4. Third moment cross covariance and cross bispectrum 
ioe) oe) 
Ryxx(T1, 72) = 2 | | h2(uj, U2) Ryx(t1— 1) Rxx{t2—u2) dujdu2, (11.76) 
therefore SYXX(W 1,2) = 2H2(W1,W2) Sxx(@1) Sxx(W2). (11.77) 
If we use X7(r) instead of Y(t), from Eqs. 11.56 and 11.57 we get 
Ryoyx(T, T2) = 2RxxT1) Rxx(T2) (11.78) 
Sxy2xx(W1,@2) = 2sxx(@1) Sxx(@2) (11.79) 
as we did in the preceding section for Eq. 11.50, inverting the variables from w ,w> into 
W1—-@2,0,+@2, 
2 Syxx (W@1 —@2,@1 + W2) = Syxx(@1, 2). (11.80) 
Therefore, the quadratic frequency response function H> (@; 2) is obtained by 
SYXX(@1,@2)  __S¥xx(@1,2) (11.81) 
25xx(W 1)5xx(@2) — Sx2xx(W1, @2) 
_ SYXX(@1—@2,@1+@2) _ S¥xx(@1—@2,@1 + @2) (11.81’) 
Sxx(@1) Sxx{W2) Sx2xx{@1 —@2,@1 + W2) 
These are the same as Eqs. 11.49, 11.55, 11.58, and 11.59 for the quadratic process and it 
means that the process to calculate the quadratic response H2 (w 1,2) is the same as for 
the quadratic nonlinear process, even for this third order nonlinear process. 
Hx(@1,@2) = 
5. Fourth moment cross correlation and trispectrum 
When Y(t) is expressed by Eq. 11.67, which includes the third order nonlinear for 
the cubic nonlinear process term, the fourth moment cross correlation can be obtained by 
the same kind of manipulation as in the quadratic nonlinear process as 
Ryxxx(t, 72,73) = E[{¥(t)—my} X(t-11) X(t—t) X(t-73)] 
26 | 
Its three—dimensional Fourier transform is called the trispectrum and is 
SYXXx(W 1,2, 03) = 6 H3(@1,W2,@3) Sxx(@1) Sxx(@2) Sxx(w3). —(11.83) 
If we use X°(z) instead of Y(t), then 
Rysyyx(T1, 72,73) = ORxx(T )Rxx(T2)Rxx(Ts) (11.84) 
Sy3 yyy(W1, 2,03) = OSxx(W1)Sxx(W2)Sxx(W3). (11.85) 
_— 8 
_— 8 
h3(q41,92,93,) Rxx(t— 41) Rxx(t2— 92) Rxx(t3 — q3) 
dqidq2zdq3. (11.82) 
8 
8 
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