scde_demo_wls_dumped.py

# -*- coding: utf-8 -*-
"""A demonstration of Frequency Estimation Based on WLS-SCDE for a dumped sinusoid.

Copyright (C) 2025 by Akira TAMAMORI

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
"""

import numpy as np
import numpy.typing as npt
from scipy.signal.windows import get_window


def calculate_observables(
    frame: npt.NDArray[np.float64], fs: float
) -> tuple[
    npt.NDArray[np.complex128],
    npt.NDArray[np.complex128],
    npt.NDArray[np.complex128],
    npt.NDArray[np.float64],
]:
    """Calculates the observables (gw, hw, kw) from a signal frame.

    These are Fourier coefficients weighted by a window function and its derivatives.
    """
    n_samples = len(frame)
    fft_size = 1 << (n_samples - 1).bit_length()

    # Use a window function whose value and derivatives are zero at the boundaries.
    # The Blackman-Harris window has excellent properties for this purpose.
    p_t = get_window("blackmanharris", n_samples)
    dt = 1.0 / fs
    dp_t = np.gradient(p_t, dt)
    d2p_t = np.gradient(dp_t, dt)

    # Calculate Fourier coefficients for the signal weighted by p(t), p'(t), and p''(t).
    gwk_full = np.fft.rfft(frame * p_t, n=fft_size)
    hwk_full = np.fft.rfft(frame * dp_t, n=fft_size)
    kwk_full = np.fft.rfft(frame * d2p_t, n=fft_size)

    freq_bins_hz = np.fft.rfftfreq(fft_size, 1 / fs).astype(np.float64)
    omega_k_full = 2 * np.pi * freq_bins_hz

    return gwk_full, hwk_full, kwk_full, omega_k_full


def select_bins_for_damped_scde(
    gwk_full: npt.NDArray[np.complex128],
    omega_k_full: npt.NDArray[np.float64],
    fs: float,
    num_bins: int = 7,
) -> npt.NDArray[np.int64]:
    """Selects a set of frequency bins around the spectral peak."""
    freq_bins_hz = omega_k_full / (2 * np.pi)
    mag_spec = np.abs(gwk_full)

    search_range = (freq_bins_hz > 50) & (freq_bins_hz < fs / 2 * 0.95)
    if not np.any(search_range):
        return np.array([], dtype=int)

    search_indices = np.where(search_range)[0]
    mag_spec_searched = mag_spec[search_indices]

    center_peak_local_idx = np.argmax(mag_spec_searched)

    half_bins = num_bins // 2
    start_local_idx = np.maximum(0, center_peak_local_idx - half_bins)
    end_local_idx = start_local_idx + num_bins

    if end_local_idx > len(mag_spec_searched):
        end_local_idx = len(mag_spec_searched)
        start_local_idx = max(0, end_local_idx - num_bins)

    selected_local_indices = np.arange(start_local_idx, end_local_idx)

    return search_indices[selected_local_indices]


def solve_complex_a_sq_wls(
    gwk: npt.NDArray[np.complex128],
    hwk: npt.NDArray[np.complex128],
    kwk: npt.NDArray[np.complex128],
    omega_k: npt.NDArray[np.float64],
) -> np.complex128 | None:
    """Solves for the complex squared frequency using Weighted Least Squares."""
    if len(gwk) < 3:
        return None

    # Equation: (gwk) * ã² = (ωk^2 * gwk + 2jωk * hwk - kwk)
    coeff = gwk
    b = omega_k**2 * gwk + 2j * omega_k * hwk - kwk

    # Weight W is the power |gwk|^2
    w = np.abs(gwk) ** 2

    # Solve the weighted normal equation: (A^{H}WA)x = (A^{H}Wb)
    coeff_h = coeff.conj()
    lhs = np.sum(coeff_h * w * coeff)
    rhs = np.sum(coeff_h * w * b)

    if np.abs(lhs) < 1e-12:
        return None

    complex_freq: np.complex128 = rhs / lhs
    return complex_freq


def separate_params_from_complex_a_sq(
    complex_a_sq: np.complex128, peak_omega: np.float64
) -> tuple[np.float64, np.float64]:
    """Separates frequency (a) and damping (ε) from the complex a^2."""
    real_part = complex_a_sq.real
    imag_part = complex_a_sq.imag

