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1D non-stationary seislet transform

When a 1D signal has a constant angular frequency, the prediction filter in equation 6 can characterize a sinusoidal signal. When the 1D signal contains a sinusoid with a variable frequency, or in other words, is non-stationary, we can replace $ Z_0$ with $ Z_t$ , $ Z_t=e^{i\omega(t)\Delta t}$ denotes the frequency modulation at time $ t$ . Let us modify equation 8 to the following form:

$\displaystyle P_t(Z) = 1/2(Z/Z_t + Z_t/Z),$ (10)

in order to best characterize the non-stationary signal. $ P_t(Z)$ denotes the prediction filter at time $ t$ . In the physical domain, the linear prediction and updating operators can be expressed as:

$\displaystyle P_t(e)$ $\displaystyle =(S_t^{(+)}(e_{t-1}) + S_t^{(-)}(e_t))/2,$ (11)
$\displaystyle U_t(r)$ $\displaystyle =(S_t^{(+)}(r_{t-1}) + S_t^{(-)}(r_t))/4,$ (12)

where $ S_t^{(+)}$ and $ S_t^{(-)}$ are operators that predict an element from its left and right neighbors by modulating each element according to their local frequency $ \omega(t)$ .

Figure 1 shows a comparison between the wavelet transform and the 1D stationary and non-stationary seislet transforms in compressing a 1D signal with smooth frequency components. The frequency ranges from 250 to 186 Hz. Both the wavelet transform and stationary seislet transform fail to compress the signal well while the non-stationary seislet transform obtains a perfectly sparse representation.

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Figure 1.
Demonstration of 1D non-stationary seislet transform for non-stationary signal. Upper: non-stationary chirp signal, frequency ranges from 250 to 186 Hz. Upper middle: compressed using 1D wavelet transform. Down middle: compressed using 1D stationary seislet transform with the frequency of 250 Hz. Down: compressed using 1D non-stationary seislet transform.
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next up previous [pdf]

Next: Estimating local frequency by Up: Method Previous: 1D seislet transform

2019-02-12