- ... direction.
^{2.1} - To determine whether a surface is
effectively flat, it may first be smoothed so that variations less
than a wavelength in size are ignored. That is, waves do not ``see''
variations on a scale much less than a wavelength.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{2.2} - In the code of Fig. 1.9, the first executable line
changes the global STK sampling-rate from 44100 Hz (the default) to
that of the input soundfile. If this is not done, there is a hidden
sampling-rate conversion using linear interpolation when the input
file is read, if its sampling rate is different from the global STK
sampling-rate. This default rate conversion could alternatively be
canceled by saying
`input.setRate(1.0);`after the input file is opened.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{2.3} - It is easy to see that the exponential function
has the property
. To show that all
differentiable functions with this property are exponentials, one can
look at the definition of the derivative of with respect to
:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... dispersion
^{2.4} *Dispersion*occurs in a traveling wave whenever the propagation speed is different at two or more frequencies.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... as
^{2.5} - Note that
Fig. 1.18 is given in direct form 2, so its describing equations are
. To make the figure
correspond exactly to the direct-form-1 difference equation, simply
move from the output arrow to the input arrow. That is, commute
the scaling by and the feedback loop so that is first.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... filter|textbf,
^{2.6} - Readers of this book are not expected to know linear algebra,
but one does not have to learn much to follow the discussion in this
section. See, for example, [422, available online] for the
basic matrix operations. An excellent linear algebra text is
[308].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{2.7} - The superscript denotes matrix
transposition (see any linear algebra text).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... line.
^{2.8} - This
structure may be quickly derived by forming a series combination of a
feedback comb filter followed by a feedforward comb filter, and
noticing that the two delay lines contain the same numbers at all
times. Therefore, the two delay lines can be replaced by a single
shared delay line.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...causal|textbf
^{2.9} - A filter is said
to be causal if its impulse response is zero for .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...fbffcf.
^{2.10} - Gerzon's starting point was Schroeder's
allpass which differs slightly from Eq. (1.13), having difference
equation [386]

with required for stability. This structure can be derived from Eq. (1.13) on page by moving the feedforward path to the other side of the input summer and deriving the new gain for . Gerzon's idea was to replace the -sample delay by a multi-input, multi-output (MIMO) allpass-filter matrix; Gerzon's ``unitary frequency-response matrices'' correspond to

*paraunitary matrix transfer functions*[426, Appendix D].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... response|textbf
^{2.11} - A (possibly complex) matrix
is said
to be
*unitary*if , where the `' superscript denotes complex-conjugate transposition: (also called the*Hermitian transpose*). A real unitary matrix is said to be*orthogonal*, and this is the case most commonly used in practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... impedance
^{2.12} -
See §J.1 for definitions and discussion regarding
*impedance*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... ear,
^{3.1} - It is also possible to use one delay
line for a single source to multiple ears, or multiple sources to one
ear.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... time
^{3.2} - Reverberation time is typically defined as
, the time, in seconds, to decay by 60 dB.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... shown
^{3.3} - A simple proof for rectangular rooms can be based on
considering a single spherical wave produced from a point source in
the room. Imagine tesselating all of 3D space with copies of the
original room (minus the source), all in the same orientation. As the
spherical wave expands, it intersects a number of rooms that is
roughly proportional to its radius squared. Since the radius is
proportional to time for an expanding spherical wavefront (),
the number of rooms containing a wave section grows as . By
adding all the rooms together (flipping every other one along and
), we obtain the acoustic field in the original reverberant room at
time , for the case of lossless wall reflections of pressure waves.
