comparing Eq. 11.21 with Eq. 11.16, and taking into account the relation 
oO 
1 
= | exp[i(@1 + @2 + w3)tdt = 0(@; + w2 + @3)], 
—-o 
| E[dZ(w1) dZ(w2) dZ(w3)] 6(@1 + @2 + W3)dw3 = Syyy(W1, 2) 
0 dw \dw2. (11.23) 
This shows that E[dZ(@,) dZ(w2) dZ(w3)] must be 0 except along the line 
@,1+W 2 +W3=0. This corresponds to the fact that, for the linear case, 
Rxx(t) = | | expo +0"rexpior - E[dZ(w) - dZw"| 
—-@® —CO 
ao 
= | expio T - Sxx(w) dw, (11.24) 
and | Fraziw 220" y0w +w')dw' = sxx(w)d(w). 
By the same logic that because of stationarity, Ryx(t) should be independent of r, there- 
fore E[dZ(w) dZ(w’)] must be zero except when w +’ = 0, i.e., only along the line 
@+o'=0 andw’=-wo. 
Because of the symmetry of Ryyx(T1,T2), 
Rxxx(T1,T2) = Rxxx(T2, 71) = Rxxx- 11, T2 — 71) = Rxxx(t2 -71,—-T1) 
= Ryxx(—T2, 7] —T2) = Ryxx(T1 — T2,-T2) (11.25) 
the spectra also have symmetry, 
SXXX(W 1, 02) = Sxxx(W2, 01) = Sxxx(W1,— W1 — W2) = Sxxx(-@1 — 2,01) 
= Sxxx(W2,-—@1 —W2) = Sxxx(—@1 —@2, 2) (11.26) 
and SXxx(@1,@2) = S * (—@j,-—@2). (11.27) 
Accordingly, when we think of the bispectrum sxxx(@1, #2) we must consider the 
third frequency w3, besides w, and w2, that makes 
@1+@2+03=0, W3=—-@|-@2. 
From Eq. 11.26, the bispectrum is found”® to be the same for six permutations for two of 
the three frequencies w 1,2, and w3. Twelve bispectra, including the conjugate bispectra 
299 
