E[X 1 X2X3X4] = My2M3q + M13M24 + M14M23, (11.69) 
E[XX2X3X4X5X6] = M12M34M56 + My2M35Ma6 + M1 2M 36M, 
+ M13M24Ms5¢6 + M13M25M36 + M13M26Mas 
+ M14M73M56 + M14M25M36 + M14M26M35 
+ M15M34M6 + M15M32Mae + M15sM36M 24 
+ M16M34M52 + Mj6M35Ma2 + M16M32Mas, (11.70) 
where m;; = E[X; X;]. These characters as shown in Eqs. 11.68, 11.69, and 11.70 were 
fully utilized in the calculations of the following together with the symmetry properties of 
hz (uy, U2) h3( qi, G2, 73) in terms of Uy, U2; 41, 92, 93. 
1. Mean of Y(t), my = E[Y(2)] 
my = | | ho(u4, U2) Ry(u2—u,) duyduz 
ho(uy, u2) | sxx(@) exp{- i(wu; + Wu2)} dw - dujduz 
—oO —c 
ll 
| 
3° 8 
_— 8 
| #@,-0) Sxx(w) dw. (11.71) 
Equation 11.71 is the same expression with Eq. 11.33’ for the quadratic process. 
2. Second order covariance function, cross spectrum 
Ryx(t) = El{¥@)— my} X¢-9)] 
= | hy(r)Rxx(t —r) dr 
+3 | | | h3(qi, 92,93) Rxx(q2— 91) Rxx(t— 93) dqidq2rdq3. (11.72) 
Therefore 
Syx(@) = Hy(@)sxx(w) + 3 | Hx@1-0,.0) Sxx(@1) Sxx(w) dw 
= Sxx(w) [H;(@) +3 | Hato,-01,0 Sxx(@ 1) dw]. (11.73) 
—-o 
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