(This is the dual of the usual
*image method*for computing the impulse response of a room [14].) Since each wave-section traverses each point exactly once in each room image (disregarding reflections, which are accounted for by different room images), the number of echoes at any point in the room during the time it takes a plane wave to traverse the room is very close to the number of wave sections in the room at the beginning of that time interval.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... shown
^{3.4} - As described in virtually any acoustics textbook, the resonant modes of a
rectangular room are given by (see,
*e.g.*, [327, pp. 284-286])

where denotes the th harmonic frequency (radian/meter) of the fundamental standing wave in the direction ( being the length of the room along ), and similarly for the and directions. Thus, the mode frequencies of a room can be enumerated on a uniform 3D Cartesian grid indexed by . The grid spacings along , , and are taken to be , , and , respectively. From Eq. (2.1), the spatial frequency of mode is given by the*distance*from the origin of the grid to the point indexed by . Therefore, the number of room modes having spatial frequency between and is equal to the number of grid points lying in the spherical annulus between radius and . Since the grid is uniform, this number grows as .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{3.5} - One definition of when late
reverberation begins is when it begins to look
*Gaussian*, since a random sum of plane waves uniformly distributed over all angles of arrival yields a Gaussian distributed pressure field. (The fact that late reverberation in a rectangular room always approaches a superposition of plane waves traveling in all directions can be seen from the ``image-method dual'' argument in §2.2.1.) One way to test for Gaussianness is to form a*histogram*of impulse response sample values over a finite window (say 10ms, or roughly 500 samples), and compare the normalized histogram to a standard Gaussian bell curve computed using the measured sample variance. A threshold can be placed on , or some number of higher order sample moments can be compared. A simpler test is to determine when roughly 30% of the samples in a frame have magnitude larger than one standard deviation ( ) [2], since the probability of this occurring in truly Gaussian sequences is erf, where erf denotes the ``error function'' (integral from to of the Gaussian probability density function with normalized variance ). For a zero-mean signal , the sample standard deviation is given by the root-mean-square (RMS level). Another desired property of ``late reverb'' is that the remaining energy in the impulse response (Schroeder's*Energy Decay Curve*(EDC) [387]), has begun what appears to be an exponential decay. This can be ascertained by fitting an exponential to the EDC using,*e.g.*, Prony's method.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MoorerReverb79.
^{3.6} - In computer music, an old trick
for making a synthesized tone sound reverberated is to randomly modulate
its amplitude envelope, and to append a low-level exponentially decaying
tail to the amplitude envelope (also modulated). This can produce a very
convincing illusion.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... filters,
^{3.7} - While
allpass filters are ``colorless'' in theory, perceptually, their
impulse responses are only colorless when they are extremely short
(less than 10 ms or so). Longer allpass impulse responses sound
similar to feedback comb-filters. For steady-state tones, however,
such as sinusoids, the allpass property gives the same gain at every
frequency, unlike feedback comb filters.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... integer.
^{3.8} - As an alternative to
the output delay values shown in Fig. 2.5,
the values
have been used.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... matrix.
^{3.9} - A second-order
*Hadamard matrix*may be defined by*e.g.*,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... diagonal.
^{3.10} - This is easy to
show by performing an expansion by minors to calculate
, always choosing to expand along the top row (for lower
triangular) or first column (for upper triangular).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{3.11} - This substitution was first applied
to complete reverberators by Jot [200,201]. The
idea is closely related to the approximate pole analysis in
[294, p. 17], [404, pp. 170-172],
and [194], and somewhat also to the allpass conformal
maps used in [295,431].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{3.12} - See
[404, pp. 171-172] for a more detailed derivation which allows
to have arbitrary phase, which results in shifting of the pole
frequencies.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Kendall84.
^{3.13} - Of course, correct angles-of-arrival
for early reflections are more straightforward to implement using an
array of loudspeakers enclosing the listener.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
sound.
^{4.1} - For sound examples, see
`http://www.harmony-central.com/Effects/Articles/Flanging/`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...notches
^{4.2} - The term
*notch*here refers to the elimination of sound energy at a single frequency or over a narrow frequency interval. Another term for this is ``*null*''.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
notches.
