音频处理Lec4：频率分析 – 黑凤梨君的个人网站

首先介绍一些正弦波的基础概念

x(t)=A \sin (2\pi f_0t+\phi)

其中

A 是幅度 amplitude
$f_0$ 是频率 frequency，单位为赫兹（每秒周期数） in hertz (cycles per second)
$\phi$ 是相位偏移 phase offset
$T=1/f_0$ 是周期 period，单位为秒 in seconds
$\omega$ 角频率 Angular frequency 为 $2\pi f$

频率分析 Frequency Analysis

引言 Intro

信号在时域中可能难以理解/分析，但其频谱应能揭示更多关于其形成方式的信息。
While a signal may be difficult to understand/analyse by looking at its time domain, its frequency spectrum should reveal much more information about how it is formed.

听觉 Hearing

高频在耳蜗基底附近产生最强响应。
High frequencies produce their strongest response near the base of the cochlea.

低频在更靠近顶端处产生最强响应。
Low frequencies produce their strongest response nearer the apex.

高频波约为 1,500–20,000 Hz。
中频波约为 600–1,500 Hz。
低频波约为 200–600 Hz。

滤波器组 Filter Bank

滤波器组是一组滤波器，每个滤波器具有不同中心频率，并通过不同频谱区域的频谱能量。
A filterbank is a set of filters arranged such that each has a different centre frequency and passes spectral energy from different spectral regions.

带通滤波器与频谱分析 Bandpass Filters – Spectrum Analysis

带通滤波器通过选定范围内的频率并衰减该范围外的频率。
A bandpass filter passes frequencies within a selected range and attenuates frequencies outside that range.

频谱分析可以使用带通滤波器（BPF）完成。
The job of spectral analysis can be carried out using bandpass filters (BPF).

默认方法是选择等间隔频率，分析仪隔离并测量每个频率周围的能量。
Pick a set of equally spaced frequencies. The analyser isolates and measures the energy around each frequency.

傅里叶分析 Fourier Analysis

频谱分析仪使用傅里叶分析原理确定信号的频率分量。
The spectrum analyser uses the principles of Fourier analysis to determine the frequency components of a signal.

对于采样数字数据，常用离散傅里叶变换（DFT），而快速傅里叶变换（FFT）是计算 DFT 的高效算法。
For sampled digital data, a discrete Fourier transform (DFT) is commonly used, and the fast Fourier transform (FFT) is an efficient algorithm for computing the DFT.

傅里叶级数 Fourier series

傅里叶级数将周期信号表示为谐波相关正弦波之和，适用于理解周期波形。
Fourier series represents periodic signals as sums of harmonically related sinusoids. Useful for understanding periodic waveforms.

傅里叶变换 Fourier transform

傅里叶变换（FT）描述连续时间信号的频率内容，作为一般频率分析概念很有用。
Fourier transform (FT) describes frequency content of continuous-time signals. Useful as a general frequency-analysis concept.

离散傅里叶变换（DFT）

离散傅里叶变换从 N 个数字样本计算有限频率系数集合，直接用于采样音频分析。
Discrete Fourier Transform computes a finite set of frequency coefficients from N digital samples. Used directly in sampled-audio analysis.