Fourier synchrosqueezed transform

Many real-world signals such as speech waveforms, machine vibrations, and physiologic signals can be expressed as a superposition of amplitude-modulated and frequency-modulated modes. For time-frequency analysis, it is convenient to express such signals as sums of analytic signals through

f(t)=∑k=1Kfk(t)=∑k=1KAk(t)ej2πϕk(t).

The phases ϕk(t) have time derivatives dϕk(t)/dt that correspond to instantaneous frequencies. When the exact phases are unknown, you can use the Fourier synchrosqueezed transform to estimate them.

The Fourier synchrosqueezed transform is based on the short-time Fourier transform implemented in the spectrogram function. For certain kinds of nonstationary signals, the synchrosqueezed transform resembles the reassigned spectrogram because it generates sharper time-frequency estimates than the conventional transform. The fsst function determines the short-time Fourier transform of a function, f, using a spectral window, g, and computing

Vgf(t,η)=∫−∞∞f(x)g(x−t)e−j2πη(x−t) dx.

Unlike the conventional definition, this definition has an extra factor of ej2πηt. The transform values are then “squeezed” so that they concentrate around curves of instantaneous frequency in the time-frequency plane. The resulting synchrosqueezed transform is of the form

Tgf(t,ω)=∫−∞∞Vgf(t,η) δ(ω−Ωgf(t,η)) dη,

where the instantaneous frequencies are estimated with the “phase transform”

Ωgf(t,η)=1j2π∂∂tVgf(t,η)Vgf(t,η)=η−1j2πV∂g/∂tf(t,η)Vgf(t,η).

The transform in the denominator decreases the influence of the window. To see a simple example, refer to Detect Closely Spaced Sinusoids with the Fourier Synchrosqueezed Transform. The definition of Tgf(t,ω) differs by a factor of 1/g(0) from other expressions found in the literature. fsst incorporates the factor in the mode-reconstruction step.

Unlike the reassigned spectrogram, the synchrosqueezed transform is invertible and thus can reconstruct the individual modes that compose the signal. Invertibility imposes some constraints on the computation of the short-time Fourier transform:

The number of DFT points is equal to the length of the specified window.

The overlap between adjoining windowed segments is one less than the window length.

The reassignment is performed only in frequency.

To find the modes, integrate the synchrosqueezed transform over a small frequency interval around Ωgf(t,η):

fk(t)≈1g(0)∫|ω−Ωk(t)|dAk(t)dt for all k.

Distinct modes must be separated by at least the frequency bandwidth of the window. If the support of the window is the interval [–Δ,Δ], then |dϕk(t)dt−dϕm(t)dt|>2Δ for k ≠ m.

For an illustration, refer to Detect Closely Spaced Sinusoids with the Fourier Synchrosqueezed Transform.