Sine and Cosine Fourier Transforms in IB Mathematics | IB数学：正弦与余弦傅里叶变换

📚 Sine and Cosine Fourier Transforms in IB Mathematics | IB数学：正弦与余弦傅里叶变换

From the vibration of a guitar string to the signal in a mobile phone, many real-world quantities behave like combinations of sine and cosine waves. The Fourier transform is the mathematical tool that measures how much of each frequency is present in a function. In this article we focus on the two half-line forms of that idea: the Fourier sine transform and the Fourier cosine transform.

从吉它弦的振动到手机中的信号，许多现实世界的量都表现得像正弦波和余弦波的组合。傅里叶变换是一种数学工具，用来衡量一个函数中含有多少不同频率的分量。本文关注这个思想的两种半直线形式：傅里叶正弦变换与傅里叶余弦变换。

1. Frequencies and Spectra | 频率与频谱

A periodic function such as sin(πt) repeats itself after a fixed period. A non-periodic function, for example a single pulse that appears once, does not. Fourier’s great insight was that even non-periodic signals can be interpreted as an infinite mixture of sine and cosine waves with all possible frequencies.

周期函数如sin(πt)会按固定周期重复。非周期函数，例如只出现一次的脉冲，则不会重复。傅里叶的伟大洞见是：即使是非周期信号，也可以被理解为具有各种可能频率的正弦波与余弦波的无限叠加。

In the sine and cosine forms of the Fourier transform, we study a function f(t) only for t ≥ 0. The variable ω represents angular frequency, often measured in radians per second. The output F(ω) is called the spectrum of f.

在正弦与余弦形式的傅里叶变换中，我们只研究定义在 t ≥ 0 上的函数 f(t)。变量 ω 表示角频率，通常以弧度每秒为单位。输出 F(ω) 被称为 f 的频谱。

2. From Fourier Series to Fourier Transform | 从傅里叶级数到傅里叶变换

You may already know that a periodic function can be expanded as a Fourier series: a sum of terms such as a₀, a₁cos(t), b₁sin(t), a₂cos(2t), b₂sin(2t), and so on. The Fourier transform is the limit of this idea as the period tends to infinity.

你可能已经知道，周期函数可以展开为傅里叶级数：一项项 a₀、a

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