



                         Signal Processing in GramoFile

      A discussion of implemented filters, framework and technical details



                                           By J. A. Bezemer
                                           J.A.Bezemer@ITS.TUDelft.NL
                                           Last update: 07/13/1998

                                           GramoFile website:
                                           http://www.opensourcepartners.nl/~costar/gramofile


INTRODUCTION

The processing of digital signals is a major part of the GramoFile
program. This processing is done by so-called filters, as in the
`Conditional Median Filter'. To allow easy programming and application of
these filters, a flexible framework has been implemented. This allows
filters to be used in a random order (even multiple times) in one single
run. Each filter has its own set of (user-changeable) parameters. This may
be visualized as shown below.


   ___________         ________          ________         ___________
  |           |       |        |        |        |proc'd |           |
  | Harddisk  |signal |        |        |        |signal | Harddisk  |
  |           |------>|Filter 1|- - - ->|Filter N|------>|           |
  | .wav file |       |        |        |        |       | .wav file |
  |___________|       |________|        |________|       |___________|

  Figure 1. The signal processing structure.


The GramoFile user interface is used to specify what filters should be
used, and in what order. The parameters of each filter may also be changed
using the user interface. This is the option `Process the audio signal' in
the Main Menu.

In this version of GramoFile, four filters have been implemented:
 - Simple Mean Filter
 - Simple Median Filter
 - Double Median Filter
 - Conditional Median Filter

In the remainder of this document these four filters will be described in
more detail. Also more information is given on the filter framework, and
how to implement new filters. 


SIMPLE MEAN FILTER

The Simple Mean Filter is the most simple filter in GramoFile. It takes
the (unweighted) mean of an adjustable number of samples, see figure 2
below. 


  input signal    x[t-3]  x[t-2]  x[t-1]  x[t]  x[t+1]  x[t+2]  x[t+3]
                         \______________        ______________/
                                        \      /
                                          Mean
                                           |
                                           V
  output signal   y[t-3]  y[t-2]  y[t-1]  y[t]  y[t+1]  y[t+2]  y[t+3]


  Figure 2. The Simple Mean Filter.


In a mathematical expression:

                      i=-N
                1     ---
       y[t] =  ----   )    x[t+i]
               2N+1   ---
                       N

where the `length of the filter' is 2N+1; in figure 2 the length is 5, so
there N=2.

The effect of this filter on an audio signal with ticks, is that the ticks
will be `spread out', and be somewhat reduced in strength. However, they
still remain clearly audible. But also the `clear' audio signal is
affected. With filter lengths greater than 5, the quality of the output
audio signal degrades rapidly. This filter behaves like a crude lowpass
filter.


SIMPLE MEDIAN FILTER

The Simple Median Filter computes the median of an adjustable number of
samples. A well-known property of a `running median' is its ability to
filter out short impulses (like ticks) in a signal.

The median of a row of numbers is the middle one if the row is sorted by
value. To find the median of the numbers 5, 3, 8, 1, 2, we sort them
first, resulting in 1, 2, 3, 5, 8, and then take the middle one, 3 (note
that the mean is different, namely 19/5 = 3.8).

The impulse-filtering property is illustrated in figure 3, where a
(distorted) signal is filtered with a running medians of length 3 and 5,
so every time the median of 3 or 5 samples is taken. 


   input signal:            median-3 filtered:       median-5 filtered:

       |      | |                       |
       |     |||| |                   |||||                    |||||
       |    ||||| |                  ||||||                   ||||||
   ||  | || ||||| || |      ||   |||||||||||||         |  |||||||||||||
   ||| |||||||||| |||||     ||||||||||||||||||||     ||||||||||||||||||||

   22105122134545042121     22111222234454422211     11211222234444422211

  Figure 3. Examples of running medians with lengths 3 and 5.


In figure 3, it is clear that all disturbances have disappeared in the
filtered signals. To be more precise, all disturbances with a length of 1
and 2, respectively, are filtered out. Generally, a median of 2N+1 samples
will filter out all disturbances with a length smaller than or equal to N.

In audio signals, ticks tend to have a length of more than, say, 5 samples
(CD-quality sampling). So when median filtering with length 11 is applied,
most ticks disappear. But also the sound itself is badly affected. Most of
the high frequencies are lost, and an annoying noise is added to the
signal.

This filter may be used to determine the best size for the median
filtering of the Conditional Median Filter.


DOUBLE MEDIAN FILTER

The Double Median Filter is described by R.R. Lawrence (et al.) in
`Applications of a Nonlinear Smoothing Algorithm to Speech Processing',
IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol.
ASSP-23, No. 6, Dec 1975. It is composed of a `normal' median filter and a
median filter that filters the error (disturbance) signal. It is depicted
in figure 4.

                 ________
    input       |        |                          +     output
    -------+--->| Median |-----+--------------------->O--------->
           |    |________|     |                      ^ +
           |                   |                      |
           |                   |         ________     |
           |                   V -      |        |    |
           +------------------>O------->| Median |----+
                              +  error  |________|


  Figure 4. Block diagram of the Double Median Filter.


The idea is, that the addition of the median of the error signal should
enhance the quality of the output signal. But during tests, there was not
too much audible difference with the Simple Median Filter.


CONDITIONAL MEDIAN FILTER

The Conditional Median Filter is described in detail by T. Kasparis (et
al.) in `Adaptive Scratch Noise Filtering', IEEE Transactions on Consumer
Electronics, Vol. 39, No. 4, Nov 1993. The basic functionality is 1)
detect ticks, 2) interpolate if there is a tick.

Ticks are mostly `composed of' high frequencies. Undisturbed audio signal
(especially from gramophone records) only has relatively low frequencies. 
This observation is used to detect ticks and scratches. The interpolation
is performed by a median filter, see above (Simple Median Filter) for its
interpolating properties. 

The block diagram of the Conditional Median Filter is shown in figure 5.


                                                 __________
  input x[t]                                    |          |  output y[t]
  ----------+---------------------------------->|  Median  |------------->
            |                                   |  Filter  |
       _____V_____                              |__________|
      |           |                                   ^
      | Highpass  |                                   |
      | Filter    |                                   | g[t]
      |___________|                                   |
            |                                    _____|_____
            | w[t]                              |           |
            +---------------------------------->|           |
            |    _____       ___________        | Gate      |
            |   |     |     |           |       |           |
            |   | |   |     | Recursive |  b[t] | Generator |
            +-->| V K |---->| Median    |------>|           |
                |_____|     | Filter    |       |___________|
                            |___________|

  Figure 5. Block diagram of the Conditional Median Filter.


The highpass filter in reality is a combination of a second-order
derivative, and a Root-Mean-Square filter. The second order derivative is

    z[t] = x[t-1] - 2 x[t] + x[t+1]

and the RMS is

                    --------------------------
                   /        i=-N
              _   /    1    ---             2
    w[t] =     | /   ----   )     ( z[t+i] )
               |/    2N+1   ---
               /             N

where 2N+1 is the `length' of the RMS operation. The resulting signal w[t]
has large values if there is a tick, and is very `quiet' otherwise.

But w[t] is not 0 if there are no ticks, it has a certain `background
noise' level. That background level is estimated by taking the median of a
large number of samples of w[t]. The best results are accomplished using a
recursive median filter, defined as

    y[t] = median ( y[t-M], ... , y[t-1], x[t], x[t+1], ... , x[t+M] )

where x[t] is the input and y[t] the output. To reduce calculation time,
the signal w[t] is decimated first (only one of K samples is considered). 
The estimated background level resulting from decimation and recursive
median filtering is b[t]. 

The gate generator is (currently) nothing more than a simple condition,
namely

             /           w[t] - b[t]
            |   1,  if   -----------  >  C
            |                b[t]
    g[t] = <
            |
            |   0,  otherwise
             \

where C is a certain factor, for example 2.5. If the gate signal g[t] is
1, the `main' median filter is `switched on', letting it compute medians,
otherwise the signal x[t] passes unchanged.

In the Properties screen of the Conditional Median Filter, several
parameters van be changed:
 - Length of the `main' median filter
 - Length of the RMS operation on z[t] in the Highpass Filter (2N+1)
 - Length of the Recursive Median Filter (2M+1)
 - Decimation factor (K)
 - Threshold for gate generation (C)

The optimal length of the `main' median filter may be found by using the
Simple Median Filter on a piece of badly disturbed audio signal. 

During tests, the Conditional Median Filter has given excellent results. 
With the default properties (21 - 9 - 11 - 5 - 2500), nearly all ticks are
filtered out, without audible losses in signal quality. But some rare
musical instruments generate tones that can fool the tick detection
algorithm, resulting in faulty interpolation. If this occurs, try to use
the properties 15 - 11 - 9 - 4 - 2500. 


THE FILTER FRAMEWORK

(Well, we're getting more technical every minute...)

Audio samples don't travel through filters out of free will. Actually,
they are pulled through by the harddisk. This is made visible in figure 6.


   ___________         ________          ________         ___________
  |           |  Q    |        |        |        |  Q    |           |
  | Harddisk  |<------|        |<- - - -|        |<------| Harddisk  |
  |           |       |Filter 1|        |Filter N|       |           |
  | .wav file |------>|        |- - - ->|        |------>| .wav file |
  |___________|    S  |________|        |________|    S  |___________|

  Figure 6. Questions and answers (Samples) in the signal processing
  structure


The destination file on the harddisk `asks' the last filter for the next
sample (that's the Q of Question). But a filter can't produce an output
sample without an input sample. So the last filter asks the previous
filter for the next sample. Those questions continue till the first filter
is reached. That filter `asks' the source file on the harddisk. Then the
first filter can compute its output sample, and passes it through to the
second filter. When all filters have computed their outputs, the
destination file eventually gets the next (processed) sample.

This process is easily implemented in C, as illustrated in listing 1
below.


    sample_t filter1()
    {
        sample_t nextsample;
        sample_t procd_sample;

        nextsample = read_from_file();
        /* Do processing */
        return procd_sample;
    }

    sample_t filter2()
    {
        sample_t nextsample;
        sample_t procd_sample;

        nextsample = filter1();
        /* Do processing */
        return procd_sample;
    }

    write_funtion()
    {
        sample_t nextsample;

        for (...)
        {
            nextsample = filter2();
            write_to_file(nextsample);
        }
    }

  Listing 1. Example implementation of questions and answers.


By calling filter2(), the write_function() effectively asks for the next
sample, and stores the answer in the variable nextsample. The filters
themselves ask the previous ones, and eventually the harddisk.

With only one sample traveling, there are only very limited possibilities
of filtering. Most filters need a few samples around the current one. For
example, the Simple Median Filter with length 3 will need samples x[t-1],
x[t], and x[t+1]. To accommodate this, so-called `buffers' have been
implemented. These buffers can `remember' and `read ahead' a few samples.
This is illustrated in figure 7.


   ___________         ________          ________         ___________
  |           |  Q    | :      |        | :      |  Q    |           |
  | Harddisk  |<------| : Filt1|<- - - -| : FiltN|<------| Harddisk  |
  |           |       |B:      |        |B:      |       |           |
  | .wav file |------>| :      |- - - ->| :      |------>| .wav file |
  |___________|    S  |_:______|        |_:______|    S  |___________|

  Figure 7. Buffers (B) in the signal processing structure. Questions
  (Q) and returned samples (S) stay the same.


When the harddisk-writing-routine asks for the next sample, the `effective
time' of the last filter is incremented by 1. So the buffer `window' of
that filter advances one time unit. The oldest sample is discarded (e.g.
the old x[t-2], now x[t-3]), and a new sample is needed (e.g. the new
x[t+2]). So that buffer asks the previous filter to supply the sample for
time-index t+2.  Then t+2 is the effective time for the last-but-one
filter. In general, all filters have different effective times. 

When starting the filtering process, all filters have an effective time
t=0. But all buffers must get filled before the first sample appears at
the output of the last filter, so first all buffers will fill up and the
effective times will be adjusted automatically. During the rest of the
process, the differences in effective times will remain constant.

But, you don't have to understand anything about effective times. That's
all embedded in the buffer routines. And it's working fine. 

When we zoom in a little, the structure becomes more clear, figure 8. 


                                 ________
                                |        |
                  +-------------|advance_|
                  |             |current_|<-------------------+
                  |    +------->|pos()   |                    |
    params        |    |     S  |________|    params          |
      |         Q |    |            |S          |             |
     _V___       _V____|_          _V_       ___V____       __|___
          | Q   |get_    |        |___| \   |        | Q   |      |
          |<----|sample_ |        |___|  |  |get_    |<----|      |<---
     Filt1|     |from_   |        |___|  |->|from_   |     |Filt 2|
          |---->|filter()|        |___|  |  |buffer()|---->|      |--->
     _____|   S |________|        |___| /   |________|   S |______|
                                 buffer

  Figure 8. Interconnection of filters and buffers.


A filter can read from its buffer using the get_from_buffer function. This
function has a parameter `offset', to specify which sample should be
returned. An offset of -2 means the x[t-2] sample. 

The function advance_current_pos has the effect of increasing the
effective time by 1. In other words, it moves the buffer `window' one time
unit further. The oldest sample is shifted out (discarded), and a brand
new one must be shifted in. To accomplish this, the function
get_sample_from_filter is called. This routine determines which filter
should be questioned for the next sample, gets and returns it to the
advance_current_pos function, that shifts it in the buffer. The
advance_current_pos function is called once at the very beginning of each
filter routine.

The first time the advance_current_pos function is called, the buffer is
empty. Advance_current_pos takes care that it is filled correctly, the
first part with zeroes (silence) and from x[t] to x[t+N] with new samples,
calling the get_sample_from_filter routine repeatedly. So, after each call
to advance_current_pos, the buffer is completely filled and up-to-date.

Each filter has its own set of parameters, that are changeable via the
GramoFile user interface.

Both filter 1 and filter 2 may be `instances' of the same filter `type'.
For example, both may be Simple Median Filters. The software routines for
the filters then are the same. The only things that are different are the
parameters and the buffers. Actually, the buffer is seen as another
parameter. During the initialization of the filtering process, a number of
parameter-sets are created in memory. The get_sample_from_filter not only
finds out which filter it should ask for the next sample, also it tells
the filter which set of parameters to use. That is why the filters are
declared like

    sample_t simple_median_filter(parampointer_t parampointer)

accepting a (pointer to a) set of parameters, and returning the computed
sample.


IMPLEMENTING NEW FILTERS

Implementing a new filter shouldn't be that difficult. You don't have to
think about reading or writing from or to a .wav file, because that has
been implemented already. As are the buffer-functions. The only thing you
have to worry about is the filtering algorith itself.

It is a good idea to look at a simple example, to get an idea of how it
works. Let's take the Simple Mean Filter (in signpr_mean.c). There are
five functions:
 - reset the parameters to some reasonable defaults
 - provide a properties-screen by which the parameters may be changed
 - initialize the buffers and other parameters, called just before the
    signal processing starts
 - delete the buffer data structures that have been used, called after the
    signal processing has ended
 - the filter itself

A variable of the parampointer_t type is just a pointer to a param_t
structure. Both are declared in signpr_general.h. As pointed out above,
the buffers are part of the parameters (3 per filter, see below). Then
there is a `filterno', that is the serial number of the filter, in other
words the position of this particular filter in the `Selected Filters'
panel in the `Signal Processing - Options' screen. Both buffers and
filterno are filled in by the init_*_filter() functions. 

The parameters that are settable in the user interface (using the
*_param_screen() functions) are mostly pre- and postlengths, and an
int and long type variable (just create more if necessary). The
`postlength' is the number of samples that a buffer must `remember', the
`prelength' is the `read-ahead' amount. Int1 is for example used for the
decimation factor, and long1 for the threshold level. There are also some
sslists, signed-short-arrays, to avoid continuous creation/deletion of
local arrays, in an attempt so speed up the signal processing process, but
you don't have to use these.

The Simple Mean Filter is a easily understandable example of a signal
processing routine, and it doesn't need any more explanation. Note that
advance_current_pos() is called first of all.

The Simple Median Filter (signpr_median.c) uses the sslists to
temporarily copy the entire buffer to be sorted, in order to determine the
median. That takes place in the median() routine that calls the qsort2()
routine, both in signpr_general.c. The very fast sorting routine was
adapted from the example by L. Ammeraal in `Ansi C', Academic Service,
Schoonhoven, The Netherlands, 2nd ed., 1993.

The Double Median Filter uses two buffers. That is needed because two
median functions are present, and the second uses the results of the
first. The block diagram from figure 4 is printed again below, now
including names for all signals.


                 ________
    x[t]        |        |  z[t]                    +       y[t]
    -------+--->| Median |-----+--------------------->O--------->
           |    |________|     |                      ^ +
           |                   |                      |
           |                   |         ________     |
           |                   V -      |        |    |
           +------------------>O------->| Median |----+
                              +   e[t]  |________| c[t]


  Figure 9. Block diagram of the Double Median Filter.


Below are example sequences of the various signals. Note that every signal
can be computed completely when all upper signals are known. This is the
way you would do it by hand.


              t<0
                 :
    x[t]       0 : 100  50  30  80  90  10  70  50  40  20  10  80  20
                 :      /       \___  ___/
                 :     /            \/
    z[t]       0 :  50 |50  50  80  80  70  50  50  40  20  20  20
     =med(x)     :     \ |                              /
                 :      \|                             /
    e[t]       0 :  50   0 -20   0  10 -60  20   0   0 | 0 -10  60
     =x-z        :                  \___  ___/         |
                 :                      \/             |
    c[t]       0 :   0   0   0   0   0  10   0   0   0 | 0   0
     =med(e)     :                                     \ |
                 :                                      \|
    y[t]       0 :  50  50  50  80  80  80  50  50  40  20  20
     =z+c        :


  Figure 10. Example signals for the Double Median Filter, when both
  median-lengths are 3. The lines point at the values needed to find the
  lowest one.


All filers for which you can find the answer in this way can be
implemented in GramoFile. But you must be inventive with the use of
buffers. Generally speaking, every computation that needs a sample other
than one directly above it, will need a buffer. With the Double Median
Filter, both z[t] and x[t] will need one. Well, we _can_ do with only one,
but that's much much slower... I've implemented the Double Median Filter
as shown below.


        ______________________________________
  x[t] |  |-4|-3|-2|-1|+0|+1|+2|+3|+4|+5|+6|  |
       |__|__|__|__|__|__|__|__|__|__|__|__|__|
                          \_____ ______/
                       |        V
                       |        |
          .------------'     Median
          |                     | z[t]
          |   __________________V_
          |  |-3|-2|-1|+0|+1|+2|+3| z[t]
          |  |__|__|__|__|__|__|__|
          |             \
          |          - | \
          |            V  `-----------.
          `----------->O              |
                    +  | e[t]         |
                       V              |
             \__________ _________/   |
                        V             |
                        |             |
                     Median           |
                        | c[t]        |
                      + V             |
                        O<------------'
                        |  +  z[t]
                        V
                       y[t]

  Figure 11. Schematic of a possible implementation of the Double Median
  Filter.


The original idea was to buffer e[t], but then we've lost z[t], which we
need to compute y[t]. This way there is only a little drawback, for we now
have to do one addition extra when we need e[t].

The second buffer is implemented in a similar way as the `normal' buffer,
it is accessible through the get_from_buffer() function. But filling the
buffer works somewhat different, as the next sample can't be asked from
the previous filter. The next sample must (generally) be computed from the
contents of the `normal' buffer. To allow this, there is a function
advance_current_pos_custom() that accepts a fill-function as parameter. 
This fill-function will be called to supply the value for the samples in
the second buffer. 

As can be seen in figure 11, the fill-function for the second buffer of
the Double Median Filter will return the median of a row of samples in the
first buffer. Which samples should be considered, is dependent on the
position of the to-be-computed sample in the second buffer. For example,
z[t] is the median of x[t-2] through x[t+2], and z[t+5] is the median of
x[t+3] through x[t+7]. The fill-function needs to know where to look, so
some parameters have to be supplied to it, namely offset and offset_zero. 
Offset is the position in the second buffer, and offset_zero is the
position of the `0'-position in the second buffer relative to the
`0'-position in the first buffer. In the case of figure 11, offset is +3
and offset_zero is 0. Offset is supplied by advance_current_pos_custom(),
and offset_zero must be supplied to advance_current_pos_custom(), that
will pass it through. 

Advance_current_pos_custom() should be called immediately after the call
to the ordinary advance_current_pos(), in order to assure up-to-date
contents of all buffers.

In the case of the Double Median Filter, the main routine (the one that is
called by get_sample_from_filter()), is double_median_filter(). The first
thing is the customary call of advance_current_pos() that updates the
first buffer. Then the second buffer may be updated, so
advance_current_pos_custom() is called. The first argument is the buffer
that should be updated, buffer2 in this case. The next argument is the
fill-function, or, to be more specific, a pointer to that function. That
pointer is declared just above double_median_filter(), and is of the
fillfuncpointer_t type, defined in signpr_general.h. The third argument is
the offset_zero. In this case, the second buffer is `fixed' at
offset_zero=0. The last argument is the omnipresent parampointer, with the
invaluable information about pre/postlengths etc.

The fill-function double_median_filter_1() is much like an ordinary
filter, only advance_current_pos() isn't called here (already done in the
main routine). But the computation and returning of the computed sample
are still there. Note that, during the computation, the offset and
offset_zero are used to determine the interval of the median. In this
case, the offset and offset_zero are added, as they will be in most cases.
Offset_zero will always be 0, but is left for completeness.

Back in double_median_filter(), operation continues normally. The
structure of figure 11 is clearly visible. The `j /= 2' is there to
prevent overflow in case sample and sample2 have different signs and large
values. Compensation after computation of the median. Yes, this is really
wrong, but then this filter is not meant for real use...

The Conditional Median Filter (signpr_cmf.c) is even more complex, as it
uses three buffers. The structure is illustrated in figure 12.


             __________________________________________________
       x[t] |  |-4|-3|-2|-1|+0|+1|+2|+3|+4|+5|+6|+7|+8|+9|  |  |
            |__|__|__|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
                                                    \__ ___/
                \___________ ____________/             V
                            V                          |
                            |                       Highpass
        ,-------------------'                          | z[t]
        |                            __________________V_
        |                      z[t] |-3|-2|-1|+0|+1|+2|+3|
        |                           |__|__|__|__|__|__|__|
        |                            \________ _________/
        |                                     V
        |                                     |
        |    ,--------------.                RMS
        |    |              |                 | w[t]
        |    |     _________V_________________V_
        |    |    |-3|-2|-1|+0|+1|+2|+3|+4|+5|+6| w[t]
        |    |    |__|__|__|__|__|__|__|__|__|__|
        |    |               \
        |    |                `---------------------.
        |    |                                      |
        |    |     \________ ___ ___ ___ ___ __/    |
        |    |              V     (decimate)        |
        |    |              |                       |
        |    |           Median                     |
        |    |         b[t] |                       |
        |    |             /|  ,--------------------'
        |    `------------' |  | w[t]
        |                   V  V
        |                 Compare
        |                    | g[t]
        V     gate           |
     Median <----------------'
        |
        |
        V
      y[t]

  Figure 12. Schematic of the implementation of the Conditional Median
  Filter.


The general structure of the implementation resembles that of the Double
Median Filter. Some custom fill-functions are used to fill the two `extra'
buffers. In particular the buffer of w[t] is necessary, due to the
recursive nature of the computation that is done with it. The recursive
median is implemented as an ordinary median, but after each computation,
the result is stored again in the w[t] buffer. This is dune with the
put_in_buffer() function (we grab w_t before putting the new value in its
place). Decimation is accomoplished by selecting only a few of the
pre-read samples of w[t], and computing the median thereof. 

In the main filter routine, cond_median_filter(), there is first the
customary call of advance_current_pos, and thereafter only one call to
advance_current_pos_custom(), for the third buffer that is fixed at
offset_zero=0. The second buffer is not fixed, but `floats' with the
to-be-computed offset in the third buffer. The second buffer is therefore
updated in the fill-function of the third buffer. The offset_zero of the
second buffer is offset + offset_zero (both of the third buffer, the
latter being 0). In the situation of figure 12, the offset in the third
buffer is +6, so the offset_zero of the second buffer also is +6. The
fill-function of the second buffer is supplied with its own offset, +3 in
the case of figure 12. The second derivative (`highpass operation') now
needs to be computed for the offset+offset_zero = 9th sample in the first
buffer.

Special care should be taken when allocating the first buffer in the
init_*_filter() functions, as there may be more places where the samples
in the buffer are accessed. In the Conditional Median Filter, the samples
in the first buffer are accessed by the main routine, that computes the
median if needed, and by the fill-function of the second buffer, that
computes the second derivative (highpass). So, the buffer must be big
enough to accommodate the median. The highpass function is different, as
it `floats'. During the first filling of the buffers (when the first
sample is asked by the harddisk), all offsets and offset_zeroes are 0, so
the highpass needs samples x[t-1], x[t] and x[t+1]. When all buffers are
filled, the middle of the highpass lies at the added values of the
`prelengths' of the second and third filter. In the case of figure 12, the
minimal prelength of the first buffer is 6 + 3 + 1 (the `right side' of
the second derivative) = 10. In that example, the median requires x[t-4]
through x[t+4], and the highpass requires x[t-1] through x[t+10], so the
first buffer must at least contain x[t-4] through x[t+10]. 

If you want to implement a new filter, it is generally a good idea to
start with a numeric example like figure 10. It gives a good overview of
the filter, and also a way to verify the correctness of the implementation
in a later phase. With such a numeric example, the drawing of a diagram
like figures 11 and 12 isn't too difficult. After that, implementing
should be straightforward.

If there are problems with either this document or the implementation of a
new filter, you can contact me. See the general `README' file for more
information. If it is an implementation problem, the inclusion of drawings
like figures 4, 5, 9, 10, 11 and 12 is highly recommended. Also, if you
have succeeded in implementing a new filter, please send the source files
and documentation to me, so that it can be included in a next release of
GramoFile.



