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.

Fourier transforms integral equation : ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx,\quad \forall \ \xi \in \mathbb {R} .$

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.

Convolution equation : $(f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .$