Historically, electronic circuits were characterized by exciting them with an oscillator’s output, and then with an oscilloscope determining how the circuit affected the magnitude and phase of its input signal. In general, this technique is only appropriate for linear circuits. However, it is also useful for non-linear circuits containing transistors when the signals are small enough that the transistors can be adequately characterized by their operating points and small linear changes about their operating points; that is, the nonlinear transistors can be accurately described using small-signal analysis. The use of this technique for characterizing electronic circuits has led to the application of frequency-domain techniques to the analysis of most any linear or weakly non-linear system, a technique that is now ubiquitous in system analysis.

Consider a linear time-invariant system with transfer function `H(s)`

being excited by an input signal having Laplace transform `X`

._{in}(s)^{1} The Laplace transform of the output signal `X`

, is given by_{out}(s)

In the time domain, assuming `X`

is the inverse Laplace transform of _{in}(t)`X`

, and _{in}(s)`h(t)`

is the inverse Laplace Transform of `H(s)`

(often called its impulse response), we have

That is, the output signal ...

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