Spectral modelling techniques spectrum analysis

Spectral modelling techniques are the legacy of the Fourier analysis theory. Originally developed in the nineteenth century, Fourier analysis considers that a pitched sound is made up of various sinusoidal components, where the frequencies of higher components are integral multiples of the frequency of the lowest component. The pitch of a musical note is then assumed to be determined by the lowest component, normally referred to as the fundamental frequency. In this case, timbre is the result of the presence of specific components and their relative amplitudes, as if it were the result of a chord over a prominently loud fundamental with notes played at different volumes. Despite the fact that not all interesting musical sounds have a clear pitch and the pitch of a sound may not necessarily correspond to the lower component of its spectrum, Fourier analysis still constitutes one of the pillars of acoustics and music. 

Chapter 6 looks lurther at using the Fourier analysis spectrum as a sound analysis tool. The techniques of spectral modeling are introduced, breaking up sounds into important perceptual and physical components. 

In postulating the stationarity of the stochastic process, very strong assumptions regarding the structure of the process are made. Once these assumptions are dropped, the process can become nonstationary in many different ways. In the framework of the spectral analysis of nonstationary processes, Priestley (see, e.g., Priestley 1999) introduced the evolutionary power spectral density (EPDS) function. The EPSD function has essentially the same type of physical interpretation of the PSD function of stationary processes. The main difference is that whereas the PSD function describes the power-frequency distribution for the whole stationary process, the EPSD function is time dependent and describes the local power-frequency distribution at each instant time. The theory of EPSD function is the only one which preserves this physical interpretation for the nonstationary processes. Moreover, since the spectrum may be estimated by fairly simple numerical techniques, which do not require any specific assumption of the structure of the process, this model, based on the EPSD function, is nowadays the most adopted model for the analysis of structures subjected to nonstationary processes as the seismic motion due to earthquakes. 

More recently, Noda has proposed the use of infrared two-dimensional correlation spectroscopy (2D-IR) to increase the information that can be extracted from a spectrum. This approach, essentially different from 2D-NMR spectroscopy, uses correlation analysis of the dynamic fluctuations caused by an external perturbation to enhance spectral resolution without assuming any line shape model for the bands. The technique was intended for the study of polymers and liquid crystals, and it has recently been applied to proteins. In the latter case, the perturbation can be achieved through changes in temperature, pH, ligand concentration and lipid-to-protein ratio. 

A second group of techniques may be called lineshape analysis. Simple methods entail the measurements of linewidths or second moments as a function of temperature. More sophisticated methods involve the analysis or the model fitting of spectral lineshapes. A prominent method is ID lineshape analysis for deuterium-labeled polymers, which is sensitive to motions in the frequency range of lO -lO s (149). The 2D wideline separation NMR (WISE) experiment permits correlation of the high resolution spectrum with the wideline spectrum, which provides dipolar information (11,150). The linewidth is a function of the frequency of the polymer motion relative to the time scale of dipolar couplings. 

While cepstral generation via inversion of tiie cepstral analysis technique is possible, it is more common in HMM synthesis to use tiie more direct technique of Mel Logarithmic Spectrum Approximation (MLSA) [229]. This technique is quicker (in that it doesn t require the expensive inverse DFT operations) and can be more accurate at modelling spectral envelopes. MLSA uses more sophisticated signal processing techniques than have so far been introduced and so is beyond our present scope. See however Imai s original paper for details of how this is performed [229]. 

As with quantitative models, outlier samples in the training set can have an unwanted influence on the discrimination ability of the model. Many of the same techniques (spectral residual plots and cluster plots) used in quantitative models can also be used to check for outliers in discriminant models. However, keep in mind the purpose of the discriminant analysis experiment to build a model that can accurately match a spectrum to the training group but allow enough variation in the model to compensate for the natural variations seen in real samples. 

Analysis of the spectrum of the H atom led to the Bohr model, the first step toward our current model of the atom. From its use by 19 -century chemists as a means of identifying elements and compounds, spectrometry has developed into a major tool of modem chemistry. The terms spectmscopy, spectrophotometry, and spectrometry refer to a large group of instrumental techniques that obtain spectra that correspond to a substance s atontic or molecular eneigy levels. (Elements produce lines, but complex molecules produce spectral peaks.) The two types of spectra most often obtained are emission and absorption spectra ... 

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