MvFIF: Multivariate Fast Iterative Filtering

• MvFIF is a data-driven method that decomposes multichannel signals into a few intrinsic mode functions and a trend by estimating a common filter length based on the signal’s rotational dynamics.
• It uses FFT-based convolution with Fokker–Planck filters, ensuring quasi-dyadic spectral resolution and quasi-orthogonal decompositions with finite-time convergence.
• The method maintains cross-channel alignment by applying a joint multivariate scale selection, making it computationally efficient for long multichannel recordings.

Multivariate Fast Iterative Filtering (MvFIF) is a fully data-driven method for decomposing multichannel, nonstationary signals into a small number of multivariate intrinsic mode functions (IMFs) plus a multivariate trend. It is the multivariate extension of Fast Iterative Filtering (FIF), which itself is the FFT-based implementation of Iterative Filtering (IF), and its defining feature is that it preserves cross-channel coupling by computing a single filter length from the joint geometry of the multichannel signal rather than decomposing each channel independently. Within the broader IF family, MvFIF retains convolution-based adaptivity, does not require a predefined basis, and provides aligned multivariate IMFs; in later work on multidimensional and multivariate decomposition, it is explicitly reused as the temporal multivariate building block of MdMvFIF (Cicone et al., 2019, Cavassi et al., 2024).

1. Position within the Iterative Filtering family

IF was introduced as an iterative, nonlinear alternative to Empirical Mode Decomposition (EMD) for decomposing a nonstationary signal into oscillatory components called IMFs plus a residual. In IF, the local moving average of a one-dimensional signal $s$ is computed by convolution with a compactly supported filter $w$,

$\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$

where $L$ is the filter half-support adaptively chosen from the signal. One IMF is extracted by repeatedly subtracting this moving average; the outer loop then proceeds on the residual until it has at most one extremum. FIF accelerates this construction by expressing convolution via the Fast Fourier Transform (FFT), exploiting the fact that convolution becomes multiplication in the Fourier domain, so repeated filtering becomes powers of a diagonal operator (Cavassi et al., 2024).

This convolutional formulation is the principal structural distinction from EMD. EMD computes its local mean through spline envelopes of maxima and minima, whereas IF/FIF replaces envelope interpolation with a linear convolution operator. Under periodic extension, the discrete convolution matrix is circulant and diagonalizable by the discrete Fourier transform (DFT), which makes the method linear at the inner-loop level, spectrally analyzable, and computationally efficient. The same spectral formulation underlies both direct and fast implementations of IF and provides the operator-theoretic basis later reused by MvFIF (Cicone, 2018).

MvFIF occupies a specific place among the IF extensions. FIF addresses scalar one-dimensional signals; MIF and FIF2 address scalar multidimensional signals by replacing the one-dimensional filter with a multidimensional filter; MvFIF addresses vector-valued one-dimensional signals by keeping one-dimensional filtering but introducing a multivariate criterion for scale selection; MdMvFIF combines the multidimensional spatial machinery of FIF2 with the multivariate temporal machinery of MvFIF to handle data varying simultaneously in space and time (Cicone et al., 2015).

2. Signal model and multivariate scale selection

MvFIF considers a multivariate time series either as $s \in \mathbb{R}^{n\times m}$, with $n$ channels and $m$ samples, or equivalently as $s \in \mathbb{R}^n \times \mathbb{R}$, where each time $t$ is associated with a vector

$v(t)=\mathbf{v}(t)\in\mathbb{R}^n.$

The decomposition has the standard IMF-plus-trend form

$w$0

with each $w$1 intended to be a multivariate oscillatory component whose channels share the same mode index and similar instantaneous frequency content (Cicone et al., 2019).

The distinctive ingredient of MvFIF is its geometry-based estimate of the common filtering scale. Instead of computing a separate filter length from the extrema of each channel, the algorithm measures the rotation of the multichannel signal vector through the scalar sequence

$w$2

Here $w$3 is the Euclidean norm and the dot denotes the standard inner product. Because each sample vector is normalized before the dot product is taken, the angle is purely directional and independent of amplitude. The sequence $w$4 therefore encodes the rotational dynamics of the multivariate trajectory in $w$5 (Cavassi et al., 2024).

The filter length $w$6 is then set as twice the average distance between extrema in $w$7. In the notation of the MdMvFIF exposition, if $w$8 denotes the extrema locations of $w$9, one writes

$\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$0

The stated purpose of this rule is to ensure that the decomposition adapts to the highest frequency rotations embedded in the signal. A frequent misconception is that MvFIF is merely channel-wise FIF; the defining correction is that channel-wise filtering is performed only after this common multivariate scale has been estimated, so IMF alignment is enforced by construction rather than left to chance (Cavassi et al., 2024).

3. Algorithmic structure

For each IMF extraction, MvFIF first forms the multivariate samples $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$1, computes the rotation-angle sequence $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$2, detects its local extrema, and determines the common filter length $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$3. It then constructs a one-dimensional filter $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$4, typically a Fokker–Planck filter, with support $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$5. The filter/window in the FIF framework is a nonnegative and even function in $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$6, $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$7, such that

$\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$8

In practice, the filters emphasized in this literature are Fokker–Planck filters, which are $\mathcal{M}(s)(x)=\int_{-L}^{L} s(x+t)w(t),$9 and compactly supported (Cicone et al., 2019).

