Continuous-Time Systems (Signals and Communication by Y.S. Shmaliy

By Y.S. Shmaliy

This paintings deals scholars in any respect degrees an outline of linear, nonlinear, time-invariant, and time-varying digital continuous-time structures. As an assemblage of actual or mathematical elements equipped and interacting to transform an enter sign to an output sign, an digital approach will be defined utilizing diversified equipment provided by way of the trendy platforms concept. To make attainable for readers to appreciate platforms, the booklet systematically covers the key foundations of the platforms idea.

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Example text

Therefore, oscillators are typically unstable at zero and stable at an equilibrium point. There are several approaches of how to ascertain stability of systems in a different sense depending on stationarity and linearity. In the sequel, we shall observe most of them in the time and transform domains as well as in the phase plane. For now, it seems in order to give the reader an idea about the seemingly most obvious input-to-output stability. The requirement for the system output not to diverge as long as the input does not diverge relates to whether the signal is bounded or not.

Fig. 16. Robust system models with uncertainty: (a) additive, (b) multiplicative, and (c) feedback. 1. An ideal integrator is given with the nominal transfer function ˜ = H(s) = 1/s. The first order real physical integrator is described with H 1/(s+a), where a is a small constant. Find the weight W for the multiplicative model (Fig. 16b). ˜ = H(1 + W Δ), we may write H˜ − 1 = W Δ. For the allowable Having H H |Δ| < 1, the relation yields an inequality ˜ H −1 H |W | that, for the transfer functions given, results in ing transfer function is thus s a+s −1 |W |.

2. Given a system (Fig. 1. Let the input be a unit-step function, v(t) = u(t). 12) shown in Fig. 1c. 12). 1 System Responses to Test Signals 47 Frequency response: The ratio of the system response to the complex exponential signal x(t) = ejωt and ejωt is the system frequency response, Frequency response = Response to ejωt . 13) Because a system is typically with memory, its frequency response is commonly a complex function. The absolute value of this function is called the magnitude response and its phase the phase response.

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