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

The phases *ϕ*_{k}(*t*) have time derivatives *dϕ*_{k}(*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

Unlike the conventional
definition, this definition has an extra factor of *e*^{j2πηt}.
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

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

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 *T*_{g}f(*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 Ω_{g}f(*t*,*η*):

where *ɛ* is
a small number.

The synchrosqueezed transform produces narrow ridges compared
to the windowed short-time Fourier transform. However, the width of
the short-time transform still affects the ability of the synchrosqueezed
transform to separate modes. To be resolvable, the modes must obey
these conditions:

For each mode, the frequency must be strictly greater
than the rate of change of the amplitude: $$\frac{d{\varphi}_{k}(t)}{dt}>\frac{d{A}_{k}(t)}{dt}$$ 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 $$\left|\frac{d{\varphi}_{k}(t)}{dt}-\frac{d{\varphi}_{m}(t)}{dt}\right|>2\Delta $$ for *k* ≠ *m*.

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