These transforms change a function of one variable into a function of another variable, and, when applied to problems in the physical sciences, the transform pairs and variables may correspond to physical quantities. We assign time and frequency as the transform variables because these are the variables associated with the filtering devices considered in this text. Accordingly, one of this chapter's goals is to relate the frequency-domain concepts introduced here to their time-domain counterparts introduced in Chapter 1.

We first discuss the mathematical aspects of these transforms and then apply the Laplace transform method to a differential equation to obtain its solution. This leads to the introduction and discussion of the transfer function, poles and zeros, and the steady-state responses-magnitude, phase, and group delay. The final section introduces a third transform, the Hilbert transform, which is useful for relating various functions of a specified system.

The basic premise underlying the Fourier and Laplace transforms is that a time function may be decomposed into a linear combination of exponentials. Furthermore, when the exponents are imaginary, the composite signal is representable as a spectrum of sinusoidal components.

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