- ...
email
^{1.1} `jos`at`ccrma`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... performance.
^{2.1} - This distinction
is also important when evaluating real-world musical instruments.
It is better to ask a skilled performer to comment on the quality of
an instrument than a listener. A good musician can make almost any
instrument sound good, while appreciating its defects and working
around them in performance.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
pianos''.
^{2.2} - For example, the
*Synthogy Ivory*(a $349 software product in 2006), ships as 40 Gigabytes on ten DVDs (three sampled pianos). Every key is sampled, with 4-10 ``velocity layers'', separate recordings with the soft pedal down, and separate ``release'' recordings, for multiple striking velocities. (Source:*Electronic Musician*magazine, October 2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... acoustic
^{2.3} - In the case of audio effects,
``acoustic'' recordings are normally replaced by ``electronic''
recordings. The same applies to the sampling of vintage electronic
instruments, such as the Fender Rhodes electric piano, etc.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... equation
^{2.4} - See
Appendix B for further discussion of Newton's laws of motion.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{2.5} *Digitizing*a system generally means converting it from continuous-time to discrete-time form. For an ODE, for example, this typically involves algebraically replacing the time differential in the ODE by a practical sampling interval , as will be discussed below and in §7.3.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... dashpot
^{2.6} - A
*dashpot*is the idealized mechanical equivalent of a*resistor*in electrical circuit theory. Its compression-velocity is proportional to applied force,*i.e.*, . Dashpots are often used to model forces due to*friction*and are typically valid over a restricted frequency range. Masses and springs are mechanical equivalents of electrical inductors and capacitors, respectively. More about these elements will be discussed below in this chapter and later in this book.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{2.7} - Transverse displacement
is displacement in a plane orthogonal to the string axis
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... beyond.
^{2.8} - It is interesting to note that in the
development of
*quantum mechanics*in the early 1900s, it was necessary to replace deterministic Newtonian dynamics with a*probabilistic*model. In quantum mechanics, probability distributions may follow deterministic trajectories as in Newtonian mechanics (see, for example, Schrödinger's equation), but they are only probability distributions, so there is no deterministic ``clock works'' at the smallest physical scales. We are fortunate to be able to use Newtonian dynamics with such great accuracy. Our difficulty will be complexity, not randomness. Paradoxically, the typical way to deal with overwhelming complexity is to model it as random (*e.g.*, filtered white noise).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{2.9} - As discussed in §1.3.6, only velocity
is
needed for the state variable of a mass, since a mass moving in one
dimension has only one degree of freedom (with energy
).
However, since it is physically reasonable to expect both velocity
and position to be needed for the state of a point mass, let's
``play that out'' and see how it goes. This also gives us a simple
*vector*example to study.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... proper
^{2.10} - See
[452, p. 133] regarding the cases
for which a
``long division'' is first performed to obtain an FIR part in
parallel with a strictly proper part.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... diagonal.
^{2.11} - More precisely,
is diagonal when the
poles are
*distinct*. A repeated pole can result in a block of having 1s along its superdiagonal.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... eigenvectors.
^{2.12} - A
*generalized eigenvector*of matrix corresponding to eigenvalue having multiplicity is a nonzero solution of .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... zero.
^{2.13} - The other law routinely used in
circuit analysis is that the
*sum of all currents entering a circuit node (connection of wires) is zero*. This kind of analysis will be revisited in §9.3.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... resistance.
^{2.14} - Note that models of damping in
practical physical systems are rarely completely independent of
frequency, as is the ideal dashpot.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP
^{2.15} - For a short online introduction to Laplace transforms, see,
*e.g.*,`http://ccrma.stanford.edu/~jos/filters/Laplace_Transform_Analysis.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... direction.
^{3.1} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...STK4.
^{3.2} - In the code of Fig.2.10, 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{3.3} - 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
^{3.4} *Dispersion*occurs in a traveling wave whenever the propagation speed is different at two or more frequencies.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... equal.
^{3.5} - The signs may differ, however. For
example, imagine
*twisting*a stretched string about a point along its length in the plane of transverse vibration. In that case, the transverse-displacement pulses propagating away from the twist have opposite sign. This opposite-sign effect happens also for*slope waves*emanating from a pluck disturbance, as we'll see in Chapter 6. (A positive pluck corresponds to a positive slope on its left and a negative slope on its right.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... law
^{3.6} - Newton's third law states that ``every
action produces an equal and opposite reaction.'' Therefore, every
``physical'' signal connection must be
*bidirectional*, in order to model both the applied force and the equal reaction force, or ``loading effect''. Bidirectional signal connections are typically called*ports*. A ``port'' (one physical connection point) in electrical circuit theory consists of two variables, typically voltage and current. In acoustics, corresponding port variables would be pressure and velocity. As discussed later (*e.g.*, for Wave Digital Filters in Appendix F), a simple 2x2 matrix linear transformation converts ``force and flow'' port variables, such as these, to ``traveling-wave component'' port variables.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
waves
^{3.7} - Loading effects can be viewed as the result of
instantaneous return waves. This equivalence is seen clearly, for
example, in Wave Digital Filters (Appendix F).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
neglected.
^{3.8} - This follows years of standard practice in analog
electric circuit design using
*voltage transfer interconnections*. For ``voltage transfer'' from one circuit-stage to another with minimized loading effects, output impedances are made as low as practical, while input impedances are set very high. For example, in analog synthesizers built during the 1970s, input impedances were typically around 100 times output impedances (*e.g.*, 100K and 1K Ohms, respectively in a typical op-amp circuit) [205].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... as
^{3.9} - Note that
Fig.2.24 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,
^{3.10} - Readers of the main chapters of this book are not
required to know linear algebra, but one does not have to learn much
to follow the discussion in this section. See, for example,
[454, available online] for the basic matrix operations. An
excellent linear algebra text is [332]. See also the lecture
videos by Strang by searching for ``linear algebra'' at
`http://ocw.mit.edu/`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{3.11} - If
is complex, transposition must include
complex conjugation,
*i.e.*, . The conjugate-transpose is called the*Hermitian transpose*of .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
1,
^{3.12} - By definition,
, so that all singular values
of an orthogonal matrix are equal to 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... line.
^{3.13} - 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
^{3.14} - A filter is said
to be causal if its impulse response
is zero for
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...fbffcf.
^{3.15} - Gerzon's starting point was Schroeder's
allpass which differs slightly from Eq.(2.15), having difference
equation [415]

