4) Spectrum and covariance functions
Related to the covariance function C(t) of a homogeneous R.F. there is another function defined by means of Fourier Transform theory.
In order to make it simple, we shall put a restriction to the class of functions where we shall investigate the property of “being a covariance function”, because in this class the theory of Fourier Transform is much simpler ; namely we shall assume that :
. (27)
Remark 7 : just to reason a little on the implication of our hypothesis (27), we can say the following.
Of course (27) does not imply that C(t) ® 0 when |t| ® ¥ , nevertheless we can say that whatever is e > 0 the set { |C(t)| > e } must have finite measure for, otherwise, the integral (27) would diverge.
Hence this set is rather ‘small’, with respect to the whole of Rn.
In practice we shall almost always consider functions C(t) that do tend to zero at infinity, implying that the underlying variables u(t+t,w) and u(t,w) do tend to become uncorrelated when | t | ® ¥.
Another technical consequence of (27) is that C(t) Î L2 too, in fact
(28)
Def. 7 : the spectrum of a homogeneous R.F. whose covariance satisfies (27) is the function
(29)
i.e. the Fourier transform of C(t)
Let us immediately note that, when t is in R1, R2, R3…. then the exponent in (29) is respectively
p×t = pt in R1
p×t = p1t1 + p2t2 in R2
p×t = p1t1 + p2t2 + p3t3 in R3
and so forth.
Some comments on Def.7 are in order.
First of all the integral (29) is well defined because of condition (27); in addition, by exploiting (27) and a famous theorem of Lebesgue, we see that
by passing to the limit under the integral in (29).
This means that we expect S(p) to be a continuous function.
In addition, due to a famous Perseval’s identity, we have
(30)
i.e. S(p) Î L2 too.
This implies that we can use the inverse Fourier Transform in (29), namely we have
(31)
Furthermore, since
ei2pp .t = cos 2pp .t+ i sin 2pp .t
and since C(t) is an even function, we find that
(32)
Formula (32) tells us that the spectrum is a real even function too,
(p) = S (p) , S(-p) = S(p) (33)
We are now in a condition to formulate the main result of this paragraph.
Theorem 2 (A.I.Khinchin): under the hypothesis (27), C(t) is a covariance function iff the spectrum S(p) is non negative
S(p) ³ 0, " p ÎRn (34)
Proof: the “only if” part of this theorem is more difficult to prove, and we ill not do it here.
So we want to prove only that if (34) is true, then C(t) is a covariance function.
By virtue of Lemma 1, the only thing to prove is that
(35)
Since we will need to use complex numbers {lj} , let us remark that when Cjk = C(tj-tk) is real and symmetric, condition (35) is perfectly equivalent to
(36)
In fact, first of all, the quadratic form (36) is real, otherwise the inequality in that equation would be meaningless.
In fact
Then, by putting lj = aj + i bj , we see that (36) is equivalent to
(37)
" {aj} {bj} .
It is clear then that (37) is the same as (36) and is equivalent to (35)
Now we can use the condition (36) with the representation (31), i.e. we compute
(38)
Equation (38) obviously entails that condition (34) implies to be positive definite.
The if part of Theorem 2 is thus proved.
Remark 8 : two consequences follow from Theorem 2, namely:
a) S(p) Î L1,
b) C(t) has to be continuous.
In fact from (31) we have
(39)
which, thanks to the positivity of S(P) implies a).
In addition since S(p) Î L1, (31) implies that
paralleling the analogous property of S(p).
In fact
(40)
and since
we can pass to the limit for h à 0 under the integral (40)
Summarizing we shall consider RF. 2nd order stationary with covariances and spectra enjoying the following properties:
a) C(t); 0 < C(0) < +¥
C(-t) = C(t)
| C(t)| £ C(0)
C(t) is positive definite
C(t)Î L1
C(t) is continuous
b) S(p); S(p) ³ 0
S(-p) = S(p)
| S(p)| £ constant
S(p)Î L1
S(p) is continuous
5) Homogeneous and isotropic R.F., their covariances and spectra
A rototranslation in Rn ( n > 1 ) is a transformation of the vectors tÎ Rn of the type
S = t + Ut (41)
where t Î Rn is the shift and U is a square matrix satisfying
U+U = I , (42)
and an auxiliary condition guaranteeing that there is no inversion of one axis.
Rototranslations, also called rigid motions, are characterised by the fact that distance between points is invariant, i.e.
|s2-s1| = |U(t2-t1)| = {[ U(t2-t1)]+[ U(t2-t1)]}1/2 ={(t2-t1) +U+U(t2-t1)}1/2 = |t2-t1| . (43)
Def. 8: a R.F. is said to be homogeneous and isotropic (HIRF) if all its distributions of any order are invariant under rototranslations, namely
(44)
holds true.
