Fourier Transform

傅里叶变换 | Fourier Transform

1. Introduction 简介

The Fourier Transform is a fundamental tool in signal processing, physics, and mathematics, used to decompose functions into their constituent frequencies.

傅里叶变换是信号处理、物理学和数学中的基本工具,用于将函数分解为其构成频率。

2. Definitions 定义

2.1 Continuous Fourier Transform 连续傅里叶变换

Forward Transform 正变换:

$$F(\omega )={\int }_{-\infty }^{\infty }f(t)e^{-i\omega t}dt$$

Inverse Transform 逆变换:

$$f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(\omega )e^{i\omega t}d\omega$$

Where 其中:

• $f(t)$ is the original function in the time domain $f(t)$ 是时域中的原始函数
• $F(\omega )$ is the Fourier transform in the frequency domain $F(\omega )$ 是频域中的傅里叶变换
• $\omega$ is the angular frequency (radians per second) $\omega$ 是角频率 (弧度/秒)

2.2 Discrete Fourier Transform (DFT) 离散傅里叶变换

For a sequence of N complex numbers $\{x_n\}$, $n=0,\dots ,N-1$:

对于 N 个复数的序列 $\{x_n\}$, $n=0,\dots ,N-1$:

Forward Transform 正变换:

$$X_k=\sum _{n=0}^{N-1}{x}_n{e}^{-i2\pi kn/N},k=0,\dots ,N-1$$

Inverse Transform 逆变换:

$$x_n=\frac{1}{N}\sum _{k=0}^{N-1}{X}_k{e}^{i2\pi kn/N},n=0,\dots ,N-1$$

3. Properties 性质

1. Linearity 线性:

\mathcal{F}{af(t) + bg(t)} = aF(\omega) + bG(\omega)

$$2.\ast \ast TimeShift时移\ast \ast :$$

\mathcal{F}{f(t-t_0)} = e^{-i\omega t_0}F(\omega)

$$3.\ast \ast FrequencyShift频移\ast \ast :$$

\mathcal{F}{e^{i\omega_0 t}f(t)} = F(\omega-\omega_0)

$$4.\ast \ast Scaling尺度变换\ast \ast :$$

\mathcal{F}{f(at)} = \frac{1}{|a|}F(\frac{\omega}{a})

$$5.\ast \ast Convolution卷积\ast \ast :$$

\mathcal{F}{f(t) * g(t)} = F(\omega)G(\omega)

$$6.\ast \ast Multiplication相乘\ast \ast :$$

\mathcal{F}{f(t)g(t)} = \frac{1}{2\pi}F(\omega) * G(\omega)

$$7.\ast \ast Parseva{l}^{\prime }sTheorem帕塞瓦尔定理\ast \ast :$$

\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |F(\omega)|^2 d\omega

## 4. Common Transform Pairs 常见变换对 1. **Delta Function δ函数**:

\delta(t) \leftrightarrow 1

其中 $\delta(t)$ 是Dirac Delta function where $\delta(t)$ is the Dirac Delta function

\delta(t) = \begin{cases}

\infty, & t = 0 \

0, & t \neq 0

\end{cases}

且 $\int_{-\infty}^{\infty} \delta(t)dt = 1$ 2. **Constant 常数**: $$1 \leftrightarrow 2\pi\delta(\omega)$$ 3. **Sine and Cosine 正弦和余弦**: $$\sin(\omega_0 t) \leftrightarrow \pi i[\delta(\omega+\omega_0) - \delta(\omega-\omega_0)]$$ $$\cos(\omega_0 t) \leftrightarrow \pi[\delta(\omega+\omega_0) + \delta(\omega-\omega_0)]$$ 4. **Exponential 指数函数**: $$e^{-a|t|} \leftrightarrow \frac{2a}{a^2 + \omega^2}$$ 5. **Gaussian 高斯函数**: $$e^{-at^2} \leftrightarrow \sqrt{\frac{\pi}{a}}e^{-\omega^2/(4a)}$$ 6. **Rectangle Function 矩形函数**: $$\text{rect}(t) = \begin{cases} 1, & |t| < \frac{1}{2} \\ 0, & \text{otherwise} \end{cases} \leftrightarrow \text{sinc}(\omega) = \frac{\sin(\omega/2)}{\omega/2}$$ ## 5. Applications 应用 1. **Signal Processing 信号处理**: - Filtering 滤波 - Compression 压缩 - Modulation 调制 2. **Image Processing 图像处理**: - Edge detection 边缘检测 - Image compression 图像压缩 3. **Differential Equations 微分方程**: - Solving PDEs 求解偏微分方程 4. **Quantum Mechanics 量子力学**: - Wave function analysis 波函数分析 5. **Acoustics 声学**: - Sound wave analysis 声波分析 ## 6. Computation Techniques 计算技巧 1. **Fast Fourier Transform (FFT) 快速傅里叶变换**: - Efficient algorithm for computing DFT - 计算DFT的高效算法 - Reduces complexity from $O(N^2)$ to $O(N\log N)$ - 将复杂度从 $O(N^2)$ 降低到 $O(N\log N)$ 2. **Windowing 窗函数**: - Reduce spectral leakage in DFT - 减少DFT中的频谱泄漏 - Common windows: Hamming, Hann, Blackman - 常见窗函数: 汉明窗、汉宁窗、布莱克曼窗 3. **Zero-padding 零填充**: - Increase frequency resolution in DFT - 增加DFT的频率分辨率 ## 7. Common Pitfalls 常见陷阱 1. Aliasing due to undersampling 欠采样导致的混叠 2. Spectral leakage in finite-length signals 有限长信号中的频谱泄漏 3. Neglecting the scaling factor in inverse DFT 忽略逆DFT中的缩放因子 4. Misinterpreting phase information 错误解释相位信息 ## 8. Derivations and Proofs 推导与证明 ### 8.1 Derivation of the Fourier Transform 傅里叶变换的推导 Starting from the Fourier series for a periodic function with period $T$: 从周期为 $T$ 的周期函数的傅里叶级数开始： $$f(t) = \sum_{n=-\infty}^{\infty} c_n e^{in\omega_0 t}$$ Where $\omega_0 = \frac{2\pi}{T}$ and $c_n = \frac{1}{T}\int_{-T/2}^{T/2} f(t)e^{-in\omega_0 t}dt$ As $T \to \infty$, $\omega_0 \to d\omega$ and $\frac{1}{T} \to \frac{d\omega}{2\pi}$, we get: 当 $T \to \infty$ 时，$\omega_0 \to d\omega$ 且 $\frac{1}{T} \to \frac{d\omega}{2\pi}$，我们得到： $$f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} (\int_{-\infty}^{\infty} f(\tau)e^{-i\omega \tau}d\tau) e^{i\omega t}d\omega$$ This gives us the Fourier transform pair: 这给出了傅里叶变换对： $$F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}dt$$ $$f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)e^{i\omega t}d\omega$$ ### 8.2 Proof of Convolution Theorem 卷积定理的证明 To prove: $\mathcal{F}\{f(t) * g(t)\} = F(\omega)G(\omega)$ Let $h(t) = f(t) * g(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau)d\tau$ Taking the Fourier transform of $h(t)$: 对 $h(t)$ 进行傅里叶变换：

\begin{align*}

H(\omega) &= \int_{-\infty}^{\infty} h(t)e^{-i\omega t}dt \

&= \int_{-\infty}^{\infty} (\int_{-\infty}^{\infty} f(\tau)g(t-\tau)d\tau) e^{-i\omega t}dt \

&= \int_{-\infty}^{\infty} f(\tau) (\int_{-\infty}^{\infty} g(t-\tau)e^{-i\omega t}dt) d\tau

\end{align*}

Let $u = t-\tau$, then $dt = du$: 令 $u = t-\tau$，则 $dt = du$： $$\begin{align*} H(\omega) &= \int_{-\infty}^{\infty} f(\tau) (\int_{-\infty}^{\infty} g(u)e^{-i\omega (u+\tau)}du) d\tau \\ &= \int_{-\infty}^{\infty} f(\tau)e^{-i\omega \tau} (\int_{-\infty}^{\infty} g(u)e^{-i\omega u}du) d\tau \\ &= (\int_{-\infty}^{\infty} f(\tau)e^{-i\omega \tau}d\tau) (\int_{-\infty}^{\infty} g(u)e^{-i\omega u}du) \\ &= F(\omega)G(\omega) \end{align*}

Thus, the convolution theorem is proved.

因此，卷积定理得证。

8.3 Derivation of the Uncertainty Principle 不确定性原理的推导

The uncertainty principle states that a function and its Fourier transform cannot both be highly concentrated.

不确定性原理指出，一个函数及其傅里叶变换不能同时高度集中。

Define the “spread” of $f(t)$ and $F(\omega )$ as:

定义 $f(t)$ 和 $F(\omega )$ 的 ” 展宽 ” 为：

Using Parseval’s theorem and the properties of Fourier transforms, we can show:

使用帕塞瓦尔定理和傅里叶变换的性质，我们可以证明：

This is the mathematical expression of the uncertainty principle.

这是不确定性原理的数学表达。

8.4 Proof of Parseval’s Theorem 帕塞瓦尔定理的证明

To prove: ${\int }_{-\infty }^{\infty }|f(t){|}^{2}dt=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|F(\omega ){|}^{2}d\omega$

Start with the inverse Fourier transform:

从逆傅里叶变换开始：

f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)e^{i\omega t}d\omega$$ Multiply both sides by $f^*(t)$ (complex conjugate of $f(t)$): 两边乘以 $f^*(t)$ ($f(t)$ 的复共轭)： $$|f(t)|^2 = f(t)f^*(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)f^*(t)e^{i\omega t}d\omega$$ Integrate both sides with respect to $t$: 对两边关于 $t$ 积分： $$\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(\omega)f^*(t)e^{i\omega t}d\omega dt$$ Change the order of integration (Fubini's theorem): 改变积分顺序（Fubini定理）： $$\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) (\int_{-\infty}^{\infty} f^*(t)e^{i\omega t}dt) d\omega$$ The inner integral is the complex conjugate of $F(\omega)$: 内部积分是 $F(\omega)$ 的复共轭： $$\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)F^*(\omega)d\omega = \frac{1}{2\pi}\int_{-\infty}^{\infty} |F(\omega)|^2 d\omega

Thus, Parseval’s theorem is proved.

因此，帕塞瓦尔定理得证。