On various definitions of the envelope of a random process

Journal of Sound and Vibration (1986) 105(3), 503-512

ON VARIOUS DEFINITIONS OF THE ENVELOPE OF A RANDOM R.

PROCESS

S. LANGLEY

College of Aeronautics, Cranfield Institute of Technology, Cranjield, Bedford MK43 OAL, England (Received 9 January 1985, and in revised form I May 1985)

Statistical properties of the envelope definitions of Rice [l], Crandall and Mark [2] and Dugundji [3] are derived and compared. It is shown that the definitions of Rice [l]

and Dugundji [3] are equivalent, which implies that the envelope of Rice [l] is independent of the choice of a central frequency. This contradicts results which have appeared in the literature [4,5] and the reason for this contradiction is explained. The envelopes of Crandall and Mark [2] and Dugundji [3] are found to have the same first order probability density function but different crossing rates and mean frequencies.

1. INTRODUCTION

An extremely useful concept in the theory of random vibrations is that of the envelope process u(t) associated with a random process x(t). Physically, if x(t) is reasonably narrow banded, the envelope process is a smooth curve joining the peaks of x(t) as shown in Figure 1. Associated with the envelope process is a phase process such that x(t) can be represented as a cosine curve having time varying amplitude (governed by the envelope process) and time varying frequency (governed by the phase process).

Figure 1. The envelope of a random process.

There exist a number of definitions for the envelope process, the three most notable of which are those due to Rice [l], Crandall and Mark [2] and Dugundji [3] (often attributed to Cramer and Leadbetter [6]). The Rice envelope [l] is based upon the expansion of the process x(t) about some central frequency o, say. This is often considered to be the classical definition of the envelope, and among other things it has been used in the past to derive the statistical properties of ocean waves [7] and second order wave forces [ 81. The envelope of Crandall and Mark [2] is an “energy envelope” and is defined in terms of x(t) and its time derivative i(t). This definition of the envelope is used in 503

0022-460X/86/060503 + 10$03.00/O

@ 1986Academic Press Inc. (London) Limited

504

R. S. LANGLEY

stochastic averaging techniques, which are used in conjunction with the Fokker-PlanckKolmogorov equation to determine the statistics of non-linear random vibration [9]. The envelope of Dugundji [3] is derived from the random process x(t) and its Hilbert transform x^(t). This definition of the envelope has been used in the first passage problem in random vibration, and the original definition has now been extended to include non-stationary random processes [ 10,111. In what follows here the similarities and differences between the different envelope definitions are discussed and various statistical properties are derived for each. It is shown that the envelope definitions of Rice [l] and Dugundji [3] are equivalent, concerning which there appears to be some confusion in the literature. A corollary of this is that the Rice envelope [l] is completely independent of the choice of the central frequency w,, which contradicts results which have appeared elsewhere [4,5]. The reason for the contradiction is that previously an error has been made in the derivation of the statistics of the Rice envelope [l], which is corrected here. Also, it is shown that although the envelope of Crandall and Mark [2] has the same density function as the other definitions, the envelope crossing rates and the statistics of the associated phase angle are different.

2. ENVELOPE DEFINITIONS FORMED FROM A COMPLEX PROCESS 2.1. ENVELOPES A random process x(t) can be written as the real part of a complex process z(t) having the form

z(t)=x(t)+iy(t),

(1)

where y(t) is some arbitrary random process. By using equation (l), x(t) can be expressed as a cosine curve having time varying amplitude and phase, as follows: x(t)=a(t)cos~(t),

d(t) = tan-’ (y/x).

a(t)=Iz(t)l=aq,

(2,3a, b) a(t) and 4(t) are known as the random envelope and phase processes associated with x(t). The random process y(t) must be chosen to give a(t) the required physical meaning, which is ideally that a(t) is a smooth curve joining the peaks of x(t), as shown in Figure 1. Suitable choices of y(t) can be determined by considering the case where x(t) is harmonic, say x(t) = A cos wt. In this case the required envelope is A, which implies that y(t) = *A sin or. This result for y(t) can be related functionally to x(t) in two ways, either as the time derivative of x(t), y = i/w, or as the Hilbert transform of x(t), y = 2. The Hilbert transform a(t) is defined as [12] d7

(4)

and it has the property of converting cos wt to sin wt and sin wt to -cos wt. Based on this argument, two definitions for the random envelope process are possible, being a,(t)=JX*+(A/W,)*,

at(t) = Jx*+

z*.

