IRCAM
31, RUE ST. MERRI
PARIS F75004
FRANCE
PUBLISHED 6/78
ABSTRACT
 One of the results of the science of estimation theory has been the
developement of the linear prediction algorithms. This allows us to
compute the coefficients of a timevarying filter which simulates the
spectrum of a given sound at each point in time. This filter has found
uses in many fields, not the least of which is speech analysis and
synthesis as well as computer music. The use of the linear predictor in
musical applications allows us to modify speech sounds in many ways,
such as changing the pitch without altering the timing, changing timing
without changing pitch, or blending the sounds of musical instruments
and voices. This paper is concerned with the fine details of the many
choices one must make in the implementation of a linear prediction
system and how to make the sound as clean and crisp as possible.
INTRODUCTION
 Linear prediction is a method of designing a filter to best
approximate, in a mean squared error sense, the spectrum of a given
signal. Although the approximation gives as results a filter valid over a
limited time, it is often used to approximate timevariant waveforms by
computing a filter at certain intervals in time. This gives a series of
filters, each one of which best approximates the signal in its
neighbourhood. The uses for such a filter are manyfold, ranging from
geological and seismological applications (Burg) to radar and sonar
(Robinson), to speech analysis and synthesis (Atal, et al, 1970, 1971,
Itakura and Saito 1971, Makhoul 1975, Markel and Gray 1976), and to
computer music (Dodge, Moorer 1977, Petersen 1975, 1976, 1977). We shall
concentrate here on the usage of linear prediction as a method of
capturing, simulating, and applying the sounds of the human voice in
highfidelity musical contexts. Even more specifically, we will
concentrate on applications using only the digital computer as the
medium.
 This paper reports the results of work done over the last two
years in searching for ways to improve the quality of speech synthesis.
The findings were determined by largely informal listening tests with
trained musicians.
MODELING OF SPEECH
 This description is taken largely from Makhoul (1977). We start by
modeling the sound of the human voice as an allpole spectrum with a
transfer function given by
 coefficients, and p is the number of poles or predictor coefficients
in the model. If H(z) is stable (minimim phase), A(z) can be
implemented as a lattice filter (Itakura and Saito) as shown in figure
1. The reflection (or partial correlation) coefficients Km in the
lattice are uniquely related to the predictor coefficients. For a stable
H(z), we must have
 H(z) can also he implemented as a lattice form as shown in figure 2,
as well as a product of first and second order sections by factoring
A(z) and combining complex conjugate roots to form second order sections
with all real coefficients as shown in figure 3. Finally, the filter
can be implemented in direct form as shown in figure 4.
 We are excluding for the time being models which include both
poles and zeros since we have not as yet investigated a satisfactory
method to compute both poles and zeros reliably.
 To actually synthesize a speech sound, one must drive this
filter with something. This is called the excitation, and it too must be
modeled to provide a reasonable representation of the speech
excitation. We usually choose the excitation to be either white noise
for unvoiced sounds, a wideband pulse train for voiced sounds, or
silence (for silence) as shown in figure 5, although this simplification
should be discussed further.
 Thus to summarize, if we wish to synthesize speech, the process
from analysis to synthesis might be the following:
 1) Extract the pitch of the original sound.
 2) Compute the linear prediction coefficients for the original
sound at selected points in time.
 3) Decide at each point in time whether the signal is voiced,
unvoiced, or silence.
 4) Compute the gain factor for the original sound.
 5) Create an excitation from the pitch and the voicedi
unvoiced/silence decision.
 6) Scale it with the computed gain factor.
 7) Filter it with the computed predictor coefficients.
 This is roughly the outline of the process from start to
finish, but the order is not necessarily rigid. For example, we may
decide not to compute a gain factor, but merely to scale the energy of
the synthesized signal to correspond to the original energy in the
signal.
 Readers wishing to know more about the subject of linear
prediction of speech should refer to the literature (Markel and Gray
1976, Makhoul 1975).
 There are numerous other decisions to be made, such as choosing
a method for doing each of these things, choosing the order of the
filter, deciding what form the filter should be in, how to interpolate
the parameters between. We will attempt to comment on each of these.
ON MUSICAL APPLICATIONS
 The most usual application of this technique is with respect to
speech communication. The idea there is to reduce the data involved in
the transmission of speech. Indeed, using linear prediction, one can
quantize the various parameters and obtain striking reductions in the
amount of data involved (Markel and Gray). For musical purposes,
however, we cannot generally afford the loss of quality implied by this
quantization. Although even in speech communication the quality is
important, it is not as critical as in the case of music production. At
each point, we must ask ourselves "Would I pay $ 5.95 for a record of
this voice?".
 In general, there is no point in directly resynthesizing a
piece of speech or singing. One could just use directly the original
segment. The only point is to be able to modify the speech in ways that
would be difficult or impossible for tiespeaker to do. These include
modifications of the driving function, such as changing the pitch or
using more complex signals, changing the timing of the speech, or
actually altering the spectral composition. Thus we will concentrate
here not only on methods that preserve the speech quality in an
unmodified reconstruction, but also that are less sensitive to
modification, that can preserve the quality over a wide range of
modifications.
BASIC DECISIONS
 The first step in the process is to detect the pitch. This is not a
simple problem, but has been well studied (Noll, Gold and Rabiner,
Sondhi, Moorer 1974). In the musical case, we have more information
beforehand than one would in the speech communication case, in that we
can allow the program some amount of information about the speaker. In
specific, if the range of frequencies can be bounded or identified
beforehand, this eliminates immediately most of the gross errors that
pitch detectors usually commit. What few gross errors remain can be
corrected automatically by heuristic means. We have found that if the
pitch at any given time can be limited to a range of only one octave,
most of the pitch detectors reported in the literature seem to work
adequately. The only question is how often should the pitch be
determined. We are currently using pitch determination every 5
milliseconds and this seems to give fine enough resolution for most
purposes.
 Next is the voicedunvoicedsilence decision. This seems to be
the most difficult part to automate. So difficult, in fact, that we have
taken to using a graphics program to allow the composer to go through
and mark the segments himself. We use a decision theoretic procedure for
the initial labeling (Atal and Rabiner 1976, Rabiner and Sambur 1977).
We then synthesize an unmodified trial replica of the original sound.
Using graphics, 150 millisecond windows of both the original and the
replica are presented. The voicedunvoicedsilence decision as
determined by the computer is listed below the images. When the
differences between the original and the replica seem to indicate an
error in the decision, it is easily corrected by hand. It takes about 15
miniutes to go through a 12second segment of speech this way, which
represents, for instance, about one stanza of a poem (between 35 and 40
words).
WHAT KIND OF PREDICTION
 Usually in speech analysis, the analysis window is stepped by a
fixed time, such as 10 or 15 milliseconds, and takes a fixed number of
samples at each step, such as 25 milliseconds worth. This has the
problem of inconsistancy. A 25 millisecond window for a male voice will
sometimes capture two speech pulses and sometimes three, depending on
the pitch and phasing of the speech. This gives a large frametoframe
variablility in the spectral estimate. The result is a "roughness" that
depends on the relation between the instantaneous pitch and the frame
width.
 One can decrease the effect of this phenomenon in several ways.
First, by use of an allpass filter, one may distort the phase of the
speech to largely eliminate the prominance of the glottal pulse
(Rabiner, et al, 1977). One can also use a larger analysis window such
that more main pulses are incorporated, such that the omission or
inclusion on one pulse does not perturb the filter so strongly. Both of
these remedies have the effect of blurring what are often quite sharp
boundaries between voiced and unvoiced sounds. The problem is that if
the analysis window overlaps significantly an unvoiced region, the
extreme bandwidth of the unvoiced signals contributes to a filter that
passes a great deal of high frequencies. If this filter is then used to
synthesize a voiced sound, a strong buzzy quality is heard. The overall
effect was that just around fricatives, the voice before and after had a
strongly buzzy quality.
 Another problem with using analysis windows larger than a
single period is that the filter begins to pick up the fine structure of
the spectrum. The fine structure is composed of those features that
contribute to the excitation, notably the pitch of the sound. At the
high order required for highquality sound on widebandwidth original
signals (we are using 55th order filters for a deep male voice with a
sampling rate of 25600 Hertz), the filter seems to capture some of the
pitch of the original signal from overlapping several periods at once.
