The Fast Fourier Transform algorithm is a way of calculating the coefficients of all constituent frequencies, real and imaginary, making up a particular waveform, sound, or function.
These coefficients are known as the Fourier series and are governed by the following equation (Cross, 1999):

            Where : x(t) is the function being analysed
            a_k is the k^th real coefficient
            b_k is the k^th imaginary coefficient
            f₀ is the fundamental frequency

The FFT takes as its input an array representing x(t) (the sound being analysed) and outputs all values of a_k and b_k between the fundamental frequency and the Nyquist Frequency. The combination of the real, imaginary and negative frequency coefficients can be used to recreate the sound in its original form. As this project is mainly concerned with the static representation of the initial set of frequencies produced by a drum hit, it will only be looking at the magnitude of each frequency summed over a finite period of time taken from the start of each drum hit. This is calculated by (Cross, 1999):

It is suggested that the time period to be observed be initially set to about 100ms, which should be large enough to capture most the harmonics created in the resonant structures of all relevant drum types.

3.2.3. Noise reduction

Section 2.1.3 discusses the sources of noise present in sampled breaks. Figure 2 shows a typical representation of a break (amplitude vs time) obtained from Recycle.