# 时间频率分析 基础

## 三角函数

### `y＝Asin（ωx＋φ)+b`

#### `A` 振幅

振幅变换:纵坐标伸长或缩短到原来的`|A|`倍

#### `ω` 相位

`T`周期 `f=1/T`频率

周期变换:横坐标伸长或缩短到原来的`1/ω`倍

#### `φ`初相

相位变换：横坐标向左或向右平移`|φ|`个单位 若由`y＝sin（ωx)`得到`y＝sin（ωx＋φ)`的图象，则向左或向右平移应平移`|φ/ω|` 个单位

#### `b`y轴方向的平移

图象上所有的点向上或向下平行移动`｜b｜`个单位

## [时间&频率](https://jingyan.baidu.com/article/cbf0e500f1ce562eaa2893f4.html)

正弦波就是一个圆周运动在一条直线上的投影。 ![](https://i.imgur.com/Lxzzq13.gif)

频域：从侧面看 ![](https://i.imgur.com/tDJPo4V.gif) 相位： 从下面看 ![](https://i.imgur.com/q4h6Bd9.png) 相位变换就是坐标轴的伸长或者是缩短。相位大小轴表示了变化的scale. ![](https://i.imgur.com/8gOEDBH.png)

时间差并不是相位差。如果将全部周期看作2Pi或者360度的话,相位差则是时间差在一个周期中所占的比例。将时间差除周期再乘2Pi,就得到了相位差。由于cos(t+2Pi)=cos(t),所以相位差是周期的,pi和3pi,5pi,7pi都是相同的相位

#### [Time frequency decompostions](https://sccn.ucsd.edu/wiki/Time_frequency_tutorial)

There are standard range of frequencies in human that have been designated using specific names. ![](https://i.imgur.com/xMdANSn.png)

Note that these frequencies are never present as sinosoids in actual EEG data. Instead, we most often always observe a mixture of such frequencies. ![](https://i.imgur.com/j3dunPa.png)

Actual EEG signal can be seen as a mixture of different frequencies. As shown below, when mixing 2Hz, 10Hz, and 20Hz signals, a complex signal may be observed. ![](https://i.imgur.com/o90Ns5F.png)

If we run a simple Fourier Transform on this data (we will see later in this document what is actually a Fourrier Transform), then we will be able to observe 3 peaks of the same amplitude at 2, 10 and 20 Hz. ![](https://i.imgur.com/qvKlEtI.png)

Now, if we take a real EEG signal, it is possible to decompose such signal into a superposition of signal at different sinusoidal signal at different frequencies. Note that not only the amplitude of such sinoisoid but also their "phase" (the horizontal offset).

![](https://i.imgur.com/RikBzGG.png)


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