The result is that even though reasonable unmodified synthesis can be
obtained, the sound deteriorates greatly when the pitch is changed.
This, then, is a case where the unique musical application of
modification implies a more substantial change from speech communication
techniques.
 The solution that we adopted was the use of pitch synchronous
analysis, where the analysis window is set to encompass exactly one
period, and it is stepped in time by exactly one period. This prevents
any fine structure representing the pitch from being incorporated into
the filter itself. It also provides that in the case of the borders
between voiced and unvoiced regions, no more than one period will
overlap the border itself. The step size and window width is not so
critical in the unvoiced portions, so we simply invent a fictitious
pitch by interpolating between the known frequencies nearest the
unvoiced region. There does remain a slow variation in the filters
presumably caused by inaccuracies in the pitch detection process. This
can be somewhat lessened by the allpass filter approach Gtabiner,et al,
1977), but it does not seem to be terribly annoying in musical
contexts.
 Note that the adoption of pitchsynchronous analysis has
implications for the type of prediction used. The most popular method is
the autocorrelation method, but its necessary windowing is not
appropriate for pitchsynchronous analysis. Some kind of covariance or
latice method is then required. What we have chosen is Burg's method
(Burg 1967) because it does correpond to the minimization of an error
criterion, the filter is unconditionally stable, and there exists a
relatively efficient computational technique (Makhoul 1977). We have
tried straight covariance methods with the result that the instabilities
of the filters are inherent at high orders in certain circumstances and
somewhat difficult to cure. One can always factor the polynomial and
replace the ailing root by its inverse, then reassemble the filter, but
besides being expensive, there is another reason to be discussed
subsequently that is even more compelling.
 There are also a number of recursive estimation techniques
(Morgan and Craig 1976, Morf 1974, Morf, et al, 1977) which allow one to
compute the coefficients from the previous coefficients and the new
signal points. This has the advantage that no division of the signal
into discrete windows is necessary. In fact, no division is possible.
The problem is, again, that if the "memory" of the recursive calculation
is short enough to track the rapid changes, such as from an unvoiced
region to a voiced region, then it also tracks the variation of spectrum
throughout a single period of the speech sound. The shortterm spectrum
changes greatly as the glottis opens and closes. If the memory of the
calculation is long enough to smooth out the intraperiod variations,
then it also tends to mix the spectra of the adjacent regions.
THE EXCITATION FUNCTION
 To resynthesize the signal, either at the original pitch or at an
altered pitch, one must synthesize an excitation function that drives
the computed filters that embodies both the pitch and the
voicedunvoicedsilence decision. The most common method is to use a
single impulse for each period in the voiced case and uniform noise of
some sort in the unvoiced case. In the case of silence, the transient
response of the filters is allowed to unwind naturally.
 The problem with the single pulse is that it is not a
bandlimited signal. In places where the pitch is changing rapidly, this
produces a roughness in the sound that is quite annoying. For this
reason, it is generally preferable to use a bandlimited pulse of some
sort (Wynam and Steiglitz 1970). One can further improve the sound by
scrambling somewhat the phases of the components of the bandlimited
pulse to prevent the highly "peaky" appearance, but this is frosting on
the cake that is not clearly perceived by most listeners. It is audible,
but it is not the dramatic transformation from harsh to me lifluous
that one might hope. We synthesize the pulse by an inverse fast Fourier
transform. This allows us to set the phases of each component
independently. We found that a slight deviation from zero phase was
desirable and easily accomplished by adding a random number into the
phase corresponding to .5 to +.5 (radians) seemed sufficient to "round
off" the peak. Since using the FFT is a somewhat expensive way to
compute the excitation, we computed it only when the frequency changed
enough that a harmonic had to be ommitted or added. The synthesized
driving signal was kept in a table and sampled at the appropriate rate
to generate the actual excitation. This provided another benefit that we
will discuss presently. Also, since recomputing the driving function
using semirandom phases can give discontinuities when changing from one
function to a new one, we used a raised cosine to round the ends of the
driving function to zero. If a DC term is present, this is known to
leave the spectrum unchanged, except for the highest harmonics, so we
are assured that the driving spectrum is exactly flat up to near the
maximum harmonic. The raised cosine was applied just at the beginning 10
percent and ending 10 percent of the function.
