US 5,448,010 AGrant
Digital Signal Processing Using Closed Waveguide Networks
Issue Date:1995-09-05
•17 Claims
•7 Drawing Sheets
Abstract
A tone generation system includes one or more digital waveguide networks coupled to one or more junctions, one of which receives a control signal for controlling tone generation. The control signal initiates and interacts with a wave signal propagating through the waveguide networks to form a tone signal. A non-linear junction may be employed which receives a signal from a waveguide, converts it in accordance with a non-linear function based upon the value of the control signal and provides it back to the waveguide. A tone signal whose pitch is determined by the wave transmission characteristics of the waveguide network is thereby produced.
Metadata
Assignee
- The Board of Trustees of the Leland Stanford Junior University
Inventor
- Julius O. Smith, III
Application Information
Application Number:US 0352675
Filing Date:1993-03-22
Priority Date:1986-05-02
Art Unit:217
Classifications
IPC:
G10H 112
Field of Search:
84622-625;629;630;633;648;661-665;675-677;699;700;707;736-741;DIG. 9;DIG. 10 ;DIG. 11;DIG. 26
Patent Drawings (7 sheets)
Description
BACKGROUND OF THE INVENTION
This invention relates to the field of digital signal processing and particularly to signal processing useful in digital music synthesis and other applications.
Digital music synthesis has attracted increased interest as data processors have undergone new developments which provide increased performance capabilities. Digital music synthesis has many applications such as the synthesis of stringed, reed and other instruments and such as the synthesis of reverberation.
In actual practice, it has been difficult to provide satisfactory models of music instruments, based upon quantitative physical models, which can be practically synthesized on a real-time basis using present-day computers and digital circuitry.
Most traditional musical instruments such as woodwinds and strings, have been simulated by additive synthesis which consists of summing together sinusoidal harmonics of appropriate amplitude, or equivalently by repeatedly reading from a table consisting of one period of a tone (scaled by an "amplitude function") to "play a note." Another method consists of digitally sampling a real musical sound, storing the samples in digital memory, and thereafter playing back the samples under digital control. FM synthesis as described, for example, in U.S. Pat. No. 4,018,121, has also been successful in synthesizing many musical sounds including brasses, woodwinds, bells, gongs, and some strings. A few instruments have been simulated by "subtractive synthesis" which shapes the spectrum of primitive input signals using digital filters.
All of the foregoing methods (with the occasional exception of subtractive synthesis) have the disadvantage of not being closely related to the underlying physics of sound production. Physically accurate simulations are expensive to compute when general finite-element modeling techniques are used.
In accordance with the above background, there is a need for techniques for synthesizing strings, winds, and other musical instruments including reverberators in a manner which is both physically meaningful and computationally efficient. There is a need for the achievement of natural and expressive computer-controlled performance in ways which are readily comprehensible and easy to use.
SUMMARY OF THE INVENTION
The present invention is a signal processor formed using digital waveguide networks. The digital waveguide networks have signal scattering junctions. A junction connects two waveguide sections together or terminates a waveguide. The junctions are constructed from conventional digital components such as multipliers, adders, and delay elements. The number of multiplies and additions determines the number of signal-scattering junctions that can be implemented in the waveguide network, and the number of delays determines the total delay which can be distributed among the waveguides interconnecting the junctions in the waveguide network. The signal processor of the present invention is typically used for synthesis of reed, string or other instruments.
The waveguides of the present invention include a first rail for conducting signals from stage to stage in one direction and a second rail for conducting signals from stage to stage in the opposite direction. The accumulated delay along the first rail is substantially equal to the accumulated delay along the second rail so that the waveguide is balanced. The first rail is connected to the second rail at junctions so that signals conducted by one rail are also conducted in part by the other rail.
Lossless waveguides used in the present invention are bi-directional delay lines which sometimes include embedded allpass filters. Losses are introduced as pure attenuation or lowpass filtering in one or both directions.
The signal processor in some applications includes a non-linear junction connected to provide an input signal to the first rail of the waveguide and to receive an output signal from the second rail of the waveguide. The non-linear junction in some embodiments receives a control variable for controlling the non-linear junction and the signals to and from the waveguide.
In one embodiment, a reed instrument is synthesized by a non-linear junction terminating a digital waveguide. A primary control variable, representing mouth pressure, is input to the non-linear junction (also controlled secondarily by embouchure variables). The junction simulates the reed while the digital waveguide simulates the bore of the reed instrument.
