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In this section, we use the following conventions:
Appendix A will be on the hetrodyne filter
Appendix B will be on the derivation of the all-pass reverberators
Appendix C will be on spectral shaping filters
There is a reference (as yet unnumbered) to the
hetrodyne filter paper
There is a reference to Chowning's FM article
There are references to Schroeder's stuff
There are references to the linear predictor (Atal and Markel)
There are references to the cepstrum (Oppenheim, Schafer, Miller)
references to Gold and Rader
references to Boxer-Thaler transform
Figure 1 is a perspective view of a Hetrodyne filter output
Figure 2 is a pseudo-sonagram (one without nasty glitches)
Figure 3 is a Chowning FM perspective view
.NEXT PAGE
.SELECT B
COMPUTATIONAL METHODS
.SELECT A
SYNTHESIS AND ANALYSIS TECHNIQUES
.SELECT 1
.BEGIN FILL ADJUST
In this section, we describe the computational techniques
used in the analysis and synthesis of musical instrument tones. Let
us begin by catagorizing the general types of synthesis.
Sound synthesis can be roughly divided into three catagories: additive
synthesis, modulation synthesis, and subtractive synthesis.
We will discuss each of these techniques together with a description
of what analysis procedures are available to help specify the
synthesis.
.END
.GROUP SKIP 2
.SELECT 3
ADDITIVE SYNTHESIS
.SELECT 1
.BEGIN FILL ADJUST
In additive synthesis we physically
model the digitized sound waveform as a sum of sinusoids with slowly
time-varying amplitudes and phases. Equation (1) summarizes this
formulation. Each of the sinusoids in equation (1) represents the
partial tones.
.END
M
(1) F%8α%1 = %6S%1 A%8j%1 sin(%4w%8j%1hα+%4q%8j%1)
j=1
Notation: F%8α%1 is the sampled, digitized waveform at time αh
h is the time between consecutive samples
A%8j%1 is the amplitude of the jth partial tone
and is assumed to be slowly varying with time
%4q%8j%1 is the phase of the jth partial tone
and is assumed to be slowly varying with time
.BEGIN FILL ADJUST
One can see that from this model, if we can determine the
parameters A%8j%1 and %4q%8j%1 of a tone from a musical instrument,
we can easily synthesize the waveform F%8α%1 from those parameters
by use of equation (1). The degree to which this form of synthesis
has been successful will be discussed in a later section. Here we
will concern ourselves strictly with the description of the computations.
Since equation (1) uses amplitudes and phases that are varying slowly
with time, technically these functions modulate the waveform, thus
blurring the division between additive synthesis and modulation synthesis.
The distinction is somewhat arbitrary. If the functions vary slowly, we
call it additive synthesis.
.END
.NEXT PAGE
.SELECT 3
ANALYSIS FOR ADDITIVE SYNTHESIS
.SELECT 1
.BEGIN FILL ADJUST
We now turn to the problem of determining the parameters
A%8j%1 and %4q%8j%1 of a musical instrument tone. To aid the
analysis, we must assume the frequencies of the partial tones,
%4w%8j%1, are nearly harmonically related. That is, there is some
frequency, %4w%1, such that %4w%8j%1 is approximately j%4w%1. We shall
call this frequency %4w%1 the fundamental frequency of the tone.
The method we have found most useful we call the "hetrodyne
filter." This is described in detail in reference [**Moorer**] and is
derived briefly in appendix A. Basically, the method is as follows:
First, compute the following two summations at each point in time α.
.END
α+N-1
(2) %9a%8j%8α%1 = %6S%1 F%8i%1sin(j%4w%80%1ih+%4f%80%1)
i=α
α+N-1
(3) %9b%8j%8α%1 = %6S%1 F%8i%1cos(j%4w%80%1ih+%4f%80%1)
i=α
From these, we calculate two more sequences:
(4) A%8jα%1 = (%9a%8jα%22%1+%9b%8jα%22%1)%21/2%1
(5) %4f%8jα%1 = atan(%9a%8jα%1/%9b%8jα%1)
.BEGIN FILL ADJUST
The summations are taken to be over one period of a sinusoid
of frequency %4w%80%1, that is, Nh%4w%80%1 = 2π. This places
somewhat of a restriction on the frequency of analysis, %4w%80%1,
because in the discrete domain, the period, N, is restricted
to integral values. This has not proved a problem in our experience.
