[9] In the power spectrum estimation, the waveform is treated as a superposition of statistically uncorrelated harmonic components, and the phase relations between the spectral components are suppressed. To extract the information regarding the presence of nonlinearities, one has to use the higher order spectra, defined in terms of the higher order moments or cumulants of the data. The third-order spectrum, commonly referred to as the bispectrum decomposes the signal's third moment over frequency. Since it is related to the skewness of a signal, it is usually used to detect the asymmetric non-linearities, such as the electrostatic decay [*Henri et al.*, 2009; *Balikhin et al.*, 2001; *Walker et al.*, 2003] and harmonic generation [*Bale et al.*, 1996]. The amplitude of the bispectrum at the bifrequency (*f*_{k}, *f*_{l}) measures the amount of coupling between the frequencies, *f*_{k}, *f*_{l} and *f*_{k+l}.

[10] The fourth-order spectrum is referred to as the trispectrum. It can be viewed as the decomposition of the signal's kurtosis over frequency. If the signal is unskewed and contains information about a symmetric process, such as the OTSI type of four wave interaction, it can be extracted using the trispectral analysis. The trispectral method has been developed [*Kravtchenko-Berejnoi et al.*, 1995a] and applied to synthetic [*Kravtchenko-Berejnoi et al.*, 1995b; *Lefeuvre et al.*, 1995] and to simulated data [*Soucek et al.*, 2003]. The trispectral analysis can detect the phase relationships among four Fourier components, which is the key information regarding four-wave interaction. In OTSI, the sidebands interact with the strong beam excited Langmuir waves satisfying the matching rules for wave numbers, frequencies and phases, simultaneously. For the OTSI type of four-wave interaction, the cumulant based trispectrum is given by [*Kravtchenko-Berejnoi et al.*, 1995a]

where *X*_{1}, *X*_{2}, *X*_{3} and *X*_{4} are the complex Fourier components of the signal corresponding to frequencies *f*_{1}, *f*_{2}, *f*_{3} and *f*_{4}, *N*(1, 2, 3, 4) = *E*[*X*_{1}*X*_{2}]*E*[*X*_{3}**X*_{4}*] + *E*[*X*_{1}*X*_{3}*]*E*[*X*_{2}*X*_{4}*] + *E*[*X*_{1}*X*_{4}*]*E*[*X*_{2}*X*_{3}*], *f*_{4} = *f*_{1} + *f*_{2} − *f*_{3}, and *E*[] is the expectation operator. The tricoherence, which is the normalized trispectrum is more useful, because it eliminates the dependence of trispectrum on the amplitude of the signals. The expressions for cumulant based square tricoherence function can be written as [*Kravtchenko-Berejnoi et al.*, 1995a]:

A unit value for the tricoherence indicates perfect coupling, a zero value indicates no coupling, and any value between zero and one indicates partial coupling. The tricoherence quantifies the fraction of the total product of powers at the frequency quartet (*f*_{1}, *f*_{2}, *f*_{3}, *f*_{1} + *f*_{2} − *f*_{3}), that is owing to cubicly phase-coupled modes. The tricoherence is zero for a Gaussian process. Even for a Gaussian process, due to statistical fluctuations, the estimate of the tricoherence from a finite data record will not be zero. Therefore, the method of periodograms is used to estimate the trispectrum and tricoherence. This involves the division of the data record into M segments; an appropriate window is applied to each segment to reduce leakage; the trispectrum as well as tricoherence are computed for each segment by using the DFT; finally, the trispectrum is averaged across segments in order to reduce the variance of the estimator. As seen fromequation (1), the trispectrum estimator is symmetric with respect to permutations of its arguments *f*_{1}, *f*_{2} and *f*_{3}. The principal domain for the interaction of the type *f*_{1} + *f*_{2} = *f*_{3} + *f*_{4} is determined as [*Kravtchenko-Berejnoi et al.*, 1995a] 0 ≤ *f*_{1} ≤ *f*_{N}, 0 ≤ *f*_{2} ≤ *f*_{1}, 0 ≤ *f*_{3} ≤ *f*_{2}, and *f*_{3} ≤ *f*_{1} + *f*_{2} − *f*_{3} ≤ *f*_{N}, where *f*_{N} is the Nyquist frequency.

[11] In this study, we have used the segment length N = 1000 (0.004 s), number of segments M = 16 and Hamming window, and calculated the tricoherence spectrum as a function of three frequencies. Since it is difficult to visualize the results in such a 3-D space, we have fixed one of its frequencies and displayed the results in two dimensions. We have made the cross-sections of the tricoherence spectrum at frequencies*f*_{D} = 29.5 kHz, *f*_{L} = 30 kHz and *f*_{U} = 30.5 kHz, corresponding to the Stokes, beam-excited Langmuir, and anti-Stokes modes, respectively. InFigure 4, we present these cross sections. The cross-section of the tricoherence spectrum at*f*_{D} = 29.5 kHz(top panel) shows: (1) a peak at (29.5 kHz, 29.5 kHz, 29.5 kHz); this is due to self-dependence and is of no interest, (2) a second peak at (30.5 kHz, 30 kHz, 29.5 kHz), which is due to the phase relation 2*ϕ*_{L} = *ϕ*_{D} + *ϕ*_{U}, where *ϕ*_{L}, *ϕ*_{D} and *ϕ*_{U}are the phases of the beam-excited Langmuir, Stokes and anti-Stokes modes, respectively; the maximum tricoherence*t*^{2} ∼ 0.53 in this case, and (3) a third peak at (30.5 kHz, 30.5 kHz, 29.5 kHz); this appears to be due to the interaction of the sidebands themselves, and it is of no interest in the present case. The cross-section of the tricoherence spectrum at*f*_{L} = 30 kHz (second panel) shows (1) a peak at (30 kHz, 30 kHz, 30 kHz), which is due to self-dependence, and (2) a second peak at (30 kHz, 30.5 kHz, 30 kHz), which is due to the phase relation 2*ϕ*_{L} = *ϕ*_{D} + *ϕ*_{U}; in this case, the maximum *t*^{2} ∼ 0.5. The tricoherence cross-section at*f*_{U} = 30.5 kHz (bottom panel) does not show any peaks. Thus, the significant tricoherences corresponding to 2*ϕ*_{L} = *ϕ*_{D} + *ϕ*_{U}in the cross-sections of the tricoherence spectrum at Stokes and beam-excited Langmuir wave frequencies provide evidence for the OTSI type of four-wave interaction.