Signal, deterministic component

Expressivity. Here we define expressivity as the variation of the spectrum and time evolution of a signal for musical purposes. That variation is usually considered to have two components, a deterministic component and a random component. The deterministic element of expressivity is the change in spectrum and time evolution controlled by the user during performance. For example, hitting a piano key harder makes the note louder and brighter (more high frequency content). The random component is the change from note to note that is not possible to control by the musician. Two piano notes played in succession, for example, are never identical no matter how hard the musician attempts to create duplicate notes. While the successive waveforms will always be identified as a piano note, careful examination shows that the waveform details are different from note to note, and that the differences are perceivable. 

Signal Modification. The decomposition approach has been applied successfully to speech and music modification [Serra, 1989] where modification is performed differently on the two deterministic/stochastic components. Consider, for example, time-scale modification. With the deterministic component, the modification is performed as with the baseline system using Equation (9.27), sustained (i.e., steady ) sine waves are compressed or stretched. For the aharmonic component, the white noise input lingers over longer or shorter time intervals and is matched to impulse responses (per frame) that vary slower or faster in time. 

The notion of sines plus noise modeling was posed and implemented by Xavier Serra and Julius Smith in the Spectral Modeling Synthesis (SMS) system. They called the sinusoidal components the deterministic component of the signal, and the leftover noise part the residual or stochastic component. Figure 6.12 shows the decomposition of a sung ahh sound into deterministic (harmonic sinusoidal) and stochastic (noise residue) components. 

Frequency Analysis. The statistical analysis of chaotic signals includes spectral analysis to confirm the absence of any spectral lines, since chaotic signals do not have any periodic deterministic component. Absence of spectral lines would indicate that the signal is either chaotic or stochastic. However, chaos is a complicated nonperiodic motion distinct from stochastic processes in that the amplitude of the high-frequency spectrum shows an exponential decline. The frequency spectrum can be evaluated using FFT-ba methods outlined the earlier sections. 

One of the more challenging unsolved problems is the representation of transient events, such as attacks in musical percussive sounds and plosives in speech, which are neither quasi-periodic nor random. The residual which results from the deterministic/stochastic model generally contains everything which is not deterministic, i.e., everything that is not sine-wave-like. Treating this residual as stochastic when it contains transient events, however, can alter the timbre of the sound, as for example in time-scale expansion. A possible approach to improve the quality of such transformed sounds is to introduce a second layer of decomposition where transient events are separated and transformed with appropriate phase coherence as developed in section 4.4. One recent method performs a wavelet analysis on the residual to estimate and remove transients in the signal [Hamdy et al., 1996] the remainder is a broadband noise-like component. 

Deterministic analysis Coupled biochemical systems Reaction kinetics are represented by sets of ordinary differential equations (ODEs). Rates of activation and deactivation of signaling components are dependent on activity of upstream signaling components. Spatially specified systems Reaction kinetics and movement of signaling components are represented by partial differential equations (PDEs). Useful for studies of reaction-diffusion dynamics of signaling components in two or three dimensions. (64-70)... 

The deterministic analysis provides the average behavior of signaling components in a network but does not include the results of fluctuations in the activation states of the signaling components (noise). Such analysis becomes essential when the volume of interest becomes in the scale of femtoliters because of the limitingly small number of molecules contained in such a small volume (1 fL of a 0.1 p.M solution is equivalent to 60 molecules). Small volumes such as these are biologically relevant and present in cellular systems quite often, for example, in the spines of dendrites and endosomal regions. Moreover,... 

In contrast to the principal component (PCA)-based approach for detection and discrimination, the cross-correlation or matched-filter approach is based on a priori knowledge of the shape of the deterministic signal to be detected. However, the PCA-based method also requires some initial data (although the data could be noisy, and the detector does not need to know a priori the label or the class of the diHerent signals) to evaluate the sample covariance matrix K and its eigenvectors. In this sense, the PCA-based detector operates in an unsupervised mode. Further, the matched-filter approach is optimal only when the interfering noise is white Gaussian. When the noise is colored, the PCA-based approach will he preferred. 

Transient signals such as earthquake seismograms may have a complicated waveform which is determined by the earthquake source, the structure of the Earth, and the properties of the seismograph. The amplitudes and phases of the harmonic components of the signal are not random numbers but follow certain mathematical relationships. Such waveforms or spectra are sometimes called deterministic, in contrast to the stochastic or random waveform of noise. However, in a strict sense, even microseis-mic noise is deterministic as well. But the difference is that its sources and propagation paths are not known well enough to predict or interpret the waveform, and therefore it is assumed that the spectral amplitudes and phases are random numbers. Again, this is a mathematical simplification that may or may not be appropriate. 

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