transformSST: Synchrosqueezed Wavelet Transform

In xemd: eXtended Empirical Mode Decomposition

Synchrosqueezed Wavelet Transform

Description

Computes the Morlet continuous wavelet transform of a sampled signal and reassigns its coefficients to frequency cells by the instantaneous frequency estimated from the transform.

Usage

transformSST(signal, samplingInterval, voiceCount = 16,
             morletCenter = 2 * pi, threshold)

Arguments

signal	Numeric vector of at least four finite values without dim, in the units of the quantity.
samplingInterval	Sampling interval \Delta t in seconds, a finite number greater than zero.
voiceCount	Whole number \ge 1 of voices per octave; default 16, the value of SynchWave.
morletCenter	Center \mu > 0 of the Morlet wavelet in normalized frequency; default 2\pi, the value of SynchWave.
threshold	Number greater than zero in the units of the signal: only coefficients with |W| > threshold are reassigned. No default; SynchWave uses sqrt(.Machine$double.eps).

Details

The record is extended by symmetric padding to T = 2^{1 + \mathrm{round}(\log_2 N)} samples, and the transform uses the scales a_j = 2^{j / \mathrm{voiceCount}}, j = 1, \ldots, (\log_2 T - 1)\,\mathrm{voiceCount}. For each coefficient above the threshold, the frequency

\frac{\mathrm{Re}\,\Omega_f(a, b)}{2\pi} = \frac{1}{2\pi} \mathrm{Im}\frac{\partial_b W_f(a, b)}{W_f(a, b)}

in Hz goes to the nearest of J logarithmic cells from 1/((N - 1)\Delta t) to 1/(2\Delta t), J the number of scales, where the coefficient adds W a^{-1/2}; the sums are divided by voiceCount. Estimates that are not positive or fall outside the cells are not accumulated, and a transform whose energy is below the machine epsilon gives a zero result. The representation recovers the signal only approximately. The threshold is in the units of the signal, so decomposeTSL has no operation for this transform.

Value

A list with

transform	Complex matrix, samples by frequency cells.
frequency	Numeric vector of the cell frequencies in Hz.

References

Daubechies, I., Lu, J. and Wu, H.-T. (2009). Synchrosqueezed wavelet transforms: a tool for empirical mode decomposition. arXiv:0912.2437.

The wavelet transform, the Morlet wavelet and the reassignment are adapted from SynchWave 1.1.2, revision 9ac80b2c3e60b417d0ee5567a66e8e64faa1a979 (MATLAB original by Eugene Brevdo; R port by Dongik Jang, Hee-Seok Oh and Donghoh Kim; the Synchrosqueezing toolbox attribution also names Hau-Tieng Wu), LGPL-2.0-or-later, which xemd distributes under GPL-3 as section 3 of the GNU Library General Public License version 2 permits; see the NOTICE file of the installed package.

See Also

decomposeEWT, computeInstantaneousAttributes

Examples

SamplingInterval <- 1 / 1000
Time <- seq_len(1000) * SamplingInterval
Signal <- cos(2 * pi * 2 * Time) + 0.25 * cos(2 * pi * 24 * Time) +
    0.0625 * cos(2 * pi * 288 * Time)
OUT <- transformSST(Signal, samplingInterval = SamplingInterval,
                    voiceCount = 16, morletCenter = 2 * pi,
                    threshold = sqrt(.Machine$double.eps))
dim(OUT$transform)
range(OUT$frequency)

xemd documentation built on Nov. 9, 2026, 1:09 a.m.