Wavelet transform time 20 century 80 A mathematical analysis method gradually developed in the middle and late S , As soon as he appeared, he was widely valued by the mathematical and engineering circles .1984 French scientists J.Molet When analyzing the local characteristics of seismic waves , Firstly, the signal is analyzed by wavelet transform , And put forward the term wavelet .

wavelet , Small waveform , Small means it has attenuation , Wave means that it has volatility , That is to say, the amplitude of wavelet has the form of oscillation between positive and negative amplitude . Wavelet theory adopts the idea of multi-resolution , Nonuniform division of time-frequency space , It enables the signal to be decomposed on a set of orthogonal basis , It provides a new way for the analysis of non-stationary signals .


Wavelet is a function or signal that satisfies the condition in function space . Wavelet analysis can analyze any fine structure of function and signal at any designated point , meanwhile , This also determines the wavelet analysis in the time-frequency analysis of non-stationary signals , The ability to localize time and frequency simultaneously .


A variable rectangle in time-frequency plane of time-frequency window of continuous wavelet , The area of his time-frequency window is related to the generating function of wavelet , This determines the special role of wavelet transform in time-frequency analysis of signals .


Characteristics of wavelet analysis ;

The time-frequency relationship of wavelet transform is restricted by the uncertainty principle . And Heng Q nature ,Q Quality factor of parent wavelet .Q= bandwidth / center frequency .

permanent Q Property is an important property of wavelet transform , It is also different from other types of wavelet transform , An important reason for its wide application . When using smaller a High frequency analysis of signal , In fact, we use high frequency wavelet to observe the signal carefully ; And the larger a Low frequency analysis of signal , In fact, we use low frequency wavelet to make a general observation of the signal .

The development and development of wavelet analysis , The difference is

1. The basic functions used in Fourier transform are unique , The functions used in wavelet analysis are not unique , The same problem is analyzed by different wavelet functions , There's a long way to go .

2. In frequency domain , Fourier transform has better localization ability , Especially for those signals with simple frequency components , Fourier transform can easily express the signal in the form of superposition of various frequency components ; But in the time domain , Fourier transform has no localization ability , It is impossible to see the shape of the original signal near any time point from the Fourier transform of the signal .

3. If the signal is interpreted through a filter , The difference between wavelet transform and short time Fourier transform is , For short time Fourier transform , The bandwidth of the band-pass filter is independent of the center frequency ; contrary , The bandwidth of the wavelet transform band-pass filter is proportional to the center frequency , That is, the filter corresponding to wavelet transform has a constant relative bandwidth .


Multiresolution analysis of wavelet transform

To roughly reveal the concept of multi-resolution, we use the metaphorical relationship between camera lens and the decommissioning of the observed scene . This step-by-step analysis of things from coarse to fine is called multiresolution analysis , Its characteristics are determined by the natural characteristics of the signal . Multiresolution analysis can be introduced from two aspects , Cost division and ideal filter bank in function space .

From the perspective of ideal filter banks , In essence, multiresolution analysis is to decompose the signal according to the frequency band , The decomposition method can divide the equal frequency band , A binary decomposition can also be used . After each stage decomposition, the frequency band of the signal is reduced by half compared with the previous stage , Therefore, each level has an extraction link , He said that he would save a little bit of data for every two points .


From the decomposition of signal , The original signal is divided into subband signals with different frequency bands by successive decomposition , If these subband signals are processed separately DFT, And DFT It's the same length , Then the frequency resolution of each subband signal is different . This kind of method that decomposes the original signal into its own preferences according to the frequency band is called “ Multiresolution analysis ”, Let's introduce the concept

Division of frequency space , permanent Q The bandpass space of sex , Consistency of filters at all levels .

From the perspective of function space division , In the case of two points Mallat Starting from the concept of multiresolution spatial decomposition of function , Establishing a connection between wavelet transform and multiresolution analysis , If we take the function of square technology as a limit case of a certain successive approximation , Then no machine approximation is the result of smoothing the signal with a certain smoothing function , But the smooth function does the scaling step by step when approaching step by step , That is to say, different resolutions are used to approach the functions to be analyzed step by step . For spaces , We can find the orthonormal basis of the corresponding space , The scaling function and wavelet function can be constructed from this . Scale function corresponds to low pass filter , Wavelet function corresponds to high pass filter .


Application of wavelet transform in speech processing

Wavelet transform plays an important role in signal analysis and processing . Simulation of auditory perception system using wavelet transform , Denoise speech signal , To clear up , Dullness judgment .

Simulation of auditory system by wavelet transform : Cochlear filter , Basement membrane completes signal analysis , The hair cells complete the conversion from mechanical vibration to electric motor ; Reduction of acoustic spectrum characteristics by side suppression network . Using generalized wavelet transform , That is to use wavelet transform and wavelet packet transform together , Processing input signal with incomplete wavelet packet transform . Flexible time-frequency analysis ability of wavelet packet algorithm , It can better conform to the frequency analysis characteristics of human ear basement membrane .

The basic idea of removing random noise is : According to the wavelet spectrum of noise and signal in different scales, it has different characteristics , Remove the noise spectrum components on the scales where the noise spectrum is dominant , In this way, the preserved wavelet is the wavelet spectrum of the original signal , Then we use wavelet transform to reconstruct the algorithm , Reconstruct the original signal . The key of wavelet transform de-noising is how to filter out the wavelet spectrum component produced by noise . Even a theorem can be embodied , With the increase of scale , The wavelet spectrum of white noise will disappear gradually , However, the wavelet transform of effective signal still has a clear performance on large scale . By observing the evolution of signal and noise wavelet spectral mode value with the increase or decrease of scale , It can distinguish white noise and transform modulus value of signal .

Used to judge voiced and unvoiced sounds

Low frequency profile , Equivalent to signal passing through low-pass filter , High frequency description details , It's like passing through a high pass filter . According to the characteristics of short-term stability of voice signal , Firstly, the speech signal is divided into frames and wavelet transform is carried out , The coefficients in wavelet domain are divided into two parts 4 Frequency bands , Calculate the average energy of each frequency band . The energy of the highest band in the wavelet domain is larger than that of other bands , The energy ratio of the lowest band to the highest band is less than 0.9, The speech signal is considered as unvoiced .

Wavelet transform can also be used for dynamic spectrum analysis . As a means of feature extraction .

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