A sound perceived by our ear may be represented as a function of time: a pressure changing over time. We have seen a pure tone that may be represented as a sinusoidal function over time: the pressure at our ear varies up and down with a sinusoidal trend.

But the same sound may also be represented as a function of frequency. We have seen how an ideal pure tone should be represented in the frequency domain as well as the frequency plot of a real pure tone generated by the speakers of our computer.

We have two different variables (t and f) and two different functions: g(t) and G(f). g(t) and G(f) represent the same physical system (a sound in our case) in two different domains. Any sound may be represented both in the time domain, with a certain g(t), and in the frequency domain, with the corresponding G(f).

Since both functions describe the same system, it is not difficult to understand that they are connected to each other. The mathematical calculation allowing to transform g(t) in the corresponding G(f) is called Fourier Transform (FT).