In another embodiment, a string instrument is synthesized. A primary control variable, representing the bow velocity, is input to the non-linear junction. The non-linear junction represents the bow-string interface (including secondary controls such as bow force, bow angle, bow position, and friction characteristics). In the stringed instrument embodiment, two digital lossless waveguides are connected to the non-linear junction. The first waveguide represents the long string portion (from the bow to the nut) and the other waveguide simulates the short string portion (from the bow to the bridge). A series of waveguides can also be used to implement the body of, for example, a violin, although in such a case there is normally no direct physical interpretation of the waveguide variables.
In particular embodiments, the reflection signal or signal coefficients introduced into the waveguides from the nonlinear junction are obtained from a table. In one embodiment, the nonlinearity to be introduced into the waveguides is f(x) where x is the table address and also the incoming signal sample in the waveguide (a travelling wave sample). In another embodiment, the values g(x)=f(x)/x are stored in the table and the table is addressed by x. Each value of g(x) addressed by x from the compressed table (where g(x) is called a coefficient) is then multiplied by x, x*g(x) which thereby produces the desired value of f(x).
In accordance with the above summary, the present invention captures the musically important qualities of natural instruments in digital music synthesis with digital processing techniques employing digital waveguides which are computationally efficient and therefore capable of inexpensive real-time operation.
The foregoing and other objects, features and advantages of the invention will be apparent from the following detailed description in conjunction with the drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 depicts a simple closed waveguide network.
FIG. 2 depicts a 3-port waveguide network.
FIG. 3 depicts a junction of two waveguides.
FIG. 4 depicts a cascade waveguide network in accordance with the present invention.
FIG. 5 depicts one embodiment of a cascade waveguide network section.
FIG. 6 depicts another embodiment of a cascade waveguide network section.
FIG. 7 depicts a third embodiment of a cascade waveguide network section.
FIG. 8 depicts a pipelined embodiment of a waveguide filter.
FIG. 9 depicts a travelling pressure wave at a general point within a waveguide section.
FIG. 10 depicts a normalized-waveguide digital filter.
FIG. 11 depicts a wave-normalized waveguide junction.
FIG. 12 depicts a transformer junction.
FIG. 13 depicts transformer-coupled waveguide junction.
FIG. 14 depicts a non-linear junction, controlled by a control variable, and connected through a plurality of ports to a plurality of waveguides.
FIG. 15 depicts a terminating non-linear junction controlled by a control variable and connected to a waveguide network.
FIG. 16 depicts further details of the non-linear junction of FIG. 9.
FIG. 17 depicts a block diagram representation of the waveguide of FIG. 9.
FIG. 18 depicts a non-linear junction connected to first and second waveguides.
FIG. 19 is a signal processor forming a music instrument using digital waveguides.
FIG. 20 is a graph of a waveform representing the data stored in the table of FIG. 16 for a reed instrument.
FIG. 21 is a graph of a waveform representing the data stored in the table of FIG. 16 for a string instrument.
DETAILED DESCRIPTION
Lossless Networks--FIG. 1
In FIG. 1 a network 10 is a closed interconnection of bi-directional signal paths 11. The signal paths 11 are called branches or waveguides, designated 11-1, 11-2, 11-3, 11-4, and 11-5 and the interconnection points are called nodes or junctions, designated 12-1, 12-2, 12-3, and 12-4. An example of a simple network is shown in FIG. 1 where each signal path is bi-directional, meaning that in each waveguide there is a signal propagating in one direction and an independent signal propagating in the other direction when a signal reaches a junction, one component is partially reflected back along the same waveguide, and other components are partially transmitted into the other waveguides connected to the junction. The relative strengths of the components of the transmitted or "scattered" signals at each junction are determined by the relative characteristic impedances of the waveguides at the junction. In FIG. 1, the waveguides 11 intersect at the junctions 12.
A lossless waveguide, such as each of the waveguides in FIG. 1, is defined specifically as a lossless bi-directional signal branch. In the simplest case, each branch or waveguide 11 in a waveguide network 10 is merely a bi-directional delay line. The only computations in the network take place at the branch intersection points (nodes or junctions). More generally, a lossless waveguide branch may contain a chain of cascaded allpass filters. For practical reverberator and other designs, losses are introduced in the form of factors less than 1 and/or low pass filters with a frequency response strictly bounded above by 1 in magnitude.
