Struct ComplexPaul 

pub struct ComplexPaul {
    pub order: u32,
}

Complex Paul Wavelet

The Paul wavelet is particularly good for detecting oscillatory patterns and modulated signals in financial time series. It has asymmetric time localization properties.

Mathematical form: ψ(t) = 2^m * i^m * m! / (π * (m + 1/2)!) * (1 - it)^(-m-1)

Fields

order: u32

Order parameter (m)

Implementations

impl ComplexPaul

pub fn new(order: u32) -> Self

Create a new Complex Paul wavelet

pub fn for_oscillations() -> Self

Create version optimized for oscillation detection

Trait Implementations

impl Clone for ComplexPaul

fn clone(&self) -> ComplexPaul

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl ContinuousWavelet for ComplexPaul

fn admissibility_constant(&self) -> f64

ComplexPaul admissibility constant.

Closed form: C_ψ = π / m for m ≥ 1. Derived from C_ψ = ∫₀^∞ |Ψ̂(ω)|²/ω dω for the analytic Paul wavelet Ψ̂(ω) ∝ ω^m · e^{−ω}; the integral evaluates exactly (via Γ(2m)/2^{2m}) to give 1/m modulo normalization-convention factors that bring the ferro-wave-normalized result to π/m.

Empirical match within 0.5% for m=4 (calibrated by icwt(cwt(cos)) across multiple frequencies — see tests/transforms/cwt_reference.rs). Depends on the 2.0.0 fix to the wavelet’s √(π·(2m)!) denominator (was √(π·Γ(m+1.5))).

m = 0 is non-admissible. The Paul wavelet at order 0 degenerates: Ψ̂(ω) ∝ ω^0 · e^{−ω} = e^{−ω} has non-vanishing Ψ̂(0), so the Calderón admissibility integral diverges logarithmically — there is no finite C_ψ. The public constructor ComplexPaul::new(0) is preserved for the existing parameter-validation contract; for icwt purposes the trait default 2π is used as a non-amplitude-correct fallback (consistent with how non-admissible wavelets are handled elsewhere — see e.g. ComplexMexicanHat for ω₀ < 2). icwt(cwt(x), ComplexPaul::new(0)) is therefore shape-correct but amplitude-biased; use m ≥ 1 for amplitude-correct reconstruction.

fn wavelet_function(&self, t: f64, scale: f64) -> Complex<f64>

Get the wavelet function value at position t for scale a

fn center_frequency(&self) -> f64

Get the center frequency of the wavelet

fn bandwidth(&self) -> f64

Get the bandwidth parameter

fn is_complex(&self) -> bool

Check if the wavelet is complex-valued

fn wavelet_function_includes_l2_norm(&self) -> bool

Whether wavelet_function already includes the CWT L2 amplitude factor 1/√a in its return value.

fn supports_frequency_domain(&self) -> bool

Check if this wavelet supports direct frequency-domain computation

fn sst_inversion_constant(&self) -> f64

Synchrosqueezing inversion constant R_ψ = ∫₀^∞ Ψ̂*(ω) dω / ω, used by crate::transform::sswt_extract_component to recover an EMD-like component from an SST ridge.

fn wavelet_fft( &self, _buffer: &mut [FftComplex<f64>], _scale: f64, _fft_size: usize, )

Compute the wavelet directly in frequency domain (if supported)

fn recommended_scales( &self, signal_length: usize, sampling_freq: f64, ) -> Vec<f64>

Get recommended scale range for a given signal length and sampling frequency

impl Debug for ComplexPaul

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Default for ComplexPaul

fn default() -> Self

Returns the “default value” for a type.