with required for stability. This structure can be derived from Eq.(2.15) 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*[452, Appendix D].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... matrix|textbf
^{3.16} - A (possibly complex) matrix
is said to be
*unitary*if , where is the*Hermitian transpose*of . A real unitary matrix is said to be*orthogonal*, which is the case most commonly used in practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... impedance
^{3.17} -
See §7.1 for definitions and discussion regarding
*impedance*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... ear,
^{4.1} - One tapped
delay line can also model one source to multiple ears, or vice versa
(Fig.5.6).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... time
^{4.2} *Reverberation time*is typically defined as , the time, in seconds, to decay by 60 dB.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... shown
^{4.3} - 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 [11].) 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. Incidentally, this construction gives, in the limit as , a derivation and vivid visualization of the*diffuse field*--the superposition of equal-amplitude traveling plane-waves arriving from all directions and having uniformally distributed random arrival phase [352,47].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... shown
^{4.4} - As described in virtually any acoustics textbook, the resonant modes of a
rectangular room are given by (see,
*e.g.*, [352, 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.(3.1), 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 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{4.5} - 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 the*diffuse field*--a superposition of plane waves traveling in all directions can be seen from the ``image-method dual'' argument in §3.2.1.) 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 ( ) [3], 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) [416]), 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... (TDL).
^{4.6} - This idea was apparently first suggested by
Schroeder in 1970 [418] and evidently first implemented
by Moorer [317].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MoorerReverb79.
^{4.7} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
examples.
^{4.8} `https://ccrma.stanford.edu/~jos/wav/FM-BrassCanon2.wav`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{4.9} - As
an alternative to the output delay values shown in Fig.3.7,
the values
have been used.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... alone.
^{4.10} - There is a Freeverb
`pd```external'' in both the`pd-extended`and`faust-pd`packages, and it appears in the`muse`sequencer,`kdemultimedia`,`clm`, and`faust`language packages, all included with Planet CCRMA. Other versions may be found by a Web search for ``Freeverb source'', but I have not found any version that isn't essentially the same June 2000 version by Jezar at Dreampoint.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
(LBCF
^{4.11} - ``Lowpass-Back-Comb-Filter''--to keep all the
comb-filter acronyms down to four characters: FFCF, FBCF, LBCF.
Author's prerogative!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...math.lib
^{4.12} - Details about
Faust may be found at
`http://faust.grame.fr/`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... diagonal.
^{4.13} - 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).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
repeats.
^{4.14} - Fritz Menzer, in ``Choosing Optimal Delays for Feedback Delay Networks'',
`http://pub.dega-akustik.de/DAGA_2014/data/articles/000025.pdf`, found that ``using mutually prime delays [in FDNs] avoids only a negligible subset of echo superpositions'', and proposed free optimization of the delay lengths to achieve desired qualities.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{4.15} - This substitution was first applied
to complete reverberators by Jot [217,218]. The
idea is closely related to the approximate pole analysis in
[317, p. 17], [432, pp. 170-172],
and [208], and somewhat also to the allpass conformal
maps used in [318,461].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...FDNJot).
^{4.16} - Note that, as
developed here, the delay-line filters
should be located
either before or after its associated delay line, while
Fig.3.10 associates it with the feedback matrix. The
difference is typically not audible.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... dc.
^{4.17} - In a Faust
implementation, this form was found to be more numerically sensitive
than that of the previous section when changing the
s in
real time (see §3.7.9).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... FOSS
^{4.18} - FOSS
Free Open Source Software
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...zita-rev1,
^{4.19} `http://kokkinizita.linuxaudio.org/linuxaudio/zita-rev1-doc/quickguide.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP,
^{4.20} `https://ccrma.stanford.edu/~jos/filters/Low_High_Shelving_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
respectively.
^{4.21} - This section describes the Faust version
`zita_rev1`, which may not be exactly the same as the C++ version`zita-rev1`, although each control parameter range is the same, and the filter-orders are the same, so at least near-equivalance is expected.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Kendall84.
^{4.22} - Of course, correct angles-of-arrival
for early reflections are more straightforward to implement using an
array of loudspeakers enclosing the listener.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{5.1} - A causal IIR filter can be used for random-access table
lookup by simply starting the filter at a point sufficiently far
away in the table and ``running it'' until it reaches steady-state
at the desired interpolation point. While this requires at least as
many multiply-adds as an FIR filter spanning the same duration, the
number of IIR filter coefficients can be much less than the number
of FIR filter coefficients required. IIR interpolators are therefore
possibly interesting for certain memory-starved VLSI applications.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... samples.
^{5.2} - 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... matlab:
^{5.3} - Say ``
`help;`'' within`maxima`and/or ```man maxima`'' in a shell to get started. Emacs users should say ``M-x maxima''.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{5.4} - To see this, consider that a unit-amplitude sinusoid at half
the sampling rate
is either
or
. Therefore,
the frequency response
of every real filter at
is
a real number
that is either positive or negative (or
zero). When
, the filter's phase delay must be an even integer
number of samples. When
, the filter delay is constrained to be
some odd number of samples. Taken together, the possible delays at
for real filters are the integers. The particular integer is
obtained by finding the limit as
. The case
can be
associated with a non-integer limit, as suggested by the line for
samples delay in Fig.4.18 above, and as easily
shown analytically for order 1 (simple linear interpolation with
coefficients
, corresponding to a desired delay of
samples) [452].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...VesaT.
^{5.5} `http://www.acoustics.hut.fi/~vpv/publications/vesan_vaitos/ch3_pt2_lagrange.pdf`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... rule,
^{5.6} - Evaluation of a polynomial
by
*Horner's rule*can be expressed as .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{5.7} - Of course, all derivatives of
must
be calculated from the underlying continuous signal
represented by the samples
and then evaluated at
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP.
^{5.8} - 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):. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
processing.
^{5.9} - The material in this section was adapted from