(536)

Equation (5) is the envelope definition of Crandall and Mark [2] while equation (6) is the envelope suggested by Dugundji [3]. The above approach offers a unified treatment of the two envelope definitions which were originally developed independently by using different arguments. In equation (5), o, represents some constant frequency, the best choice of which is yet to be determined. It can be seen that a,(t) is guaranteed to follow the peaks of x(t) since the envelope and the process coincide at a peak (i = 0). The

ENVELOPE

OF

A RANDOM

505

PROCESS

extent to which a*(t) follows the peaks of x(t) will depend upon how closely related a(t) is to a(t). For a stationary Gaussian process, this can be assessed by forming the joint density function ~(2, x) and thus the conditional density function p(x^/k) = ~(2, ~?)/p(i). It is easily shown that ~(a/_%) is Gaussian having a mean and variance given by [13]

J3~/4 = ((+z/~,W,

Var [a/x] = a:( 1 - p’),

(7a, b)

where a: = E[a’], 0: = E[i*] and p = E[&]/ (T,(T*.It can be shown [12] that E[f*] = mo, E[i*] = m, and E[fi] = -WI,, where m, is the nth spectral moment of the single sided spectrum of x(f), S,,(o): aJ m, = o”S,,(w) dw. (8) I0 Equation (7) then yields E[ft/l]

= -(m,/m,)i,

Var [i/a]

= moq2,

(9a, b)

where q is a parameter which measures the extent to which x(t) is narrow banded: q* = I-

m:/ mom*.

(10)

From equation (9) it can be seen that when 1= 0 the mean value of 2 is zero and the variance depends upon q. For a narrow banded process for which q is small it can be concluded that 2 will be approximately zero when 1= 0, from which equation (6) implies that a,(t) will follow the peaks of x(r). As x(t) becomes more broad banded it can be expected that a*(t) will bear less relationship to the peaks of x(t). Concerning the choice of the frequency w, appearing in equation (5) there are two obvious candidates, the mean frequency of x(t) given by w1 = m,/ m, and the mean zero crossing frequency given by o. = a. By referring to equation (5) it can be seen that E[a:( t)] = m,+ m,/wz. For w, = w. this would give E[a:( t)] = 2mo, which is in agreement with the case where x(t) is harmonic, in that the average value of the envelope squared (A*) is equal to twice the average value of the process squared (2 x ;A*). Another argument for using w, = w. is that the statistics of a,(t) are considerably more simple than for the case w, = w,. For these reasons o, = o. is used here, although it is noted that Nigam [ 131 has used w, = w, for a narrow banded process. A result which follows from using w, = w. is that a,(t) and u2( t) have the same mean squared values, which can be proved from equation (5) and (6). The statistics of a,(t) and a*(t) are discussed in sections 5 and 4 respectively. 2.2.

THE

ENVELOPE

OF

RICE

Rice [l] has presented a definition of an envelope process which is based on the expansion of a stationary random process x(t) about some central or carrier frequency w,. Firstly, x(t) is expressed as a Fourier series in the time interval t:0+ T,

x(t)=C{u,, cosw,t+b, sin WJ}, n

(11)

where w, = (2~n/ T). Since x(t) is a random process, each realization of x(t) on this time interval must be different. The coefficients a, and b, must therefore be random variables when viewed across a large number of realizations of x( t), or as more commonly stated, they must be random variables in the ensemble sense. If x(t) is Gaussian with zero mean then it can be shown that as T + CDa, and b, are independent zero mean Gaussian random variables whose mean squared values are related to the spectrum of

