Fourier Transformation

The transformation of a signal from the time domain into a representation of the component frequencies and phases is known as Fourier analysis or spectrum analysis, and is modelled with the Fourier Transformation. The output of the Fourier transform is a Fourier transform. The Fourier theorem states that complex signal can be analysed as the sum, for each point in time, of simple sinusoid signals:

1. The simple sinusoid is a continuous sine wave; it can only be approximated by real signals, which are always complex to some degree.
2. Complex signals can always be modelled as a sum of simple signals (Fourier 's Theorem):
   1. Tones are sums of simple signals which are harmonically related, Tones ar e typically sonorants, particularly vowels, in speech.
   2. Noises are sums of simple signals which are not harmonically related. Noi ses are typically obstruents in speech.
   3. Superimposed tones and noises are typically the voiced obstruents i n speech, in which the voicing component has the property of being a tone, and t he obstruent component has the quality of being a noise.

The Fourier transformation in its mathematical form is valid under the following conditions:

1. The signal is continuous. For discrete signals, the transformation takes a simpler form.
2. The signal is periodic and therefore by implication infinite in length. For actual speech signals, the assumption is made that their periodicity is stable over short windows, and that this periodicity extends arbitrarily far in either temporal direction; this idealisation is supported by the heuristic of amplitude modulating the input signal segment with a complete cosine period of the length of the segment (a raised cosine window), i.e. raising the cosine to the positive value range, and multiplying with the input signal segment.

Cross-correlation is the method which basically underlies implementations of the Fourier transformation: signals of varying frequency and phase are correlated with the input signal, and the degree of correlation in terms of frequency and phase represents the frequency and phase spectrums of the input signal. The Fourier transformation then no longer represents the signal magnitude as a function of time, but as a pair of functions: as a function of frequency and as a function of phase.

The phase domain is generally not considered further in linguistically or phonetically oriented speech signal processing, except in areas such as stereophonic perception, for instance in connection with spatial orientation in resolving the `cocktail party effect' (i.e. the ability to track a conversation in the middle of several similar competing signals). If the original input signal is the be reconstituted exactly, then both frequency and phase must be involved in the inverse transformation. However, the perceptual impression is the same, even if the phase is normalised to an identical initial value for all frequency components.

The formulation of the Fourier Transformation for discrete signals is known as the Discrete Fourier Transformation (DFT).