^{4.3} - The author discovered this by looking at the circuit
for the MXR phase shifter in 1975.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... filters,
^{4.4} - Moog has built a
12-stage phaser of this type [54] and up to 20 stages (10
notches) have been noted [228].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP
^{4.5} - Available online at
`http://ccrma.stanford.edu/~jos/filters/Analog_Allpass_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{4.6} - Saying that the frequency is the
*break frequency*for the one-pole term is terminology from*classical control theory*. Below the break frequency, , and above, . On a log-log plot, the amplitude response may be approximated by a slope-zero line at height from dc to , followed by an intersecting line with negative slope of 20 dB per decade for all higher frequencies. At the break frequency, the true gain is down 3dB from , but far away from the break frequency, the piecewise-linear approximation is extremely accurate. Such an approximate amplitude response is called a*Bode plot*. Bode plots are covered in any introductory course on control systems design (also called the design of ``servomechanisms'').. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... first
^{4.7} - The ``first'' write-pointer is defined as the
one writing farthest ahead in time; it must
*overwrite*memory, instead of summing into it, when a circular buffer is being used, as is typical.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Leslie,
^{4.8} `http://en.wikipedia.org/wiki/Leslie_speaker`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
inversion:
^{5.1} - All wave phenomena involve two physical state
variables--one force-like and the other velocity-like. When one of
these variables reflects from a termination with a sign inversion, the
other reflects with no sign inversion, and vice versa. In acoustic
systems, the force-like variable is pressure, and the velocity-like
variable is either particle velocity (in open air) or volume velocity
(in acoustic tubes--see §E.14 for definitions). In
electromagnetic systems, the state variables are electric and magnetic
field strengths or voltage and current. In a mass-spring oscillator,
we may choose the velocity and acceleration of the mass as the
coordinates of our state space, or position and velocity. For
transverse waves on vibrating strings, it is usually preferable to use
transverse force and velocity waves as described above.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...fMovingTermb,
^{5.2} - This
diagram can be seen
*animated*along with Figure 4.3 at`http://ccrma.stanford.edu/~jos/swgt/movet.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...,
^{5.3} - In somewhat more detail,

so that , and .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...fsstring
^{5.4} - We should use the notation
for
this loop-filter, since it depends on the string length (in
samples). The dependence of
on is suppressed for
simplicity of notation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP.
^{5.5} `http://ccrma.stanford.edu/~jos/filters/Allpass_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...OppenheimAndSchafer,JOSFP.
^{5.6} `http://ccrma.stanford.edu/~jos/filters/Transposed_Direct_Forms.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP,
^{5.7} `http://ccrma.stanford.edu/~jos/filters/Filters_Preserving_Phase.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...invfreqz
^{5.8} - In
Matlab, the Signal Processing Tool Box is required, and in Octave, the
Octave-Forge package is needed.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...KlapuriMohonk05,KlapuriSAP03,Klapuri01.
^{5.9} - Klapuri's publication home page:
`http://www.cs.tut.fi/~klap/iiro/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
instrument.
^{5.10} - Electric guitars with magnetic pickups have
nearly rigid terminations, but even then, coupling phenomena are
clearly observed, especially above the sixth partial or so.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... be
^{5.11} `http://ccrma.stanford.edu/~jos/filters/Finding_Eigenvalues_Practice.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... law,
^{5.12} - A function is said
to be
*odd*if .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... function:
^{5.13} - See
`http://www.trueaudio.com/at_eetjlm.htm`for further discussion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SB.
^{5.14} - SynthBuilder is now a proprietary
in-house tool at Analog Devices, Inc. See Appendix A for a
description of the STK software prototyping environment that many of
us use today. In particular, the
`Mandolin.cpp`patch in the STK distribution is based on commuted synthesis. Some of us are also using the STK package in conjunction with`pd`[335] in order to obtain drag-and-drop graphical user interface support and a large library of MIDI processing tools.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... D.
^{6.1} - The PC88 stretching begins in the fourth
octave, while the measured Steinway stretching is greatest in the
first octave (reaching nearly -20 cents relative to ``nominal tuning''
for the first note A0) and picking up again, going sharp, in the sixth
octave. See
`http://www.precisionstrobe.com/apps/stretchdata/stretchdata.html`for the measured curves.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
impulse.
^{6.2} - We use the term ``pulse'' to refer to a filtered
impulse, while the term ``impulse'' will always denote an isolated
nonzero sample at some amplitude.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... excited.
^{6.3} - In other
words, to obtain maximum signal-to-noise ratio in a finite dynamic
range, it is better numerically to amplitude-scale at the final output
rather than prior to any recursive digital filters (
*e.g.*, at the excitation before the string). However, such output scaling will distort the relative amplitude of one note with respect to a previous note that is still ringing in the string. For this reason, one might use some scaling both before and after the string, using powers of 2 scaling for one of them to avoid a second multiply.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... manner
^{6.4} - Implementation of a ``soft pedal'' can be done by
suppressing the input to one of the delay lines, as if the hammer were
``shifted over'' a little so as to miss one of the strings.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... signals.
