The Laplace Transform

The two main techniques in signal processing, convolution and Fourier analysis, teach that a
linear system can be completely understood from its impulse or frequency response. This is a
very generalized approach, since the impulse and frequency responses can be of nearly any shape
or form. In fact, it is too general for many applications in science and engineering. Many of the
parameters in our universe interact through differential equations. For example, the voltage
across an inductor is proportional to the derivative of the current through the device. Likewise,
the force applied to a mass is proportional to the derivative of its velocity. Physics is filled with
these kinds of relations. The frequency and impulse responses of these systems cannot be
arbitrary, but must be consistent with the solution of these differential equations. This means that
their impulse responses can only consist of exponentials and sinusoids. The Laplace transform
is a technique for analyzing these special systems when the signals are continuous. The ztransform
is a similar technique used in the discrete case.
CHAPTER
The Laplace Transform
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The two main techniques in signal processing, convolution and Fourier analysis, teach that a
linear system can be completely understood from its impulse or frequency response. This is a
very generalized approach, since the impulse and frequency responses can be of nearly any shape
or form. In fact, it is too general for many applications in science and engineering. Many of the
parameters in our universe interact through differential equations . For example, the voltage
across an inductor is proportional to the derivative of the current through the device. Likewise,
the force applied to a mass is proportional to the derivative of its velocity. Physics is filled with
these kinds of relations. The frequency and impulse