`https://ccrma.stanford.edu/~jos/resample/`, which is an updated online version of [464].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... kHz.
^{5.10} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Bartlett70,SmithAllpassFlanger,Bode84,Anderton85,Dattorro97,Keen99.
^{6.1} - Flanging was apparently invented independently several
times. According to
`http://www.geofex.com/Article_Folders/phasers/phase.html`, the flanging technique was discovered by Phil Spectre in the 1950s in the context of trying to ``thicken'' a vocal track by summing two copies of a track variably delayed in this way. His first use of flanging as an*effect*is said to be in his song ``The Big Hurt''. According to historian Mark Lewisohn [288], as summarized at`http://en.wikipedia.org/wiki/Flanging`(accessed March 28, 2010), the term ``flanging'' was coined by John Lennon of the Beatles to describe what EMI engineer Ken Townsend called ``Artificial Double Tracking'' (ADT). ADT was said to be developed by Townsend in response to John Lennon's request to try to eliminate the work of recording vocal tracks twice in order to ``double'' them. George Martin has said to have explained the effect as follows: ``Now listen, it's very simple. We take the original image and we split it through a double-bifurcated sploshing flange with double negative feedback.'' [288]. This suggests that negative regenerative feedback was used in the original stereo flanging technique (§2.6.2). The first Beatles track using flanging is said to have been ``Tomorrow Never Knows'' on the Revolver album (recorded April 6, 1966). Perhaps the most obvious early example of flanging was in the popular song ``Itchykoo Park'' by The Small Faces (1967); this instance is said to have originated with engineer George Chkiantz at Olympic Studios in London. The song featured a foreground drum-roll on a snare with strong flanging throughout, and the vocals were flanged as well.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... sound.
^{6.2} - For sound
examples, see
`http://www.harmony-central.com/Effects/Articles/Flanging/`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...notches
^{6.3} - 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*''.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Bartlett70,
^{6.4} - According to
`http://www.eventide.com/About/History.aspx`, a dual 200 ms delay-line for simulating flanging was called the*Instant Phaser*. This was the first commercial product circa 1970 made by Eventide Corp. (better known as makers of the*Harmonizer*).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... notches.
^{6.5} - The
author discovered this first-hand by looking at the circuit for the
MXR phase shifter in 1975.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SmithEtAlDAFx02.
^{6.6} `https://ccrma.stanford.edu/~jos/doppler/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...eq:dopplershift).
^{6.7} - If the tape travels in a loop,
then Fig.5.4 provides a model for the
*Echoplex*(Maestro, 1960), which consists of a tape loop with a fixed write-head and movable read-head.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... first
^{6.8} - 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,
^{6.9} `http://en.wikipedia.org/wiki/Leslie_speaker`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... accurate.
^{6.10} - See also
the Hammond Leslie FAQ at