^{6.5} - In the Synthesis Tool Kit
[86], vectorized signals are implemented by the
`tickFrame()`member function, replacing the single-sample`tick()`function.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... complementary|textbf.
^{7.1} - To show this, let
denote the reflection transfer function (``reflectance'') of the bell
for pressure waves, and the transmission transfer function
(``transmittance''). Then, as discussed in §K.2 and
§G.8, linearity and energy conservation demand that
*z*transform of the pressure signal transmitted through the bell, and denotes the reflected pressure wave. Similarly, and denote the transmitted and reflected velocity waves ( flips the reflected velocity wave to that it flows in the same direction as the transmitted velocity wave), and and denote the pressure and velocity waves*incident*on the bell. All of these*z*transform have omitted arguments , where denotes the sampling interval in seconds.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
closure.
^{7.2} - For operation in fixed-point DSP chips, the
independent variable
is generally confined to the interval . Note that having the
table go all the way to zero at the maximum negative pressure
is not physically reasonable (0.8 would be more reasonable), but it has the
practical benefit that when the lookup-table input signal is about to clip,
the reflection coefficient goes to zero, thereby opening the feedback
loop.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...exponentially
^{9.1} - If the flare of the bell is expressed as
, where denotes the horn radius at
position along the bore axis, then is called the
*flare constant*of the bell.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... pressure
^{9.2} - As always
in this book, by ``air pressure'' we mean the
*excess air pressure*above ambient air pressure. In the case of brass instruments, excess air pressure is created by the muscles of the lungs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... mouthpiece.
^{9.3} - When
the kinetic energy of a jet is converted back into air pressure, this
is called
*pressure recovery*. We assume, following [92] and others, that pressure recovery does not occur in this model. Instead, the kinetic energy of the ket is assumed to be dissipated in the form of turbulence (vortices and ultimately heat). Note that air flow is therefore assumed inviscid within the mouth and between the lips, but viscous around the jet in the mouthpiece.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Putland.
^{9.4} - For velocity waves, the flare may be
*hyperbolic*[45].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
below).
^{A.1} - A shorter method for unzipping and unpacking is
`zcat delay.tar.gz | tar -xvf -`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Makefiles
^{A.2} - Only STK makefiles are regenerated by
`configure`--the files`myproj/Makefile`and`myproj/Makefile.proj`must be edited by hand, if needed.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
option.
^{A.3} - If you encounter any ``missing statements'' or variables while
single-stepping
`gdb`, try turning optimization*off*altogether by removing the`-O2`option from`CFLAGS`in the`Makefile`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... <Enter>.
^{A.4} - The notation
`M-x`, pronounced ``meta x,'' means to hold down the`Alt`key and press the`x`key. Alternatively, one can type the`Esc`key*followed by*the`x`key. In either case,`emacs`will prompt for a command line input. The notation`<Enter>`means, of course, the`Enter`key (sometimes labeled`Return`).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... below.
^{B.1} - For further introductory information
regarding compiling, linking, and debugging programs in the UNIX
environment, see http://cslibrary.stanford.edu/107/.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... folder).
^{B.2} - It is good practice to use your own
folder,
*e.g.*,`myprojects`instead of`projects`, so that you can simply copy`myprojects`into new releases of the STK (and start debugging all over again - ).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... message
^{C.1} - In
`C++`, ``sending message to object '' amounts simply to executing function in struct . This is sometimes referred to*static binding*, as opposed to*dynamic binding*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...dAlembert.
^{D.1} - For a short biography of d'Alembert, see
`http:``//www-groups.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... (1685-1731)
^{D.2} `http:``//<`*ibidem*>/Taylor.html. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... (1707-1783)
^{D.3} `http:``//<`*ibidem*>/Euler.html. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
eyeglasses.
^{D.4} `http:``//www.mathphysics.com/pde/history.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... vibrations.
^{D.5} `http:``//www.stetson.edu/~efriedma/periodictable/html/B.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...ZadehAndDesoer,Kailath80,Depalle.
^{D.6} `http:``//ccrma.stanford.edu/~jos/PianoString/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...strain.
^{D.7} `http:``//www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke.cfm`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
public.
^{D.8} `http:``//cnx.rice.edu/content/m0050/latest/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...FettweisMain.
