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： ei2mut =cos(2ut)+isin(2nut) 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{Uk}: All Functions {ek(t)} Harmonics Transform ak =f.ek ak =f.el2mukt f(t)ek(t)dt = f(t)euk!dt Inverse f(t)=∑akek(t) f()=∑ake2mst 17
傅里叶级数 17 对于一个有限集合
傅里叶变换 将uk换为频率u,则有 Fourier Series Fourier Transform Transform ak =f.ei2nukt F(u)=f.ei2rut f(teu!dt f(t)eiut dt Inverse f0=∑ae2:f= F(u)e2mu du 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): F(U)=f(t)e-i2xut dt F(u)is the Fourier Transform of f(t):F(u)=F(f(t)) To turn the weights back into the signal(invert the transform): f(t)=F(u)ei2sut du f(t)is the Inverse Fourier Transform of F(u):f(t)=F-1(F(u)) 19
傅里叶变换 19
傅里叶变换 How to handle the complex numbers: F(四)=f0e2ct f(t)[cos(-2πut)+isin(-2πut)】dt f(t)cos(-2xut)dt+ if(t)sin(-2πut)at Real-valued Integral Real-valued Integral So,all we're really doing is projecting onto a cosine (one integral) projecting onto a sine(the other integral) encoding the result as a complex number 20
傅里叶变换 20