`http://www.theatreorgans.com/hammond/faq/files/hammond-faq.pdf`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SmithEtAlDAFx02.
^{6.11} `https://ccrma.stanford.edu/~jos/doppler/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...dAlembert
^{7.1} - See §A.1 for more about the
history of the wave equation and its traveling-wave solution.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... derivation)
^{7.2} - Note that
and
, as
defined, are traveling-wave components of the force
*acting to the right*on the string. That is, their sum is physically the force that the string-segment to the*left*of position applies (in the upward direction) to the string-segment to the*right*of point . In other words, denotes the vertical force applied by the left segment to the right segment at time and position ; thus, it ``acts to the right'', even though its traveling-wave components, and , travel to the left and right, respectively, at speed . The net physical force acting to the right at each point is exactly canceled by an equal and opposite force acting to the left at each point . See §C.7.2 for a detailed derivation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... inversion:
^{7.3} - 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), as described in §6.2 above. 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
force and velocity waves as described above.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...fMovingTermb,
^{7.4} - This
diagram can be seen
*animated*along with Figure 6.4 at`http://ccrma.stanford.edu/~jos/swgt/movet.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... bound.
^{7.5} - Our model becomes invalid as the slope becomes
large. In particular, the string tension
obviously increases as the
string length increases. Here we assume
is constant.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...,
^{7.6} - In somewhat more detail,

so that , and .

