# Fourier analysis: the simplification of a complex waveform into simple component sine waves of different amplitudes and frequencies

### Abstract

Fourier analysis is the simplification of a complex waveform into simple component sine waves of different amplitudes and frequencies.^{1} A discussion on Fourier analysis necessitates reiteration of the physics of waves. A wave is a series of repeating disturbances that propagate in space and time.^{2}

- Frequency: the number of oscillations,
^{2}or cycles per second. It is measured in Hertz and denoted as 1/time or s^{-1}. - Fundamental frequency: the lowest frequency wave in a series. It is also known as the first harmonic. Every other wave in the series is an exact multiple of the fundamental frequency.
^{2} - Harmonic: whole number multiples of the fundamental frequency.
^{1} - Amplitude: the maximum disturbance or displacement from zero caused by the wave. This is the height of the wave.
^{2} - Period: time to complete one oscillation.
^{2} - Wavelength: physical length of one complete cycle.
^{2}This can be between two crests or two troughs. The higher the frequency, the shorter the wavelength. - Velocity: frequency x wavelength.
- Phase: displacement of one wave compared to another, described as 0°–360°.

A sine wave is a simple wave. It can be depicted as the path of a point travelling round a circle at a constant speed, defined by the equation ‘y = sinx’.^{3} Combining sine waves of different frequency, amplitude and phase can yield any waveform,^{3} and, conversely, any wave can be simplified into its component sine waves.

Fourier analysis is a mathematical method of analysing a complex periodic waveform to find its constituent frequencies (as sine waves). Complex waveforms can be analysed, with very simple results. Usually, few sine and cosine waves combine to create reasonably accurate representations of most waves. Fourier analysis finds its anaesthetic applications in invasive blood pressure, electrocardiogram (ECG) and electroencephalogram (EEG) signals, which are all periodic waveforms.^{2} It enables monitors to display accurate representations of these biological waveforms.

Fourier analysis was developed by Joseph Fourier, a mathematician who analysed and altered periodic waveforms.^{2} It is done by computer programmes that plot the results of the analysis as a spectrum of frequencies with amplitude on the y-axis and frequency on the x-axis.

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