Appendix B: Calculation Procedures of Triple Correlation, Bispectrum, and Examples

Triple Correlation and Bispectrum Estimation

Definitions of triple correlation and bispectrum for continuous signal x(t) are given in Chapter 6. However, calculation of both triple correlation and bispectrum is normally achieved in the discrete domain. Thus, the discrete triple correlation is estimated as follows:

where x(kdt) is the discrete version of x(t), k is integer number, dt = 1/f_s is the sampling period, f_s is the sampling frequency, and the delay variables are also discretized as, m = 0,1,2…, N/2-1.

Similarly, the discrete bispectrum is estimated by the discrete Fourier transform of C₃:

where m,n are integers and the frequency variables N/2−1 and the frequency resolution, N is the total number of samples in each computation window.

In this thesis, the following steps are used to estimate the bispectrum:

•  A discrete process or signal is divided into M computation frames in which the number of samples N in each frame chosen is 1024. The sampling time dt is properly selected to ensure that the significant frequency components of x(kdt) are in the range from –B to B, where B = 1/(2dt).

FIGURE B.1
(a) Triple correlation and (b) magnitude bispectrum (Bottoms: corresponding contour representations).

•  The triple correlation of each data frame is computed by using (B.1). The result obtained is an array with the size 512 × 512. Each value in the array is represented for the amplitude of the triple correlation.

•  The bispectrum of each frame is then calculated by using the discrete Fourier transform (B.2). Thus, the bispectrum is also an array with the same size. Finally, the bispectrum is averaged over M data frames via the following expression:

•  Both the triple correlation and the bispectrum can be displayed in a three-dimensional graph as shown in Figure B.1 in which the magnitude is normalized in a logarithmic scale. However, the contour representation is selected to display effectively the variation in the bispectrum structure.

Properties of Bispectrum

Important properties of the bispectrum are briefly summarized as follows [1]:

•  The bispectrum is generally complex. It contains both magnitude and phase information that is important for signal recovery as well as identifying nonlinear response and processes.

•  The bispectrum has the lines of symmetry f₁ = f₂, 2f₁ = –f₂ and 2f₂ = –f₁ corresponding to permutation of the frequencies f₁, f₂.

•  The bispectrum of a stationary, zero-mean Gaussian process is zero. Thus, a nonzero bispectrum indicates a non-Gaussian process.

•  The bispectrum suppresses linear phase information or constant phase shift information.

•  The bispectrum is flat for non-Gaussian white noise and zero for Gaussian white noise.

Bispectrum of Optical Pulse Propagation

In this section, propagation of optical pulses through optical fiber as an example is characterized and analyzed by the triple correlation and the bispectrum. Figures B.2 and B.3 show, respectively, the triple correlations and the bispectra of the 6.25 ps Gaussian pulse propagating at different lengths of the optical fiber with the second-order group velocity dispersion (GVD) coefficient b₂ = –21.6 ps²/km. Figures B.4 and B.5 show, respectively, the triple correlations and the bispectra of the 6.25 ps super-Gaussian pulse propagating at different lengths of the same fiber.

More important, the triple correlation can detect easily the asymmetrical distortion of the pulse that is impossible in an autocorrelation estimation. Figures B.6 and B.7 show, respectively, the triple correlations and the bispectra of the super-Gaussian pulse propagating through the fiber with the third-order dispersion coefficient b₃ = 0.133 ps³/km.

FIGURE B.2
Triple correlations of Gaussian pulse propagating in the fiber with b₂ = –21.6 ps²/km at different distances: (a) z = 0, (b) z = 50 m, (c) z = 650 m, (d) z = 1 km (Insets: the waveforms in time domain).

FIGURE B.3
Corresponding bispectra of Gaussian pulse propagating in the fiber at different distances: (a) z = 0, (b) z = 50 m, (c) z = 650 m, (d) z = 1 km (Insets: the corresponding phase bispectra).

FIGURE B.4
Triple correlations of a super-Gaussian pulse propagating in the fiber with b₂ = –21.6 ps²/km at different distances: (a) z = 0, (b) z = 50 m, (c) z = 100 m, (d) z = 300 m (Insets: the waveforms in time domain).

FIGURE B.5
Corresponding bispectra of the super-Gaussian pulse propagating in the fiber at different distances: (a) z = 0, (b) z = 50 m, (c) z = 100 m, (d) z = 300 m (Insets: the corresponding phase bispectra).

FIGURE B.6
Triple correlations of a super-Gaussian pulse propagating in the fiber with b₂ = 0, b₃ = 0.133 ps³/km at different distances: (a) z = 100 m, (b) z = 500 m (Insets: the waveforms in time domain).

FIGURE B.7
Corresponding bispectra of the super-Gaussian pulse propagating in the fiber at different distances: (a) z = 100 m, (b) z = 500 m (Insets: the corresponding phase bispectra).

Reference

1. C. L. Nikias and M. R. Raghuveer, Bispectrum Estimation: A Digital Signal Processing Framework, Proc. IEEE, 75, 869–891, 1987.