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

- ...fsstring
^{7.7} - 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.
^{7.8} `http://ccrma.stanford.edu/~jos/filters/Allpass_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...OppenheimAndSchafer,JOSFP.
^{7.9} `http://ccrma.stanford.edu/~jos/filters/Transposed_Direct_Forms.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP,
^{7.10} `http://ccrma.stanford.edu/~jos/filters/Filters_Preserving_Phase.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...invfreqz
^{7.11} - In
Matlab, the Signal Processing Tool Box is required, and in Octave, the
`octave-signal`MacPort package is needed. The Linux`octave`package already includes this (at least on Fedora 13).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...KlapuriMohonk05,KlapuriSAP03,Klapuri01.
^{7.12} - Klapuri's publication home page:
`http://www.cs.tut.fi/~klap/iiro/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
time-slices:
^{7.13} - Note that this derivation also holds if the
power `2' is replaced by an arbitrary
, thereby supporting a
generalization from the EDR to what might be called the ``LPDR'' using
a kind of
norm on the remaining decay, where the EDR is
regarded as the
case.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{7.14} - The delay-line length
is only the ``quasi-period''
in samples when the phase delay associated with
can be
neglected.
is never a true ``period'' because the synthesized
signal is decaying from one block to the next.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
instrument.
^{7.15} - 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
^{7.16} `http://ccrma.stanford.edu/~jos/filters/Finding_Eigenvalues_Practice.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... parts.
^{7.17} - 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.
^{7.18} - 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.(6.19).) However, in guitar-amplifier distortion simulation, even-order terms are considered quite important [398].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{7.19} - The convolution theorem for Fourier transforms states that
convolution in the time domain corresponds to pointwise multiplication
in the frequency domain [452]. 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.
^{7.20} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... transform|textbf,
^{8.1} - For a short online introduction to Laplace transforms, see,
*e.g.*,`http://ccrma.stanford.edu/~jos/filters/Laplace_Transform_Analysis.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... relation:
^{8.2} - Of course, here we should call it the ``force
divider'' relation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... mass.''
^{8.3} - 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP.
^{8.4} - Available online at
`http://ccrma.stanford.edu/~jos/filters/Graphical_Amplitude_Response.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{8.5} - To avoid the introduction of
half a sample of delay by the approximation, the first-order finite
difference may be defined instead as
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP).
^{8.6} - In
continuous time, the order is incremented once for each
independently moving mass or spring. In discrete time, the order is
increased by one when a sample of delay is added to the system
state, and the number of multiplies needed to implement a digital
simulation is bounded by twice the order plus one.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP,
^{8.7} `http://ccrma.stanford.edu/~jos/filters/State_Space_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...blt).
^{8.8} - Normally one or more output signals
are defined as linear
combinations of the state vector
,
*viz.*, . However, we can define the state itself as the output for now, and form any desired linear combinations separately.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...FranklinEtAl98.
^{9.1} - Estimating transfer functions based on
input-output measurements is called ``system identification''
[290,432]--used in advanced automatic control
applications.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{9.2} - For
convenience, we typically define the discrete-time counterpart of a
continuous-time function as though the sampling interval were
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
response.
^{9.3} - Frequency-domain aliasing is discussed
in [454].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP:
^{9.4} `http://ccrma.stanford.edu/~jos/filters/Partial_Fraction_Expansion.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP).
^{9.5} - The
*strictly proper*constraint is natural in practice because the frequency responses of typical real-world systems generally roll off at least -6 dB per octave. Notice that Eq.(8.1) becomes as , which is a dB/octave roll off. Setting and yields a dB/octave roll-off, and so on. However, this depends on the physical units chosen for the input and output signals of the system. For example, consider an ideal nonzero mass driven by a force; while the resulting velocity and displacement go to zero as frequency goes to infinity (for any finite applied force), the acceleration does not roll off, being proportional to applied force by Newton's 2nd law.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... modes.
^{9.6} - Modal synthesis
could well be renamed ``Bernoulli synthesis,'' as Daniel Bernoulli
was quite alone in advocating the concept of seeing general
vibration as a superposition of ``simple'' sinusoidal
vibrations. This view was resisted by Euler and d'Alembert at the
time [103].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP
^{9.7} `http://ccrma.stanford.edu/~jos/filters/Modal_Representation.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... response,
^{9.8} - A
``causal frequency response''
is the Fourier transform of
a causal impulse response
(
*i.e.*, for all ). Extensions to finitely noncausal spectra are straightforward: Time-shift the desired impulse response to make it causal, perform the filter design, then reverse-time-shift the filter numerator if needed.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{9.9} - A
*stability margin*may be specified, for example, by requiring all poles to satisfy , where determines the stability margin. In particular, with this specification on the poles, the impulse response must decay asymptotically at least as fast as [452].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...convex
^{9.10} - Since convex functions bulge upward, like
the top of a circle, it makes sense to
*maximize*them in general. Since our error-surface formulation is always*minimized*with respect to the filter parameters, we could call it ``concave'' or ``convex from below''. In the optimization domain, however, a ``convex'' region is one that contains all of its chords, thus it applies to either case.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{9.11} -
cannot go to infinity
since the constraint
and the stability constraint
imply that
ln
is zero-mean by the argument
principle [299,329].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Octave.
^{9.12} - Octave's
`remez`function and Matlab's`firpm`function (Signal Processing Toolbox) have a special mode for designing FIR differentiators that are optimal in the Chebeshev sense, while`invfreqz`minimizes equation error, which is arguably not as good. We focus on`invfreqz`in this chapter because it offers the level of generality needed for fitting digital filters to arbitrary measured frequency responses.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{9.13} - When working with
`invfreqz`, it is often helpful to multiply the desired spectrum by a linear phase term [452] corresponding to an added delay in the impulse response [432]. The filter designed by`invfreqz`will typically be stable if sufficient delay is added to the desired frequency response in this way. Thus, if a filter designed by`invfreqz`is found to be unstable, multiply the desired frequency response by a steeper negative-slope linear-phase term and repeat the design.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... data.
^{9.14} - The number of poles and zeros needed for a
reasonable fit were determined empirically. No zeros gives a poor
overall match, and only two poles yields a significantly less sharp
resonant peak.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
responses.
^{9.15} - Ideally, they are also constrained to produce
*stable*filters as well, but`invfreqz`does not offer a stability constraint, and so stability should be checked whenever specifying one or more poles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP.
^{9.16} `https://ccrma.stanford.edu/~jos/filters/Creating_Minimum_Phase_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SB.
^{9.17} - SynthBuilder is now a proprietary
in-house tool at Analog Devices, Inc. See [86] 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`[358] in order to obtain drag-and-drop graphical patch construction and a large library of MIDI processing tools.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SmithCommutedFromPASP,MattiGuitar,MattiAndSmith96.
^{9.18} - Extracts of
this material were published previously in [230].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... volume.
^{9.19} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...allpass1phaser.
^{9.20} - This is the
basic architecture of the MXR phase shifter as well as the
*Univibe*used by Jimi Hendrix [243], and described in detail at (`http://www.geofex.com/Article_Folders/univibe/uvfrindx.htm`).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
filters,
^{9.21} - Moog has built a 12-stage phaser of this type
[59] and up to 20 stages (10 notches) have been noted
[244].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP
^{9.22} - Available online at

