Finite-Difference Schemes (FDSs) aim to solve differential
equations by means of finite differences. For example, as discussed
in §C.2, if
denotes the displacement in meters of a vibrating
string at time
seconds and position
meters, we may approximate
the first- and second-order partial derivatives by
where
![$ T$](http://www.dsprelated.com/josimages_new/pasp/img42.png)
denotes the time
sampling interval and
![$ X$](http://www.dsprelated.com/josimages_new/pasp/img453.png)
denotes the
spatial
sampling interval. Other types of finite-difference schemes
were derived in Chapter
7 (§
7.3.1), including a look at
frequency-domain properties. These
finite-difference approximations
to the partial derivatives may be used to compute solutions of
differential equations on a discrete grid:
Let us define an abbreviated notation for the grid variables
and consider the ideal
string wave equation (cf, §
C.1):
![$\displaystyle y''= \frac{1}{c^2}{\ddot y} \protect$](http://www.dsprelated.com/josimages_new/pasp/img372.png) |
(D.2) |
where
![$ c$](http://www.dsprelated.com/josimages_new/pasp/img125.png)
is a positive real constant (which turns out to be wave
propagation speed). Then, as derived in §
C.2, setting
![$ X=cT$](http://www.dsprelated.com/josimages_new/pasp/img380.png)
and substituting the finite-difference approximations into
the ideal
wave equation leads to the relation
everywhere on the time-space grid (
i.e., for all
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
and
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
). Solving
for
![$ y_{n+1,m}$](http://www.dsprelated.com/josimages_new/pasp/img3379.png)
in terms of displacement samples at earlier times yields an
explicit finite-difference scheme
for string displacement:
![$\displaystyle y_{n+1,m}= y_{n,m+1}+ y_{n,m-1}- y_{n-1,m} \protect$](http://www.dsprelated.com/josimages_new/pasp/img4453.png) |
(D.3) |
The FDS is called
explicit because it was possible to solve for
the state at time
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
as a function of the state at earlier times (and
any other positions
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
). This allows it to be implemented as a time
recursion (or ``
digital filter'') which computes a solution at time
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
from solution samples at earlier times (and any spatial
positions). When an explicit FDS is not possible (
e.g., a non-
causal
filter is derived), the discretized differential equation is said to
define an
implicit FDS. An implicit FDS
can often be converted to an explicit FDS by a rotation of coordinates
[
55,
481].
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