Fourier series and the Fourier transform (ii)

Fourier series and the Fourier transform (a)

Book connected to the back, on the back gives the Fourier expansion of the function:

$f(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty }\left [{\color{Red} \int_{-\infty }^{\infty }f(t)e^{-j\omega_{i}t}dt} \right ]e^{j\omega_{i}t}d\omega$

Note For periodic functions, the lower limit becomes a cycle interval of integration are integer multiples of the establishment, such as for a period of $2\pi$ function are:

$f(t)=\frac{1}{2\pi}\int_{-\pi }^{\pi }\left [ \int_{-\pi }^{\pi }f(t)e^{-j\omega_{i}t}dt \right ]e^{j\omega_{i}t}d\omega=\frac{1}{2\pi}\int_{0 }^{2\pi }\left [ \int_{0}^{2\pi }f(t)e^{-j\omega_{i}t}dt \right ]e^{j\omega_{i}t}d\omega$

Not to prove here.

We note that the red part of the house style ${\color{Red} \int_{-\infty }^{\infty }f(t)e^{-j\omega_{i}t}dt}$ in the points after the variable $t$ has nothing to do, but about $\omega_{i}$ function $F(\omega)$ .

In fact ${\color{Red}F(\omega)= \int_{-\infty }^{\infty }f(t)e^{-j\omega_{i}t}dt}$ that is a function of $f(t)$ the Fourier transform, and

${\color{Red} f(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty }\left [{\color{Red} F(\omega)} \right ]e^{j\omega_{i}t}d\omega}$ This transformation is the inverse transform. So far Fourier transform be deduced finished, but there are a lot Fourier transform is defective, if the $f(t)$ emergence of some break points in the domain of definition, the Fourier transform is meaningless, but not elaborated here.

In computer graphics and signal Fourier transform learning in many applications, but can only deal with discrete subject to computer calculations, so we need to transform the discrete observation transformation:

${\color{Red}F(\omega)= \int_{-\infty }^{\infty }f(t)e^{-j\omega_{i}t}dt}$

We need to find a discrete summing transformation to approximate the integral transformation. Function is provided $f(t)$ for the period $2\pi$ of the periodic function for integrating $\frac{1}{2\pi}\int_{0}^{2\pi}f(t)e^{-j\omega_{i} t}$ , using the definition of definite integral, we divided the integration domain $(0,2\pi)$ into several parts by a length of $\frac{2\pi}{N}$ a portion, approximate:

$\int_{0 }^{2\pi }f(t)e^{-j\omega_{i}t}dt\approx\frac{1}{N}\sum_{n=0}^{N-1} f(\frac{1}{N}n)\ \cdot e^{-j\omega_{i} \frac{2\pi}{N}n}$

We make $K=\omega_{i}$ , $Y_{n}=f(\frac{2\pi}{N}n)$ , $W = e ^ {j \ frac {2 \ pi} {N}}$

Finally get

$F(K)=\frac{1}{N}\sum_{n=0}^{N-1}Y_{n}W^{-nK}$

We used to approximate the inverse transform, by the ${\color{Red} f(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty }\left [{\color{Red} F(\omega)} \right ]e^{j\omega_{i}t}d\omega}$

Approximated $Y_{n}=\sum_{m=0}^{M-1} F(K)} W^{nK}$

Fourier series and the Fourier transform (ii)

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