 The production of the noise for the unvoiced regions does not
seem to be highly critical. We are using Gaussian noise (Knuth 1969).
 One might ask why we attempt to synthesize the driving
function. Why not use the residual of the original signal directly? This
indeed has the advantage that there is no pitch detection involved and
no voicedunvoicedsilence decision at all. The problem is that then for
musical purposes, one must be able to modify the residual itself. There
exist methods for doing this using the phase vocoder as a modification
tdol (Moorer 1976, Portnoff 1976). There is even a recent study about
making the phase vocoder more resistant to degradation from modification
(Allen 1977).
 The problem is that to produce the residual, it is often
necessary to amplify certain parts of the spectrum that might have been
very weak in the original. The very definition of "whitening" the signal
is to bring all parts of the spectrum up to a uniform level. There are
inevitable weak parts of the spectrum  nasal zeros or some such. If
there is precious little energy at a certain band of frequencies, then
the whitening process will simply amplify wnatever noise was present in
the recording process. If this resulting noise then falls under a strong
resonance when the prediction filter is then applied, this filter then
just amplifies that noise. The perceptual effect is that the signal
looses its "crispness" and becomes "fuzzy", and sometimes even downright
noisy.
CROSSSYNTHESIS
 In discussing excitation functions, one must mention a particular
application that has been found quite useful for the production of new
musical timbres, and that is the operation of crosssynthesis (Petersen
1975, 1976, 1976). Here we use another musical sound as excitation,
rather than attempting to model the speech excitation. What results is a
bizzare but often interesting combination of the source sound and the
speech sound. In this manner we can realize the sounds of "talking
violin" or "talking trumpet", in a manner of speaking. In fact, if one
uses the musical signal directly as excitation, quite often the result
is not highly intelligible. This is because most musical signals are not
spectrally flat wideband signals, but instead have complicated
spectra. For this reason, it is usually good to whiten the source sound.
One can do this also with a loworder linear predictor as shown in
figure 6. The error signal of a 4th to 6th order linear prediction
process is usually sufficiently whitened to improve the intelligibility
greatly, but at the expense of the clarity of the original musical
source. In fact, one can choose from a continuum of sounds between the
original musical source and the speech sound. Depending on the
compositional goals, one might choose something more instrumentlike and
scarcely intelligible, progressing through stages of increasing
intelligibility, or whatever.
 One can use any sound as excitation with varying results. For
instance, if the source sound has very band limited spectral
caracteristics, such as an instrument with a very small number of
haromonics like a flute, the whitening process will just amplify
whatever noise happens to be present in the recording process, producing
an effect somewhat like a "whispering" instrument, where the pitch and
articulation of the instrument is clearly audible, but the speech sounds
distinctly whispered.
 One can also deliberately defeat the pitch synchronicity of the
analysis to capture the fine structure of the spectrum. If one then
filters a wide band sound, such as the sound of ocean waves, one can as
the order increases impose the complete sound of the voice on the
source. We can in this manner realize something like the sounds of the
sirens on the waves, or the "singing ocean". In general, this sort of
effect takes a very high order filter. For example, if the vocal signal
were in steadystate, it would require one secondorder section for each
harmonic of the signal. Thus for a low male voice of 40 to 60
harmonics, an order of 80 to 120 would be required. Indeed, our
experiments have shown that as the order approaches 100 (or 50 for
female voice), the pitch of the vocal sound becomes more and more
apparant.
 Using the lattice form for the synthesis filter gives us a
convenient way of adjusting the order of the filter continuously. Since
with the lattice methods, the first N sections (coefficients) of the
filter are optimal for that order, we can just add one section after
another to augment the order. Setting coefficients to zero, starting
from the highest, still produces an optimal filter of a lower order.
 This is not true of the direct form or the factored form.