It is clear that such a definition does apply only to processes over R2, R3,….
As for Def.4 , (45) means that all the transformed fields v(t,w) = u (t+Ut,w) do have the same distribution.
The following Lemma 3 is an almost obvious consequence of Def.8.
Lemma 3 : if u(t,w) is HIRF, satisfying the hypotheses of Remark 8, then
(45)
and
(46)
From the fact that u(t,w) is homogeneous we already know that (45) is true and that, assuming for the sake of simplicity,
.
Now we know that v(t,w) = u(Ut,w) must have the same distribution as u(t,w) and then the same covariance function; but this means that
(47)
As (47) has to hold for every rotation U , we have that Cu(t) must be function of |t| only, since it is constant an spheres where |t| = const , i.e.
, (48)
which is the same as (46).
Def. 9 : a R.F. u(t,w) will be said to be weakly homogeneous and isotropic (with the same acronym HIRF) if its average and covariance function do satisfy (45) and (46).
Once again a weakly HIRF needs not to be a strong HIRF, but for the case of gaussian fields for which the properties of average and covariance are immediately translated into analogous properties of the distributions.
Now we ask ourselves whether it is possible to characterize the property HI by means of the spectrum. This is done in the next Lemma.
Lemma 4 : u(t,w) is a HIRF iff
Su(p) = S(|p|) . (49)
In addition, when the parameter space is Rn the relation between the two functions, defined on R+, C(r) , S(q) (r ³ 0, q ³ 0) is
(50)
where Jn(x) are Bessel functions of the first kind.
In fact by definition, recalling that Cu(t) = C(r) where r = |t| ,
(51)
Now let U be any rotation that brings the vector p into a vector aligned with the n-th axis, en ; so we have
Up = qen , q = |p| . (52)
Equation (52) implies (remember that U+U = I )
p = U+(qen) = q (U+en) ,
p· t = q(U+en) · t = q en · (U t) (53)
We substitute (53) in (51) to get
(54)
and we change variable by putting
Ut = h . (55)
We notice that
r = | t | = | h |
and that
dnt = dn h
because the volume element is invariant under rotations; hence (54) becomes
(56)
(r=|h|).
Equation (56) already proves (49).
Now we can put
h = r s , |s| = 1 , en· s = cos q (57)
and we can compute the integral (56) in spatial coordinates.
Let us recall that in spherical coordinates
dnh = rn-1dr dn-1s (58)
where dn-1s is the “area” element of the n-D sphere (i.e. the unit sphere in Rn).
Since we are strongly interested in the cases n = 2 and n = 3 let us state that
d1s = dq
d2s = sin q dq dl
as shown in the following Fig.3
Fig.3
Substituting (57) and (58) in (56), yields
(59)
Now (50) follows for n = 2 by recalling that by definition
(60)
so that, in this case,
K2(qr) = J0(qr) (61)
The inverse formula is easily proved by using the inverse Fourier transform.
As for the case n = 3 , we compute directly from the formula of the area element d2s,
(62)
This proves in this case the second of (50).
The first is proved again by observing that for even-real functions the Fourier transform and its anti-transform become perfectly symmetrical and as a matter of fact a pure cosine transform (cfr. (32) ).
Remark 9 : the J0(x) function, which plays a very important role for 2-D RF, is an even function quite similar to a cosine, with the main difference that it goes to zero when x ® ¥ according to the asymptotic law
(63)
An idea of the behaviour of J0(x) is given in Fig.4
Fig.4
A warning is in order at this point on the fact that our definition of J0(x) is a natural consequence of our normalization of the Fourier integral obtained by putting the factor 2p in the exponential.
In other classical text books where this is not done, a slightly different definition is used, namely
; (64)
a comparison with (60) shows that
Remark 10: it has to be strongly underlined that a function C(t) can be a covariance function in Rn but not a covariance function in Rn+1, while the converse is always true.
This can be understood by generalizing the following reasoning between R1 and R2 ; let {u(t,w)} be H.I. in R2 and consider a “section” of such a R.F. , i.e. the values of {u(t,w)} when t runs over a line.
Since a line in R2 has equation
t = a + e l ( (a , e ) Î R2, |e| = 1 , l Î R1 )
and since clearly
|t1 - t2| = | l 1- l 2| · |e| = | l 1- l 2|
we immediately see that the 1D covariance
C(| l 1- l 2|) = C ( |t1 - t2| )
can be derived from the 2D covariance.
This means that the 2D covariance function C (| t |) can also be considered as 1D covariance function.