The need for instantaneous frequency (IF) is a real one for data from non-stationary and nonlinear processes. If the process is non-stationary, the frequency should be ever changing, albeit at a slow rate. Then, there is a need for frequency value as a function of time, for the value will not be constant throughout. For the nonlinear cases, the frequency is definitely modulating not only among different oscillations, but also within one period. A detailed explanation of IF is disclosed in related patent application Ser. No. 10/615,365, filed on Jul. 8, 2003, entitled “Computing Instantaneous Frequency by Normalizing Hilbert Transform”, inventor Norden E. Huang, which is incorporated by reference and assigned to the same assignee as this application. A computer implemented method of computing IF is disclosed in U.S. Pat. Nos. 5,983,162, 6,381,559, 6,311,130, all of which are also incorporated by reference.
The above-disclosed method of computing IF includes two essential steps and the associated presentation techniques of the results. The first step is a computer implemented Empirical Mode Decomposition to extract a collection of Intrinsic Mode Functions (IMF) from nonlinear, nonstationary signals. The decomposition is based on the direct extraction of the energy associated with various intrinsic time scales in the signal. Expressed in the IMF's, they have well-behaved Hilbert Transforms from which instantaneous frequencies can be calculated. The second step is the Hilbert Transform of the IMF. The final result is the Hilbert Spectrum. Thus, the method can localize any event on the time as well as the frequency axis. The decomposition can also be viewed as an expansion of the data in terms of the IMF's. Then, these IMF's, based on and derived from the data, can serve as the basis of that expansion. The local energy and the instantaneous frequency derived from the IMF's through the Hilbert transform give a full energy-frequency-time distribution of the data, which is designated as the Hilbert Spectrum.
However, there is a need for an alternative method to the Hilbert Transform in constructing an energy-frequency-time distribution. The Hilbert Transform is an intensive calculation process that requires powerful computers beyond many end users, including ordinary engineers and scientists, resources. Furthermore its results are not intuitive. If a signal to be analyzed has a surge-like behavior, i.e. the signal contains a very high amplitude for a short period time such as sound signal of a gunshot, the method disclosed in the above references may not generate reasonable results due to limitations imposed by spline fitting, which is used in the EMD in constructing extrema envelopes of the signal. The problem with spline fitting applied to the situation just mentioned, might result in both overshoot/undershoot problems and eventually divergence of IMF'S.
Zero Crossing points are the point where the voltage polarity of a waveform changes from negative to positive (or vice-versa) as it crosses the zero axis. In the field of signal processing, for a subject signal, the zero crossing points are detected and counted for analyzing the signal or making a decision for further actions. For example, zero crossing points are used to define bandwidth of a signal, which is assumed to be stationary and Gaussian. The bandwidth can be defined in terms of spectral moments as follows. The expected number of zero crossings per unit time is given by             N      0        =                  1        π            ⁢                        (                                    m              2                                      m              0                                )                          1          2                      ,where the expected number of extrema per unit time is given by             N      1        =                  1        π            ⁢                        (                                    m              4                                      m              2                                )                          1          2                      ,in which mi is the ith moment of the spectrum Therefore, the parameter, v, defined as                     N        1        2            -              N        0        2              =                            1                      π            2                          ⁢                                                            m                4                            ⁢                              m                0                                      -                          m              2              2                                                          m              2                        ⁢                          m              0                                          =                        1                      π            2                          ⁢                  v          2                      ,offers a standard bandwidth measure. For a narrow band signal v=0, the expected numbers of extrema and zero crossings have to equal. However, the previous equation defines the bandwidth in the global sense; it is overly restrictive and lacks precision at the same time as there might be more than two extrema between two zero crossing points. Consequently, the bandwidth limitation on the Hilbert transform to give a meaningful instantaneous frequency has never been firmly established.