TABLE OF CONTENTS We can analyze linear systems in many different ways by taking advantage of the property of linearity, where by the input is expressed as a sum of simpler components. The system response to any complex input can be found by summing the system's response to these simpler components of the input. In time-domain analysis, we separated the input into impulse components. In the frequency-domain analysis in Chapter 4, we separated the input into exponentials of the form e st (the Laplace transform), where the complex frequency s= ? + j ?. The Laplace transform, although very valuable for system analysis, proves somewhat awkward for signal analysis, where we prefer to represent signals in terms of exponentials e j ? t instead of e st. This is accomplished by the Fourier transform. In a sense, the Fourier transform may be considered to be a special case of the Laplace transform with s= j ?. Although this view is true most of the time, it does not always hold because of the nature of convergence of the Laplace and Fourier integrals.
In Chapter 6, we succeeded in representing periodic signals as a sum of (everlasting) sinusoids or exponentials of the form e j ? t. The Fourier integral developed in this chapter extends this spectral representation to aperiodic signals.
Applying a limiting process, we now show that an aperiodic signal can be expressed as a...
