![]() ![]() The action of the system on an input e” is multiplication by the transfer function H(s). We define the transfer function: H(s) = ∫ h( τ ) e^st dτ. The system output will be given by: y(t) = H = h(t) * x(t) ∫ h( τ ) x(t- τ) dτ. Let us consider applying an input of the form x(t) = e^st to an LTI system with impulse response h(t). ![]() The bilateral Laplace transform offers insight into the nature of system characteristics such as stability, causality, and frequency response, The primary role of the Laplace transform in engineering is the transient and stability analysis of causal LTI systems described by differential equations. ![]() The unilateral Laplace transform is a convenient tool for solving differential equations with initial conditions. The Laplace transform comes in two varieties: The transfer function generalizes the frequency response characterization of an LTI system’s input-output behavior and offers new insights into system characteristics. Hence, the output of an LT system is obtained by multiplying the Laplace transform of the input by the Laplace transform of the impulse response, which is defined as the transfer function of the system. As with complex sinusoids, one consequence of this property is that the convolution of time signals becomes multiplication of the associated Laplace transforms. For example, we shall see that continuous-time complex exponentials are eigenfunctions of LTI systems. Many of these properties parallel those of the Fourier Transform. The Laplace transform possesses a distinct set of properties for analyzing signals LTI systems. ![]()
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