Throwing out one coefficient requires changing all the other
coefficients to render the filter optimal again. Indeed, instead of just
turning a coefficient on or off in the latice form, we may also turn it
up or down. That is to say when we add a new coefficient, we may add it
gradually, starting at zero, and slowly advancing to its final value
(presumably precomputed). This allows us to "play" the order of the
filter, causing the vocal quality to strengthen and fade at will in a
continuous manner.
 Crosssynthesis between musical instruments and voice seems to
make the most sense if the two passages are in some way synchronized. We
have done this in two different ways to date: one is to record some
speech, poetry, or whatever, performed by a professional speaker to
achieve the desired presentation, then using synchronized recording and
playing, either through a multitrack tape recorder or through digital
recording techniques, have the musician(s) play musical passages exactly
synchronized with the vocal sounds. This takes a bit of practice for
the musician, in that speech sounds in English are not typically exactly
rythmically precise, but can nonetheless be done quite precisely. The
other avenue is to record the music first to achieve some musical
performance goal, then have the speaker synchronize the speech with the
musical performance. Either one of these approaches achieves
synchronicity at the expense of naturalness in one or the other of the
performances, vocal or musical, but renders the combination much more
convincing.
INTERPOLATION
 To make the resulting speech as smooth as possible, virtually
everything must change smoothly from one point to the next. For
instance, if the filter coefficients are changed abruptly at the
beginning of a period, there is a perceivable roughness produced. If we
wish to interpolate the filter coefficients, however, we must be careful
about the choice of a filter structure.
 We can envision at least three filter structures: direct form,
factored form, and latice form. In factored form, the filter is realized
using first and second order sections. The direct form is just a single
highorder tapped delay line. The problem with the direct form is that
it's numerical properties are somewhat less than ideal and that one
cannot necessarily interpolate directly the coefficents. If you
interpolate linearly between the coefficients of two stable potynomials,
the resulting intermediate polynomials are not necessarily stable.
Indeed, if the roots of the polynomials are very similar, the
intermediate polynomials will probably be stable, but if the roots are
very different, the intermediate polynomials are quite likely to be
unstable. Thus the direct form is not suitable for interpolation without
further thought.
 The factored form can be interpolated directly in a stable
manner. In a secondorder section representing a complex conjugate pole
pair, the stability depends largely on the term of delay two. As long as
this term is less than unity, the section will probably be stable,
depending on the remaining term. There are two problems remaining,
though, in the use of the factored form. The first is that the
polynomial must be factored. With 55th order polynomials, this is a
nontrivial task. There is no estimation technique known that can
produce the linear prediction filter in already factored form. Although
factoring polynomials is an established science, it is still quite
timeconsuming, especially with high order. In addition to that, one
must also group the roots such that each section changes only between
roots that are very similar. Since there is no natural ordering of the
roots, one must invent a way of so grouping them. We have tried
techniques of minimizing the Euclidian distance on the Zplane between
pairs of roots, and this seems to give reasonable results, except in
certain cases when, for instance, real roots collide and form complex
conjugate pairs. There is no telling at any given time how many real
roots a polynomial will have, and quite often there are one or two real
roots that move around in seemingly random fashion. The factored form
does, however, have one strong advantage, which is that it is very clear
how to directly modify the spectrum at any given point. Since the roots
are already factored, it is quite clear which sections control which
parts of the spectrum. Moreover, when the roots are interpolated, they
form clear, welldefined patterns that have welldefined effects on the
spectrum. Except for the inefficiencies involved in factoring and
ordering the roots, the factored form seems ideal.
 With the latice form, there is no problem in interpolation. The
reflection coefficients can be interpolated directly without fear of
instabilities because the condition for stability is simply that each
coefficient be of magnitude less than unity. When one interpolates,
however, between reflection coefficients of two stable filters, the
roots follow very complex paths, thus if the filters are not already
very similar, one can only expect that the intermediate filters will be
only very loosely related to the original filters.
 If one wishes to modify the spectrum, however, one must convert
the reflection coefficients into direct form and then factor the
polynomial. This can be done without problem with the expendature of
sufficient quantities of computer time, but the inverse process,
converting the direct form back into reflection coefficients, cannot be
done accurately. The only process for doing so is highly numerically
unstable (Markel and Gray), so that for higher orders, it simply cannot
be done in reasonable amounts of time. Thus, once factored, the
polynomial must stay factored for all time henceforth.
 For our own synthesis system, we currently use the latice form
because it uses directly the output of the analysis technique and
because interpolation can be used easily on the reflection coefficients.
 Filter coefficients are, of course, not the only things that
must be interpolated. The frequency must also be continuously
interpolated for the most smooth sounding results. This is where the
advantage of using table lookup for the excitation occurs. With table
lookup, one can continuously var% the rate at which the table is
scanned. If one uses interpolation on the table itself, the resulting
process can be made very smooth indeed. Again, the table must be
regenerated each time the frequency changes significantly, but this
seems to occur seldom enough to allow the usage of the FFT for
generating the excitation function.
AMPLITUDE CONTROL
 The amplitude of the synthetic signal should be controlled to
produce a loudness contour that corresponds as much as possible to the
original loudness. As mentioned by Moorer (1976), what would be ideal is
some kind of direct loudness normalization, using possibly a model of
human loudness perception (Zwicker and Scharf 1965). Unfortunately, this
computation is so unwieldy as to render it virtually useless at this
time, so some other methods must be chosen.
 Atal (Atal and Hanauer 1971) used a method of normalization of
energy such that the energy of the current frame (period) is scaled to
correpond exactly to the original energy. Although this sounds like the
right thing to do, it has several problems. First is just the way it is
calculated. The filter has presumably been run on the previous frame and
now has a nonzero "memory". That means that even with zero input this
frame, it will emit a certain response that will presumably die away. We
seek, then, to scale the excitation for this frame such that the
combination of the remaining response from the previous frame and the
response for this frame (starting with a fresh filter this frame) will
have the correct energy. Since the criterion is energy, a squared value,
this reduces to the solution of a quadratic equation for the gain
factor. The problem comes when the energy represented by the tail of the
filter response from the previous frame already exceeds the desired
energy of this frame. In this case, the solution of the quadratic is, of
course, complex. What this means is that the model being used is
imperfect. Either the filter or the excitation is not an accurate model
of the input signal. This is possible since the modeling process,
especially for the excitation, is not an exact procedure. There are even
instabilities that can result in the computation of the gain. For
instance, if the response from the last frame is large, but not quite as
large as the desired energy, then a very small value of gain will be
computed. That means that in the next frame, there will be very little
contribution from the previous frame and the gain factor will be quite
large. As the model deteriorates, this oscillation in the gain increases
until no solution is possible. The only hope is that this occurs
sufficiently rarely as to not be a detriment. Experience, however, seems
to indicate the contrary: that this failure in modeling is something
that happens even in quite normal speech and must be taken into
account.Beides all that, even if you do normalize the energy, the
perceived loudness will often be found to change noticably over the
course of the utterance. This is especially true during voiced
fricatives, although the theoretical explanation for this phenomenon is
not clear at this time.
 The method of amplitude control that we have chosen is
twofold: for crosssynthesis, we choose a twopass postnormalization
scheme that computes the energies of the original signal and the
synthesized signal. The synthesized signal is then multiplied by a
piecewiselinear function, the breakpoints of which are the gain factors
required to normalize the energy at the points where the energies were
computed.
 For resynthesis of the vocal sounds, we use an openloop method
of just driving the filter with an excitation that correponds in energy
to the energy of the error signal of the inverse filter. This is, of
course, only an approximation because the excitation never corresponds
to the actual error signal, but in practice it seems to produce the
smoothest most naturally varying sounds. Note also that this does not
guarantee any correspondance between the energies of the original and
the synthetic signals. With the autocorrelation method of linear
prediction, the error energy is easily obtained as an automatic result
of the filter computation. For other methods, it is generally necessary
to actually apply the filter to the original signal to obtain the error
energy. As with all other parameters, we interpolate the gain in a
continuous piecewiselinear manner throughout the synthesis.
WHERE TO FROM HERE?
 Problems remain in certain areas, such as the synthesis of nasal
consonants and the voiced/unvoiced/silence decision. With nasal
consonants, it is theorized that the presence of the nasal zero must be
simulated in the filter. This cannot be entirely true because some
nasals can be synthesized quite well and some cannot. Additional work
must be done to try to distinguish the features of the nasals that do
not adapt well to the linear prediction method and decide what is to be
done about them. Some amount of work has been done on the simultaneous
estimation of poles and zeros (Stieglitz 1977, Tribolet 1974) and we
will be very interested to examine the results in critical listening
tests. The voiced/unvoiced/silence decision may well require hand
correction for the forseeable future.
 These techniques have been embodied in a series of programs
that allow the composer to specify transformations on the timing, pitch,
and other parameters in terms of piecewiselinear funttions that can be
defined directly in terms of their breakpoints, graphically or
implicitly in terms of resulting contours of time, pitch, or whatever.
More work must be done in arranging these in a more convenient package
for smoothly carrying the system through from start to finish without
excessive juggling and hitormiss estimation.
 FIGURE CAPTIONS
 Figure 1  Latice form of inverse filter.
 Figure 2  Latice form of allpole filter. The filter is
unconditionally stable if all the coefficients are of magnitude less
than one.
 Figure 3  Factorization of allpole filter into second order
sections.
 Figure 4  Direct form for the realization of an allpole filter.
 Figure 5  Schema of the synthesis of speech using as excitation
either a pulse train or white noise.
 Figure 6  Diagram of cross synthesis. The source signal, X(n),
might be a musical instrument. Its spectrum is whitened by a loworder
optimum inverse filter, then filtered by a highorder allpole filter
representing the spectrum of another signal, Y(n), which is presumably a
speech signal of some kind
 REFERENCES
 ALLEN (J.B.), Short Term Spectral Analysis, Synthesis, and
Modification by Discrete Fourier Transform, IEEE Trans. on Acoustics,
Speech, and Signal Processing, vol ASSP25, · 3 June 1977, pp 235238
 ATAL, (B.S.), SCHROEDER, (M.R.), Adaptive Predictive Coding of
Speech Signals, Bell Syst. Tech. J., vol. 49, 1970, pp19731986.
 ATAL, (B.S.), HANAUER, (S.L.), Speech Analysis and Synthesis by
Linear Prediction of the Speech Wave, J. Acoust. Soc. Amer., vol. 50,
pp 637655, Feb. 1971.
 ATAL, (B.S.), RABINER, (L.K., A Pattern Recognition Approach to
VoicedUnvoicedSilence Classification with Applications to Speech
Recognition,
IEEE Trans. Acoust., Speech, and Signal Processing, vol. ASSP24,
pp2Ol211,
June 1976.
 BURG, (J.P.), Maximum Entropy Spectral Analysis, presented at
the 37th Annual Meeting Soc. Explor. Geophy., Oklahoma city, OK, 1967.
 DODGE, (C.), Synthetic Speech Music, Composer's Recordings,
Inc., New York, CRISD348, 1975 (disk).
 GOLD, (B.), RABINER, (L.R.), Parallel Processing Techniques for
Estimating Pitch Periods of Speech in the Time Domain, J. Acoust. Soc.
Amer., vol 46,
*2, August 1969, pp442448.
 ITAKURA and SAITO, Digital filtering techniques for speech
analysis and synthesis, presented at the 7th International Congreos on
Acoustics, Budapest, 1971, Paper 25CI.
 MAKHOUL, (John), Lattice Methods for Linear Prediction, IEEE
Trans. on Acoustics, Speech, and Signal Processing, vol ASSP25, *5,
October 1977, pp423428.
 MAKHOUL (J.), Linear Prediction: A Tutorial Review, Proceedings
of the IEEE, Vol. 63, April 1975, pp561580.
 MARKEL, (J.D.), GRAY, (A.H.), Linear Prediction of Speech,
SpringerVerlag, Berlin Heidelberg, 1976.
 McGONEGAL, (C.A.), RABINER (L.R.), ROSENBERG (A.E.), A
Subjective Evaluation of Pitch Detection Methods Using LPC Synthesized
Speech , IEEE Trans. on Acoustics, Speech, and Signal Processing, vol
ASSP25, *1 June 1977, pp221229
 MOORER, (J.A.), The Optimum Comb Method of Pitch PeriQd
Analysis of
Continuous Digitized Speech, IEEE Trans. Acoust., Speech, and Signal
Processing, vol. ASSP22, October 1974, pp330338.
 MOORER, (J.A.), The Synthesis of Complex Audio Spectra by Means
of Discrete Summation Formulas, J. Aud. Eng. Soc., Vol 24, *9, November
197 , pp717727.
 MOORER (J.A.), Signal Processing Aspects of Computer Music: A
Survey, Proc. of the IEEE, vol 65, *8~ August 1977, pp11081137.
 MORF, (M.), Fast Algorithms for Multivariable Systems, PhD
thesis, Dept. of Electrical Engineering, Stanford University, Stanford
California, 1974.
 MORF, (M.), VIEIRA, (A), LEE (D.T.), KAILATH, (T), Recursive
Multichannel
Maximum Entropy Method, Proc. 1977 Joint Automatic Control Conf.,
San Francisco, California, 1977.
 MORGAN, (D.R.), CRAIG (S.E.), RealTime Adaptive Linear
Prediction Using the Least Mean Square Gradient Algorithm, IEEE Trans.
on Acoustics, Speech, and Signal Processing, vol ASSP24, *6, December
1976, pp494507.
 NOLL (A.M.), Cepstrum Pitch Determination, J. Acoust. Soc.
Amer., vol 41, February 1967, pp293309.
 PETERSEN, (T.L.), Vocal Tract Modulation of Instrumental Sounds
by Digital Filtering. presented at the Music Computation Conf. II,
School of Music, Univ. Illinois, UrbanaChampaign, Nov. 79, 1975.
 PETERSEN, (T.L.), Dynamic Sound Processing, in Proc. 1976 ACM
Computer Science Conf. (Anaheim, California), February 1012, 1976.
 PETERSEN (T.L.), AnalysisSynthesis as a Tool for Creating New
Families of Sound, presented at the 54th Conv. Audio Eng. Soc. (Los
Angeles, California), May 47, 1976.
 PORTNOFF, (M.R.), Implementation of the Digital Phase Vocoder
Using the
Fast Fourier Transform, IEEE Trans. on Acoustics, Speech, and Signal
Processing, vol ASSP24, *3, June 1976, pp243248.
 RABINER (L.R.), CHENG, (M.J.), ROSENBERG, (A.E.), McGONEGAL,
(C.A.)1
A comparative Performance Study of Several Pitch Detection Algorithms,
IEEE Trans. on Acoustics,Speech, and Signal Processing, vol Assp24,
*5, October 1976, pp399418.
 RABINER (L.R.), SAMBUR, (M.R.), Application of an LPC Distance
Measure to the VoicedUnvoicedSilence Detection Problem, IEEE Trans. on
Acoustics Speech, and Signal Processing, vol ASSP25, *4, August 1977,
pp338343.
 RABINER, (L.R.), ATAL, (B.S.), SAMBUR, (M.R.), LPC Prediction
Error 
Analysis of Its Variation with the Position of the Analysis Frame, IEEE
Trans. on Acoustics, Speech, and Signal Processing, vol ASSP25, *5,
October 1977, pp434442.
 ROBINSON, (E.A.), Statistical Communication and Detection,
Hafner, New York, 1967.
 SONDHI, (M.M.), New Methods of Pitch Extraction, IEEE Trans.
Audio Electroacoust., vol AU16, June 1968, pp262266.
 STEIGLITZ, (K.), On the Simultaneous Estimation of Poles and
Zeros in Speech Analysis, IEEE Trans. Acoust., Speech and Signal
Processing, vbl. ASSP25, *3, June 1977, pp229234.
 TRIBOLET, (J.M.), Identification of Linear Discrete Systems
with
Applications to Speech Processing, MS Thesis, MIT Department of
Electrical Engineering, January 1974.
 ZWICKER, (E.), SCHARF, (B.), A Model of Loudness Summation,
Psychol. Rev., vol. 1, 1965, pp326.
 WINHAM, (G.), STEIGLITZ (K), Input Generators for Digital Sound
Synthesis, J. Acoust. Soc. Amer., vol 47, *2 (part 2), pp665666, 1970.