506

R. S. LANGLEY

[ll. In fact E[u;] = E[6:] = 27rS,(w,)/ T, where S,(w) is the single sided spectrum of x(t). If x(t) is non-Gaussian then the statistical distribution of a, and b, is much more complicated, although the representation of equation (11) is still valid. Equation (11) can be re-written as

x(t)

x(t)=C{a,cos[(w,-w,)t+w,t]+b,sin[(o,-w,)t+w,t]}, n

(12)

where o, is any fixed frequency, usually chosen as a frequency near to the peak of the spectrum S,(w). Rearranging equation (12) gives x(t) = ZE(t) cos w,t - Z,(t) sin o,?,

(13)

Zc(t)=C{a,cos(w,-o,)t+b,sin(w,-w,)t}, n

(14)

Z,(t)=C{a,sin(w,-w,)t-b,cos(w,-o,)t}. n

(15)

Equation (13) can now be written such that x(t) is represented as a cosine curve having time dependent amplitude and frequency: x(t)=a(t)cos[w,t+e(t)],

a2(t)=z~(t)+z;(t),

d(t)=tan-‘[Zs(t)/Zc(t)]. (16-18)

a(t) as given by equation (17) is Rice’s definition of the envelope of x(t). The above steps have followed closely the procedure used by Rice [l]. In order to compare this envelope with those appearing in the previous section, it is helpful to re-derive equation (17) in a slightly different way. Firstly, equation (11) can be written in the form x(r) = Re {r(t)),

z(t)=C(a,-ib,)e’“n’. n

(19a, b)

Now it follows from equation (19b) that Im {z(t)} = 1 {a, sin w,t - b, cos o,t} = Z(t), ”

(20)

and thus z(f)=x(f)+iXl(t).

(21)

Also, equations (12) and (13) can be written as x(t) = Re {z(t) e-‘“r’ e’“r’},

~(t)=Re{[Z,(t)+iZ,(t)]e’“~‘},

(22,23)

from which it follows that Z,(t)+iZ,(t)=z(t)eP~‘.

(24)

Combining equations (17), (21) and (24) then gives u’(t)=Z~(t)+Z~(t)=]z(t)e-iU~‘]2=x2(t)+~2(f).

(25)

Comparing equations (25) and (6) then shows that the envelope definitions of Rice [ll and Dugundji [3] are equivalent, from which it follows that the envelope of Rice must be independent of w,. Although this was in fact stated in the original paper of Dugundji [3] and is suggested by remarks made by Cramer and Leadbetter [6] there appears to be some confusion about this in the literature. Tayfun [14], in applying the Rice envelope to ocean waves, suggested that the random phase process is dependent on the choice of w,, although the fact that the envelope is independent of w, implies that the phase process must also be independent of w,. Lin [4] stated that the Rice envelope is highly dependent

ENVELOPE

OF

A RANDOM

PROCESS

507

upon w,. In a revised version of this work [5] he stated that the first order statistics p(a) are independent of the central frequency but that the second order statistics p(a, ci) are very sensitive to the central frequency which is used. The error which led to this conclusion is indicated in the following section.

3. THE STATISTICS OF THE ENVELOPE OF RICE In this section an expression is derived for the joint probability density function (jpdf) p( a, d, 8, 6) for the case when x(t) is Gaussian with zero mean, where a and fI are defined by equations (17) and (18). The method used is to start with an expression for p(Z,, Z,, Z,, is) =p(I) and to transform to the variables (a, ri, 0, 8). From equations (14) and (15) it can be concluded that if a, and b, are zero mean Gaussian random variables then I is a zero mean Gaussian vector process with jpdf p(I) = (1/4a2\S1”2) exp { -41TS11},

(26)

where S is a 4 x 4 correlation matrix given by S = EIIIT]. This matrix can be found by noting from equations (21) and (24) that Z,(t) = -x(t)

Z,(t) = x(t) cos w,r + sZ(t) sin o,t,

sin w,t + i(t) cos w,l. (27,28)

Further, Papoulis [ 121 has shown that E[x*] = E[$*] = m,,, E[b] = 0, E[ii] = -E[&] = -ml and E[1*] = E[i*] = m2. Using these results in equations (27) and (28) and their time derivatives leads to { m0

0

0

0

s=

0 M2

M2\

0 -M2 0

m.

