In our music synthesizer, each bell will be represented by a simple harmonic oscillator. This chapter solves the equations for a continuous-time oscillator, and Chapter 3 does the same for the discrete-time version. Comparing the two cases will tell us nearly everything about the differences between discrete-time and continuous-time signal processing.

But why study a simple harmonic oscillator? Why not start out with something more complicated? As it turns out, even the most complicated linear, time-invariant systems in both the continuous-time and discrete-time worlds can be understood as a collection of interacting simple harmonic oscillators. The simple harmonic oscillator is central to linear system theory.

Why the *linear* and *time-invariant* restrictions? The restriction to linear systems is essential because of the enormous complexity of even simple nonlinear systems. Nonlinear systems are the topic of much current research, popularly known as chaos theory, catastrophe theory, and fractals.

The time-invariant restriction is less critical, but still necessary unless some other restriction is imposed on the time variations. Mixing is an example of a linear (for the proof that it is linear, see Section 2.1), time-dependent process that can be easily modeled in digital signal processing. More complicated time-dependent systems are, in general, difficult to model and are not generally treated as part of digital signal processing. ...

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