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.

You must list out $i,k,x,t,f$ individually, and state individually what they are explicitly.

For example:

where:

• $i$ represents imaginary numbers,
• $k$ is so and so,
• $x(t)$ is the strength of the signal over time,
• $t$ represents time,
• $f(x)$ or $x(f)$ represents the frequency function....

This must apply to all equations ...

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

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

-From The Fourier Series and Fourier Transform Demystified

In this equation f(x) is the time function we're calculating the Fourier series for. Then we times it by exponential$e^{i\omega x}$.