A closed lossless network preserves total stored signal energy. Energy is preserved if, at each time instant, the total energy stored in the network is the same as any other time instant. The total energy at any time instant is found by summing the instantaneous power throughout the network waveguides 11. Each signal sample within the network contributes to instantaneous power. The instantaneous power of a stored sample is the squared amplitude times a scale factor, g. If the signal is in units of "pressure", "force", or equivalent, then g=1/Z, where Z is the characteristic impedance of the waveguide 11 medium. If the signal sample instead represents a "flow" variable, such as volume-velocity, then g=Z. In either case, the stored energy is a weighted sum of squared values of all samples stored in the digital network 10.
N-Port Network--FIG. 2
In FIG. 2, an N-port network 14 is shown in which for N=3, three waveguides, called ports, leave the network with one port 15 designated for input and two ports 16-1 and 16-2 designated for output. Such a structure is suitable, for example, for providing stereo reverberation of a single channel of sound. Note, however, that really in FIG. 2 there are three inputs (15, 16-1, 16-2) and three outputs (15, 16-1, 16-2) because in an N-port, each waveguide connected to the network provides both an input and an output since each waveguide is bi-directional.
An N-port network 14 of FIG. 2 is lossless if at any time instant, the energy lost through the outputs, equals the total energy supplied through the inputs, plus the total stored energy. A lossless digital filter is obtained from a lossless N-port by using every port as both an input and an output. This filter is the general multi-input, multi-output allpass filter.
An N-port network 14 is linear if superposition holds. Superposition holds when the output in response to the sum of two input signals equals the sum of the outputs in response to each individual input signal. A network is linear if every N-port derived from it is linear. Only linear networks can be restricted to a large and well-understood class of energy conserving systems.
Lossless Scattering--FIG. 3
Consider a parallel junction of N lossless waveguides of characteristic impedance Z.sub.i (characteristic admittance .GAMMA..sub.i =1/Z.sub.i) as depicted in FIG. 3 for N=2.
If in FIG. 3 the incoming traveling pressure waves are denoted by P.sub.i.sup.+, where i=1, . . . , N, the outgoing pressure waves are given by Eq. (1) as follows:
where P.sub.j in Eq. (1) is the resultant junction pressure given as follows: ##EQU1##
For N=2,
P.sub.j =.alpha..sub.1 P.sub.1.sup.+ +.alpha..sub.2 P.sub.2.sup.+
.alpha..sub.1 =(2.GAMMA..sub.1)/(.GAMMA..sub.1 +.GAMMA..sub.2)
.alpha..sub.2 =2-.alpha..sub.1
Define the reflection coefficient by k=.alpha..sub.1 -1, then from Eq. 1, ##EQU2##
Thus, we have, for N=2,
which is the one-multiplier lattice filter section (minus its unit delay). More generally, an N-way intersection requires N multiplies and N-1 additions to obtain P.sub.j, and one addition for each outgoing wave, for a total of N multiplies and 2N-1 additions.
The series flow-junction is equivalent to the parallel pressure-junction. The series pressure-junction or the parallel flow-junction can be found by use of duality.
Cascade Waveguide Chains--FIG. 4
The basic waveguide chain 25 is shown in FIG. 4. Each junction 26-1, 26-2, . . . , 26-i, . . . , 26-M enclosing the symbol k.sub.i (t) denotes a scattering junction characterized by k.sub.i (t). In FIG. 4, the junction 26-i typically utilizes multipliers(M) 8 and adders(+) 7 to form the junction. In FIG. 4, the multipliers 8-1, 8-2, 8-3 and 8-4 multiply by the factors [1+k(i)], [-k.sub.i (t)], [1-k.sub.i (t)], and [k.sub.i (t)], respectively. An alternative junction implementation 26'-i of FIG. 13 requires only one multiply. The junction 26-2 in FIG. 4 corresponds, for example, to the junction 12 in FIG. 3. Similarly, the delays 27-1 and 27-2 in FIG. 4 correspond to the branches 15 and 16, respectively, in FIG. 3. The Kelly-Lochbaum junctions 26-i and one-multiply junction 26'-i (see FIG. 13) or any other type of lossless junction may be used for junction 26. In particular, the two-multiply lattice (not shown) and normalized ladder (FIG. 11) scattering junctions can be employed. The waveguide 25 employs delays 27 between each scattering junction 26 along both the top and bottom signal paths, unlike conventional ladder and lattice filters. Note that the junction 26-i of FIG. 4 employs four multipliers and two adds while junction 26'-i of FIG. 13 employs one multiply and three adds.
