Discrete Wavelet Transfom for Nonstationary Signal Processing

Abstract

In engineering, digital signal processing techniques need to be carefully selected according to the characteristics of the signals of interest. The frequency-based and time-frequency techniques have been frequently mentioned in some literature (Cohen, 1995). The frequencybased techniques (FBTs) have been widely used for stationary signal analysis. For nonstationary signals, the time-frequency techniques (TFTs) in common use, such as shorttime Fourier transform (STFT), wavelet transform (WT), ambiguity function (AF) and wigner-ville distribution (WVD), etc., are usually performed for extracting transient features of the signals. These techniques use different algorithms to produce a time-frequency representation for a signal. The STFT uses a standard Fourier transform over several types of windows. Waveletbased techniques apply a mother wavelet with either discrete or continuous scales to a waveform to resolve the fixed time-frequency resolution issues inherent in STFT. In applications, the fast version of wavelet transform, that is attributed to a pair of mirror filters with variable sampling rates, is usually used for reducing the number of calculations to be done, thereby saving computer running time. AF and WVD are quadratic time-frequency representations, that use advanced techniques to combat these resolution difficulties. They have better resolution than STFT but suffer from cross-term interference and produce results with coarser granularity than wavelet techniques do. Of the wavelet-based techniques, discrete wavelet transform (DWT), especially its fast version, is usually used for encoding and decoding signals, while wavelet packet analysis (WPA) are successful in signal recognition and characteristic extraction. AF and WVD with excessive transformation durations are obviously unacceptable in the development of real-time monitoring systems. In applications, the FBTs were typically used in noise and vibration engineering (Brigham, 1988). They provide the time-averaged energy information from a signal segment in frequency domain, but remain nothing in time domain. For nonstationary signals such as vehicle noises, some implementation examples are the STFT (Hodges & Power, 1985), WVD, smoothed pseudo-WVD (Baydar & Ball, 2001) and WT (Chen, 1998). In particular, the WT as “mathematical microscope” in engineering allows the changing spectral composition of a nonstationary signal to be measured and presented in the form of a time-frequency map and thus, was suggested as an effective tool for nonstationary signal analysis.