- ...
email
^{1.1} -
`http://www.w3k.org/cgi-bin/register.cgi?email_jos`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... analysis.
^{2.1} - Testing a filter by
sweeping an input sinusoid through a range of frequencies is often
used in practice, especially when there might be some distortion that
also needs to be measured. There are particular advantages to using
*exponentially swept sine-wave analysis*[24], in which the sinusoidal frequency increases exponentially with respect to time. (The technique is sometimes also referred to as*log-swept sine-wave analysis*.) Swept-sine analysis can be viewed as a descendant of*time-delay spectrometry*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...real
^{2.2} - We may define a
*real filter*as one whose output signal is real whenever its input signal is real.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
sinusoid|textbf.
^{2.3} - Some authors refer to
as a
*complex exponential*, but it is useful to reserve that term for signals of the form , where . That is, complex exponentials are more generally allowed to have a non-constant*exponential amplitude envelope*. Note that all complex exponentials can be generated from two complex numbers, and ,*viz.*, . This topic is explored further in [84].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
Octave,
^{3.1} - Users of Matlab will also need the Signal Processing
Toolbox, which is available for an additional charge. Users of
Octave will also need the free ``Octave Forge'' collection, which
contains functions corresponding to the Signal Processing Toolbox.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
language.
^{3.2} - In an effort to improve the matlab language, Octave
does not maintain 100% compatibility with Matlab. See
`http://octave.sf.net/compatibility.html`for details.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...filter
^{3.3} - Say
`help filter`in Matlab or Octave to view the documentation. In Matlab, you can also say`doc filter`to view more detailed documentation in a Web browser.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... filter:
^{3.4} - These adjectives will be
defined precisely in Chapters 4 and 5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
1.
^{3.5} - As we will learn in §5.1,
`A(1)`is the coefficient of the*current output sample*, which is always normalized to 1. The actual feedback coefficients are .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...,
^{3.6} - The notation
denotes the
*half-open interval*--the set of all real numbers between and , including but not .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... execution.
^{3.7} - As a fine point, the fastest known FFT for
power-of-2 lengths is the
*split-radix FFT*--a hybrid of the radix-2 and radix-4 cases. See`http://cnx.org/content/m12031/latest/`for more details.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT.
^{3.8} `http://ccrma.stanford.edu/~jos/mdft/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
matlab
^{4.1} - The term ``matlab'' (uncapitalized) will refer here to
*either*Matlab or Octave [82]. Code described as ``matlab'' should run in either environment without modification.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... completeness.
^{4.2} - Most plots in
this book are optimized for Matlab. Octave uses
`gnuplot`which is quite different from Matlab's handle-oriented graphics. In Octave, the plots will typically be visible, but the titles and axis labels may be incorrect due to the different semantics associated with statement ordering in the two cases.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... frequency:
^{4.3} - As always,
radian frequency
is related to frequency
in Hz by the
relation
. Also as always in this book, the sampling
rate is denoted by
. Since the frequency axis for digital
signals goes from
to
(non-inclusive), we have
, where
denotes a half-open interval. Since the
frequency
is usually rejected in applications, it is more
practical to take
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
``echo''.
^{4.4} - The minimum perceivable delay in audio work depends
very much on how the filter is being used and also on what signals are
being filtered. A few milliseconds of delay is usually not
perceivable in the monaural case. Note, however, that delay
perception is a function of frequency. One rule of thumb is that, to
be perceived as instantaneous, a filter's delay should be kept below a
few cycles at each frequency. A near-worst-case test signal for
monaural filter-delay perception is an impulse (pure click). (A
worst-case test would require some weighting vs. frequency.) Delay
distortion is less noticeable if all frequencies in a signal are
delayed by the same amount of time, since that preserves the original
waveshape exactly and delays it as a whole. Otherwise
*transient smearing*occurs, and the ear is fairly sensitive to onset synchrony across different frequency bands.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{5.1} - One might argue that
nonlinear filters must be considered a special case of time-varying
filters, because any variation in the filter coefficients must occur
over time, and in the nonlinear case, this variation simply happens to
occur in a manner that depends on the input signal sample values.
However, since a constant signal (dc) does not vary over time, a
nonlinear filter may also be time-invariant. As we will see in
this chapter, the key test for nonlinearity is whether the filter
coefficients change as a function of the input signal.
A
*linear*time-varying filter, on the other hand, must exhibit the*same*coefficient variation over time for*all*input signals.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... space
^{5.2} - A set of vectors
(or
)
is said to form a
*vector space*if and for all , , and for all scalars (or ) [84,73].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... purposes.
^{5.3} -
For more about the mathematics of linear vector spaces, look into
*linear algebra*[58] (which covers finite-dimensional linear vector spaces) and/or*operator theory*[56] (which treats the infinite-dimensional case). The mathematical treatments used in this book will be closer to*complex analysis*[14,43], but with some linear algebra concepts popping up from time to time, especially in the context of matlab examples. (The name ``matlab'' derives from ``matrix laboratory,'' and it was originally written by Cleve Moler to be an interactive desk-calculator front end for a library of numerical linear algebra subroutines (`LINPACK`and`EISPACK`). As a result, matlab syntax is designed to follow linear algebra notation as closely as possible.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... domain.
^{6.1} - The term ``difference
equation'' is a discrete-time counterpart to the term ``differential
equation'' in continuous time. LTI difference equations in discrete
time correspond to
*linear*differential equations with*constant coefficients*in continuous time. The subject of*finite differences*is devoted to ``discretizing'' differential equations to obtain difference equations [96,3].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
scheme|textbf.
^{6.2} - The term ``explicit'' in this context means that the
output
at time
can be computed using only
*past*output samples , , etc. When solving*partial*differential equations numerically on a grid in 2 or more dimensions, it is possible to derive finite difference schemes which cannot be computed recursively, and these are termed*implicit finite difference schemes*[96,3]. Implicit schemes can often be converted to explicit schemes by a change of coordinates (*e.g.*, to*modal coordinates*[86]).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... filter.
^{6.3} - Instead of defining
the impulse response as the response of the filter to
, a
unit-amplitude impulse arriving at time zero, we could equally well
choose our ``standard impulse'' to be
, an
amplitude-
impulse arriving at time
. However, setting
and
makes the math simpler to write, as we will see.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT
^{6.4} `http://ccrma.stanford.edu/~jos/mdft/Convolution.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT
^{6.5} `http://ccrma.stanford.edu/~jos/mdft/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... causal,
^{7.1} - In a causal filter (§5.3),
each output sample is computed using only current and past input
samples--no future samples. A
*causal signal*is similarly zero before time zero ( ). An LTI filter is causal if and only if its impulse response is a causal signal.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
points
^{7.2} - When
is a filter transfer function (
*i.e.*, is a filter impulse response), these singularities are called*poles*of the transfer function, as will be defined in §6.6 below. Analogously, one can speak of ``poles'' in the*z*transform of a signal containing exponential components of the form .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
follows:
^{7.3} - Each `
' in these equations should be interpreted
as `
'.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... terms,
^{7.4} - By the fundamental theorem of
algebra, a polynomial
of any degree can be completely factored
as a product of first-order polynomials, where the zeros may be complex.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...residuez
^{7.5} - Matlab Signal Processing
Toolbox or Octave Forge collection--see also
§J.5 (p. ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...biquads.
^{7.6} - A
*biquad*is simply a second-order filter section--see §B.1.6 for details.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
function|textbf.
^{7.7} - The case
is called a
*proper transfer function*, and is termed*improper*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... filter
^{7.8} - In physical models, such a superposition of
identical resonances is often called
*degeneracy*[86].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... coefficients
^{7.9} - We would like to use the
term
*residue*here instead of*coefficient*, but strictly speaking the coefficient in the pole-term is the*residue*of the pole only when (multiplicity one); when is any integer other than 1, the residue is zero (see any book on complex analysis and/or Cauchy's Residue Theorem). For , we will try to remember to call the `` th coefficient'' of the pole .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
on:
^{7.10} -
These closed-form sums were quickly computed using the
free symbolic mathematics program called
`maxima`running under Linux, specifically by typing`factor(ev(sum(m+1,m,0,n),simpsum));`followed by`factor(ev(sum(%,n,0,m),simpsum));`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...B2.
^{7.11} - Since convolution is
*commutative*, either operand to a convolution can be interpreted as the filter impulse-response while the other is interpreted as the input signal. However, in the matlab`filter`function, the operand designated as the input signal (3rd argument) determines the length to which the output signal is truncated.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT.
^{8.1} - Some elementary
review regarding signals and spectra is given in Appendix A.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... ripple
^{8.2} -
A
*passband*may be defined as any frequency band that the filter is trying to ``pass''--*i.e.*, not trying to suppress. For example, in a lowpass filter with cut-off frequency , the passband is the interval . A lowpass filter typically also has a*stopband*that the filter*is*designed to suppress. For practical realizability, there should be a*transition band*between a passband and stopband. In some simple filters, such as Butterworth filters introduced in §7.6.4 below, there are passbands but no stopbands; instead, the stopband is replaced by a*roll-off*, typically specifiable in dB/octave. (The rolloff region can be viewed as a transition band between a passband and a ``stop point'' such as some number of zeros at for lowpass filters, or for highpass filters. Within a passband or stopband, the amplitude response may exhibit*ripple*, that is, it may oscillate about the desired band gain, as discussed further in §7.6.4 below. A*ripple specification*sets a maximum deviation limit on the ripple.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...freqzDemoTwo.
^{8.3} - The ``multiplot''
created by the
`plotfr`utility (§J.4) cannot be saved to disk in Octave, although it looks fine on screen. In Matlab, there is no problem saving multiplots to disk.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{8.4} - The quantity
is known as the
*para-Hermitian conjugate*of the polynomial . It coincides with the ordinary complex conjugate along the unit circle, while elsewhere in the -plane, is replaced by and only the coefficients of are conjugated. A mathematical feature of the para-Hermitian conjugate is that is an*analytic*function of while is not.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{9.1} - See
§6.2 and
§8.7 for related discussion.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... grow.
^{9.2} - As discussed in §6.8.5, the
impulse response of a
*repeated*pole of multiplicity at a point on the unit circle may grow with amplitude envelope proportional to .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...PASP.
^{9.3} -
See,
*e.g.*,`http://ccrma.stanford.edu/~jos/pasp/Passive_Reflectances.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
time-constant|textbf
^{9.4} - Decay time constants were introduced in Book I
[84] of this series (``Exponentials''). The
*time constant*is formally defined for exponential decays as the time it takes to decay by the factor . In audio signal processing, exponential decay times are normally defined instead as or , etc., where ,*e.g.*, is the time to decay by 60 dB. A quick calculation reveals that is a little less than seven time constants ( ).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT
^{10.1} `http://ccrma.stanford.edu/~jos/mdft/Two_s_Complement_Fixed_Point_Format.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...GrayAndMarkel75,MG,PASP.
^{10.2} -
`http://ccrma.stanford.edu/~jos/pasp/Conventional_Ladder_Filters.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...:
^{10.3} - The notation ``
'' is an abbreviation for ``modulo
''
commonly used in the branch of mathematics known as
*number theory*.. Two integers and are said to be equal modulo if there exists an integer such that . Thus, is equal to modulo because . These integers are also equal to , , , , and so on.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... plane.
^{10.4} - To
plot the poles and zeros for the example of §7.5.2, one can
say (in matlab)
`plot(roots(B),'o',roots(A),'x')`-- or say`help zplane`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
`tf2sos`|textbf^{10.5} - In
Matlab, the Signal Processing Toolbox is required for second-order
section support. In Octave, the free Octave-Forge add-on collection
is required.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Octave.
^{10.6} - The Matlab Signal Processing Toolbox has even
more
`sos`functions--say ```lookfor sos`'' in Matlab to find them all.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
bandwidths
^{10.7} - See §E.6 for a definition of half-power
bandwidth.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... series.
^{10.8} - In this
particular case, there is an even better structure known as a
*ladder filter*that can be interpreted as a*physical model*of the vocal tract [48,86].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...pfe).
^{10.9} - In practice, it is not critical to get the
biquad numerators exactly right. In fact, the vowel still sounds ok
if all the biquad numerators are set to 1, in which case, nulls are
introduced between the formant resonances in the spectrum. The ear is
not nearly as sensitive to spectral nulls as it is to spectral peaks.
Furthermore, natural listening environments introduce nulls quite
often, such as when a direct signal is mixed with its own reflection
from a flat surface (such as a wall or floor).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...even.
^{11.1} - In the complex
case, the zero-phase impulse response is
*Hermitian*,*i.e.*,

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

- ... algorithm.
^{11.2} - See the function
`firpm`in the Matlab Signal Processing Toolbox and`remez`in the Octave Forge collection.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
Matlab).
^{11.3} - This information updated on 2018-02-20 for MacPorts
GNU Octave, version 4.2.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... response.
^{11.4} - In the context of
*statistical signal processing,*we can say that the impulse response has been replaced by its*autocorrelation*, and the complex frequency response has been replaced by its magnitude squared (``power response'').. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{12.1} - Another way to show that all minimum-phase filters and
their inverses are causal, using the Cauchy integral theorem from
complex variables
[14], is to consider a Laurent series expansion of the transfer
function
about any point on the unit circle. Because all poles
are inside the unit circle (for either
or
), the
expansion is one-sided (no positive powers of
). A Laurent
expansion about a point on the unit circle interprets unstable poles
as noncausal exponentials in the time domain, which ``decay'' in the
direction of negative time, as discussed and illustrated in
§8.7.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...).
^{12.2} - The
allpass
as defined has magnitude
over the unit
circle instead of
as is usually defined for allpass gains. To
normalize the allpass gain to
, we can define
instead.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... phase,
^{12.3} - The convolution of two minimum phase
sequences is minimum phase, since this just doubles each pole and zero
in place, so they remain inside the unit circle.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{12.4} - We can loosely
define
*nonparametric signal processing*as performing array operations on signals and spectra, as opposed to working with parametric representations such as poles and zeros. Generally speaking, nonparametric signal processing is typically more robust than parametric signal processing [87].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...tmps.
^{12.5} - A Mathematica notebook for this
purpose was written by Andrew Simper:
`http://www.vellocet.com/dsp/MinimumPhase/MinimumPhase.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT.
^{A.1} -
`http://ccrma.stanford.edu/~jos/mdft/Sinusoids_Exponentials.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT.
^{A.2} -
`http://ccrma.stanford.edu/~jos/mdft/Discrete_Fourier_Transform_DFT.html`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... overflow.
^{B.1} - A small chance of
overflow remains because sinusoids at different frequencies can be
delayed differently by the filter, causing an increased peak amplitude
in the output due to phase realignment.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...resonance
^{B.2} - A
*resonance*may be defined as a local peak in the amplitude response of a filter, caused by a pole close to the unit circle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... case,
^{B.3} - In the case of complex coefficients
,
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... applications.
^{B.4} - For the reader with some background
in analog circuit design, the dc blocker is the digital equivalent of
the analog
*blocking capacitor*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Zolzer
^{B.5} - The form of Equations
(B.11) and (B.12) work well for
*first-order*shelf filters. For higher (odd) orders, it is better to use a Butterworth band-split, as has been used in Faust's`filter.lib`since June 2012 (see Appendix K). (This footnote added in the 4th printing.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... response,
^{B.6} - See §8.2
in Chapter 8.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... virtual
^{B.7} - The term
*virtual analog synthesis*refers to digital implementations of classic analog synthesizers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...causal|textbf
^{C.1} - Recall that a filter is said to be
causal if its impulse response
is zero for
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... have,
^{C.2} - Note that the
time-domain norm
is
*unnormalized*(which it must be) while the frequency-domain norm is*normalized*by . This is the cleanest choice of norm definitions for present purposes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Filters
^{C.3} - The remainder of this appendix is
relatively advanced and can be omitted without loss of continuity in
what follows.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Filters
^{C.4} - The remainder of this appendix is
relatively advanced and can be omitted without loss of continuity in
what follows.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...causal
^{D.1} - A signal
is said to be
causal if it is zero for all
. A system is said to be causal if
its response to an input never occurs before the input is received;
thus, an LTI filter is a causal system whenever its impulse response
is a causal signal.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
order,
^{D.2} - The
*order of a pole*is its multiplicity. For example, the function has a pole at of order 3.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... there''.
^{D.3} - Note that, mathematically,
our solution specifies that the mass position is zero prior to time
0. Since we are using the unilateral Laplace transform, there is
really ``no such thing'' as time less than zero, so this is
consistent. Using the
*bilateral*Laplace transform, the same solution is obtained if the mass is at position for all negative time , and the driving force imparts a*doublet*having ``amplitude'' at time 0 ,*i.e.*, , and all initial conditions are taken to be zero (as they must be for the bilateral Laplace transform). A*doublet*is defined as the time-derivative of the*impulse*signal (defined in Eq.(E.5)). In other words, impulsive inputs at time 0 can be used to set up arbitrary initial conditions. Specifically, the input slams the system into initial state at time 0.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{E.1} - To show this, differentiate the
squared-magnitude frequency response
with respect to
, equate to zero, and solve for
. You can see from
checking the denominator of the derivative that the result holds
whenever
,
*i.e.*, as long as there as any amount of decay in the impulse response (any nonzero bandwidth).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
respectively.
^{E.2} **Exercise:**Determine and and check your result by performing the Laplace transform and comparing to Eq.(E.7).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
filters.
^{F.1} - A short tutorial on
*matrices*appears in [84], available online at`http://ccrma.stanford.edu/~jos/mdft/Matrices.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT.
^{F.2} `http://ccrma.stanford.edu/~jos/mdft/Matrix_Formulation_DFT.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...MDFT
^{F.3} `http://ccrma.stanford.edu/~jos/mdft/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... properties.
^{F.4} -
`http://en.wikipedia.org/wiki/Circulant_matrix`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... follows:
^{F.5} - While this
example is easily done by hand, the matlab function
`tf2ss`can be used more generally (``transfer function to state space'' conversion).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
way.
^{F.6} - The methods discussed in this section are intended for
LTI system identification. Many valued guitar-amplifier modes, of
course, provide highly
*nonlinear distortion*. Identification of nonlinear systems is a relatively advanced topic with lots of special techniques [24,17,97,4,86].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
derive.
^{F.7} - There are many possible definitions of pseudoinverse
for a matrix
. The Moore-Penrose pseudoinverse is perhaps most
natural because it gives the
*least-squares solution*to the set of simultaneous linear equations , as we show later in this section.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Golub.
^{F.8} - Say
`help slash`in Matlab.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... matrix,
^{G.1} - A short tutorial on
*matrices*appears in [84], available online at`http://ccrma.stanford.edu/~jos/mdft/Matrices.html`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
elsewhere.
^{G.2} *I.e.*, , where is the identity matrix, and denotes the discrete-time impulse signal (which is 1 at time and zero for all ).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... matrix.
^{G.3} - To emphasize something is a matrix, it is
often typeset in a
**boldface**font. In this appendix, however, capital letters are more often used to denote matrices.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... series,
^{G.4} - Let
, where
is a square matrix.
Then
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{G.5} - Equivalently, a causal transfer function
contains a delay-free path whenever
, since
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
model.
^{G.6} - An exception arises when the model may be
*time varying*. A time varying matrix, for example, will cause time-varying zeros in the system. These zeros may momentarily cancel poles, rendering them unobservable for a short time.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
quasi-harmonic
^{G.7} - The overtones of a vibrating string are never
exactly harmonic because all strings have some finite
*stiffness*. This is why we call them ``overtones'' instead of ``harmonics.'' A perfectly flexible ideal string may have exactly harmonic overtones [55].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... form.
^{G.8} - As
of this writing, this function does not exist in Octave or Octave
Forge, but it is easily simulated using
`sos2tf`followed by`tf2ss`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
matlab.
^{G.9} - Specifically, this example was computed using
Octave's
`tf2ss`. Matlab gives a different but equivalent form in which the state variables are ordered in reverse. The effect is a permutation given by`flipud(fliplr(M))`, where`M`denotes the matrix`A, B`, or`C`. In other words, the two state-space models are obtained from each other using the similarity transformation matrix`T=[0 0 1; 0 1 0; 1 0 0]`(a simple permutation matrix).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... numerically.
^{G.10} - If the Matlab
Control Toolbox is available, there are higher level routines for
manipulating state-space representations; type ``
`lookfor state-space`'' in Matlab to obtain a summary, or do a search on the Mathworks website. Octave tends to provide its control-related routines in the base distribution of Octave.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{G.11} - In general,
we can write an order
Jordan block
corresponding to eigenvalue
as
*binomial theorem*,

where denotes the

*binomial coefficient*(also called `` choose '' in probability theory). Thus,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
matrix.
^{H.1} - Recursive filters are brought into this framework in
the time-invariant case by dealing directly with their impulse
response, or the so called
*moving average*representation. Linear time-varying recursive filters have a matrix representation, but it is not easy to find. In general one must symbolically implement the equation and collect coefficients of .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... dc).
^{I.1} - In other
words, matching leading terms in the Taylor series expansion of
about
determines the poles as a function of the
zeros, leaving the zeros unconstrained. It is shown in
[64] that any filter of the form
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Tustin,
^{I.2} - A. Tustin, ``A
method of analysing the behaviour of linear systems in terms of time
series,'' J. Inst. Elect. Engrs., Part IIA, Automatic Regulators and
Servo Mech vol. 94, no. 1, May 1947,
pp. 130-142
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... causal
^{I.3} -
is said to be
*causal*if for .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Octave.
^{J.1} - On a Red Hat Fedora
Core Linux system,
`octave-forge`is presently in ``Fedora Extras'', so that one can simply type`yum install octave-forge`at a shell prompt (as`root`).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
complex):
^{J.2} - Thanks to Matt Wright for contributing the original
version of this example.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... language|textbf
^{K.1} - The Faust home page is
`http://faust.grame.fr/.`Faust is included in the Planet CCRMA distribution (`http://ccrma.stanford.edu/planetccrma/software/`). The examples in this appendix have been tested with Faust version 0.9.9.2a2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... environments.
^{K.2} - Faust
``architecture files'' and plugin-generators are currently available
for Max/MSP, PD [65,31], VST, LADSPA, ALSA-GTK, JACK-GTK,
and SuperCollider, as of this writing.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{K.3} - A ``
`with`'' block is not required, but it minimizes ``global name pollution.'' In other words, a definition and its associated`with`block are more analogous to a C function definition in which local variables may be used. Faust statements can be in any order, so multiple definitions of the same symbol are not allowed. A`with`block can thus be used also to override global definitions in a local context.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... background.
^{K.4} - Facility with basic C++ programming is also assumed for
this appendix.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... signals,
^{K.5} - A
*causal signal*is any signal that is zero before time 0 (see §5.3).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...cpgr.dsp.
^{K.6} - The
`faust2firefox`script (distributed with Faust version 0.9.9.3 and later) can be used to generate SVG block diagrams and open them in the Firefox web browser.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... response:
^{K.7} - This
specific output was obtained by editing
`cpgrir-print.cpp`to replace`%8f`by`%g`in the print statements, in order to print more significant digits.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{K.8} - In many cases the signal processing in Faust can occur within
a ``foreign function'' written in C or C++ and used as a ``black
box'' within Faust, like the
`cos()`function in Fig.K.5. However, this approach is presently limited because foreign functions can have only`float`and`int`argument types, and they can only return a`float`each sample. It is possible to set up persistent state in a foreign function by means of static variables, but this does not generalize easily to multiple instances. Therefore, more general extensions may require direct modification of the generated C++, which usually obsoletes the Faust source code.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... patch,
^{K.9} - All manually
generated
`.dsp`files and`pd`patches in this appendix are available at`http://ccrma.stanford.edu/realsimple/faust/faustpd.tar.gz`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...cpgrui~-help.pd,
^{K.10} - In
`pd`, a dynamically loadable module (`pd`plugin) is called an*abstraction*. (This is distinct from the*one-off subpatch*which is encapsulated code within the parent patch, and which resides in the same file as the parent patch [66].) It is customary to document each abstraction with its own ``help patch''. The convention is to name the help patch ``name-help.pd'', where ``name'' is the name of the abstraction. Right-clicking on an object in`pd`and selecting ``Help'' loads the help patch in a new`pd`window.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...loadbang
^{K.11} - The
`loadbang`object sends a ``bang'' message when the patch finishes loading.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...synth.pd.
^{K.12} - On a Linux system with
Planet CCRMA installed, the command ``
`locate synth.pd`'' should find it,*e.g.*, at`/usr/share/doc/faust-pd-0.9.8.6/examples/synth/synth.pd`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{K.13} - After running
`jack-rack`, the LADSPA plugin was added by clicking on the menu items ``Add / Uncategorised / C / Constant_Peak_Gain_Resonator''. If`jack-rack`does not find this or other plugins, make sure your`LADSPA_PATH`environment variable is set. A typical setting would be`/usr/local/lib/ladspa/:/usr/lib/ladspa/`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...jack-rack-connect.
^{K.14} - Sound routings such as this may be
accomplished using the ``Connect'' window in
`qjackctl`. In that window, there is an Audio tab and a MIDI tab, and the Audio tab is selected by default. Just click twice to select the desired source and destination and then click ``Connect''. Such connections can be made automatic by clicking ``Patchbay'' in the`qjackctl`control panel, specifying your connections, saving, then clicking ``Activate''. Connections can also be established at the command line using`aconnect`from the`alsa-utils`package (included with Planet CCRMA).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Receptor|textbf.
^{K.15} - The
Receptor is a hardware VST plugin host designed for studio work and
live musical performance. While it only supports Windows VST
plugins, it is based on a Red Hat Linux operating system using
`wine`for Windows compatibility. The VST plugin described in this section was tested on system version 1.6.20070717 running on Receptor hardware version 1.0. This system expects VST-2.3 plugins, and so VST-2.4 plugins cause a warning message to be printed in the Receptor's system log. However, v2.4 plugins seem to work fine in the 2.3 framework. There was a competitor to the Receptor called Plugzilla that supported both VST and LADSPA plugins, but Plugzilla no longer appears to be available.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... input.
^{K.16} - Pd must have at least one MIDI-input port
defined at startup for this to work. For example, a typical
`~/.pdrc`file might contain the following startup options for`pd`:`-jack -r 48000 -alsamidi -midiindev 1 -midioutdev 1 -audiooutdev 1 -outchannels 2 -path /usr/lib/pd/...`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .