IntroductionHiérarchie de section : Signal Analysis > Signal Analysis Introduction > Preliminary Notions > Fourier Transform

Jean Baptiste Joseph Fourier (1768 – 1830) was a french mathematician and physicist who defined the Fourier series, which can apply to vibrations. As far as we are concerned, he demonstrated that any **time varying function** can be divided in** single periodic signals**.

Decomposition of a Signal

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Application

The **Fourier Transform** is an ** algorithm** that can be used for the decomposition a sequence of values – an digital audio signal, for instance – into components of different frequencies. Hence, it can be applied to analyse the spectral components of a sound. The **Fast Fourier Transform** is a variante of the Fourier Transform, which allows the fast calculus of the components.

Window Parameters and Analysis Resolution

Because the analysis is done step by step, it is based on a** window**. This window is applied to a number of **signal samples**, which determines its **width, or size**.

Windows are applied successively in time to the signal until the whole file is analysed. The number of samples of each window is also divided in bars, or "bins". The number of bins is proportional to the number of samples in the window : it is the **FFT size**, which reminds us of the main principle of the Fourier Series. Consequently, the window size and the FFT size determine the resolution of the representation in **frequency** and **time**.

Overlapping Windows

To perform an efficient analysis, windows **overlap**. This "space" between each window is called the **window** **step**. To avoid clicks when overlapping the windows, the windows have a **windowing curve**, which defines the **window type**. These parameters interact with the **sample rate** of the signal.

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