^{D.9} - Derivation:
`http://ccrma.stanford.edu/~jos/pasp/Wave_Digital_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
Belevitch.
^{D.10} `http:``//www.ieee.org/organizations/history_center/oral_histories/transcripts/fettweis.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Flanagan72,FlanaganAndRabinerEds,RabinerAndSchafer78,SchaferAndMarkel79,OShaughnessy87,DPH93,Keller94.
^{D.11} - The
overview [285] of early speech production models is
freely available online, thanks to the Smithsonian Speech Synthesis
History Project.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... formulation.
^{D.12} `http:``en.wikipedia.org/wiki/Oliver_Heaviside`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... phenomenon.
^{D.13} `http:``www.microwaves101.com/encyclopedia/history.cfm`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{D.14} - Scattering
theory:
`http://ccrma.stanford.edu/~jos/pasp/Scattering_Impedance_Changes.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... method.
^{D.15} - ``Bicycle Built for
Two'' by Kelly, Lochbaum, and Matthews,
1961:
`http://ccrma.stanford.edu/~jos/wav/daisy-klm.wav`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... completion.
^{D.16} `http:``//www.bell-labs.com/news/1997/march/5/2.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...PRCT.
^{D.17} `http:``www.cs.princeton.edu/~prc/SingingSynth.html`(includes sound examples). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SchellengA,SchellengB,CremerC.
^{D.18} `http:``//www.zainea.com/Oscilationsofbowedstring.htm`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
simple
^{E.1} - While this formula seems fairly simple now, in
Newton's day, it was necessary to invent
*calculus*before it could be stated in this way.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... velocity.
^{E.2} - The only
massless particles are the photon and graviton (presuming it exists),
and massless particles can only travel at the speed of light. By
Einstein's famous formula , we know that a particle's mass is
proportional to its
*energy*at rest.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
constant|textbf.
^{E.3} - The gravitation constant is given in
*International Standard Units (``SI units'')*by

where the following

*physical units*abbreviations are used:

These physical units for come directly from Newton's second law of motion (§E.1).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...force|textbf.
^{E.4} - From
a modern point of view, all forces are ``mediated'' by some particle,
and there are only four basic forces: The
*electromagnetic force*is mediated by the*photon*, the*strong nuclear force*is mediated by the*gluon*, the*weak nuclear force*is mediated the and bosons, and*gravity*is thought to be mediated by the*graviton*, although gravity is not yet incorporated into the*Standard Model*of theoretical physics. Only the electromagnetic and gravitational forces are encountered in everyday life, since the strong and weak nuclear forces decay exponentially on the scale of an atomic nucleus. The approximate relative strengths of the strong:weak:EM:gravity forces are . At very high energies, the weak and EM forces unify, according to the Standard Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... force.
^{E.5} - In this example, the
mass-spring system is in
*equilibrium*(not moving), so all forces in the system must sum to zero. Equilibrium also must hold if the whole system is traveling with a constant velocity; in other words, it is the lack of*relative*motion that matters.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... displacements.
^{E.6} - To show
this quantitatively, consider two electrons, each having electric
charge , separated by meters. Then the Coulomb force between
the electrons is proportional to . Adding a small
perturbation to yields a new force proportional to
*binomial expansion*to obtain the approximation. (The notation means ``terms order of '', and such terms approach zero as fast as approaches zero.) Thus, the effective spring constant connecting the two electrons is , when .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... yielding
^{E.7} - A more formal methodology for arriving
at differential equations by applying Newton's law and Hooke's law is
described in Appendix N. In the present example, consideration of the
underlying physics should convince you that the signs in Eq. (E.4)
are correct. For example, when is positive, the spring must push
the mass back to the left. Therefore,
means the mass
will be accelerated to the left.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
time:
^{E.8} - See,
*e.g.*, [426, p. 235-236] for a derivation, and Appendix N, specifically §N.3.6, for related analysis.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
object.
^{E.9} - In SI units,
*work*is in units called*Joules*(abbreviated J). Thus, Joules are Newtons times meters.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... considered.
