**Fourier transformation.**

Following **Fourier**, any periodic function y(t) which repeats itself with a period T:

y(t) = y (t + nT), where n = 0, ±1, ±2, ±3...

can be expanded in a **Fourier** series:

y(t) = y_{0} + S_{n} [C_{n }cos((n**2p/**T) ^{. }t) + S_{n }sin((n**2p/**T) ^{. }t)) ]

Where C_{n} and S_{n} are the appropriate Fourier coefficients. The term T is the period of the expansion.

Using the **Euler equation exp( i t) = cos(t) + i sin(t)** the last

y(t) = S_{n} [E_{n }exp((n**2p/**T) ^{. }t) ]

If angular frequency is used in the expansion, the term (n**2p/**T) ^{. }t) yields just nw ^{. }t. instead of n**2p**f ^{.} t, which makes the equations simpler.

**Fourier transformation** is used to obtain the values of coefficients C_{n}, S_{n} or E_{n } and therefore reveal the periodicity of a function since the result gives us the amplitudes and the frequencies of the simple harmonic (sin and cos) functions which, when combined, produce the complex periodic function y(t). This process is also called **Fourier analysis **of a complex
function.

**Inverse Fourier transformation **is used to obtain the complex periodic function y(t) if the frequencies and corresponding amplitudes are already known. It can be called **Fourier synthesis** of a complex function from simple periodic ones.