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论文要内容如下(有点乱，但是不影响内容)：（谢谢大家的积极回应）
Performance of Wavelet Transform and Empirical Mode Decomposition in Extracting Signals Embedded in Noise
Examples
The discussion herein will call upon concepts associated with the analytic signal z(t) and the IF: the derivative of the analytic signal’s phase .The instantaneous frequency has traditionally been identified from the analytic signal generated by the Hilbert transform HT, though this method is not capable of handling multicomponent analyses. EMD+HT and CWT have this capability, though they decompose multicomponent signals and generate the analytic signal in fundamentally different ways.
In the case of EMD+HT, EMD is used to decompose the signal into its intrinsic mode functions IMFs and then the HT is subsequently applied to generate the analytic signal. For the CWT, an analytic parent wavelet is used e.g., Morlet wavelet, yielding wavelet coefficients that are directly proportional to the analytic signal at the stationary points or ridges of the time-frequency map.
Before presenting the results, it should be emphasized that they are not achieved without a proper understanding of each approach and its implementation. For instance, it is important to note that the resolution characteristics of the Morlet wavelet analysis are dictated by the central frequency parameter f0, according to relationships discussed in Kijewski and Kareem 2003, and greatly impact the wavelet’s ability to detect nonlinear characteristics Kijewski-Correa and Kareem 2007 and to isolate closely spaced time or frequency components Kijewski-Correa and Kareem 2006. In this study, to preserve the capability to track time varying features, a Morlet wavelet with central frequency f0=1 Hz is used. Larger central frequency values f0=5 Hz essentially approach a Fourier-like representation Kijewski-Correa and Kareem 2006. It should also be emphasized that EMD+HT results are presented in the form of a Hilbert spectrum, which plots the amplitude of the Hilbert-transformed IMFs as a function of time and IF. These results are compared to the wavelet instantaneous frequency spectrum WIFS Kijewski-Correa and Kareem 2006, which presents a comparable representation, in contrast to the scalogram comparisons presented in Huang et al. 1998. Finally, EMD was applied under the following conditions: The maximum iteration number for each sifting was chosen as 1,000 and the number of successive sifting steps that produce the same number of extrema and zero crossings was limited to 5. Note that other sifting criteria may yield some variations in the IMFs obtained.In the examples which follow, the signal-to-noise ratio SNR is defined as
SNR =xN 1
where x=standard deviation of the signal and N=standard deviation of the additive white noise drawn from a standard normal distribution. A noise embedded case SNR1 will be considered as the “worst case” scenario, comparable to the noise levels investigated in other IF studies Boashash 1992. Low-noise examples SNR=10 are also provided so that the performance of the methods can be enveloped between two noise extremes.
Constant Frequency Sinusoid Embedded in Noise The first noise-embedded signal is a unit amplitude, 1 Hz sinusoid SNR=0.707. Interestingly, analysis of this signal by EMD yields 6 IMFs, while an EMD analysis of the additive noise signal by itself yielded 7 IMFs. The IMFs are omitted for brevity but can be found in Kijewski-Correa and Kareem 2005. The instantaneous frequency components associated with each IMF for the noise-embedded signal are shown in Fig. 1a. Notice the mixing of frequency content between the first and second IMF due to EMD being “. . . as a filter bank of overlapping band-pass filters”Flandrin et al. 2004. Other studies Olhede and Walden 2004;Kijewski-Correa and Kareem 2005, 2006 noted the implication of such mode-mixing and its influence on the quality of estimated IFs. For comparison, the EMD+HT analysis of the additive noise signal by itself is presented in Fig. 1c. Notice the similarities to Fig. 1a, with again the presence of mode mixing and a distribution of energy content over the entire time-frequency map. Thus,the 1 Hz sinusoid cannot be extracted from the large amplitude additive noise. As the sifting operation of EMD+HT is based on spline fits to envelope functions, the fact that the signal is so grossly overcome by noise implies that any decomposition based on envelope functions will likely capture only the signal components contributing to that envelope—in this case dominated by noise.
The same signal is now analyzed by CWT in Fig. 1b. Like its Hilbert counterpart, there is a rich distribution of energy over the map, but with coefficients dominant near 1 Hz forming a continuous wavelet ridge. For comparative purposes, the same wavelet analysis is conducted on the additive noise signal by itself and the results are shown in Fig. 1d. Note the lack of continuous ridge in the vicinity of 1 Hz and instead the sole presence of the intermittent noise distributed over the time-frequency plane. The real and imaginary components of the analytic signal extracted from this wavelet ridge are shown in Fig. 2a. Note that the amplitude of the analytic signal is somewhat distorted due to the noise;however, the quadrature shift and thus phase is preserved. The IF estimated from the wavelet analytic signal is shown in Fig. 2band its statistics are presented in Table 1.
Since no IMF captured the embedded sinusoid, an EMD+HT IF estimate for the sinusoid cannot be provided for comparison.However, the IF derived from a direct application of the HT to the noise-laden signal is provided in Fig. 2c for reference, and it statistics are similarly summarized in Table 1. Note the high degree of variability in the estimated IF law. Also provided for comparison is an analysis on the same sinusoid, but now under low noise see SNR=10 in Table 1. For this low-noise case, EMD produced 5 IMFs, with the first solely carrying the extracted sinusoid, whose IF is estimated and reported in Table 1. Interest- ingly, the IF estimated from a direct application of HT to this sinusoid with SNR=10 also provided in Table 1 performs slightly better than the EMD+HT result. This underscores the inaccuracies in the resulting IMFs, even in the presence of very low noise levels. Finally, note that the wavelet produces lesser coefficients of variation COVs and its mean IF shows no sensitivity to noise level.
Discussion
The performance of EMD in the presence of noise can be explained by the fact that its sifting procedure is based upon signal envelopes that are highly distorted by noise and thus negatively impact EMD’s ability to capture the embedded signal’s scales.
Thus, the resulting bases or IMFs are themselves derived from noise, impacting the ability to accurately isolate a frequencymodulated FM wave and estimate its IF with low variance. The examples herein demonstrate that analytic parent wavelets such as the Morlet wavelet are better suited to achieving high similitude with FM waves, despite the presence of large amplitude noise.
The scales of signals are not impacted by noise to the same extent as signal envelopes. Thus, transforms that seek similitude in scale and do not derive their bases from signal envelopes perform better, explaining the superior results obtained by CWT.
Conclusions
This study provided an evaluation of CWT and EMD+HT in extracting signals embedded in both high and low noise levels. Whereas both approaches can capture the instantaneous frequency in a mean sense, irregardless of the noise level, Hilbert transformbased approaches demonstrate a higher coefficient of variation that increases with the noise level, particularly in the case of a quadratic chirp. This performance is attributed to the fact that their bases are derived from the noise-contaminated data. It was also noted that in high noise situations, there is considerable mixing of the embedded signal over the IMFs. As such, when noise is very high, a signal may not be isolated by EMD and its IF law cannot be estimated. Even in low noise simulations, IMFs were somewhat distorted and actually yielded IF estimates of higher variance than a direct application of the Hilbert transform. Thus signal extraction and reconstruction from the empirical bases of EMD+HT can be problematic as noise levels increase; thus wavelet transforms may provide a more reliable alternative for such analyses.
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