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.

 

[tacman]

for the question and for taking the time to read my article on the Fourier Transform. To answer your question, the Fourier Transform is a mathematical tool that allows us to break down a complex signal into its individual sinusoidal components. This is because any signal, no matter how complex, can be represented as a combination of sinusoidal waves with different frequencies, amplitudes, and phases.

In other words, the Fourier Transform allows us to analyze a signal in terms of its frequency content. By looking at the Fourier Transform of a signal, we can see how much of each frequency component is present in the signal. This is why we can say that the Fourier Transform tells us "how much sinusoid" there is in the waveform at a given frequency "w".

As for its application in linear circuit analysis, the Fourier Transform is a powerful tool that allows us to analyze the behavior of circuits in the frequency domain. By taking the Fourier Transform of a circuit's input and output signals, we can determine the frequency response of the circuit, which tells us how the circuit will behave at different frequencies. This is crucial in designing and analyzing electronic circuits, as different components and systems have different frequency responses.

I hope this helps to clarify the concept of the Fourier Transform and its application in circuit analysis. Thank you for your interest and happy learning!