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but the first term on the right-hand side is just a2 A </>(t). Therefore
Ë ô(t) = à2À ô(t) - 2Pa (83)
or, for À Ô 0,
m =g2(A x2Pla)m- (84)
The solution to (83) has four possible forms corresponding to
(i) A = 0;
(ii) 0 < A < —;
..... , 2 P
(in) A = —;
ã ÷ ^ 2?
(IV) Ë > — .
We can show that the integral equation cannot be satisfied for (i), (iii), and (iv). (Cf. Problem 3.4.1.)
For (ii) we may write
2 — a2(A — 2 Pja)
b =----------—^--------—> 0 < b2 < 00. (86)
ô(0 = Cleibt + c2e~jbt. (87)
Substituting (87) into (80) and performing the integration, we obtain
[r^-(a + jb)T rnP-(a-jb)T~l rr~-ia-jb)T (.a + jbW]
-l + —-----------— - e4^e --------Ë (88)
a + jb a — jb J I -a + jb —a — jb j
We can easily verify that if cx ô ±c2, (88) cannot be satisfied for all time. For Ci = —c2 we require that tan bT = — b\a. For cx = c2 we require tan bT = alb. Combining these two equations, we have
(tan bT + (tan bT - = 0. (89)
The values of b that satisfy (89) can be determined graphically as shown in Fig. 3.8. The upper set of intersections correspond to the second term in (89) and the lower set to the first term. The corresponding eigenvalues are
A|-7TV '-‘•2......................... <90>
Observe that we have ordered the solutions to (89), < b2 < b3 < • • •. From (90)
we see that this orders the eigenvalues Ax > A2 > A3 • • •. The odd-numbered solutions correspond to ñ ³ = c2 and therefore
1 è * nr ^ ^ T (j
Rational Spectra 189
Fig. 3.8 Graphical solution of transcendental equation.
The even-numbered solutions correspond to Ci = — c2 and therefore
Ô³(t) = ---Ã-----\ ,¦ sin bit, -T < t < T (/even). (92)
sin 2Ü³Ò\Óã ‘ 2biT )
We see that the eigenfunctions are cosines and sines whose frequencies are not harmonically related.
Several interesting observations may be made with respect to this example:
1. The eigenvalue corresponding to a particular eigenfunction is equal to the height of the power density spectrum at that frequency.
2. As T increases, bn decreases monotonically and therefore An increases monotonically.
3. As bT increases, the upper intersections occur at approximately (i — 1) tt/2 [/ odd] and the lower intersections occur at approximately
190 3.4 Homogeneous Integral Equations and Eigenfunctions
(i — 1) n/2 [/ even]. From (91) and (92) we see that the higher index eigenfunctions are approximately a set of periodic sines and cosines.
TVL j sin 2&,TV% °OS P IT *]’ T < t < T (i odd). \ 2btT J
sin 26,7V’sm [( IT *]’ ã - ã - r even>-I V 2btT )
This behavior is referred to as the asymptotic behavior.
The first observation is not true in general. In a later section (p. 204) we shall show that the An are always monotonically increasing functions of T. We shall also show that the asymptotic behavior seen in this example is typical of stationary processes.
Our discussion up to this point has dealt with a particular spectrum. We now return to the general case.
It is easy to generalize the technique to arbitrary rational spectra. First we write Sx(a>) as a ratio of two polynomials,
Looking at (83), we see that the differential equation does not depend explicitly on T. This independence is true whenever the spectrum has the form in (93). Therefore we would obtain the same differential equation if we started with the integral equation
A^(0 = Kx(t — u)<j>(u)du, —oo < t < oo. (94)
J — 00
By use of Fourier transforms a formal solution to this equation follows immediately:
ËÔÎ) = Sx(<o) Ô(» = ÔÎ) (95)
0 = [AD(a>2) - N(oj2)] Ô(». (96)