M,

0

0 -M,

M,

9

(29)

I

where M2 = m, - corm0 and M, = m2 - 2u,m, + o;mo. Lyon [ 151performed this transformation for a process whose spectrum was symmetric about the chosen frequency w,, meaning that M2 = 0 and Zc, Is, Z, and Z, were all statistically independent. Later [4] the assumption of statistical independence was applied to a process having a general spectrum, meaning that an erroneous value of M,=O was used. This led to conclusions different from the ones reached below regarding the importance of the choice of w,. The transformation of variables is performed as follows [13], ~(a, d, 6 6) = Idet[S(Z,, Z,, Zc, Z,)lS(a, d, 6

&lb(I) = IJlpU),

(30)

where the partial derivatives involved in the above determinant can be found by noting from equations (17) and (18) that Zc(t) = a(t) cos e(t) and Z,(t) = a(t) sin 0(t). After some algebra it is found that I.ZI= a*. The expression for p(I) can be simplified by noting from equation (29) that the variables (I,, Z,) are uncorrelated from (I,, Z,) which means that p(I) =p(Z,, i,)p(Z,, i,). This avoids inverting the matrix S to evaluate p(I). The final result is then given by equation (30) with I written in terms of (a, ci, 0, 8). Performing the necessary algebra gives

p(a,d,e,f_Q=

‘*

4r2m0m2q

*exp{

-S[($)+(-&-)(b*+a*[i--z+wr)]},

(31)

R. S. LANGLEY

508

where q is the spectral parameter given by equation (10). Integrating over 8 and ti gives Aa, b) = (alGqm0) a result which is independent

exp

~-~~~~2/~~~+~~zlq2~~~1~,

(32)

of w,. integrating equation (31) over a and b gives

p(B, ~)=(1/4~qm,J~){(l/m,)+(l/q2m2)[~-(m,/m,)+0,]2}~3’2.

(33)

Integrating over 4 shows that 8 has a uniform distribution p( 0) = 1/(2~r) valid on a range of 2~. Integrating equation (33) over 0 gives p( 4) as 27r times the right-hand side of this equation, from which it can be shown that 4 has a mean value of (m,/ m,) - w,. Thus the phase process e(t) introduced in equations (16) to (18) is not independent of the central frequency w,. However, if equation (16) is re-written as x(t)=a(t)cos$(t),

(34)

where $(t) = w,t + e(t), then it follows from the above results that 4 has a mean value of (ml/m,) and a distribution given by p(4) = (I/2qm0Jm,){(I/m0)+

(I/q2m2)[lir -

~~l~012~-3’2,

(35)

and is thus independent of w,. 4 can be regarded physically as the “instantaneous” frequency of the random process x(t). In conclusion, the description of x(t) given by Rice [l] as equation (16), or equivalently equation (34), is completely independent of the choice of central frequency w,. Although the cosine term in equation (16) contains a term w,t this is adjusted for by the fact that e has a mean value of (ml/m,) - 0,. The true “carrier frequency” of the Rice envelope is then always w1 = (m,/mJ.

4. THE STATISTICS

OF THE ENVELOPE

OF DUGUNDJI

If x(t) is Gaussian with zero mean then x, 2, f and $ form a Gaussian vector process, x = (x, 2, i, $T, say, which has a jpdf given by p(x) = (1/4~*1R[l’~) exp { -ix’R-‘x}.

