EEG signal processing - wavelet transform series

Time domain signal analysis

​ Time domain signal analysis is often based on phase 、 energy , Even cross frequency coupling .

​ The common time domain signal analysis methods are ERPS, That is, multi-channel brain wave mean filtering , This method needs to pay attention to the need for baseline standardization , Put all the data on the same scale , Separate task related activities from background activities , It tends to be more normal distribution .

​ But the time domain analysis method has some shortcomings ：

1. Jitter and non phase locked activity cannot be observed ;
2. There is limited analysis that can be done
3. Poor signal-to-noise ratio and low statistical efficiency

Stability

​ Stationarity is defined as ： Whether the statistical characteristics of time series signals have similar characteristics over time .EEG The signal is a highly non-stationary signal .

​ Wavelet transform is such a process ： First, take the original signal as the input signal , Through a set of orthogonal wavelet bases, it is decomposed into high-frequency part and low-frequency part , Then the low-frequency part is used as the input signal , And wavelet decomposition , Get the high-frequency part and low-frequency part of the next stage , And so on . As the number of wavelet decomposition increases , The higher the resolution in frequency domain . This is multiresolution analysis

Continuous wavelet transform

​ Continuous wavelet transform is a square integrable function f(t) And a wavelet function with good local properties in time and frequency domain ψ(t) Inner product ：

$$

Wf(a, b)={a, b}>=dfrac{1}{sqrt a}psi^*(dfrac{t-b}{a})dt\

psi_{a, b}(t)=dfrac{1}{sqrt a}psi(dfrac{t-b}{a})$$

$psi_{a, b}(t)$ It's mother wavelet $psi(t)$ A family of functions generated by displacement and expansion , It is called wavelet basis function or simply ** Wavelet basis **.

​ $psi(t)$ The time domain waveform of has “ Attenuation ” and “ Volatility ”, That is, its amplitude has positive and negative oscillations ; Look at the set from the spectrum ,$psi(w)$ In one “ Small ” In the frequency band , have “ Bandpass ”.

Discrete wavelet transform

​ In practice, scale factor $a$ And displacement factor $b$ Discrete processing , take ：

$$

a = a0^m, b=nb0a_0^m$$

​ Discrete wavelet form ：

$$

psi{m, n}(t)=dfrac{1}{a0^m}psi(dfrac{t-nb0a0^m}{a0^m})=dfrac{1}{a0^m}psi(a0^{-m}t-nb0)\

wf(m, n)={a, b}>=f{-infty}^{+infty}f(t)psi{m, n}^*(t)dt$$

​ Wavelet analysis Time series S Decompose into low frequency information a1 And high frequency information d1 Two parts , In decomposition , Low frequency a1 Information lost in the high frequency d1 Capture . In the next level of decomposition , And will be a1 Decompose into low frequencies a2 And high frequency d2 Two parts , Low frequency a2 Information lost in the high frequency d2 Capture . And so on , Can be further decomposed .

The length of the result after wavelet transform is equal to the length of wavelet + Signal length -1

Be careful ：

1. The sampling rate of wavelet must be consistent with the data sampling rate
2. The wavelet must be at the center , Prevent resulting phase shift

Wavelet packet transform

​ Wavelet packet decomposition Not only the low-frequency part is decomposed , And the high-frequency part is decomposed . therefore , Wavelet packet decomposition is a more widely used wavelet decomposition method , Applied to signal decomposition 、 code 、 Denoising 、 Compression and so on .

Remember the parent wavelet in wavelet packet transform $Phi(t)$ by $mu0^0(t)$ , Mother wavelet $Psi(t)$ by $u0^1(t)$ , The superscript represents the decomposition series of the wavelet packet , The subscript indicates the position of the wavelet packet in its stage .

$$

left{

begin{aligned}

mu{2n}^{L-1}(t)=sumkhkmu n^L(t-k), mu{n}^{L}(t-k)=mu{n}^{L-1}(2t-k)\

mu{2n+1}^{L-1}(t)=sumkgkmu n^L(t-k), mu{n}^{L}(t-k)=mu{n}^{L-1}(2t-k)\

end{aligned}

right.$$

among , $hk$、$gk$ Is the same as wavelet transform ,$mu$ Is wavelet packet

application

​ Because of the sparse coding characteristic of wavelet decomposition , It can be used for data compression , The main methods are ： The signal is decomposed by wavelet , And set the smaller wavelet coefficient to zero . Equivalent to will not be important （ The features are not obvious ） Information component removal , Achieve the effect of data reduction .

​ Wavelet decomposition can also be used for signal filtering , The main methods are ： The signal is decomposed by wavelet , And set the wavelet coefficients above a specific series to zero . It is equivalent to removing high-resolution information components , Achieve the effect of data smoothing .

​ Wavelet decomposition can also be used for signal de-noising , The main methods are ： The signal is decomposed by wavelet , And by setting a threshold , Set the wavelet coefficients below the threshold to zero . It is equivalent to removing the noise part with low proportion in the signal .