函数的基:调和函数 调和函数可以作为输入函数的一组正交基: cos(2Tut)+i sin(2rut) The real part is a cosine of frequency u The imaginary part is a sine of frequency u 16
函数的基: 调和函数 16 调和函数可以作为输入函数的一组正交基：
傅里叶级数 对于一个有限集合{4k} All Functions ek(t) Harmonics feer Transform ak=f ek 2丌ukt f(t)ek(t)dt =/( 2丌ukt at Inverse f()=∑a0e())=∑ ak e cut k k 17
傅里叶级数 17 对于一个有限集合
傅里叶变换 和将换为频率u,则有 Fourier series Fourier transform Transform 2丌ukt f(t) dt f(t)e nverse (t)=∑ e/2TUk t 2丌u 18
傅里叶变换 18 将uk换为频率u，则有
傅里叶变换 To get the weights(amount of each frequency) -12Tut F(u)is the Fourier Transform of f(t): F(u=F((t)) To turn the weights back into the signal (invert the transform F(u)e/2r ut du f(t) is the Inverse Fourier Transform of F(u: f(t)=F(F(u 19
傅里叶变换 19
傅里叶变换 How to handle the complex numbers: F(u 12ut e f(t)[cos(-2T ut)+i sin(-2 ut) dt f(t) cos(-2Tut)dt +i/f(t)sin(-2rut)dt Real-valued Integral Real-valued Integral So, all were really doing is projecting onto a cosine(one integral) e projecting onto a sine( the other integral) encoding the result as a complex number
傅里叶变换 20