Waveguide Variations--FIGS. 4-14
Reduction of junction 26 to other forms is merely a matter of pushing delays 27 along the top rail around to the bottom rail, so that each bottom-rail delay becomes 2T seconds (Z.sup.-2T) instead of T seconds Z.sup.-T. Such an operation is possible because of the termination at the right by an infinite (or zero) characteristic impedance 6 in FIG. 4. In the time-varying case, pushing a delay through a multiply results in a corresponding time advance of the multiplier coefficient.
Imagine each delay element 27 in FIG. 4 being divided into halves, denoted by a delay of T/2 seconds. Then any waveguide can be built from sections such as shown in FIG. 5.
By a series of transformations, the two input-signal delays are pushed through the junction to the two output delays. A similar sequence of moves pushes the two output delays into the two input branches. Consequently, we can replace any waveguide section of the form shown in FIG. 5 by a section of the form shown in FIG. 6 or FIG. 7.
By alternately choosing the structure of FIGS. 6 and 7, the structure of FIG. 8 is obtained. This structure has some advantages worth considering: (1) it consolidates delays to length 2T as do conventional lattice/ladder structures, (2) it does not require a termination by an infinite characteristic impedance, allowing it to be extended to networks of arbitrary topology (e.g., multiport branching, intersection, and looping), and (3) there is no long delay-free signal path along the upper rail as in conventional structures--a pipeline segment is only two sections long. This structure, termed the "half-rate waveguide filter", appears to have better overall characteristics than any other digital filter structure for many applications. Advantage (2) makes it especially valuable for modeling physical systems.
Finally, successive substitutions of the section of FIG. 6 and reapplication of the delay consolidation transformation lead to the conventional ladder or lattice filter structure. The termination at the right by a total reflection (shown as 6 in FIG. 4) is required to obtain this structure. Consequently, conventional lattice filters cannot be extended on the right in a physically meaningful way. Also, creating network topologies more complex than a simple series (or acyclic tree) of waveguide sections is not immediately possible because of the delay-free path along the top rail. For example, the output of a conventional structure cannot be fed back to the input.
Energy and Power
The instantaneous power in a waveguide containing instantaneous pressure P and flow U is defined as the product of pressure and flow as follows:
where,
define the right-going and left-going power, respectively.
For the N-way waveguide junction, we have, using Kirchoff's node equations, Eq. (6) as follows: ##EQU3##
Thus, the N-way junction is lossless; no net power, active or reactive, flows into or away from the junction.
Quantization Effects
While the ideal waveguide junction is lossless, finite digital wordlength effects can make exactly lossless networks unrealizable. In fixed-point arithmetic, the product of two numbers requires more bits (in general) for exact representation than either of the multiplicands. If there is a feedback loop around a product, the number of bits needed to represent exactly a circulating signal grows without bound. Therefore, some round-off rule must be included in a finite-precision implementation. The guaranteed absence of limit cycles and overflow oscillations is tantamount to ensuring that all finite-wordlength effects result in power absorption at each junction, and never power creation. If magnitude truncation is used on all outgoing waves, then limit cycles and overflow oscillations are suppressed. Magnitude truncation results in greater losses than necessary to suppress quantization effects. More refined schemes are possible. In particular, by saving and accumulating the low-order half of each multiply at a junction, energy can be exactly preserved in spite of finite precision computations.
Signal Power in Time-Varying Waveguides
The convention is adopted that the time variation of the characteristic impedance does not alter the traveling pressure waves P.sub.i.sup..+-.. In this case, the power represented by a traveling pressure wave is modulated by the changing characteristic impedance as it propagates. The actual power becomes inversely proportional to characteristic impedance:
This power modulation causes no difficulties in the Lyapunov theory which proves absence of limit cycles and overflow oscillations because it occurs identically in both the finite-precision and infinite-precision filters. However, in some applications it may be desirable to compensate for the power modulation so that changes in the characteristic impedances of the waveguides do not affect the power of the signals propagating within.