Once $L$0 and $L$1 are fixed, the multivariate part of the algorithm is complete and the fast filtering step proceeds channel by channel with the same filter for every channel. If $L$2 denotes the $L$3-th channel, the inner-loop iterate is represented in the Fourier domain by

$L$4

After the stopping criterion is met, the current multivariate IMF is assembled as

$L$5

and the residual is updated by subtracting this multichannel component from the current signal (Cicone et al., 2019).

The inner-loop stopping rule is joint across channels: $L$6 The outer loop continues until the number of extrema of $L$7 is less than $L$8, at which point the residual is treated as the trend. This joint stopping criterion is important because it synchronizes convergence across channels rather than allowing different channels to stop at different scales (Cicone et al., 2019).

The computational complexity of MvFIF is

$L$9

where $s \in \mathbb{R}^{n\times m}$0 is the number of channels and $s \in \mathbb{R}^{n\times m}$1 is the signal length. This is the multichannel analogue of univariate FIF’s quasi-linear complexity in the sample size and is the main computational reason MvFIF can scale to long multichannel recordings (Cavassi et al., 2024).

4. Spectral formulation, convergence, and decomposition properties

The fast formulation of MvFIF inherits the spectral structure of scalar IF/FIF. Under periodic extension, the discrete convolution matrix $s \in \mathbb{R}^{n\times m}$2 associated with the filter is circulant, hence diagonalizable by the DFT: $s \in \mathbb{R}^{n\times m}$3 where $s \in \mathbb{R}^{n\times m}$4 is the Fourier matrix and $s \in \mathbb{R}^{n\times m}$5 is diagonal with entries given by the DFT of the filter. In scalar FIF, repeated moving-average subtraction therefore corresponds to repeated multiplication by $s \in \mathbb{R}^{n\times m}$6; in MvFIF the same diagonal operator is applied to every channel, because all channels share the same filter. This is why the method is both fast and analytically tractable (Cicone et al., 2018).

The main convergence statement for MvFIF is finite-time convergence. Given a periodic multivariate signal $s \in \mathbb{R}^{n\times m}$7, a filter vector $s \in \mathbb{R}^{n\times m}$8 derived from a symmetric filter convolved with itself, a filter length $s \in \mathbb{R}^{n\times m}$9 equal to the double average distance between subsequent extrema in $n$0, and a tolerance $n$1, the first IMF is

$n$2

where $n$3 is the minimal integer such that

$n$4

The same paper also states that MvFIF produces a uniquely defined decomposition for a given signal and parameter set, in contrast to projection-based multivariate EMD variants whose output depends on the sampling of projection directions (Cicone et al., 2019).

Several empirical structural properties are associated with MvFIF. When applied to multichannel white Gaussian noise, the extracted IMFs form a quasi-dyadic filterbank: their average power spectral densities occupy bands whose center frequencies are roughly equally spaced on a log-frequency axis. The IMFs are also quasi-orthogonal in the sense that the correlation matrix between distinct IMFs is nearly diagonal. In the same study, MvFIF is reported to be robust to noise perturbation even when the number of channels is increased considerably (Cicone et al., 2019).

The scalar IF literature adds a more granular perspective on frequency resolution. In ideal conditions, IF can resolve two stationary frequencies as close as one DFT bin apart in the discrete periodic setting, provided the filter zero aligns with the high-frequency component. The practical limitation is not the filtering operator itself but scale selection: when a lower-frequency component has sufficiently large amplitude, extrema-based scale estimation may treat the signal as one frequency. This suggests that MvFIF inherits both the high theoretical resolving power of IF/FIF and the practical sensitivity of scale estimation, although in MvFIF the scale is estimated from multivariate rotation rather than scalar extrema alone (Cicone et al., 2021).

5. Relation to adjacent methods and frequent misreadings

MvFIF is most clearly understood by distinguishing three separate axes in the IF family: scalar versus multivariate, one-dimensional time versus multidimensional space, and iterative filtering versus envelope-based or variational decompositions. The following summary captures the role of the main related methods.

Method	Data type	Defining mechanism
FIF	scalar 1D signal	1D convolution filter, FFT acceleration
MvFIF	multichannel 1D signal	common $n$5 from multivariate rotation, then 1D FIF per channel
FIF2 / MIF	scalar multidimensional signal	multidimensional moving average with multidimensional filter
MdMvFIF	space-time multivariate data	FIF2-like spatial step plus MvFIF-like temporal step

MvFIF should not be conflated with FIF2. FIF2 operates on a single multidimensional spatial field $n$6 with a multidimensional filter supported in $n$7; MvFIF operates on a multivariate one-dimensional time series $n$8 with a scalar one-dimensional filter and a rotation-based scale estimate. MdMvFIF combines these two ideas by treating each spatial field at fixed time as a high-dimensional vector and then applying the MvFIF temporal rule to its time evolution (Cavassi et al., 2024).

Relative to EMD and its multivariate variants such as MEMD, the fundamental substitution is envelope interpolation versus convolution-based moving averages. The IF/FIF/MvFIF line does not require any a priori assumptions on the signal, replaces spline-based envelopes with compactly supported filters, and retains a clear mathematical formulation. Relative to MEMD and FA-MVEMD, MvFIF does not rely on projections on the $n$9-sphere, so it avoids the dependence of the decomposition on the choice or sampling of projection directions. Relative to MVMD and MSSA, it does not require an a priori choice of the number of modes or optimization regularization parameters (Cicone et al., 2019).

A second common misreading is to treat MvFIF as “FIF on each channel with a shared post hoc labeling.” The literature states the opposite: the common filter length is estimated first from the joint signal geometry and then enforced across channels, which is precisely why the method guarantees aligned frequency scales. A third misreading is to view MvFIF as a joint space-time method; that role belongs to MdMvFIF, whose temporal step is MvFIF-like but whose spatial step is FIF2-like (Cavassi et al., 2024).

6. Empirical behavior, applications, and later extensions

The 2019 MvFIF study reports synthetic and real-data experiments demonstrating several characteristic behaviors. In a constructed bivariate nonstationary signal containing a common $m$0 Hz component in both channels, MvFIF places that $m$1 Hz component in IMF $m$2 of both channels, with the correct relative phase, whereas channel-wise FIF misaligns it by putting the same oscillation in IMF $m$3 of one channel and IMF $m$4 of the other. In four-channel white Gaussian noise with $m$5 realizations and $m$6 samples, the method exhibits the quasi-dyadic filterbank structure already noted. In ensembles with $m$7, $m$8, and $m$9 channels, each of length $s \in \mathbb{R}^n \times \mathbb{R}$0 and decomposed into $s \in \mathbb{R}^n \times \mathbb{R}$1 IMFs plus trend, the reported computation times are approximately $s \in \mathbb{R}^n \times \mathbb{R}$2 s, $s \in \mathbb{R}^n \times \mathbb{R}$3 s, and $s \in \mathbb{R}^n \times \mathbb{R}$4 s, while the filterbank structure and quasi-orthogonality remain essentially unchanged as the number of channels grows (Cicone et al., 2019).

Real-data case studies include bivariate oceanographic float trajectories, multichannel EEG, and trivariate geomagnetic field measurements. For the North Atlantic float example, MvFIF yields $s \in \mathbb{R}^n \times \mathbb{R}$5 multivariate oscillatory modes plus a trend, with the IMFs represented as rotating trajectories in time. For EEG from an alcoholic subject with $s \in \mathbb{R}^n \times \mathbb{R}$6 electrodes, sampling frequency $s \in \mathbb{R}^n \times \mathbb{R}$7 Hz, and $s \in \mathbb{R}^n \times \mathbb{R}$8 s trials, the method produces decompositions into $s \in \mathbb{R}^n \times \mathbb{R}$9 IMFs plus trend for both $t$0- and $t$1-channel configurations; the cited runtimes are about $t$2 s for $t$3 channels and $t$4 s for $t$5 channels. For ESA Swarm magnetic field data with $t$6 samples and trivariate $t$7, $t$8, and $t$9 components, the cited runtime is $v(t)=\mathbf{v}(t)\in\mathbb{R}^n.$0 s. In these studies MvFIF is consistently faster than MEMD, FA-MVEMD, and MVMD, and the provided implementation of MEMD cannot handle more than $v(t)=\mathbf{v}(t)\in\mathbb{R}^n.$1 channels in the EEG experiment (Cicone et al., 2019).

The later MdMvFIF framework makes the role of MvFIF especially explicit. Given a space-time signal $v(t)=\mathbf{v}(t)\in\mathbb{R}^n.$2, MdMvFIF first performs a spatial FIF2-like filtering with a multidimensional Fokker–Planck filter and then performs a multivariate temporal step by treating each field $v(t)=\mathbf{v}(t)\in\mathbb{R}^n.$3 as a high-dimensional vector. The temporal rotation is measured through

$v(t)=\mathbf{v}(t)\in\mathbb{R}^n.$4

and the time-filter half-support is again computed as the double average distance between subsequent extrema in $v(t)=\mathbf{v}(t)\in\mathbb{R}^n.$5. In this sense, MdMvFIF generalizes MvFIF from multichannel time series to time series of spatial fields while preserving the same geometric principle for temporal scale selection (Cavassi et al., 2024).

The main limitations stated across this literature are also consistent. FFT-based FIF and MvFIF assume periodicity at the boundaries, so nonperiodic data generally require pre-extension to reduce edge artifacts. Strongly nonstationary monocomponents may undergo mode splitting, as in other adaptive decomposition methods. For close-frequency separation, the operator itself is not the bottleneck; the critical issue is the reliability of the scale-selection rule. These constraints delimit MvFIF’s operating regime without altering its central status as a fast, convolution-based, multivariate decomposition method with aligned IMFs, finite-time convergence, and a uniquely defined decomposition (Cicone et al., 2021).