理論的な連続フーリエ変換(CFT)と離散フーリエ変換(FFT)を比較検証

# Program Name: Fourier_Transform_Analysis_Simulator_V3
# Creation Date: 2026-01-09
# Purpose: Interactive simulation with PDF export capability

# 1. README
def print_documentation():
    print("=========================================================================")
    print("【フーリエ変換インタラクティブシミュレータ / Fourier Transform Simulator】")
    print("=========================================================================")
    print("■ 目的 (Purpose)")
    print("  矩形パルス、ガウス関数、指数減衰信号の3つの波形について、")
    print("  理論的な連続フーリエ変換(CFT)と離散フーリエ変換(FFT)を比較検証します。")
    print("")
    print("■ 支配方程式 (Governing Equations)")
    print("  1. 矩形パルス (Rectangular Pulse):")
    print("     x(t) = 1 (|t| <= W/2)  ->  X(f) = W * sinc(f * W)")
    print("  2. ガウス関数 (Gaussian Function):")
    print("     x(t) = exp(-a * t^2)   ->  X(f) = sqrt(pi / a) * exp(-(pi * f)^2 / a)")
    print("  3. 指数減衰信号 (Exponential Decay):")
    print("     x(t) = exp(-a * t) * u(t) -> X(f) = 1 / (a + j * 2 * pi * f)")
    print("")
    print("■ 使い方 (Usage)")
    print("  1. [Model Parameters] で波形とパラメータ(aまたはW)を選択。")
    print("  2. [Observed / Input Data] でサンプリング設定を入力。")
    print("  3. [Run Simulation] で計算とプロットを実行。")
    print("  4. [Save PDF] で結果をPDFとしてダウンロード。")
    print("=========================================================================")

# 2. Setup Code
# !pip install numpy matplotlib ipywidgets
import numpy as np
import matplotlib.pyplot as plt
import datetime
import ipywidgets as widgets
from google.colab import files
from IPython.display import display, clear_output

# Global setting
np.random.seed(42)
plt.style.use('seaborn-v0_8-muted')

# 3. TheoreticalModel
class FourierModel:
    def __init__(self):
        self.history = []

    def validate(self, params):
        if params['fs'] <= 0 or params['n_points'] <= 0:
            print("【Error】サンプリング周波数と点数は正の値である必要があります。")
            print("[Error] Fs and N must be positive.")
            return False
        if params['param_a'] <= 0:
            print("【Error】パラメータ(a/W)は0より大きい必要があります。")
            print("[Error] Parameter a/W must be > 0.")
            return False
        return True

    def calculate(self, params):
        fs = params['fs']
        n = params['n_points']
        wave_type = params['wave_type']
        a = params['param_a']
        
        # Time axis (Centered)
        t = np.linspace(-n/(2*fs), n/(2*fs), n, endpoint=False)
        dt = t[1] - t[0]
        
        # 1. Time Domain Signal
        if wave_type == "Rectangular":
            x = np.where(np.abs(t) <= a/2, 1.0, 0.0)
        elif wave_type == "Gaussian":
            x = np.exp(-a * (t**2))
        else: # Exponential Decay
            x = np.where(t >= 0, np.exp(-a * t), 0.0)
            
        # 2. FFT Calculation
        # Shift to align t=0 with index 0 for FFT
        x_fft_input = np.fft.ifftshift(x)
        X_fft_raw = np.fft.fft(x_fft_input) * dt
        X_fft = np.fft.fftshift(X_fft_raw)
        freqs = np.fft.fftshift(np.fft.fftfreq(n, d=dt))
        
        # 3. Theoretical CFT
        f_theo = np.linspace(freqs.min(), freqs.max(), 1000)
        if wave_type == "Rectangular":
            X_theo = a * np.sinc(f_theo * a)
        elif wave_type == "Gaussian":
            X_theo = np.sqrt(np.pi / a) * np.exp(-(np.pi * f_theo)**2 / a)
        else: # Exp Decay
            X_theo_complex = 1 / (a + 1j * 2 * np.pi * f_theo)
            X_theo = np.abs(X_theo_complex)

        return t, x, freqs, np.abs(X_fft), f_theo, np.abs(X_theo)

    def print_calculation_steps(self, params):
        print("\n--- Theoretical Formula / 理論式 ---")
        w_type = params['wave_type']
        val = params['param_a']
        if w_type == "Rectangular":
            print(f"X(f) = {val} * sinc(f * {val})")
        elif w_type == "Gaussian":
            print(f"X(f) = sqrt(pi / {val}) * exp(-(pi * f)^2 / {val})")
        else:
            print(f"|X(f)| = 1 / sqrt({val}^2 + (2 * pi * f)^2)")
        print("FFT normalized by dt to match CFT amplitude.")

# 4. Visualizer
class Visualizer:
    def __init__(self):
        self.fig = None
        self.ghost = None

    def plot(self, t, x, f_fft, X_fft, f_theo, X_theo, wave_type, use_ghost=False):
        self.fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 5))
        
        # Plot 1: Time Domain
        ax1.plot(t, x, label='Current', color='dodgerblue', linewidth=2)
        if use_ghost and self.ghost:
            ax1.plot(self.ghost['t'], self.ghost['x'], '--', color='gray', alpha=0.4, label='Previous')
        ax1.set_title(f'Time Domain: {wave_type}')
        ax1.set_xlabel('Time [s]')
        ax1.set_ylabel('Amplitude')
        ax1.grid(True)
        ax1.legend()

        # Plot 2: Frequency Domain
        ax2.plot(f_fft, X_fft, 'o', markersize=3, label='FFT Result', color='crimson', alpha=0.6)
        ax2.plot(f_theo, X_theo, label='Theory (CFT)', color='black', linewidth=1)
        ax2.set_title('Frequency Domain: Magnitude')
        ax2.set_xlabel('Frequency [Hz]')
        ax2.set_ylabel('|X(f)|')
        ax2.set_xlim(-10, 10)
        ax2.grid(True)
        ax2.legend()
        
        plt.tight_layout()
        plt.show()
        self.ghost = {'t': t, 'x': x}

# 5. UIBuilder
class App:
    def __init__(self):
        self.model = FourierModel()
        self.viz = Visualizer()
        self.out = widgets.Output()
        
        # [Model Parameters]
        self.wave_type = widgets.Dropdown(options=["Rectangular", "Gaussian", "Exp Decay"], value="Rectangular", description="Signal Type")
        self.param_a = widgets.BoundedFloatText(value=1.0, min=0.01, max=10.0, step=0.1, description="a / Width")
        
        # [Observed / Input Data]
        self.fs = widgets.FloatText(value=50.0, description="Fs [Hz]")
        self.n_pts = widgets.IntText(value=1024, description="Points [N]")
        
        # [Environment / Options]
        self.btn_run = widgets.Button(description="Run Simulation / 実行", button_style='primary')
        self.btn_reset = widgets.Button(description="Reset / リセット", button_style='warning')
        self.btn_save = widgets.Button(description="Save PDF / PDF保存", button_style='success')
        self.btn_log = widgets.Button(description="Show Log / 履歴表示")
        self.btn_clear = widgets.Button(description="Clear Log / 履歴消去")
        
        self.btn_run.on_click(self.run)
        self.btn_reset.on_click(self.reset)
        self.btn_save.on_click(self.save)
        self.btn_log.on_click(self.log)
        self.btn_clear.on_click(self.clear)

    def start(self):
        display(widgets.VBox([
            widgets.HTML("<b>[Model Parameters]</b>"), self.wave_type, self.param_a,
            widgets.HTML("<b>[Observed / Input Data]</b>"), self.fs, self.n_pts,
            widgets.HTML("<b>[Environment / Options]</b>"),
            widgets.HBox([self.btn_run, self.btn_reset, self.btn_save]),
            widgets.HBox([self.btn_log, self.btn_clear]),
            self.out
        ]))

    def run(self, _):
        with self.out:
            clear_output(wait=True)
            p = {'wave_type': self.wave_type.value, 'param_a': self.param_a.value, 'fs': self.fs.value, 'n_points': self.n_pts.value}
            if not self.model.validate(p): return
            try:
                res = self.model.calculate(p)
                self.model.print_calculation_steps(p)
                self.viz.plot(*res, p['wave_type'], use_ghost=True)
                self.model.history.append({"time": datetime.datetime.now().strftime("%H:%M:%S"), "params": p})
                print("【Success】計算完了 / Calculation completed.")
            except Exception as e:
                print(f"【Error】{e}")

    def reset(self, _):
        self.wave_type.value = "Rectangular"; self.param_a.value = 1.0
        self.fs.value = 50.0; self.n_pts.value = 1024
        self.viz.ghost = None; self.model.history = []
        with self.out:
            clear_output(); print_documentation()
            print("【Reset】リセット完了 / Reset done.")

    def save(self, _):
        if self.viz.fig is None:
            with self.out: print("【Warning】プロットがありません / No plot.")
            return
        fn = f"fourier_analysis_{datetime.datetime.now().strftime('%Y%m%d_%H%M%S')}.pdf"
        self.viz.fig.savefig(fn, format='pdf')
        with self.out:
            print(f"【Info】PDFをダウンロードします: {fn}")
            files.download(fn)

    def log(self, _):
        with self.out:
            print("\n--- History / 履歴 ---")
            for entry in self.model.history: print(entry)

    def clear(self, _):
        self.model.history = []
        with self.out: print("【Log】履歴消去完了 / Log cleared.")

# 6. Initialization
if __name__ == "__main__":
    print_documentation()
    App().start()