(36)

The correlation matrix R = E[xxT] can be shown from the results of Papoulis [12] to be

R=

(37)

The jpdf p(a, ci, 4, c$), where a(t) and (6(t) are the random envelope and phase processes which were introduced in equations (2) and (3) (with y(t) = x^(t)), can be found from the following transformation: p(a, d, d, d) = ldet [a( x, 4 i, X”)lW b, 4,

ddllPw = IJlPb4

(38)

Upon using x(t) = u(t) cos 4(t) and a(t) = u(t) sin (p(t) to evaluate the partial derivatives appearing in the above determinant, it can be shown that IJI = u2. Noting that p(x) = p(x, $p(i, 2) and performing the necessary algebra gives

p(a,~,+,d)=4r2mom2q lz22exP{

-~[($)+(-$-J(b2+a2[$-~~)]}.

(39)

Integrating equation (39) over (1,and 4 gives an identical result for p(u, ci) to that given by equation (32). The above derivation of this function offers an alternative approach to

ENVELOPE

OF A RANDOM

509

PROCESS

that used by Naess [16] who stated that no simple derivation has been found in the literature. Integrating equation (39) over a, ci and 4 gives p(d) to be the same as ~(4) of equation (35), demonstrating again the equivalence of the envelopes of Dugundji and Rice. From equation (32) the mean rate at which the envelope crosses a level a with positive slope can be calculated as [4]

J

cc

v,:=

ci) dd = -$$@==)exp[-i($J], &

+(a,

0

(40)

-

which can be shown to have a maximum when a = dmo which is given by (vz),,,

= (l/&)Jm,lmoq

e-1’2.

(41)

Now integrating equation (32) over ci shows that a(t) has a Rayleigh distribution with mean value m. From equation (40) it follows that the envelope crossing rate is a maximum at a level different from the mean, which implies that the envelope process itself is by nature broad banded. This agrees with results given by Rice [l] who calculated the peak distribution of the envelope for a special case and found that a significant number of peaks occur below the mean value of the envelope. It can also be noted that equation (40) agrees with an asymptotic result for large a given by Rice [l] for the mean rate at which peaks occur above the level a. The rate at which the random process x(t) crosses a level x is known to be

J

00

v:=

~$(x,i)di=~

’

&$xp

[

(42)

+(f)]

0

which has a maximum when x = 0, giving (v:),,, = (1/27r)JmJ mo. A measure of the rate at which the envelope varies as compared to the process x(t) is then (v&,,/(

v:),,,

=&q

e-l’* = 1.52q.

(43)

It follows that the envelope process will be slowly varying for narrow banded processes for which q is small. Given a value of q, equation (43) can be used to give a rough estimate of the applicability of the concept of a “slowly varying envelope” to a particular random process. 5. THE STATISTICS OF THE ENVELOPE OF CRANDALL AND MARK From equations (2) and (3) it can be seen that the following relations hold for the envelope definition of Crandall and Mark [2] (y = i-/oo) x(t)=a(r)cos4(t),

Differentiating

i(t)/w,

= a(t) sin 4(t).

(44,45)

these two equations gives jl/w, = d sin 4 + ai cos 4.

i=cicos+-adsind,

(46347)

From equations (44)-(47) it can be seen that, unlike the envelope definition of Dugundji [3], for this definition of the envelope a, ri, 4 and 4 are not independent variables. In particular, from equations (45) and (46) it follows that 4 = -wo+(ci/a)

cot 4.

(48)

By using equation (48), equations (44), (45) and (47) can now be written as x = a cos 4,

i/we=

a sin 4,

%/uo= ci(sin ++cot

4 cos 4)-aaoocos

4.

(49-51)

510

R. S. LANGLEY

statistics of the envelope of Crandall and Mark can now be determined by using equations (49)-(51) to transform the jpdf of the variables x = (x, 1/w,, ?/w,,)~ to the new variables (a, ri, 4). In the case where x(t) is Gaussian the variables (x, x!/w~, g/w,) have a joint Gaussian distribution given by The

p(x) = [1/(2~r)~‘~]P]“*] exp { -fxTP-lx},

(52)

where the correlation matrix P = E[xxT] can be calculated to be 0

ma P=

0

i-

m2/

-m2/00 0

m2/& 0

~0

.

mdd

(53)

i

As discussed in section 2 the frequency w,=$-T-m2 m, is used here. The transformation of variables is performed as follows: ~(a, d, 4) = ldet [6(x, x/ 00, Vwo)/8(a, Upon using equation, it by noting p(z?/w,)p(x,

b,

4)llPW = IJIm.

