By Philippe de Larminat

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**Additional info for Analysis and control of linear systems**

**Example text**

24. 25. 26. 6. 1. Fourier transform Any signal has a reality in time and frequency domains. Our ear is sensitive to amplitude (sound level) and frequency of a sound (low or high-pitched tone). These time and frequency domains, which are characterized by variables that are opposite to one another, are taken in the broad sense: if a magnitude evolves according to a distance (atmospheric pressure according to altitude), the concept of frequency will be homogenous, contrary to a length. The Fourier transform is the mathematical tool that makes it possible to link these two domains.

The coefficients of the numerator and denominator polynomials are real due to their physical importance in the initial differential equation. Hence, the numerator roots, called zeros, and the denominator roots, called transfer function poles, are conjugated real or complex numbers. If x(t ) and y(t ) are causal functions, the Laplace transform of the differential equation entails terms based on initial input values x(0), x ′(0), x ( m −1) (0) and output values y (0), y ′(0), y ( n −1) (0) ; the concept of state will make it possible to overcome this dependence.

T → +∞ h(t ) ~ t n −1 Holomorphy Directly impossible harmonic analysis impossible directly except for a simple pole at the origin H(p) has poles on the right of the imaginary axis. t → +∞ h(t ) ~ ekt * k>0 * Holomorphy Possible analysis if the system is introduced in a stable looping and H ( jω ) = H ( p) p= jω H(p) has poles on the imaginary axis. 1. Unit-step responses, transfer functions and existence conditions of the frequency response Transfer Functions and Spectral Models 21 Frequency responses are generally characterized according to impulse ω = 2π j f and data H ( jω ) and Φ ( jω ) grouped together as diagrams.