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

by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

正弦波就是一个圆周运动在一条直线上的投影。

频域：从侧面看 相位： 从下面看 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.

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

There are standard range of frequencies in human that have been designated using specific names.

Note that these frequencies are never present as sinosoids in actual EEG data. Instead, we most often always observe a mixture of such frequencies.

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.

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.

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).