What is Wavelet Synchrosqueezing?

The wavelet synchrosqueezed transform is a time-frequency analysis method that is useful for analyzing multicomponent signals with oscillating modes. Examples of signals with oscillating modes include speech waveforms, machine vibrations, and physiologic signals. Many of these real-world signals with oscillating modes can be written as a sum of amplitude-modulated and frequency-modulated components. A general expression for these types of signals with summed components is

where $$A_k(t)$$ is the slowly varying amplitude and $${\varphi }_k(t)$$ is the instantaneous phase. A truncated Fourier series, where the amplitude and frequency do not vary with time, is a special case of these signals.

The wavelet transform and other linear time-frequency analysis methods decompose these signals into their components by correlating the signal with a dictionary of time-frequency atoms [1]. The wavelet transform uses translated and scaled versions of a mother wavelet as its time-frequency atoms. Some time-frequency spreading is associated with all of these time-frequency atoms, which affects the sharpness of the signal analysis.

The wavelet synchrosqueezed transform is a time-frequency method that reassigns the signal energy in frequency. This reassignment compensates for the spreading effects caused by the mother wavelet. Unlike other time-frequency reassignment methods, synchrosqueezing reassigns the energy only in the frequency direction, which preserves the time resolution of the signal. By preserving the time, the inverse synchrosqueezing algorithm can reconstruct an accurate representation of the original signal. To use synchrosqueezing, each term in the summed components signal expression must be an intrinsic mode type (IMT) function. For details on the criteria that constitute IMTs, see [2].

Wavelet Synchrosqueezing Algorithm

The synchrosqueezing algorithm uses these steps.

As described, synchrosqueezing uses the continuous wavelet transform (CWT) and its first derivative with respect to translation. The CWT is invertible and since the synchrosqueezed transform inherits the CWT invertibility property, the signal can be reconstructed.

Illustration: Instantaneous Frequency Extraction

You can express the CWT of the function $$f(t)$$ with respect to the wavelet $$\psi (t)$$ in terms of the inverse Fourier transform of the product of the function's Fourier transform and the wavelet's Fourier transform:

where the bar over $$\widehat{\psi }(s\omega )$$ denotes complex conjugate.

Consider a simple complex exponential, $$f(t)=e^{2\pi i\alpha t}$$. To extract the instantaneous frequency of $$f(t)$$:

Required Component Separation

With synchrosqueezing the signal components must be IMTs that are well separated in the time-frequency plane. If this requirement is met, you can track the trajectory of the instantaneous frequencies along a curve. The curves show the location of the maximum energy as it varies over time for each signal mode. See wsstridge for a description of the trajectory curves algorithm.

This inequality defines the required separation criteria:

where $${\varphi }^{\text{'}}$$ is the instantaneous frequency and d is a positive separation constant [2]. To determine this required separation, suppose a bump wavelet, x, has a Fourier transform with support in the range $$\left[{\epsilon }_x-\Delta ,{\epsilon }_x+\Delta \right]$$. Because the bump wavelet has a center frequency of $$\frac{5}{2\pi }$$ Hz, use $$\left[\frac{5}{2}\pi -\frac{1}{2},\frac{5}{2}\pi +\frac{1}{2}\right]$$ as the interval. Then solve $$\Delta <{\epsilon }_x\frac{d}{1+d}$$ for d to get $$d>\frac{1}{4}$$ for the bump wavelet.

To show this separation requirement for the bump wavelet, consider a signal composed of $$\cos (2\pi (0.1t))+\sin ((2\pi (0.2t))$$. Using the bump wavelet to obtain the CWT, the instantaneous phase of the cosine is $${\varphi }_1(t)=0.1t$$, and the instantaneous frequency is the first derivative, 0.1. Likewise, for the sine component, the instantaneous frequency is 0.2. The separation inequality, $$\left|0.1\right|\ge \frac{1}{4}\left|0.3\right|$$, is true. Therefore, the two signal components are IMT functions and are separated enough to use the synchrosqueezed transform.

If you use higher frequencies, such as 0.3 and 0.4 for the instantaneous frequencies, the inequality is $$\left|0.1\right|\ge \frac{1}{4}\left|0.7\right|$$, which is not true. Because these signal components are not well-separated IMTs the signal, $$\cos (2\pi (0.3t))+\sin ((2\pi (0.4t))$$, is not appropriate for use with the synchrosqueezed transform.

Examples

load quadchirp
Fs = 1000;
[wt,f] = cwt(quadchirp,"bump",Fs);
subplot(2,1,1)
hp = pcolor(tquad,f,abs(wt));
hp.EdgeColor = "none";
xlabel("Time (secs)")
ylabel("Frequency (Hz)")
title("CWT of Quadratic Chirp")
subplot(2,1,2)
wsst(quadchirp,Fs,"bump")

t = 0:2000;
x1 = cos(2*pi*.025*t);
x2 = cos(2*pi*.05*t);
x3 = cos(2*pi*.20*t);
x4 = cos(2*pi*.225*t);
x =x1+x2+x3+x4;
[sst,f] = wsst(x);
contour(t,f,abs(sst))
xlabel("Time")
ylabel("Normalized Frequency")
title("Inadequate High-Frequency Separation")

x4 = cos(2*pi*.3*t);
x =x1+x2+x3+x4;
[sst,f] = wsst(x);
contour(t,f,abs(sst))
xlabel("Time")
ylabel("Normalized Frequency")
title("Adequate High-Frequency Separation")

t = 0:2000;
y1 = chirp(t,0.4,1000,0.25);
y2 = chirp(t,0.1,1000,0.25);
y = y1+y2;
wsst(y,'bump')
xlabel('Samples')
h1 = line([583 583], [0 0.5]);
h2 = line([1417 1417], [0 0.5]);
h1.Color='white';
h2.Color='white';