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Sinusoidal Solution
It is well-known that the solution to (3.1), if is constrained to be real-valued, and if
, has the form
|
(3.3) |
where clearly, one must have
|
(3.4) |
Another way of writing (3.3) is as
|
(3.5) |
where
|
(3.6) |
is the amplitude of the sinusoid, and is the initial phase.
From the sinusoidal form (3.5) above, it is easy enough to deduce that
|
(3.7) |
which serves as a convenient bound on the size of the solution for all , purely in terms of the initial conditions and the system parameter
. Such bounds, in a numerical sound synthesis setting, are extremely useful, especially if a signal such as is to be represented in a limited precision audio format. Note however, that the bound above is obtained only through a priori knowledge of the form of the solution itself (i.e., it is a sinusoid). As will be shown in the next section, it is not really necessary to make such an assumption.
The use of sinusoidal, or more generally complex exponential solutions to a given system in order to derive bounds on the size of the solution was developed, in the discrete setting, into a framework for determining numerical stability of a simulation algorithm, also known as von Neumann analysis [210,102]. Much more will be said about this from Chapter 5 onwards.
Next: Energy
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Stefan Bilbao
2006-11-15