`http://ccrma.stanford.edu/~jos/filters/Analog_Allpass_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{9.23} - 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 3 dB from , but far away from the break frequency, the piecewise-linear approximation is very accurate. Such a plot of the log-frequency-response versus log-frequency (the real part being log-magnitude and the imaginary part being phase, plotted separately) is called a*Bode plot*. Bode plots are covered in any introductory course on control systems design (also called the design of ``servomechanisms''), and see also the Wikipedia page for ``Bode plot.''. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... response:
^{10.1} - The
corresponding impulse response is
, where
occurs at
time 0, and the transfer function is
. In practice, this filter is typically
implemented in causal form,
*i.e.*, starting at time 0, and one sample of delay is subtracted from a delay-line preceding or following the filter.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
law,
^{10.2} - A function
is said to be
*odd*if .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... function:
^{10.3} - See
`http://www.trueaudio.com/at_eetjlm.htm`for further discussion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... output.
^{10.4} - 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.
^{10.5} - The matlab function
`cohere()`can be used to compute the coherence function between two signals across multiple physical measurements (one per non-overlapping frame).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... phase.
^{10.6} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{10.7} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... below.
^{10.8} - In
particular, as illustrated in Fig.9.14 below, we can
formulate the initial mass momentum as being supplied by an external
impulsive force at time zero.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... transform
^{10.9} `http://ccrma.stanford.edu/~jos/filters/Introduction_Laplace_Transform_Analysis.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP.
^{10.10} `http://ccrma.stanford.edu/~jos/filters/Differentiation.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... function,
^{10.11} -
for
and 1 for
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{10.12} - In acoustic tubes, it is the
longitudinal-velocity wave variable that changes sign when the
direction of propagation is changed, because the air particles then
move in the opposite direction along the tube axis
. The
pressure, on the other hand, being a scalar field, does not
fundamentally change sign when the propagation direction toggles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... string.
^{10.13} - Since we
are in continuous time, a notation more consistent with
Chapter 6 would be
and
, for
, as we reserved the
superscripts
for the discrete-time case in that chapter. Notation has thus been
changed for this example.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Suzuki,Boutillon88,HallAndAskenfelt88,ChaigneAndAskenfeltII,BorinAndDePoli,Stulov95,GiordanoAndMillis01.
^{10.14} `http://www.acs.psu.edu/drussell/Piano/NonlinearHammer.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
model).
^{10.15} - See,
*e.g.*, [350] for a more elaborate model in which release is calculated from the computed geometry of the plectrum.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... that.
^{10.16} - Of course, a
truly complete plucking model would allow the plectrum to move in 3D
space, at least within a plucking plane, and the collision detection
would determine when the string (a point whirling about in the
plucking plane) intersects the plectrum.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... D.
^{10.17} - 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... order,
^{10.18} - It is
valid to neglect the reed mass when the first reed resonance is well
above the fundamental frequency of the played note, as is normally
the case.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
closure.
^{10.19} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... motion|textbf
^{10.20} `https://ccrma.stanford.edu/realsimple/travelingwaves/Helmholtz_Motion.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...exponentially
^{10.21} - 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
^{10.22} - 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.
^{10.23} - When
the kinetic energy of a jet is converted back into air pressure, this
is called
*pressure recovery*. We assume, following [98] and others, that pressure recovery does not occur in this model. Instead, the kinetic energy of the jet 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.
^{10.24} - For velocity waves, the flare may be
*hyperbolic*[50].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...dAlembert,Darrigol07.
^{A.1} - For a short biography of d'Alembert, see
`http:``//www-groups.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

- ...Darrigol07.
^{A.5} `http:``//www.stetson.edu/~efriedma/periodictable/html/B.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

- ...Flanagan72,FlanaganAndRabinerEds,RabinerAndSchafer10,SchaferAndMarkel79,OShaughnessy87,DPH01,Keller94.
^{A.11} - The
overview [309] of early speech production models is
freely available online, thanks to the Smithsonian Speech Synthesis
History Project.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

- ...PainterAndSpanias2000.
^{A.18} - In particular, CELP is used in the
free, open-source speech codec called
*Speex*(`http://www.speex.org`).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SchellengA,SchellengB,CremerC.
^{A.19} `http:``//www.zainea.com/Oscilationsofbowedstring.htm`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
excitation.
^{A.20} - The Karplus-Strong U.S. patent 4,649,783A,
``Wavetable-modification instrument and method for generating musical
sound,'' has the following abstract: A musical instrument employing
probabilistic wavetable-modification method of producing musical
sound. A randomly initialized wavetable which is periodically accessed
to provide an output signal which determines the musical sound. The
output signal from the wavetable is probabilistically modified and
stored back into the wavetable as modified data. The modified data,
after a delay, is accessed from the wavetable and thereby becomes a
new output signal. This process is periodically repeated whereby each
new output signal is stored (after possibly being modified) back into
the wavetable to produce rich and natural musical sound.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
simple
^{B.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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... constant.
^{B.2} - The gravitation constant is given in
*International Standard Units (``SI units'')*by

where the following

*physical units*abbreviations are used:

The physical units for newtons follow from Newton's second law of motion , and the physical units for follow immediately from Eq.(B.2).