Consider an arbitrary point in the i.sup.th waveguide at time t and distance x=c.tau. measured from the left boundary, as shown in FIG. 9. The right-going pressure is P.sub.i.sup.30 (x,t) and the left-going pressure is P.sub.i.sup.- (x,t). In the absence of scaling, the waveguide section behaves (according to our definition of the propagation medium properties) as a pressure delay line, and we have P.sub.i.sup.+ (x,t)=P.sub.i.sup.+ (0,t-.tau.) and P.sub.i.sup.- (x,t)=P.sub.i.sup.- (0,t+.tau.)=P.sub.i.sup.- (cT,t-T+.tau.). The left-going and right-going components of the signal power are [P.sub.i.sup.- (x,t)].sup.2 /Z.sub.i (t) and [P.sub.i.sup.+ (x,t)].sup.2 /Z.sub.i (t), respectively.
Below, three methods are discussed for making signal power invariant with respect to time-varying branch impedances.
Normalized Waveguides
Suppose the traveling waves are scaled as the characteristic impedance changes in order to hold signal power fixed. Any level can be chosen as a reference, but perhaps it is most natural to fix the power of each wave to that which it had upon entry to the section. In this case, it is quickly verified that the proper scaling is:
In practice, there is no need to perform the scaling until the signal actually reaches a junction. Thus, we implement
where
This normalization is depicted in FIG. 10. In FIG. 10, each of the multipliers 8 multiplies the signal by g.sub.i (t) as given by Eqs. (9). In the single-argument notation used earlier, Eqs. (9) become
This normalization strategy has the property that the time-varying waveguides (as well as the junctions) conserve signal power. If the scattering junctions are implemented with one-multiply structures, then the number of multiplies per section rises to three when power is normalized. There are three additions as in the unnormalized case. In some situations (such as in the two-stage structure) it may be acceptable to normalize at fewer points; the normalizing multiplies can be pushed through the scattering junctions and combined with other normalizing multiplies, much in the same way delays were pushed through the junctions to obtain standard ladder/lattice forms. In physical modeling applications, normalizations can be limited to opposite ends of a long cascade of sections with no interior output "taps."
To ensure passivity of a normalized-waveguide with finite-precision calculations, it Suffices to perform magnitude truncation after multiplication by g.sub.i (t). Alternatively, extended precision can be used within the scattering junction.
Normalized Waves
Another approach to normalization is to propagate rms-normalized waves in the waveguide. In this case, each delay-line contains
We now consider P.sup..+-. (instead of P.sup..+-.) to be invariant with respect to the characteristic impedance. In this case,
The scattering equations become
or, solving for P.sub.i.sup..+-.,
But,
whence
The final scattering equations for normalized waves are
where
can be viewed as the sine and cosine, respectively, of a single angle .theta..sub.i (t)=sin.sup.-1 [k.sub.i (t)] which characterizes the junction. FIG. 11 illustrates the Kelly-Lochbaum junction as it applies to normalized waves. In FIG. 11, the multipliers 8-1, 8-2, 8-3, and 8-4 multiply by the factors [1-k.sub.i .sup.2 (t)].sup.1/2, -k.sub.i (t), [1-k.sub.i.sup.2 (t)].sup.1/2, and k.sub.i (t), respectively. In FIG. 11, k.sub.i (t) cannot be factored out to obtain a one-multiply structure. The four-multiply structure of FIG. 11 is used in the normalized ladder filter (NLF).
Note that normalizing the outputs of the delay lines saves one multiply relative to the NLF which propagates normalized waves. However, there are other differences to consider. In the case of normalized waves, duals are easier, that is, changing the propagation variable from pressure to velocity or vice versa in the i.sup.th section requires no signal normalization, and the forward and reverse reflection coefficients are unchanged. Only sign-reversal is required for the reverse path. Also, in the case of normalized waves, the rms signal level is the same whether or not pressure or velocity is used. While appealing from a "balance of power" standpoint, normalizing all signals by their rms level can be a disadvantage. In the case of normalized delay-line outputs, dynamic range can be minimized by choosing the smaller of pressure and velocity as the variable of propagation.