If the partial tones are nearly harmonically related, if the
parameters of the tone vary slowly with time, and if %4w%80%1 is
equal to the fundamental frequency of the tone, then A%8jα%1 and
%4f%8jα%1 will indeed be approximations to the actual amplitudes and
phases of the partials of the tone under analysis.
To review the process, we must evaluate the
equations (2), (3), (4), and (5) for each of the partials of a
tone, over the entire time interval spanned by the tone. The output
A%8jα%1 and %4f%8jα%1 may then be used in equation (1) to synthesize a new
tone that hopefully retains much of the character of the original.
.END
.NEXT PAGE
.SELECT 3
MODULATION SYNTHESIS
.SELECT 1
.BEGIN FILL ADJUST
The second method of synthesis is modulation synthesis. Here
we produce the tone by varying the amplitudes and phases of a small
number of sinusoids. As noted before, the feature which distinguishes
modulation synthesis from additive synthesis is that one or more
of the parameters is no longer
restricted to be a slowly time varying function.
Modulation synthesis is generally classified as synthesis by amplitude
modulation or by frequency modulation, or by some combination of the
two. We will restrict ourselves to frequency modulation here, because
we have made significant discoveries in this field.
A more complete description of
frequency modulation synthesis is given in [**Chowning**]. We shall
describe briefly the essence of the method. The basic equation is
shown in equation (6).
.END
(6) F%8α%1 = A sin(%4w%8c%1αh + I sin(%4w%8m%1αh))
.BEGIN FILL ADJUST
For the purpose of this description, we will restrict the modulating
waveform to a pure sinusoid. I is called the "modulation index." For
useful synthesis, we allow the amplitude A and the modulation index I
to vary slowly with time.
.END
We may expand equation (6) as follows:
(7) F%8α%1 = A{J%80%1(I) sin(%4w%8c%1αh)
+ J%81%1(I)[sin((%4w%8c%1+%4w%8m%1)αh)-sin((%4w%8c%1-%4w%8m%1)αh)]
+ J%82%1(I)[sin((%4w%8c%1+2%4w%8m%1)αh)+sin((%4w%8c%1-2%4w%8m%1)αh)]
+ J%83%1(I)[sin((%4w%8c%1+3%4w%8m%1)αh)-sin((%4w%8c%1-3%4w%8m%1)αh)]
+ . . . }
.BEGIN FILL ADJUST
where J%8i%1(I) is the ith Bessel function of the first kind as a
function of the modulating index, I. Thus we see that the tone
represented by the waveform of equation (6) consists of a series of
sinusoidal componants whose frequencies are represented by the sum of the
carrier frequency, %4w%8c%1, and integral (positive and negative) multiples
of the modulating frequency, %4w%8m%1. As the modulation index increases,
the amplitudes of the partials change in complex ways, but the general trend
is that energy is shifted away from the carrier frequency. When the modulation
index is zero, equations (6) and (7) degenerate to a pure sinusoid at
the carrier frequency. As the modulation index increases, more energy is
transferred to partials representing larger and larger integral multiples
of the modulation frequency.
If the carrier frequency and the modulating frequency are integral
multiples of some other frequency, %4w%1, then the partials form a harmonic
sequence. If the carrier and modulating frequencies are not so related,
then the partials are inharmonic. Both of these cases have use in
synthesis.
To produce useful musical tones, the amplitude A and the modulating
index I are selected to be slowly time varying functions, usually very
simple in form. The details of the functions used to produce interesting
tones will be given in a later section. We will only discuss some
of the spectral implications that are not entirely obvious from
equation (7).
One interesting result occurs with the terms where %4w%8c%1-k%4w%8m%1 is
less than zero. Since this is equivalent to a positive frequency with
a phase shift, one can see that there are contributions which might
not be obvious at first glance. Suppose %4w%8c%1 = %4w%8m%1 for simplicity.
In this case, the partial at %4w%8c%1 will have amplitude (J%80%1(I)-J%82%1(I)),
the partial at 2%4w%8c%1 will have amplitude (J%81%1(I)+J%83%1(I)), and
so on. We call these negative frequencies "reflected sidebands,"
a term from the technology of FM broadcasting. Consider the case where
%4w%8m%1 = 2%4w%8c%1. Here, the only nonzero componants will be at
odd multiples of %4w%8c%1, with amplitudes again being sums of pairs
of bessel functions.
%4w%8c%1 does not always represent the fundamental frequency of the
tone. For instance, if %4w%8c%1 = 2%4w%8m%1, then a harmonic
series based on %4w%8m%1 is produced.
If we take the modulating function to be not just a single
sinusoid, but a sum of sinusoids, the resulting expansion is similar
to equation (7). The partials will be at the set of frequencies
consisting of the sum of all integral multiples of the modulating
frequencies added to the carrier frequency. The amplitudes of the
partials will be products of the Bessel functions of the modulation
indices. We can produce combinations of harmonic and inharmonic spectra
simply by the choice of the modulating frequencies.
.END
.GROUP SKIP 1
.SELECT 3
FM PREDICTIVE ANALYSIS
.SELECT 1
.BEGIN FILL ADJUST
There does not seem to be any meaningful way to determine the index
and amplitude functions for FM synthesis directly from a digitized
musical instrument tone. This being the case, we must rely upon the
intuition of the researcher to come up with suitable functions. There
are, however, tools we have given the researcher to aid in exploring
the consequences of various choices of index and amplitude functions.
The most powerful technique is an interactive graphical program which
allows the user to design amplitude and index functions,
and using those functions
evaluate equation (7). The program makes a perspective plot of the amplitudes
of the partials as a function of time. Figure 3 shows such a plot.
After the display is produced, the user can then go back and alter his
function definitions to try to converge on the desired spectral characteristics.
.END
.GROUP SKIP 1
.SELECT 3
SUBTRACTIVE SYNTHESIS
.SELECT 1
.BEGIN FILL ADJUST
The last form of synthesis is called subtractive synthesis.
The procedure here is to take a simple signal with a wide bandwidth,
such as a pulse train or a band-limited sawtooth wave, and apply
spectral shaping filters to produce the desired partial tone
amplitudes. We have not as yet used this form of synthesis, but
intend to do so in the near future. This is the type of synthesis
most commonly used in vocoders. We may thus assimilate the techniques
of analysis and synthesis of human speech and apply them in a more
general context. Two of the most useful methods seem to be the linear
predictor [**Atal, Markel**] and the homomorphic vocoder [**Oppenheim,
Schafer, Miller**].
.END
.GROUP SKIP 1
.SELECT 3
ANALYSIS FOR SUBTRACTIVE SYNTHESIS
.SELECT 1
.BEGIN FILL ADJUST
Generally, the technique is described as follows. A musical
instrument tone is analysed at discrete intervals, for instance,
every 5 milliseconds. At each analysis point, we compute a filter
whose frequency response approximates the spectral shape of the input
waveform in the interval around the analysis point. To resynthesize
the signal, we filter a pulse train, updating the filter parameters
at each analysis point. In the case of the linear predictor, the
filter is an all-pole filter. For the homomorphic vocoder, the filter
is an all-zero filter.
Since these methods are well documented in the literature [**refs**], we shall
not attempt to explain them here.
We anticipate that the linear predictor will be useful for
analysing percussion instrument tones such as drum and cymbal. In
these cases, the excitation might be modeled as band-limited noise,
rather than an impulse train, the spectral shaping being applied by
the filter produced by the linear prediction algorithm. Although the
same thing could be done with the homomorphic vocoder, a difficult
convolution is then required.
.END
.NEXT PAGE
%A
GRAPHICAL TECHNIQUES
%1
.BEGIN FILL ADJUST
As aids to the researcher, we have prepared several different
ways of displaying the results of analysis. The output of the
hetrodyne filter can, of course, be displayed as a number of
amplitude and phase plots, but it is more useful to view them
together. One method is the use of perspective plots. Figure 1 shows
such a plot of the amplitudes of the partials of a tone. The total
duration of the tone is about 500 milliseconds. The first harmonic is
in the background of the picture, the highest is in the foreground.
This aids in discovering correlations between the harmonics and
detecting features, such as formant peaks. We also have an
interactive program using the techniques of computer graphics where a
plot such as the one in figure 1 is pictured as being "suspended" in
space. The observer then by typing commands to the program can "fly"
around the plot and view it from different angles. This is very
helpful in getting a picture of the behavior of the partials of a
tone.
Since the perspective plot discards the detailed phase
information, another way of looking at the output of the hetrodyne
filter is shown in figure 2. This was inspired by sonagram plots.
Here, the thickness of each bar is proportional to the log of the
amplitude of that harmonic. The vertical position represents its
instantaneous frequency, as determined from the phase drift of the
harmonic.
Having had so much success with displays of this form, a
program for display of dynamic FM spectra was written. This program
allows the user to define the amplitude and modulation index
functions and then evaluate equation (7) to produce a display of the
spectrum of the waveform as a function of time. Figure 3 shows such a
plot. The sub-plots above show the particular amplitude function and
index function. Since there is no way of analysing a tone to directly
produce amplitude and index functions that is known at present, one
must invent suitable functions. This program has been a great aid in
developing the insight necessary to do so.
.END
.NEXT PAGE
%A
ENVIRONMENTAL SIMULATION
%1
.BEGIN FILL ADJUST
The only techniques we will discuss here are reverberation
and spectral shaping. Schroeder [**reference**] discussed methods of
producing natural sounding reverberation. We have extended his work
somewhat. For reasons which will be discussed in subsequent sections,
we have been recently working with flat, or un-colored reverberation.
The basic unit in such a scheme is the all-pass network. We use two
different forms of the all-pass.
The derivations and formulas for the all-pass networks we use are
explained in appendix B. Basically, the first form is a first order
unit whose impulse response is a pulse train with exponentially
decaying amplitude.
The second all-pass is a second order unit whose impulse response is a
pulse train whose amplitude is a damped sinusoid.
At first glance, this might
seem to be an undesirable property because one might perceive the
frequency of this sinusoid as a spurious tone. We have found
conditions, however, under which the oscillatory nature of
the second order unit may be used to great advantage.
To produce a full reverberation unit, several of the basic
unit reverberators are cascaded. The parameters of each unit
(decay time, pulse spacing) are adjusted to produce a smoothly
decaying impulse response.
We have not yet attempted any spectral shaping, but we plan
to use several digital filters in so doing. The simplest filters
we intend to use are the digital resonator and anti-resonator.
The resonator can, of course, be used to simulate formant peaks
in the spectrum.
Two other filters which are useful for producing multiple spectral resonances
or anti-resonances that are at harmonically related frequencies are the
two versions of the comb filter, an all zero version and an all pole version.
They each have a natural frequency at integral multiples of some base
frequency. This can be used in a variety of ways. For instance, if one
passes white noise through a comb resonator, depending on the Q of
the resonance, one can produce nearly periodic waveforms. The comb
filter with only zeros can perfectly anihilate a periodic input if
the delay in the comb is identical to an integral multiple of the
period of the input waveform.
The formulas for these filters are presented in appendix C.
.END
.NEXT PAGE
%B
APPENDIX A: THE HETRODYNE FILTER
%1
.BEGIN ADJUST FILL
In this section, we will compute the response of the hetrodyne
filter as defined by equations (2), (3), (4), and (5) to a
sinusoid of constant amplitude and phase. We do this by substituting
for F%8i%1 in equations (2) and (3) the function sin(%4w%1ih).
We may compute the summations without error by use of the summation
calculus [**Hamming**]. Using the fact that Nh%4w%80%1 = 2π, we may
calculate A%8jα%1 and %4f%8jα%1 explicitly.
.END
.GROUP
1 1 1
(A1) A%8jα%1 = --- sin%22%1(%4w%1Nh/2)%9{%1----------------- + -----------------
4N%22%1 sin%22%1[(%4w%1+j%4w%80%1)h/2] sin%22%1[(%4w%1-j%4w%80%1)h/2]
2cos[j%4w%80%1h-2%4f%1]
+ --------------------------------%9}%1
sin[(%4w%1+j%4w%80%1)h/2] sin[(%4w%1-j%4w%80%1)h/2]
.APART
.BEGIN FILL ADJUST
The expression for %4f%8jα%1 is a bit long and is thus not included here.