^{E.10} - Ok, there
*does*exist energy fluctuation on the scale of ``Heisenberg uncertainty''. That is, it is possible to have a violation in energy conservation by an amount over a time duration , provided is sufficiently small (on the order of Planck's constant ). This can be considered a form of the Heisenberg uncertainty principle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... potential:
^{E.11} - This was first pointed out
by Rayleigh [327, p. 39].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... gases:
^{E.12} -
`http://www.grc.nasa.gov/WWW/K-12/airplane/sound.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{G.1} - Any function of , where denotes time and denotes
position along a ``waveshape'', may be interpreted as a fixed
waveshape traveling to the
*right*(positive direction), with speed . Similarly, any function of may be seen as a waveshape traveling to the*left*(negative direction) at speed meters per second. In both cases, denotes a position along the waveshape, and denotes time. For any fixed , a ``snapshot'' of the waveshape may be seen by evaluating along .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... velocity|textbf
^{G.2} - The term
*phase velocity*is normally used when it differs from the*group velocity*, as in stiff, dispersive strings (§G.6). In the present context, the phase velocity and group velocity are the same, so the term ``wave velocity'' is unambigous here. See the analogous terms*phase delay*and*group delay*in [426] for more details about the difference.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
scattering.
^{G.3} - Here it is assumed that and in
the Kelly-Lochbaum junction can be computed exactly from in the
number system being used. This is the case in two's complement
arithmetic as is typically used in practice.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
``right-going.''
^{G.4} - In the acoustic tube literature which
involves only a cascade chain of acoustic waveguides, is
taken to be traveling to the
*right*along the axis of the tube [275]. In classical network theory [33] and in circuit theory, velocity (current) at the terminals of an -port device is by convention taken to be positive when it flows*into*the device.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... found
^{G.5} `http://ccrma.stanford.edu/~jos/filters/Similarity_Transformations.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... magnitude:
^{G.6} - While the
literature seems to mention this property only for prime numbers ,
it is straightforward to show that it holds in fact for all positive odd
integers . Prime values of have advantages, however, when harmonics
of the ``design frequency'' are considered.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... samples.
^{I.1} - Classically, an interpolation
of discrete data points always required
;
that is, the interpolating function must always pass through the given
data points. More recently, however, particularly in the field of
computer graphics, interpolation schemes such as
*Bezier splines*have been defined which do not always pass through the known points.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{I.2} - Of course, all derivatives of must
be calculated from the underlying continuous signal
represented by the samples and then evaluated at .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP.
^{I.3} - To see this, note that the integral of the
group delay around the unit circle is simply minus the
*winding number*of the phase response (the number of zeros minus the number of poles inside the unit circle):. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... kHz.
^{I.4} - We arbitrarily define the % guard band as a
percentage of half the sampling rate actually used, not as % of the
desired kHz bandwidth which would call for a kHz sampling rate.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... transform|textbf,
^{J.1} - For a short online introduction to Laplace transforms, see,
*e.g.*,`http://ccrma.stanford.edu/~jos/filters/Laplace_Transform_Analysis.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... mass.''
^{J.2} - We say that the driving point of every mechanical
system must ``looks like a mass'' at sufficiently high frequency
because every mechanical system has at least some mass, and the
driving-point impedance of a mass goes to infinity with frequency.
However, in a completely detailed model, the contact force between
objects should really be the
*Coulombe force*, which ``looks like a spring''. In other words, mechanical interactions are ultimately*electromagnetic*interactions, and it is theoretically possible for a driving force on an atom to be so small and fast that it can vibrate the outermost electron orbital without moving the nucleus appreciably, thus ``looking like a spring'' in the high-frequency limit. We will not be concerned with atomic-scale models in this book, and will persist in treating masses and springs in idealized form.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{J.3} - To avoid the introduction of
half a sample of delay by the approximation, the first-order finite
difference may be defined instead as
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{J.4} - Note that the
matched transformation would be more precisely scaled if it were
defined as
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... case.
^{J.5} - In the
case of complex impedances, the matched impedance case becomes
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... output.
^{K.1} - So-called
``electric-acoustic'' guitars, such as the Godin electric-acoustic,
use piezoelectric crystals in the bridge to measure string force at
the string endpoint. Electric violins, such as the Zeta Violin,
typically use this approach as well.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... measurements,
^{K.2} - The Matlab function
`cohere()`can be used to compute the coherence function between two sampled spectra.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... phase.