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

- ...force|textbf.
^{B.3} - 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 by 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... force.
^{B.4} - 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.
^{B.5} - 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
^{B.6} - A more formal methodology for arriving
at differential equations by applying Newton's law and Hooke's law is
described in Appendix F. In the present example, consideration of the
underlying physics should convince you that the signs in Eq.(B.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:
^{B.7} - See,
*e.g.*, [452, p. 313-316] for a derivation, and Appendix F, specifically §F.3.6, for related analysis.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
object.
^{B.8} - In SI units,
*work*is in units called*joules*(abbreviated J ). Thus, joules are newtons times meters. Power in watts is defined as joules per second. One joule is the energy dissipated in one second by an electric current of one ampere (coulomb per second) through a resistance of one ohm. For a list of physical units and their abbreviations, see, for example,`http://physics.nist.gov/cuu/Units/units.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... considered.
^{B.9} - 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
general.
^{B.10} - Conservation of energy and momentum are unified in
what is called the ``four-momentum'' of spacetime,
,
often used in special relativity calculations (see,
*e.g.*,`http://en.wikipedia.org/wiki/Four-momentum`). Thus, energy (divided by the speed of light ) is just the 0th component of the four-momentum.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... space.
^{B.11} - Additionally, mechanics problems are
formulated using the
*principle of least action*, in which the*action*is defined as the time integral of the*Lagrangian*(usually the kinetic energy minus potential energy). Equations of motion, including Newton's second law, are obtained by finding the path which minimizes this action integral. The two main resulting formulations for the equations of motion (in addition to Newton's formulation) are called the*Lagrangian*and*Hamiltonian*formulations, and they generalize more cleanly to constrained motion, more general*phase-space*coordinate systems, relativistic invariance, and problems in quantum mechanics. Another nice implication of the principle of least action is that the conservation of energy follows from homogeneity of time, and conservation of momentum from isotropy of space [272]. We will be fine with the classical Newtonian formulation here.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... points:
^{B.12} - We denote by
the set of all points in 3D Euclidean space. Thus
has three coordinates, usually denoted
,
, and
, or (as we
will use below),
,
, and
, all real scalars
(
).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
rotating
^{B.13} - We are generally concerned with the rotation of a
rigid body about some axis passing through its center of
mass. Therefore, we will not distinguish between ``rotating,'' as
the earth does to produce night and day, and ``revolving,'' as the
earth does around the sun to produce the seasons. Thus, the
individual mass particles of a rotating rigid body will be said to
*rotate*about the axis of rotation, even though ``revolve'' might win more points on a middle-school exam.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...,
^{B.14} - To create a rigid collection of point masses,
we can imagine them to be interconnected by ``massless rigid rods''
or, equivalently, ideal springs having infinite stiffness.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
density
^{B.15} - We let
be a force
*density*(a linear, spatial, force density, in newtons per meter) in order that it can be impulsive along as well as . We could have instead chosen a force applied at a particular point .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... frame|textbf
^{B.16} - At this stage, a
better name for ``body-fixed frame'' would be
*center-of-mass frame*. However, later on, we will see that not only do we need to follow the center-of-mass with the origin of our body-fixed frame, but its coordinate axes will also need to remain pointing in the so-called*principal directions*(defined in §B.4.16).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... time.
^{B.17} - There is no problem with
conservation of momentum here. We will see in §B.4.13
that, for a rotating mass
at radius
, the angular momentum
is always
times the
instantaneous linear momentum
. Thus, in our problem,
converting angular momentum
to linear momentum
(in
the body-fixed frame) verifies conservation of momentum when we add
it to the linear momentum of the center of mass, which was also
found to be
. In other words, the total momentum is still
,
as initially delivered to the system.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... vector.
^{B.18} - Another analogy is the
*right-hand screw*, in which the screw drives in the direction of the vector when it is turned clockwise from behind, as is standard.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...matrix!determinant|textbf:
^{B.19} `http://mathworld.wolfram.com/CrossProduct.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... rule|textbf.
^{B.20} - Rotate
to
by
``pushing'' with the fingers of the right hand, taking the smallest
angle possible, and your thumb points in the positive direction for
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... them:
^{B.21} - We are using ``
''
(optionally, when it looks better) to denote
*scalar multiplication*here, while in vector calculus, would normally denote a*dot product*of the two vectors and , but we are writing the dot product as . (We could also use the ``inner-product'' notation for real vectors [454].) We could say that `` '' means ordinary multiplication when appearing between two scalars, and it means the dot product when used between two vectors. Between two matrices it would of course mean matrix multiplication.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...conserved.
^{B.22} - Angular momentum is also conserved on the
smallest scales, such as
*orbital angular momentum*and the*spin*of fundamental particles such as an electron. Thus, conservation of angular momentum is a fundamental invariant in physics, along with conservation of energy and conservation of linear momentum. Since we live in 3D, that makes*seven*conserved quantities pertaining to motion of a mass through empty space: its kinetic energy, the three components of the linear momentum of its center of mass, and the three components of its angular momentum if it is rotating about its center of mass.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... identity
^{B.23} - In
`maxima`, one can verify this identity with the following input (the output of which is ):`load(vect);`

A:[A1,A2,A3]; B:[B1,B2,B3]; C:[C1,C2,C3];

expand(express((A ~ (B ~ C)) - (B * (C . A) - C * (A . B))));