As we consider the limit as %4w%1 approaches j%4w%80%1, we find great
simplification of the results. Let us define %4Dw%1 as (%4w%1-j%4w%80%1).
.END
.GROUP
1
(A2) lim A%8jα%1 = ---%9{%10 + N%22%1 + 0%9}%1 = 1/4
%4w%1→%4w%80%1 4N%22%1
sin%9{%12%4w%80%1h[(N-1)/2+α]%9}%1 + N sin%9{%4Dw%1h[(N-1)/2+α]%9}%1
(A3) lim %9a%8jα%1/%9b%8jα%1 = ----------------------------------------------
%4w%1→%4w%80%1 cos%9{%12%4w%80%1h[(N-1)/2+α]%9}%1 + N cos%9{%4Dw%1h[(N-1)/2+α]%9}%1
.APART
.GROUP
If N>>1 then (A3) reduces greatly.
(A4) lim %9a%8jα%1/%9b%8jα%1 = tan%9{%4Dw%1h[(N-1)/2+α]%9}%1
%4w%1→%4w%80%1
.APART
.NEXT PAGE
.BEGIN FILL ADJUST
Thus we see that in the limit, A%8jα%1 approaches one quarter of the
amplitude of the input sinusoid and %4f%8jα%1 is a term related to
the difference in frequency of the input sinusoid with the analysis
frequency. With instruments with partials whose frequencies deviate
from the ideal, this provides a dynamic estimation of those frequencies.
Notice that A%8jα%1 is no longer a function of α, the time parameter,
but %4f%8jα%1 is. Fortunately, %4f%8jα%1 is a linear function of α
whose slope is simply %4Dw%1h.
Figure A1 shows a plot of A%8jα%1 as indicated in equation (1) for a
range of values of %4w%1. In this case, %4w%80%1 is 2π(125 Hz) and
j is 4. We see that there is a zero of transmission at all integral
multiples of %4w%80%1 except the j%2th%1 multiple.
This technique is useful as long as the amplitudes and phases
of the partials of the input waveform change slowly with time. If
the frequencies of the partials deviate from integral multiples of the fundamental
by too great an amount, further error may be introduced.
.END
.NEXT PAGE
%B
APPENDIX B: UNIT REVERBERATORS
%1
.BEGIN FILL ADJUST
A simple all-pass filter is the pole-zero pair, symmetrically
located about the j%4w%1 axis. The complex frequency response in the
continuous case is shown in equation (B1).
.END
S-%4s%1
(B1) H(S) = ----
S+%4s%1
.BEGIN FILL ADJUST
We may convert this to a digital filter by use of the bilinear transform.
.END
(%4s%1+1)Z%2-1%1 + (%4s%1-1)
(B2) T(Z) = - ---------------
(%4s%1-1)Z%2-1%1 + (%4s%1+1)
.BEGIN FILL ADJUST
Since this is an all-pass, the magnitude of T(Z) anywhere on the unit
circle is the same. This being the case,we may raise Z in equation (B2) to any power
without altering the magnitude of T(Z) on the unit circle. If
we raise Z to the power of -m, the frequency response will cycle m
times as we go around the unit circle.
.END
(%4s%1+1)Z%2-m%1 + (%4s%1-1)
(B3) T(Z) = - ---------------
(%4s%1-1)Z%2-m%1 + (%4s%1+1)
Which implies the recurrance relation:
(B4) Y%8n%1 = {(%4s%1-1)X%8n%1+(%4s%1+1)X%8n-m%1-(%4s%1-1)Y%8n-m%1}/(%4s%1+1)
.BEGIN FILL ADJUST
This is essentially the unit reverberator used by Schroeder, except
we have realized it in the cannonical form here, thus saving one
multiplication over the form used by Schroeder. This has a frequency
response that is identically constant around the unit circle. We will
call this the "first-order" unit reverberator, although technically
it is of order m.
There is one important generalization of this reverberator.
