Linear time-invariant filters

The function Y(t) also can be visualized as the result of passing the impulse train N (t) through a linear, time-invariant filter whose impulse response is This observation coupled with the fact that the... 

The reader unfamiliar with the notion of a linear, time-invariant filter can profit from W. M. Siebert s article in E. J. Baghdady, Lectures on Communication System Theory, McGraw-Bftll Book Co., New York, 1901. 

Now, our previous result shows that F0(f), being the result of passing X(t) through the linear, time-invariant filter h0(t )> must have a gaussian first-order distribution therefore,... 

In this connection, it should be carefully noted that, even if X(t) is not a gaussian process, the mean and the autocorrelation function of the output of a linear, time-invariant filter are related to the mean and autocorrelation function of the input process according to Eqs. (3-293) and (3-294).64 This is an important fact of which use will be made in the next section. 

With the aid of the power density spectrum, we can now give a complete description of how a linear, time-invariant filter affects the frequency distribution of power of the input time function X(t). To accomplish this, we must find the relationship between the power... 

Linear Prediction (LP) is another technique for source filter separation, in which we use the techniques of LTI filters to perform an explicit separation of source and filter. In LP we adopt a simple system for speech production, where we have an input source x[n, which passed through a linear time invariant filter h[n], to give the output speech signal y n. In the time domain this is ... 

The conventional filtered white-noise process f(t, k) is the stationary response of a linear time-invariant filter subjected to a white-noise process. White-noise w t) is a stationary random process in time that has a zero mean and a constant spectral density for all frequencies. The response of a linear filter to a white-noise process may be calculated by using the Duhamel convolution integral, and hence the general formulation of a filtered white-noise process can be written in the following form ... 

Harmonic Analysis of Random Processes.—The response Y(t) of a linear, time-invariant electrical filter to an input X(t) can be expressed in the familiar form 66... 

Because allpass filters are linear and time invariant, they commute like gain factors with other linear, time-invariant components. Fig. 10.15 shows a diagram equivalent to Fig. 10.14 in which the allpass filters have been commuted and consolidated at two points. For computability in all possible contexts (e.g., when looped on itself), a... 

Our first task is to build a model where the complex vocal apparatus is broken down into a small number of independent components. One way of doing this is shown in Figure 11.1b, where we have modelled the lungs, glottis, pharynx cavity, mouth cavity, nasal cavity, nostrils and lips as a set of discrete, coimected systems. If we make the assumption that the entire system is linear (in the sense described in Section 10.4) we can then produce a model for each component separately, and determine the behaviour of the overall system fi om the appropriate combination of the components. While of course the shape of the vocal tract will be continuously varying in time when speaking, if we choose a sufficiently short time fi ame, we can consider the operation of the components to be constant over that short period time. This, coupled with the linear assumption then allows us to use the theory of linear time invariant (LTI) filters (Section 10.4) throughout. Hence we describe the pharynx cavity, mouth cavity and lip radiation as LTI filters, and so file speech production process can be stated as the operation of a series of z-domain transfer functions on the input. 

Linear, time invariant (LTI) A system is said to be LTI if superposition holds, that is, its output for an input that consists of the sum of two inputs is identical to the sum of the two outputs that result from the individual application of the inputs the output is not dependent on the time at that the input is applied. This is the case for an FIR filter with fixed coefficients. 

Al-Haj Ali et al. [5,6] developed different types of linear time invariant models by system identification, which adequately represent the fluidized-bed drying dynamics. MBC techniques such as IMC and model predictive control (MPC) were used for the designing of the control system. Simulations with multivariable MPC strategy provided robust, fast, stable, and non-oscillatory closed loop responses. A stationary form of Kalman filter was designed to estimate the particle moisture content (state observer). Performance studies showed that the Kalman filter provided satisfactory estimates even in the presence of significant noise levels and inaccurate initial states feed to the observer. 

Digital, time-invariant linear filters operate on an input signal x[ ] and produce an output signal y[ ],... 

Linear filter Throughout we have assiuned that the system operates as a time invariant linear (LTI) filter of the type described in Chapter 10. While it is well known that there are many non-linear processes present in vocal tract soimd propagation, in general the linear model provides a very good approximation to these. 

The steady-state optimal Kalman filter can be generalized for time-variant systems or time-invariant systems with non-stationary noise covariance. The time-varying Kalman filter is calculated in two steps, filtering and prediction. For the nonlinear model the state estimate may be relinearized to compensate the inadequacies of the linear model. The resulting filter is referred to the extended Kalman filter. If once a new state estimate is obtained, then a corrected reference state trajectory is determined in the estimation process. In this manner the Filter reduces deviations of the estimated state from the reference state trajectory (Kwon and Wozny, 1999 Vankateswarlu and Avantika, 2001). In the first step the state estimate and its covariance matrix are corrected at time by using new measurement values >[Pg.439]

We assume that the process being identified is stable, linear and time invariant and can be accurately represented by a reduced nth order FSF model. For an arbitrary process input u k) and the measured process output y k), the frequency sampling filter model can be written as... 

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