(54)

equations (49)-( 5 1) to calculate the partial derivatives appearing in the above can be shown that ]JI = a(cosec $1. The remaining algebra can be simplified that i/o0 is uncorrelated from x and jl/oo, meaning that p(x) = i/w,) and there is no need to invert the matrix P. The final result is 2

4

4)

=

(2T)3,2moaE

ew

-

k

t

II’

c-i2 cosec’ C#I

1K 1

alcosec ~$1 Aa,

2U2E2

(55)

where u2 = (m4mo)/ m2 and E is a second spectral width parameter given by ~~=l-rn~/rn~rn~.

(56)

Integrating equation (55) over ci gives

exp { - (a2/2mo)l,

~(a, 9) = (42rm0)

(57)

from which it can be concluded that a has a Rayleigh distribution with mean value m and 4 h as a uniform distribution valid over a range of 27~. This is in exact agreement with the envelope of Dugundji. The mean rate at which the envelope crosses a level a with positive slope is given by v;:=

cc J bp(

a,

JJ 2?r

ci)

dri =

0

0

m

bp(u, ci, 4) dd d&

(58)

0

By using equation (55) and performing the integration over ci first, equation (58) can be evaluated to give V: = [4/(2~)~‘~]Gs(u/&) This equation higher order ratio between Dugundji [3]

bears a striking resemblance spectral moments are involved the mean crossing rate of the can be seen to be independent

exp { -f(u’/m,)}.

(59)

to equation (40), with the difference that and the parameter E has replaced q. The envelopes of Crandall and Mark [2] and of the level u and given by

(YAxf/(~~)D = cu~)W(&/S)

=WdWd=?.

(60)

As an example, consider a narrow band random process having a spectrum of width 24 centred on a frequency w. Correct to order (A/w)‘, the moments of this spectrum are and m,=2Aw4[1+2(A/w)2]. Insertgiven by m. = 24, m, = 2A0, m2 =2A~~[l+~(A/o)~] ing these expressions into equations (10) and (56) gives e2=4/3(A/w)’ and

ENVELOPE

OF

A RANDOM

PROCESS

511

qL = 1/3(A/~)~, f rom which it follows that E = 2q. Equation (60) then predicts that the ratio of the two crossing rates is (4/m) = 1.27. Thus for this particular type of spectrum the crossing rates of the Crandall and.Mark envelope are around 25% higher than those of the Dugundji envelope. Certain random processes, such as the response of a linear system to white noise, have a theoretically infinite value of ma. In these cases equation (59) predicts that the envelope crossing rates are infinite. One way of avoiding this result could be to truncate the response spectrum at a frequency which is high enough to give a good estimate for m, and m2, but which yields a finite value of mq. Another way might be to replace the response spectrum by an “equivalent” band limited rectangular spectrum, as indicated by Lin [4]. It can be noted that, providing mo, m, and m, are finite, no such problem arises with the envelope of Dugundji [3]. The statistics of &Jfor the envelope of Crandall and Mark can be determined by firstly transforming p(a, d, 4) to the variables (a, &4):

Using equation (48) to evaluate the partial derivative gives the result a’( 4 + CIJ~)’ sec2 4 2U2E2

II. (62)

Integrating the above equation over a and 4 gives

It can be seen from the above that the mean value of $ is -wo, which means that the envelope of Crandall and Mark has a “carrier frequency” of w. = G, as opposed to the value W, = ml/m0 of the envelope of Dugundji. Equation (63) can be integrated to give the cumulative probability function of 4 in the form (64)

k(m), as given by equation (65b), is the complete elliptic integral of the first kind whose values are tabulated in standard mathematical tables [17].

