Fourier transform of each segment for all three processes and then estimates the cross-bispectrum by taking the average of triple products of Fourier coefficients over M segments, that is ∑x(o)xo)x"on+2) (1157) x12 3 where X (o) is the Fourier transform of the mth segment of x(n)l, and*indicates the complex conjugate. The indirect method computes the third-order cross-cumulant sequence for all segments Cm(=∑增小)(m+A)r(+ (1158) where t is the admissible set for argument n. The cross-cumulant sequences of all segments will be averaged to give a resultant estimate k,1) (1159) xrx The cross-bispectrum is then estimated by taking the Fourier transform of the third-order cross-cumulant (,1) (115.10) Since the variance of the estimated cross-bispectrum is inversely proportional to the length of each segment omputation of the cross-bispectrum for processes of finite data length requires careful consideration of both the length of individual segments and the total number of segments to be used. The cross-bispectrum can be applied to determine the level of cross-QPC occurring between x,(n) and Ix2(n)) and its effects on x(n). For example, a peak at Bx, x2*(O1,o2) suggests that the energy component O2 of [,(n) is generated due to the QPC betw of x(n)). In theory, the absence of QPC will generate a flat cross-bispectrum. However, due to the finite data length encountered in practice, peaks may appear in the cross-bispectrum at locations where there is no significant cross-QPC. To avoid improper interpretation, the cross-bicoherence index, which indicates the significance level of cross-QPC, can be computed as follows: (115.11) P(,P(OP(o, +02 where Pr (o) is the power spectrum of process x(n)). The theoretical value of the bicoherence index ranges In situations where the interest is the presence of QPC and its effects on ix( n)l, the cross-bispectrul equations can be modified by replacing Ix(n)) and x(n)) with (x(n)) and (x,(n)I with n(n)b, that is, Bn(o,0)=∑xo)y"(o) X"(o1+0 (115.12) e 2000 by CRC Press LLC© 2000 by CRC Press LLC Fourier transform of each segment for all three processes and then estimates the cross-bispectrum by taking the average of triple products of Fourier coefficients over M segments, that is, (115.7) where Xj m(w) is the Fourier transform of the mth segment of {xj (n)}, and * indicates the complex conjugate. The indirect method computes the third-order cross-cumulant sequence for all segments: (115.8) where t is the admissible set for argument n. The cross-cumulant sequences of all segments will be averaged to give a resultant estimate: (115.9) The cross-bispectrum is then estimated by taking the Fourier transform of the third-order cross-cumulant sequence: (115.10) Since the variance of the estimated cross-bispectrum is inversely proportional to the length of each segment, computation of the cross-bispectrum for processes of finite data length requires careful consideration of both the length of individual segments and the total number of segments to be used. The cross-bispectrum can be applied to determine the level of cross-QPC occurring between {x1(n)} and {x2(n)} and its effects on {x3(n)}. For example, a peak at Bx1x2x3(w1, w2) suggests that the energy component at frequency w1 + w2 of {x3(n)} is generated due to the QPC between frequency w1 of {x1(n)} and frequency w2 of {x2(n)}. In theory, the absence of QPC will generate a flat cross-bispectrum. However, due to the finite data length encountered in practice, peaks may appear in the cross-bispectrum at locations where there is no significant cross-QPC. To avoid improper interpretation, the cross-bicoherence index, which indicates the significance level of cross-QPC, can be computed as follows: (115.11) where Pxj(w) is the power spectrum of process {xj (n)}. The theoretical value of the bicoherence index ranges between 0 and 1, i.e., from nonsignificant to highly significant. In situations where the interest is the presence of QPC and its effects on {x(n)}, the cross-bispectrum equations can be modified by replacing {x1(n)} and {x3(n)} with {x(n)} and {x2(n)} with {y(n)}, that is, (115.12) B M xxx XXX m m M m m 123 12 1 1 12 23 1 2 1 ww w w w w , * ( ) = ( ) ( ) ( + ) = Â C k l x nx n kx n l xxx m m n m m 123 12 3 ( , ) = Â ( ) ( + ) ( + ) et C kl M C kl xxx xxx m m M 123 123 1 1 ( , , ) = ( ) = Â B C kl xxx xxx jk l lk 123 123 1 2 w w 1 2 w w a a a a ( , , ) = ( ) - + ( ) = -= - ÂÂ bic B PP P xxx xxx xx x 123 123 12 3 1 2 1 2 1 2 12 w w w w w w ww , , ( ) = ( ) ( ) ( ) ( + ) B M xyz XY X m m M m m ww w w w w 12 1 1 2 12 1 , * ( ) = ( ) ( ) ( + ) = Â