Note that the cross-product is called the*wedge product*in`maxima`, and is denoted by a tilde (`~`).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... balanced|textbf.
^{B.24} -
`http://www.ph.man.ac.uk/~mikeb/lecture/pc167/rigidbody/principal.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...JOSFP,
^{B.25} `https://ccrma.stanford.edu/~jos/filters/Diagonalizing_State_Space_Model.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
revolution|textbf.
^{B.26} `http://mathworld.wolfram.com/SolidofRevolution.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... vector:
^{B.27} - Here we are using '
' to
denote the ``dot product'' (or scalar product, or inner product),
since that is the traditional notation in physics. Since it appears
between two vectors, this usage is unambiguous. Thus, we have the
following equivalent notations:
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... frame
^{B.28} - An
*inertial frame*means an*unaccelerated*frame. That is, the coordinate system is not spinning or accelerated in any way. (Even gravity is neglected.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... equations:
^{B.29} `http://www.physicsforums.com/library.php?do=view_item&itemid=182`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... potential:
^{B.30} - This was first pointed out
by Rayleigh
[352, p. 39].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... gas|textbf.
^{B.31} - Air has been measured to contain 75.54% nitrogen and
23.10% oxygen, both of which are diatomic gases. Thus, dry air is
about 99% diatomic.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Fitzpatrick.
^{B.32} - This is a
rare example in which quantum mechanics must be considered in an
acoustic calculation!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{B.33} -
`http://www.grc.nasa.gov/WWW/K-12/airplane/sound.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... air.
^{B.34} - Air is predominately
*diatomic*due to nitrogen ( , 78%) and oxygen ( , 21%) comprising 99% of the Earth's atmosphere.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT.
^{B.35} `http://ccrma.stanford.edu/~jos/mdft/Projection.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... relation.
^{B.36} - Do not be misled by the name
``dispersion relation'' into thinking that wave propagation is
dispersive in this case--it is not. Dispersion occurs when
changes as a function of frequency; here it is a
constant. Thus, the dispersion relation implies wave propagation
dispersion when
is a
*nonlinear*function of .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{C.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
^{C.2} - The term
*phase velocity*is normally used when it differs from the*group velocity*, as in stiff, dispersive strings (§C.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 [452] for more details about the difference.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{C.3} - A detailed derivation of the usual stiff-string wave equation
Eq.(C.32) is given by Morse in [320] or [321].
Derivations of more elaborate wave equations including rotary inertia
and shear effects are given in Graff [170, pp. 180-195]
(``flexural waves in thin rods''). See also Kolsky [263].
See Fletcher and Rossing [145] regarding stiff
piano strings, and Cremer [95] regarding stiffness effects
in violin strings (principally the prevention of sharp corners in the
string).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... wave.
^{C.4} - The sum of the left- and right-going components,
, equals the
*net*force acting to the right.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... junction.
^{C.5} - When the wave impedance
is complex, the junction effectively has state, so that energy, but not
necessarily power, is conserved. See §C.18 for an example.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... coefficient.
^{C.6} - We
previously have denoted reflection coefficients by
and
,
where `r' stands for ``reflection.'' We now start using
because
it is most commonly used in the literature for ladder and lattice
digital filters [299]. Be careful, however, because
is also
the standard notation for
*wave number*(spatial radian frequency) in the field of*acoustics*,*i.e.*, where denotes*wavelength*(spatial period in meters). Normally will not be subscripted when it denotes wavenumber, but we saw even that in §C.8.1 above.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
scattering.
^{C.7} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... waveguides.
^{C.8} - This
section is adapted from [438,437].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
``right-going.''
^{C.9} - 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 [299]. In classical network theory [35] 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
^{C.10} `http://ccrma.stanford.edu/~jos/filters/Similarity_Transformations.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... magnitude:
^{C.11} - 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...CoxEtAl06.
^{C.12} - Footnote added 2018-01-28:
New ultra-thin quadratic residue diffusers, using acoustic resonance
in place of propagation delay to achieve desired phase shifts, thereby
reducing the depth of the diffuser array by 90%, are described at
`https://journals.aps.org/prx/pdf/10.1103/PhysRevX.7.021034`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
algorithms.
^{C.13} - See [105] for discussion of a wider
variety of digital sine generation methods.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... algorithm,
^{C.14} - 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:

The magic circle algorithm is easily shown to be equivalent to the lowpass output of the infinite- second-order

*digital state variable filter*in ``Chamberlin form'' (defined in,*e.g.*, [479,79]).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
amplitude),
^{C.15} - 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
^{C.16} `http://ccrma.stanford.edu/~jos/filters/State_Space_Models.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... velocity.
^{C.17} - More precisely,
it could be expressed as
, but this introduces
a term that is second-order in
that would ultimately be
dropped.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... network.
^{F.1} - 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
^{F.2} - 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,
^{F.3} - 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.
^{F.4} - Note that the wave
variables are now labeled in element-centric notation as opposed to
adaptor-centric notation:

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

- ... limit.
^{F.5} - 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''.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...SmithLAC08.
^{H.1} `http://ccrma.stanford.edu/realsimple/faust_strings`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .