Basic mathematics

The operations required for quantifying signals range from the simple arithmetic operations to basic trigonometry, logarithms and exponentials:

1. Basic operations:
   1. Addition is used in the definition of complex source signals.
   2. Multiplication is used in the definition of signal scaling (amplification and attenuation), and in both amplitude and frequency modulation.
2. Functions: a system is modelled as a function which maps an input signal, for each point in time, on to an output signal.
3. Series: the Fourier series of signals, in which frequencies of signals in the series are multiplied by integer multiples of the fundamental frequency, and their amplitudes are divided by the same integer multiples, defines the basic harmonic signal at the speech signal source.
4. Vectors: segments in the speech signal stream (windows) can be represented as vectors; likewise, properties present in parallel in a modulated or other complex signal can be modelled as a vector of properties.
5. Geometry and trigonometry: the basic geometry and trigonometry required is those of the right-angled triangle, in order to relate sines and cosines, and to define the complex numbers which represent the three dimensions of a signal, frequency, phase and amplitude.
6. Logarithms: the scales of frequency and of intensity in human hearing are often modelled using a logarithmic scale (e.g. a scale of semitones for frequency, or the decibel scale for intensity). Likewise, the notion of entropy or information in a signal is measures logarithmically.
7. Exponentials: the use of exponentials is central to the description of the Fourier transformation and other transformations used in speech signal processing.