith resp e ct to th e <p’s, t being h e ld c o n s ta n t.
a n d con seq u en tly th e c e n tra l lim it theorem * m a y be u se d if w ( f ) = 0 for f > F . Since we m a y m ake F as large as we please b y choosing e sm all enough, we m a y cover as large a fre q u en c y ran g e as we wish. F o r th is reaso n we w rite oo in place of F .
N ow th a t th e c e n tra l lim it th e o re m h as to ld us th a t th e d is trib u tio n of I( t) , as given b y (2.8-6), ap p ro ach es a n o rm al law , th e re rem a in s o n ly th e p ro b lem of finding th e average a n d th e s ta n d a rd d e v ia tio n :
I ( t) = X i + X2 + • • • + Xn
A = x \ + x\ + • • • + x N = 2 w ( f n) A f cos2 («„ t — tpn)
( 3 . 1 - 4 )
N
< 4(A/ ) 1/2 i [ w ( f ) ] m d f Jo
If we assum e t h a t th e in te g rals a re p ro p er, th e ra tio B A 3/2 —> 0 as N —> oo}
I if) — y j Cn CO S (core t — i p f ) — 0
N
P {t) = Ç Cn COS2 ( u n t — <Pr) ( 3 . 1 - 5 )
* Section 2.10.
M A T H E M A T I C A L A N A L Y S I S OF R A N D O M N O I S E 49
T his gives th e p ro b a b ility d en sity (3.1-3). H ence th e tw o rep resen tatio n s lead to th e sam e re su lt in th is case. E v id e n tly , th e y will continue to lead to id en tical resu lts as long as th e c e n tra l lim it th eo rem m a y be used. In th e fu tu re use of th e re p re se n ta tio n (2.8-6) we shall m erely assum e th a t the c e n tra l lim it th eo rem m a y be applied to show t h a t a n o rm al d istrib u tio n is ap p ro ach ed . W e shall o m it the w ork corresponding to eq u a tio n s (3.1-4).
T h e c h a racteristic fu n ctio n for th e d istrib u tio n of I{t) is
W e require the tw o dim ensional d istrib u tio n in w hich the first variab le is th e noise c u rre n t I ( t ) a n d th e second v aria b le is its value I ( t -f- r) a t some
expect from th e analogy w ith section 3.1. T h e second m om ents of this d istrib u tio n are
T h e expression for M2 is in line w ith our definition (2.1-4) for th e correla
tio n fu n ctio n :
I n order to g e t th e d istrib u tio n from th e rep rese n tatio n (2.8-6) we w rite (3.1-6)
3 .2 Th e Dis t r i b u t io n o f I (t) a n d I (t + r)
la te r tim e r. I t tu rn s o u t th a t th is d istrib u tio n is n o rm al24, as we m ig h t
M2 — '/'0
M2 = + r)
(3.2-1)
=
xPT = iP(t) -- L im it ~ f + r ) dt ( 2 .1 - 4 ) Jo
N
T i I (/) ^ ^ Cn COS (cCn t (fn') 1
J2 = I ( t + r ) = cn cos (cov t — <pn con r) 1
24 I t seems th a t the first person to obtain this distribution in connection w ith noise was H. Thiede, Elec. Nachr. Tek. 13 (1936), 84-95.
F ro m th e c e n tra l lim it th e o re m for tw o dim ensions it follows th a t h a n d 12