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英文书《Fast Fourier Transforms, Second Edition》附带的MATLAB实例

The latest version of FAS has some new features which were not described
in the User's Manual in FAST FOURIER TRANSFORMS.  We will describe these
new features below, they are the following:

    1. Diffrac --- a new filter choice for series

    2. Animate --- a new filter choice for series

    3. Animation --- a new choice on the graphs menu

    4. Saving remarks.

Additional information can also be obtained from the online help for FAS.

1. Diffrac

   This choice will automatically perform (in one step) the calculation
of diffraction patterns described in Chapter 4, section 3 of FAST FOURIER
TRANSFORMS.  You will be asked for a  Wave constant  and a  Time constant.
The procedure then performs both a Frescos filtering and a Fressin filtering
and adds the sum of the squares of these two filtered series. Thus, for
example, you can produce the electron diffraction patterns described in
Chapter 4, section 3, in one step.
   Note: When using 8192 points this procedure requires 560 KB of free
memory (540 KB, as cited in the User's Manual, is insufficient for this
procedure when 8192 points are used).


2. Animate

   This is a filter procedure which allows you to create animations, a
sequence of graphs parameterized by time which display like the still
frames of a movie.
   When you make this choice the following menu appears:

           Cos     Sinc      Heat     User     Diffrac

The  Cos  and  Sinc  choices allow you to create animations of vibrating
strings; they operate in much the same way as the  Cos  and  Sinc  filters
described in FAST FOURIER TRANSFORMS.  You will be asked for a  Wave constant
(which is the constant  c  in the wave equation).  Instead of a single time
constant, however, you will be asked for time data in the following form:

     Enter time data (start, end, number of frames):

followed be a blinking cursor.  You then enter three numbers, separated by
commas, specifying the time constants for the start and end of the animations
and the number of frames (still images) to be used.  For example, if you
enter  0, 2, 1000  then the animation will begin at  t = 0  and will end at
t = 2.  Each frame in the animation will be created using a value of time
which changes by the increment 2/1000.  (Note: The number of frames must
be a two byte integer.  That is, you cannot use an integer greater than
32767  for the number of frames.)  The animation will proceed in one of
four ways, called animation modes.  You choose an animation mode from the
following menu:

         Forward     Backward      Circle forward     Loop backward

These modes are similar to the ones described in the User's Manual for
Graphbook animations.  If you choose  Forward  then the animation begins
at the start time and finishes at the end time.  While if you choose
Backward  then the animation begins at the end time, and moves backward
in time by negative increments, finishing at the start time.  Circle forward
and  Loop backward  are similar to the first two modes, respectively, but
when the finishing time is reached the animation treats this time as
identical to the beginning time and proceeds in a time loop.  These last
two modes are useful for animating vibrating strings.
   As mentioned above, the  Cos  and  Sinc  choices are similar to the
Cos  and  Sinc choices described in FAST FOURIER TRANSFORMS and they are
useful for producing animations of vibrating strings.  The choice  Heat
is useful for producing an animation of a temperature profile in a heat
conduction problem.  You will be asked to enter the Diffusion constant
(which is the diffusion constant defined in FAST FOURIER TRANSFORMS,
e.g. the diffusion constant for copper is 1.14).  You will also be asked
to choose an animation mode and enter time data as described above.
   The  User  choice is similar to the  User  choice on the filter menu
(described in section 5 of Chapter 4 of FAST FOURIER TRANSFORMS) but in
this case the time variable  t  in the filter function is a variable
which will assume values ranging from the beginning time to the finishing
time as the animation proceeds.  For example, you might try animating
the example of a string vibrating due to an applied force (described in
Example 4.7 of FAST FOURIER TRANSFORMS).  First, choose to do a Sine series
using 128 points over the interval [0,10].  Then, enter the function

       f(x) = 0.2(2 < x < 3)

Choose  32  harmonics and choose  Animate  from the filter menu and  User
from the animate menu.  Then, choose  Forward  mode and enter  0, 4, 1024
as the time data.  Enter the following function as your filter function:

       f(x)= (sin(wt)-wtsinc(10mxt))/((10pi mx)^2-w^2)\m=32\w=20.00001pi

Use the online help to create this function.  Just press the  F1  key
when you are at the ``f(x) = '' prompt and select  Resonating string  from
the topics list.  Before starting the animation be sure to change the
Y-interval to [-.002, .002].

    The final choice on the  Animate  menu is  Diffrac  (Note: If you
are using  8192  points, then this choice will only appear when you started
FAS with at least 560 KB of free memory.)  This choice behaves similarly
to the  Diffrac  filter described above.  You will be asked for the  Wave
constant (e.g. for electron diffraction, specify the value .578).  You will
also be asked to choose an animation mode and to enter your time data as
described above.  Here is an interesting example.  Select  Fourier series,
then select  Sine series  and choose  256  points.  For your interval,
specify [0, 64], and for your function enter

              f(x) = exp(-.1(x-32)^2)
   
Then specify 64 harmonics (this produces a partial sum whose 2-Norm
difference from  f(x)  is less than 9 X 10^{-12}). After that you
then choose to do a filter and choose  Animation  and  Diffrac  from
the resulting menus.  Enter  .578  as the Wave constant and the following
time data 

                   0, 2000, 1000

After the initial graph is plotted (for time  t = 0) you should then
change the Y-interval to [-.1, 1.4] and select  Go.  The resulting
animation is very interesting.


3. Animation

    This choice is a new choice on the  Graphs  menu.  This menu will now
appear as the following menu:

       Draw graphs     View graphbook     Animation

If you choose  Animation  then you can create a series of graphs parameterized
by time (as described above).  For example, try choosing  128  points and
an interval of  [0,pi].  Choose  Circle forward  mode and enter  0, 2pi, 32
as your time data.  Then enter the following function

      f(x) = cos(t)sin(x)

You will produce an animation of a vibrating string (in its fundamental mode).
You might also try some of the examples in the online help (press the  F1
key when you are at the ``f(x) = '' prompt).


4. Saving remarks

    You can now save and retrieve any remarks you create to accompany graphs.
In addition to the choices  Modify  and  Create, the choices  Save  and
Load  will also appear as menu choices when you choose  Rem.  When you
select  Save  you will be asked to enter a file name.  If you enter a valid
DOS file name (not including an extension), the extension  .DCM  will be
appended to this file name to create the actual file name.  The remarks
that you have created will then be saved in this file.  You can retrieve
previously saved remarks using the  Load  choice.