Time Delay Estimation Based On Higher-order Spectrum

The main mathematic tool in treating non-minimum phase signals,nonlinear signals and non-Gaussian signals is higher-order cumulants orhigher-order spectra. During 1960s, researchers in mathematics, statistics,hydrokinetics, signal treating and other fields began to research higher-ordercumulants. But it is fully developing after 1980s. After a few years research,higher-order cumulants is obtained lots of application in radar, sonar,communication, oceanography, astronomy, electromagnetism, plasma, physicalgeography, biomedicine and other fields. Higher-order spectra have been givenlots of attention lately due to their ability to preserve information ofnon-Gaussian stationary random processes. Higher-order cumulants have moreadvantage than correlation function. Hence, all the signals treated by correlationwithout satisfied results can be treated by higher-order cumulants. The basic approach to the solution of the time delay estimation problem isto shift X (n) with respect to Y(n) and compare similarities between the tworecords at each shift. The best match will occur at a shift equal to D . Assumingthat the noise sources are zero-mean independent stationary random process, thefundamental operation adopted to "compare similarities" between {X (n)} and{Y(n)} is the cross correlation rxy (Ï„ ) which peaks at Ï„ = D . However, inpractical application problems due to finite length data records and not exactlyindependent noise sources, the rxy (Ï„ ) does not necessarily show a peak at thetime delay position. In practical application problems the signal {S(n)} can be regarded as 54non-Gaussian process and the noise sources w1(n), w2 (n) be regarded aszero-mean stationary Gaussian signal, then the similarities between { X ( n )}and {Y(n)} could also be "compared" in higher-order spectrum domains suchas the bispectrum. One of the fundamental properties of higher-order spectra isthe fact that for Gaussian processes only, all polyspectra of order greater thantwo are identically zero (in theory). Hence, when the signal is non-Gaussianstationary process and the additive noise to the signal is stationary Gaussian,there might be certain advantages estimating signal parameters in higher-orderspectrum domains. This article provides cross correlation least-square methods, matlabbispectrum methods, direct bispectrum methods, bispectrum least-squaremethods and parameter bispectrum methods estimating time delay. Crosscorrelation least-square methods use FFT translating time series into frequencyseries and using phase data computing time delay. Matlab bispectrum methods,direct bispectrum methods, bispectrum least-squares methods, parameterbispectrum methods all use higher-order spectrum estimating time delay, the firstthree methods use FFT to get bispectrum phase data estimating time delay, whileparameter bispectrum methods directly compute bispectrum by time domainsequence and use linear equation's least-square estimating time delay. In thisarticle, studies show that different methods in estimating time delay make outdifferent results. In this article, two algorithms are optimized. In cross correlationleast-square algorithm and bispectrum algorithm, phase unwrapping,phasetranslating and parameter are used to enhance time delay estimation precision.By computing, cross correlation algorithm and parameter bispectrum methodssuppress noise signals better than other methods in treating zero-mean colored 55signals added cross-correlation colored noise signal and multi-time-delay signals.In all these bispectrum methods, direct bispectrum methods show moreadvantage than matlab bispectrum methods and parameter bispectrum methodsin suppressing noise signals. Cross correlation least-square methods showsbigger error in treating cross correlation noise signals, the error relates with thephase data's linearity. When time-dela...