If we begin with an all-pass filter which has complex conjugate
poles rather than a single real pole, we realize a reverberator that
differs significantly in character from the one in equation (B3).
We begin with the following filter, again shown in the continuous
case:
.END
(S%22%1-2%4s%1S+%4s%22%1+%4w%80%22%1)
(B5) H(S) = ---------------
(S%22%1+2%4s%1S+%4s%22%1+%4w%80%22%1)
.NEXT PAGE
.BEGIN FILL ADJUST
This is again an all-pass. Let us transform it via the bilinear
transform and substitute a unit delay of m as was done above.
.END
C%81%1Z%2-2m%1+C%82%1Z%2-m%1+C%83%1
(B6) T(Z) = --------------
C%83%1Z%2-2m%1+C%82%1Z%2-m%1+C%81%1
Which, in turn, implies the recurrance relation:
(B7) Y%8n%1 = (C%83%1X%8n%1+C%82%1X%8n-m%1+C%81%1X%8n-2m%1-C%82%1Y%8n-m%1-C%83%1Y%8n-2m%1)/C%81%1
where C%81%1 = %4w%80%22%1+%4s%22%1+2%4s%1+1
C%82%1 = 2%4w%80%22%1+2%4s%22%1-2
C%83%1 = %4w%80%22%1+%4s%22%1-2%4s%1+1
.BEGIN FILL ADJUST
This, then, is another unit reverberator of a different character.
Equations (B2) and (B6) describe our unit reverberators. We place
different combinations of these unit reverberators in series to
produce a complete reverberator.
It should be noted that with the substitution of Z%2-m%1
for Z%2-1%1, the frequency of the sinusoid for the second order
reverberator (equation (B6)) becomes %4w%80/m%1. The decay time likewise
becomes m/%4s%1. The same is true about the first order reverberator.
.END
.NEXT PAGE
%B
APPENDIX C: SPECTRAL SHAPING FILTERS
%1
.BEGIN FILL ADJUST
We use several basic filters. We shall describe only
the ones that are different from the more common low-pass
and high-pass networks, as these are documented extensively
in the literature.
.END
.SKIP 1
.SELECT 3
SIMPLE RESONATORS AND ANTI-RESONATORS
.SELECT 1
.BEGIN FILL ADJUST
Let us begin with the digital resonator. This is most simply
expressed as follows:
.END
1-qZ%2-1%1
(C1) T(Z) = ----------------------
1-2r cos(%4w%80%1h)Z%2-1%1+r%22%1Z%2-2%1
With recurrance relation
(C2) Y%8n%1 = X%8n%1-qX%8n-1%1+2r cos(%4w%80%1h)Y%8n-1%1-r%22%1Y%8n-2%1
.BEGIN FILL ADJUST
This has a resonant frequency of %4w%80%1. r determines the Q of the
resonator. q is a zero of transmission. This resonator is used
extensively by Gold and Rader [**refs**], and thus will not be examined
further here.
Another filter that is useful is the anti-resonance,
or notch filter. In the continuous case, the transfer function
is given by:
.END
S%22%1+2%4d%1S+%4d%22%1+%4w%80%22%1
(C3) H(S) = ------------
S%22%1+2%4s%1S+%4s%22%1+%4w%80%22%1
.BEGIN FILL ADJUST
To prevent warping of the frequency axis, the Boxer-Thaler
transformation is used [**ref**]:
.END
.GROUP
C%81%1+C%82%1Z%2-1%1+C%83%1Z%2-2%1
(C4) T(Z) = --------------
C%84%1+C%85%1Z%2-1%1+C%86%1Z%2-2%1
.APART
.NEXT PAGE
.GROUP
With its related recurrance formula:
(C5) Y%8%1 = (C%81%1X%8n%1+C%82%1X%8n-1%1+C%83%1X%8n-2%1-C%85%1Y%8n-1%1-C%86%1Y%8n-2%1)/C%84%1
where C%81%1 = h%22%1(%4d%22%1+%4w%80%22%1)+12h%4d%1+12
C%82%1 = 10h%22%1(%4d%22%1+%4w%80%22%1)-24
C%83%1 = h%22%1(%4d%22%1+%4w%80%22%1)-12h%4d%1+12
C%84%1 = h%22%1(%4s%22%1+%4w%80%22%1)+12h%4s%1+12
C%85%1 = 10h%22%1(%4s%22%1+%4w%80%22%1)-24
C%86%1 = h%22%1(%4s%22%1+%4w%80%22%1)-12h%4s%1+12
.APART
.BEGIN FILL ADJUST
This gives a magnitude-frequency response that is unity everywhere
except near %4w%80%1. The strength at %4w%80%1 is determined by
the values of %4s%1 and %4d%1. If both %4s%1 and %4d%1 are small
compared to %4w%80%1, then the magnitude of H(S) at S=j%4w%80%1
will be approximately %4d%1/%4s%1. If %4d%1 is zero, the magnitude
goes to identically zero at %4w%80%1. If %4d%1=%4s%1, then we have
an all-pass network like the one used in appendix B as the
second order reverberator.