^{K.3} - Zero-phase filters can normally be
implemented in practice because there are pure delay lines preceding and
following the reflectance filter, and taps can be introduced in the
delay-line preceding the reflectance to implement noncausal terms in the
impulse response.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{K.4} - To
prove this, note that the roots of are the reciprocals of the
roots of , since the conformal map
exchanges
interior of the unit circle with the exterior of the unit circle.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
disk.
^{K.5} - The strict
*outer disk*is defined as the region in the extended complex plane.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... wave.
^{N.1} - When there are more than two
ports at a junction, the `' superscript will denote a wave
traveling
*into*the junction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... network.
^{N.2} - Wave
digital filters are not the same thing as digital waveguide networks,
however, because the wave variables in a wave digital filter have a
compressed frequency spectrum (from the bilinear transform), while the
signals in a digital waveguide network have a bandlimited spectrum
which is not warped.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
waveguide
^{N.3} - Here, it is perhaps most concrete to think in terms
of electrical equivalent circuits, so that the mass is an inductor and
a ``waveguide'' is a ``transmission line.''
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... mass,
^{N.4} - The phase delay,
for example, of the reflectance
of an ideal spring is
given by
After the bilinear transform , the phase delay becomes

*i.e.*, four samples per period, we find that one sample of delay does in fact correspond to a quarter-cycle of delay.The same holds true for the group delay of the spring reflectance:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...tankwdf.
^{N.5} - Note that the wave
variables are now labeled in element-centric notation as opposed to
adaptor-centric notation:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... limit.
^{N.6} - Note that the mass and
velocity limits are tied together such that constant. This
information is lost in the final limits because the expression
is ``indeterminate''.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
algorithms.
^{O.1} - See [97] for discussion of a wider
variety of digital sine generation methods.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... algorithm,
^{O.2} - The
``magic circle'' algorithm is so named because it generates closed
curves in the plane even when numerical precision is
very low. For this reason, it has long been used as an algorithm
for drawing ``circles'' (actually ellipses) in computer graphics.
The algorithm is derived naturally by numerically integrating the
derivative relationships between sine and cosine:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
amplitude),
^{O.3} - When is constant, the time constant
of the ensuing exponential growth or decay may be found by
solving
for to obtain
ln, where
denotes the sampling interval, and
ln denotes the natural
logarithm of .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... analysis
^{O.4} `http://ccrma.stanford.edu/~jos/filters/State_Space_Models.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... modeling,
^{P.1} - The
general field in which transfer function descriptions are estimated
for physical systems is called ``system identification''
[268,404], and it is used extensively in industrial
automatic control applications.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{P.2} - For convenience, we typically define the
discrete-time counterpart of a continuous-time function as though the
sampling interval were .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SmithCommutedFromPASP,MattiGuitar,MattiAndSmith96.
^{Q.1} - Extracts of
this material were published previously in [213].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... volume.
^{Q.2} - These remarks apply to playback of the
excitation only. The string provides further filtering which could be
taken into account. However, to a first approximation, the string
filtering is like a ``sampling'' of the excitation spectrum, with a
gentle roll-off at high frequencies due to the lowpass loop
filtering. Also, the string is a highly variable filter with many
settings. It is therefore reasonable to ignore the string filtering
when determining an optimal preemphasis for excitation-table
quantization.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... parts.
^{R.1} - It is not necessary to
perform a Taylor series expansion to separate out the even and odd
parts of a function. Instead, the even part of can be computed
as
and the odd part as
. It is easily checked that is even, is
odd, and .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... mapping.
^{R.2} - Note that
memoryless nonlinearities used for distortion simulation are typically
*odd functions*of instantaneous amplitude, such as . (A good practical choice is given in Eq. (R.3).) However, in guitar-amplifier distortion simulation, even-order terms are considered quite important [367].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{R.3} - The convolution theorem for Fourier transforms states that
convolution in the time domain corresponds to pointwise multiplication
in the frequency domain [426]. The
*dual*of the convolution theorem states just the opposite: Pointwise multiplication in the time domain corresponds to convolution in the frequency domain. Thus, if the spectrum of is , then the spectrum of is , where ``'' denotes the convolution operation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... spectra.
^{R.4} - The
basilar membrane of the ear (which is rolled up inside the
snail-shaped cochlea of the ear) effectively performs a real-time
Fourier analysis which is ``felt'' by nerve cells leading to the
brain.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .