Fourier Transform and Power/Phase analysis

傅里叶变换简介^[1]

经傅里叶变换生成的函数 ${\hat {f}}$ 称作原函数 $f$ 的傅里叶变换，亦称**频谱**。通常情况下， $f$ 是实数函数，而 ${\hat {f}}$ 则是复函数，用一个复数来表示振幅和相位。

一般情况下，若“傅里叶变换”一词不加任何限定语，则指的是“连续傅里叶变换”（连续函数的傅里叶变换）。定义傅里叶变换有许多不同的方式。本文中采用如下的定义：（连续）傅里叶变换将可积函数 $f:\mathbb {R} \rightarrow \mathbb {C}$ 表示成复指数函数的积分或级数形式。即：
$$
{\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }dx = \int _{-\infty }^{\infty }f(t)\ e^{-2\pi ift }dt
$$

• $ξ$, $f$为任意实数

自变量x表示时间（以秒为单位），变换变量 $ξ$ 表示频率（以赫兹为单位）。在适当条件下，${\hat {f}}$ 可由逆变换（inverse Fourier transform）由下式确定 $f$：
$$
f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi i\xi x}d\xi
$$

• $x$ 为任意实数

对于傅里叶系数（Fourier Coefficient）的解释[^Source]：

• Amplitude spectrum is magnitude of Fourier coefficients
• Phase spectrum is angle of Fourier coefficients

计算能量与功率^[2]

对于信号 $f(t)$ ，其能量 $E$ 的定义为：
$$
E = \lim_{T \rightarrow \infty} \int^{T}_{-T} \lvert{f(t)}\rvert^{2}dt
$$

其功率 P 的定义为：

$$
P = \lim_{T \rightarrow \infty} \frac{1}{2T} \int^{T}_{-T} \lvert{f(t)}\rvert^{2}dt = \frac{E \text{ (Total Energy)}}{\text{Total time}}
$$
可以用物理学中电阻的功率计算来便于理解：

对于电阻R，施加电压$f(t)$，在区间(-∞,+∞)上，其能量 $E$ 就是：
$$
E = \frac{1}{R} \int^{\infty}_{-\infty} \lvert{f(t)}\rvert^{2}dt
$$
在信号处理的过程当中，取R的值为1，即为该信号的能量计算公式。

计算相位

相位简介^[3]

相位（英语：Phase）又称位相、相、相角，是描述信号波形变化的度量。

计算

对于傅里叶变换之后的数据（Fourier Coefficient，为一个复数），可以通过求该数据的相位角来获得其相位（Phase）。

在MATLAB中可以使用angle()函数实现：

N = 10; % Length of the sequence
fft_data = fft(data); % 应用快速傅里叶变换
phase = angle(fft_data(1:N/2+1); %计算相位

计算功率谱密度

开始之前需要注意的地方：

• Two limitations of the Fourier-transform based frequency representation
  1. Changes in frequency structure over time are difficult to visualize
  2. EEG data violate the stationarity assumption of Fourier analysis

PSD简介

Power Spectral Density (PSD): A Power Spectral Density (PSD) is the measure of signal’s power content versus frequency. A PSD is typically used to characterize broadband random signals. The amplitude of the PSD is normalized by the spectral resolution employed to digitize the signal.

谱密度估计 (Spectral density estimation)^{<span class=”hint–top hint–rounded” aria-label=”Spectral density estimation - Wikipedia}

[^Source]: Fourier coefficients - YouTube“>[4]

Many other techniques for spectral estimation have been developed to mitigate the disadvantages of the basic periodogram. These techniques can generally be divided into non-parametric, parametric, and more recently semi-parametric (also called sparse) methods。

1. non-parametric spectral density estimation

Following is a partial list of non-parametric spectral density estimation techniques:

• Periodogram, the modulus squared of the discrete Fourier transform
• Bartlett’s method is the average of the periodograms taken of multiple segments of the signal to reduce variance of the spectral density estimate
• Welch’s method a windowed version of Bartlett’s method that uses overlapping segments
• Multitaper is a periodogram-based method that uses multiple tapers, or windows, to form independent estimates of the spectral density to reduce variance of the spectral density estimate
• Least-squares spectral analysis, based on least squares fitting to known frequencies
• Non-uniform discrete Fourier transform is used when the signal samples are unevenly spaced in time
• Singular spectrum analysis is a nonparametric method that uses a singular value decomposition of the covariance matrix to estimate the spectral density
• Short-time Fourier transform
• Critical filter is a nonparametric method based on information field theory that can deal with noise, incomplete data, and instrumental response functions

2. parametric spectral density estimation

Below is a partial list of parametric techniques:

• Autoregressive model (AR) estimation, which assumes that the nth sample is correlated with the previous p samples.
• Moving-average model (MA) estimation, which assumes that the nth sample is correlated with noise terms in the previous p samples.
• Autoregressive moving average (ARMA) estimation, which generalizes the AR and MA models.
• MUltiple SIgnal Classification (MUSIC) is a popular superresolution method.
• Maximum entropy spectral estimation is an all-poles method useful for SDE when singular spectral features, such as sharp peaks, are expected.

Reference

1. 傅里叶变换 - 维基百科，自由的百科全书 (wikipedia.org) ↩
2. 能量信号和功率信号的分别 - 知乎 (zhihu.com) ↩
3. 相位 - 维基百科，自由的百科全书 (wikipedia.org) ↩
4. Spectral density estimation - Wikipedia[^Source]: Fourier coefficients - YouTube ↩