Artice M. David: The Fourier Transform

[goodness52200]

Hello everyone ^^

Why I can say "The Fourier transform tells us " how much sinusoid" there is in the waveform at a given frequency "w""

Form Linear circuit analysis by Artice M. David

thanks a lot

 

[quasar987]

It is meant in the following sense...

if you take the Fourier transform of a function F(x), the result is a function A(w), right?

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}F(x)e^{-iwx}dx=A(w)$$

Well it turns out that this function A(w) is such that

$$F(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}A(w)e^{iwx}dw$$

i.e. A(w) is the amplitude of the sine wave $e^{iwx}$ of frequency w. So the bigger A is for a given w, the bigger the contribution of this sine wave in the integral making up F(x).

You must think in terms of analogy with Fourier series.