The 1D Fourier transform is a mathematical procedure that allows a signal to be decomposed into its frequency components. On the left side, the sine wave shows a time varying signal. On the right side, you can observe its equivalent in the frequency domain. The sine wave corresponds to a plot at the same frequency as the sine wave and same amplitude as the maximal amplitude of the sine wave.

# Single frequency

The following animation demonstrates the relation between time domain and frequency domain about a sine wave representing a single frequency sound. Moving scroll bars change frequency and amplitude of the sound so that you can see modifications in time and frequency domain. To describe a sine wave, we need its amplitude and frequency, but also its phase. The following animation illustrates the consequence of a phase change on the sine wave (shift in time domain).

# Multiple frequencies

The Fourier transform is a mathematical procedure that decomposes a signal into a sum of sine waves of different frequencies, phases and amplitude. The human ear does the same processing with sounds, which are analyzed as a spectrum of elementary frequencies. Knowing frequency, amplitude and phase of each sine wave, it is possible to reconstruct the signal (inverse Fourier transform).

# Sounds

Even if a signal is very complex, the Fourier transform will always be able to decompose it into its frequency components from which we will reconstruct a signal very similar to the original. (Representing a signal exactly would require an infinite number of frequency components, which is not possible in practice).

Even music can be decomposed by a Fourier Transform. This is employed in spectrum analyser (left side). Note that low frequency components have the highest amplitude.