Fourier Transform

Fourier Transform(Q6520159)

Fourier Transform is the next level of the Fourier Series, it comes up with a way to approximate hole functions by using exponentials $e^{ikx}$ that means, unlike Fourier Series can only approximate a function on an interval, now we can approximate functions that are infinitely long.

Example for Fourier transform: We have a signal called $x(t)$ we will represent it in terms of the time domain. We also can represent it in another way which is called $x(f)$ we will represent it in terms of the frequency domain and This is why we called transformation. Fourier Transform is an equivalent representation of the signal.

Fourier Series equation $f(\omega )=\int f(x)e^{-i\omega x}dx$

exponential term $e^{i\omega x}=\cos({\omega x})+i\sin({\omega x}),$

• $f(\omega )$ is the subject of the equation
• $f(x)$ is the time function we calculating the Fourier Series for.
• $i$ represents imaginary numbers,
• $e^{i\omega x}$ is a exponential term
• $\int$ is a integral

-From The Fourier Series and Fourier Transform Demystified