    # Im(ã^2) ≈ 2ωε
    if peak_omega < 1e-6:
        epsilon = np.float64(0.0)
    else:
        epsilon = imag_part / (2 * peak_omega)

    # Re(ã^2) ≈ a^2 + ε^2
    a_sq = real_part - epsilon**2
    if a_sq < 0:
        return np.float64(np.float64(np.nan)), np.float64(np.float64(np.nan))

    a: np.float64 = np.sqrt(a_sq)
    return a, epsilon


def estimate_damped_sinusoid(
    frame: npt.NDArray[np.float64], fs: float
) -> tuple[np.float64, np.float64]:
    """Estimates the frequency (fo) and damping factor (ε) of a single damped sinusoid.

    This function is a refactored version with separated responsibilities.

    Args:
        frame (np.ndarray): The input signal frame.
        fs (int): The sampling frequency in Hz.

    Returns:
        tuple[float, float]: The estimated (fo, ε) pair.
                             Returns (np.float64(np.nan), np.float64(np.nan))
                             if estimation fails.
    """
    if len(frame) < 50 or np.var(frame) < 1e-10:
        return np.float64(np.nan), np.float64(np.nan)

    # Step 1: Calculate the necessary observables (gw, hw, kw) from the signal.
    gwk_full, hwk_full, kwk_full, omega_k_full = calculate_observables(frame, fs)

    # Step 2: Select the most informative frequency bins around the spectral peak.
    selected_indices = select_bins_for_damped_scde(
        gwk_full, omega_k_full, fs, num_bins=7
    )

    if selected_indices.size < 3:
        return np.float64(np.nan), np.float64(np.nan)

    # Step 3: Solve for the complex squared frequency using WLS.
    complex_a_sq = solve_complex_a_sq_wls(
        gwk=gwk_full[selected_indices],
        hwk=hwk_full[selected_indices],
        kwk=kwk_full[selected_indices],
        omega_k=omega_k_full[selected_indices],
    )

    if complex_a_sq is None:
        return np.float64(np.nan), np.float64(np.nan)

    # Step 4: Separate the physical parameters (a, ε) from the complex solution.
    index: np.int64 = selected_indices[np.argmax(np.abs(gwk_full[selected_indices]))]
    peak_omega = omega_k_full[index]
    a, epsilon = separate_params_from_complex_a_sq(complex_a_sq, peak_omega)

    if np.isnan(a):
        return np.float64(np.nan), np.float64(np.nan)

    # Convert angular frequency to Hz and perform final validation.
    f0_hz = a / (2 * np.pi)
    if not 50 < f0_hz < fs / 2:
        return np.float64(np.nan), np.float64(np.nan)

    return f0_hz, epsilon


def main() -> None:
    """Perform demonstration."""
    fs = 44100
    duration = 0.1
    f0_true = 440.0
    epsilon_true = 10.0  # damped coeff.
    noise_level = 0.05

    t = np.arange(int(fs * duration)) / fs
    test_signal = np.exp(-epsilon_true * t) * np.cos(2 * np.pi * f0_true * t)
    f0_est, epsilon_est = estimate_damped_sinusoid(test_signal, fs)
    print("--- Damped Sinusoid Estimation (original) ---")
    print(f"True values:      f0 = {f0_true:.2f} Hz, epsilon = {epsilon_true:.2f}")
    print(f"Estimated values: f0 = {f0_est:.2f} Hz, epsilon = {epsilon_est:.2f}")

    print("")
    test_signal += np.random.randn(len(test_signal)) * noise_level  # add small noise
    f0_est, epsilon_est = estimate_damped_sinusoid(test_signal, fs)
    print("--- Damped Sinusoid Estimation (noise) ---")
    print(f"True values:      f0 = {f0_true:.2f} Hz, epsilon = {epsilon_true:.2f}")
    print(f"Estimated values: f0 = {f0_est:.2f} Hz, epsilon = {epsilon_est:.2f}")


if __name__ == "__main__":
    main()