6. CONCLUSIONS In this paper the statistical properties of a number of diff erent definitions of the envelope and phase processes associated with a random process x(t) have been considered. The main conclusions which can be made are as follows, where unless otherwise stated it is assumed that x(t) is stationary and Gaussian. (1) The definitions of Rice [l] and Dugundji [3] are equivalent, a corollary of which is that the Rice envelope is independent of the choice of a central frequency. Although this fact has been reported elsewhere [3] various results have appeared in the literature [4,5] which contradict this statement. The errors which led to this contradiction have been pointed out and corrected here. The proof of equivalence of the two envelopes rests on algebra alone and does not depend upon x(t) being Gaussian.

512

R. S. LANGLEY

(2) The extent to which the envelope of Dugundji [3] follows the peaks of x(t) can be assessed from equation (9), which gives the mean and variance of x^(t) when x(t) has a specified value (in particular i = 0). (3) The envelope definitions of Dugundji [3] and Crandall and Mark [2] lead to the same jpdf of the random envelope and phase processes, p(a, 4). (4) The mean or carrier frequency of the envelope of Dugundji [3] is o, = m,/m,, while that of the envelope of Crandall and Mark [2] is w. = G. (5) The mean crossing rates of the envelopes of Dugundji [3] and Crandall and Mark [2] are given by equations (40) and (59) respectively. In general it can be expected that the mean crossing rates of the latter will be the greater of the two. The extent to which it is possible to state that the envelope is “slowly varying” can be assessed from equation (43) which gives the ratio of the maximum mean crossing rate of the envelope of Dugundji [3] to the zero crossing rate of x(t).

REFERENCES 1. S. 0. RICE 1954 in Selected Papers on Noise and Stochastic Processes (Editor N. Wax). New York: Dover. Mathematical analysis of random noise. 2. S. H. CRANDALL and W. D. MARK 1963 Random Vibration in Mechanical Systems. New York: Academic Press. 3. J. DUGUNDJI 1958 IRE Transactions on Information Theory IT-4, 53-57. Envelopes and pre-envelopes of real wave forms. 4. Y. K. LIN 1967 Probabilistic 7’heory ofStructural Dynamics. London: McGraw-Hill. 5. Y. K. LIN 1976 Probabilistic Theory ofSfructuraI Dynamics. New York: Krieger, second edition. 6. H. CRAMER and M. R. LEADBE’ITER 1967 Stationary and Related Stochastic Processes. New York; John Wiley. 7. M. S. LONGUET-HIGGINS 1974 Journal of Geophysical Research 80, 2688-2694. On the joint distribution of the periods and amplitudes of sea waves. 8. R. S. LANGLEY 1984 Applied Ocean Research 6, 182-186. The statistics of second order wave forces. 9. J. B. ROBERTS 1976 Journal of the Engineering Mechanics Diuision American Society of Civil Engineers 102, 851-866. First passage probability for non-linear oscillators. 10. J. N. YANG 1972 Journal ofStructural Mechanics 1, 231-248. Nonstationary envelope process and first excursion probability. 11. S. KRENK, H. 0. MADSEN and P. H. MADSEN 1983 Journal ofEngineering Mechanics 109, 263-278. Stationary and transient response envelopes. 12. A. PAPOULIS 1984 Probability, Random Variables andSfochastic Processes. New York: McGrawHill, second edition. 13. N. C. NIGAM 1983 Introduction lo Random Vibrations. London: MIT Press. 14. M. A. TAYFUN 1984 Ocean Engineering 11, 245-264. Nonlinear effects of the distribution of amplitudes of sea waves. 15. R. H. LYON 1961 Journal of the Acoustical Society ofAmerica 33, 1395-1403. On the vibration statistics of a randomly excited hard spring oscillator II. 16. A. NAESS 1982 Applied Ocean Research 4, 181-187. Extreme value estimates based on the envelope concept. 17. M. ABRAMOWITZ and I. A. STEGUN (Editors) 1965 Handbook of Mathematical Functions. New York: Dover.