Deterministic signals

The convolution or smoothing function, h f), used in moving averaging is a simple block function. However, one could try and derive somewhat more complex convolution functions giving a better signal-to-noise ratio with less deformation of the underlying deterministic signal. 

It is impossible to predict the amplitude of a stochastic signal at a certain time in the future in contrast to a deterministic signal like a sine wave. Only a statistical description, for instance by distribution functions and autocorrelation functions, can be given. Host kinds of noise have a stochastic character. 

A special kind of random noise, pseudo random noise, has the special property of not being really random. After a certain time interval, a sequence, the same pattern is repeated. The most suitable random input function used in CC is the Pseudo Random Binary Sequence (PRBS). The PRBS is a logical function, that has the combined properties of a true binary random signal and those of a reproducible deterministic signal. The PRBS generator is controlled by an internal clock a PRBS is considered with a sequence length N and a clock period t. It is very important to note that the estimation of the ACF, if computed over an integral number of sequences, is exactly equal to the ACF determined over an infinite time. 

The improved performance of the multiscale approach is due to the ability of orthonormal wavelets to approximately decorrelate most stochastic processes, and compress deterministic features in a small number of large wavelet coefficients. These properties permit representation of the prior probability distribution of the variables at each scale as a Gaussian or exponential function for stochastic and deterministic signals, respectively. Consequently, computationally expensive non-parametric methods need not be used for estimating the probability distribution of the coefficients at each scale. If the probability distribution of the contaminating errors and the prior can be represented as a Gaussian, the multiscale Bayesian approach provides... 

Deterministic Signals (Synchronized to Another Stimulus Signal or Perturbation) in Noise... 

Detecting a known (deterministic) signal in a noisy environment... 

The simple cross-correlation estimator is used extensively in the form of a matched filter implementation to detect a finite number of known signals (in other words, simultaneous acquisition of multiple chaimels of known signals). When these deterministic signals are embedded in white Gaussian noise, the matched filter (obtained from cross-correlation estimate at zero lag, k = 0, between the known signal sequence and the observed noisy signal sequence) gives the optimum detection performance (in the Bayes sense ). 

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. 

Puszynski, K., Lachor, P., Kaidynska, M., Smieja, J. Sensitivity analysis of deterministic signaling pathways models. Bull. Pol. Acad. Tech. 60,471-479 (2012)... 

For the present purposes the deterministic function generator yields a constant signal x - 0, which means the summation output is identical with that of the noise generator. 

Transforms are important in signal processing. An important objective of signal processing is to improve the signal-to-noise ratio of a signal. This can be done in the time domain and in the frequency domain. Signals are composed of a deterministic part, which carries the chemical information and a stochastic or random part which is caused by deficiencies of the instmmentation, e.g. shot noise... 

In this chapter we discuss techniques both for signal enhancement and signal restoration. Techniques to model or to reconstruct the deterministic part of a digital signal in the presence of noise are discussed in Chapter 41. 

As said before, there are two main applications of Fourier transforms the enhancement of signals and the restoration of the deterministic part of a signal. Signal enhancement is an operation for the reduction of the noise leading to an improved signal-to-noise ratio. By signal restoration deformations of the signal introduced by imperfections in the measurement device are corrected. These two operations can be executed in both domains, the time and frequency domain. 

Ideally, any procedure for signal enhancement should be preceded by a characterization of the noise and the deterministic part of the signal. Spectrum (a) in Fig. 40.18 is the power spectrum of white noise which contains all frequencies with approximately the same power. Examples of white noise are shot noise in photomultiplier tubes and thermal noise occurring in resistors. In spectrum (b), the power (and thus the magnitude of the Fourier coefficients) is inversely proportional to the frequency (amplitude 1/v). This type of noise is often called 1//... 

Daw, C. S., and Harlow, J. S., Characteristics of Voidage and Pressure Signals from Fluidized Beds using Deterministic Chaos Theory, Proc. 11th Int. Conf. FluidizedBedComb., 2 777 (1991)... 

The above model consists of two main parts a scheduler and the plant to schedule. Since the scheduler defines exactly when to start and finish the tasks, the behavior of the entire system is deterministic. Running both parts, the scheduler and the plant, corresponds to a simulation in which one single behavior of the composed system is obtained. Obviously, this situation requires the presence of a scheduler which knows the (optimal) schedule. If such a scheduler is absent, then the resource automata in the plant receive no signals. Scheduling of a plant can be understood as the task of finding a scheduler automaton which provides the optimal schedule with respect to an optimization criterion. In the sequel it is assumed that a scheduler does not exist and the automata model is designed as shown in Figure 10.4. 

In most problems involving boundary conditions, the boundary is assigned a specific empirical or deterministic behavior, such as the no-slip case or an empirically determined slip value. The condition is defined based on an averaged value that assumes a mean flow profile. This is convenient and simple for a macroscopic system, where random fluctuations in the interfacial properties are small enough so as to produce little noise in the system. However, random fluctuations in the interfacial conditions of microscopic systems may not be so simple to average out, due to the size of the fluctuations with respect to the size of the signal itself. To address this problem, we consider the use of stochastic boundary conditions that account for random fluctuations and focus on the statistical variability of the system. Also, this may allow for better predictions of interfacial properties and boundary conditions. 

External noise denotes fluctuations created in an otherwise deterministic system by the application of a random force, whose stochastic properties are supposed to be known. Examples are a noise generator inserted into an electric circuit, a random signal fed into a transmission line, the growth of a species under influence of the weather, random loading of a bridge, and most other stochastic problems that occur in engineering. In all these cases clearly (4.5) holds if one inserts for A(y) the deterministic equation of motion for the isolated system, while L(t) is approximately but never completely white. Thus for external noise the Stratonovich result (4.8) and (4.9) applies, in which A(y) represents the dynamics of the system with the noise turned off. 

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. 

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