The magnitude-frequency response of the filter in equation (C4) is
plotted in figure C1 for five different values of %4d%1/%4s%1. The
value of %4w%80%1 was 2π(500 Hz), and %4s%1 was 2π(50 Hz). The
ratios %4d%1/%4s%1 were 4, 2, 1, .5, and .25.
.END
.GROUP SKIP 1
.SELECT 3
THE COMB FILTERS
.SELECT 1
.BEGIN FILL ADJUST
The comb filter comes in four forms, two of which are all-zero filters
and two of which are all-pole filters. The two all-zero filters
have the following recurrance relations:
.END
.GROUP
(C6) Y%8n%1 = X%8n%1+X%8n-m%1
(C7) Y%8n%1 = X%8n%1-X%8n-m%1
.APART
.BEGIN FILL ADJUST
These have magnitude frequency responses as follows:
.END
(C8) |T(%9e%1↑j%4↑w%1↑h)| = {[1+cos(m%4w%1h)]%22%1+sin%22%1(m%4w%1h)}%21/2%1
(C9) |T(%9e%1↑j%4↑w%1↑h)| = {[1-cos(m%4w%1h)]%22%1+sin%22%1(m%4w%1h)}%21/2%1
.BEGIN FILL ADJUST
It is clear that equation (C8) is zero at m%4w%1h = (2n+1)π and equation
(C9) is zero at m%4w%1h = 2nπ. Thus, applying the filter specified by
equation (C7) to a perfectly periodic waveform of period mh/n
of arbitrary harmonic content will exactly anihilate the waveform. The
filter of equation (C6) is somewhat more subtle. It will anihilate, for instance,
the odd harmonics of a waveform of period 2mh.
.END
.NEXT PAGE
.BEGIN FILL ADJUST
By placing a constant in the recurrance relations, we may move the
zeros off the j%4w%1 axis:
.END
.GROUP
(C10) Y%8n%1 = X%8n%1+gX%8n-m%1
(C11) Y%8n%1 = X%8n%1-gX%8n-m%1
.APART
.BEGIN FILL ADJUST
This causes the frequency response to approach zero, but never become
identically zero unless g is exactly unity.
We may invert the spectrum of these filters by placing the delay
in the feedback path, making recursive filters of these two:
.END
(C12) Y%8n%1 = X%8n%1-gY%8n-m%1
(C13) Y%8n%1 = X%8n%1+gY%8n-m%1
.BEGIN FILL ADJUST
The fact that the signs are reversed in the two equations is no
accident. Equation (C12) does indeed correspond to equation (C10),
and likewise equation (C13) to (C11). These two filters are like the
previous ones except that they have resonances where the others
had anti-resonances.
The magnitude-frequency responses for the filters in equations (C7)
and C(11) were plotted in figures C2 and C3 respectively
for 1/(mh) = 400 Hz and for four different values
of g, which were .25, .5, .75, and 1. It might be noted that the
graphs are truncated at 20 db. Actually, for g=1, the response in figure
C2 goes to zero, which would require our plot to extend to -∞. Likewise,
the response in figure C3 goes to +∞.
We have somewhat arbitrarily placed a maximum excursion on the plot at
+ or - 20 db.
.END
.NEXT PAGE
.SELECT B
REFERENCES
.SELECT 1
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