Cycle analysis summation

The simple sine waves used for illustration reveal their periodicity very clearly. Normal sounds, however, are much more complex, being combinations of several such pure tones of different frequencies and perhaps additional transient sound components that punctuate the more sustained elements. For example, speech is a mixture of approximately periodic vowel sounds and staccato consonant sounds. Complex sounds can also be periodic the repeated wave pattern is just more intricate, as is shown in Fig. 1.105(a). The period identified as Ti appHes to the fundamental frequency of the sound wave, the component that normally is related to the characteristic pitch of the sound. Higher-frequency components of the complex wave are also periodic, but because they are typically lower in amplitude, that aspect tends to be disguised in the summation of several such components of different frequency. If, however, the sound wave were analyzed, or broken down into its constituent parts, a different picture emerges Fig. 1.105(b), (c), and (d). In this example, the analysis shows that the components are all harmonics, or whole-number multiples, of the fundamental frequency the higher-frequency components all have multiples of entire cycles within the period of the fundamental. 

Hadamard Spectroscopy. If. instead of the exit slit, an irregular arrangement of slits of various widths is used, and an arrangement in the form of a comb-like second slit is moved in front of this, for each relative position a different summation of intensities at various wavelengths is obtained on a nonmultiplex detector. By using convolution functions. spectra can be computed rapidly for a cycle of different slit positions. Since for mathematical deconvolution so-called Hadamard matrices are used, the principle is known as Hadamard spectroscopy. It has gained some importance in waste gas analysis [50]. 

For the analysis and interpretation of the experimental results, in the first instance the software should perform a pre-processing step of raw data, extracting the required number of complete cycles and, optionally, the sequence of evenly-spaced interpolated points. Successively, the software gives forth the relevant Fourier analysis. The adopted numerical procedure is based on the above-mentioned discrete Fourier transformation (DFT) algorithm (i.e., the summation over all points in Eq. (31)). As shown in the previous paragraph, the algorithm allows any number n and any particular values of periods NA / n (frequencies,/ ) to be selected, among all the possible discrete values, for the analysis of observed data. 

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