Transformer-Coupled Waveguides
Still another approach to the normalization of time-varying waveguide filters is perhaps the most convenient of all. So far, the least expensive normalization technique is the normalized-waveguide structure, requiring only three multiplies per section rather than four in the normalized-wave case. Unfortunately, in the normalized-waveguide case, changing the characteristic impedance of section i results in a changing of the reflection coefficients in both adjacent scattering junctions. Of course, a single junction can be modulated in isolation by changing all downstream characteristic impedances by the same ratio. But this does not help if the filtering network is not a cascade chain or acrylic tree of waveguide sections. A more convenient local variation in characteristic impedance can be obtained using transformer coupling. A transformer joins two waveguide sections of differing characteristic impedance in such a way that signal power is preserved and no scattering occurs. It turns out that filter structures built using the transformer-coupled waveguide are equivalent to those using the normalized-wave junction described in the previous subsection, but one of the four multiplies can be traded for an addition.
From Ohm's Law and the power equation, an impedance discontinuity can be bridged with no power change and no scattering using the following relations:
Therefore, the junction equations for a transformer can be chosen as
where, from Eq. (14)
The choice of a negative square root corresponds to a gyrator. The gyrator is equivalent to a transformer in cascade with a dualizer. A dualizer is a direct implementation of Ohm's law (to within a scale factor) where the forward path is unchanged while the reverse path is negated. On one side of the dualizer there are pressure waves, and on the other side there are velocity waves. Ohm's law is a gyrator in cascade with a transformer whose scale factor equals the characteristic admittance.
The transformer-coupled junction is shown in FIG. 12. In FIG. 12, the multipliers 8-1 and 8-2 multiply by g.sub.i (t) and 1/g.sub.i (t) where g.sub.i (t) equals [Z.sub.i (t)/Z.sub.i-1 (t)].sup.1/2. A single junction can be modulated, even in arbitrary network topologies, by inserting a transformer immediately to the left (or right) of the junction. Conceptually, the characteristic impedance is not changed over the delay-line portion of the waveguide section; instead it is changed to the new time-varying value just before (or after) it meets the junction. When velocity is the wave variable, the coefficients g.sub.i (t) and g.sub.i.sup.-1 (t) in FIG. 12 are swapped (or inverted).
So, as in the normalized waveguide case, the two extra multipliers 8-1 and 8-2 of FIG. 12 provide two extra multiplies per section relating to the unnormalized (one-multiply) case, thereby achieving time-varying digital filters which do not modulate stored signal energy. Moreover, transformers enable the scattering junctions to be varied independently, without having to propagate time-varying impedance ratios throughout the waveguide network.
In FIG. 13, the one-multiply junction 26'-i includes three adders 7-1, 7-2, and 7-3, where adder 7-3 functions to subtract the second rail signal, P.sub.i.sup.- (t), from the first rail signal, [P.sub.i-1.sup.+ (t-T)][g.sub.i (t)]. Junction 26'-i also includes the multiplier 8 which multiplies the output from adder 7-3 by k.sub.i (t). FIG. 13 utilizes the junction of FIG. 12 in the form of multipliers 8-1 and 8-2 which multiply the first and second rail signals by g.sub.i (t) and 1/g.sub.i (t), respectively, where g.sub.i (t) equals [(1-k.sub.i (t))/(1+k.sub.i (t))].sup.1/2.
It is interesting to note that the transformer-coupled waveguide of FIG. 13 and the wave-normalized waveguide (shown in FIG. 11) are equivalent. One simple proof is to start with a transformer and a Kelly-Lochbaum junction, move the transformer scale factors inside the junction, combine terms, and arrive at FIG. 11. The practical importance of this equivalence is that the normalized ladder filter (NLF) can be implemented with only three multiplies and three additions instead of four multiplies and two additions.
The limit cycles and overflow oscillations are easily eliminated in a waveguide structure, which precisely simulates a sampled interconnection of ideal transmissions line sections. Furthermore, the waveguide can be transformed into all well-known ladder and lattice filter structures simply by pushing delays around to the bottom rail in the special case of a cascade, reflectively terminated waveguide network. Therefore, aside from specific round-off error and time skew in the signal and filter coefficients, the samples computed in the waveguide and the samples computed in other ladder/lattice filters are identical (between junctions).
The waveguide structure gives a precise implementation of physical wave phenomena in time-varying media. This property is valuable in its own right for simulation purposes. The present invention permits the delay or advance of time-varying coefficient streams in order to obtain physically correct time-varying waveguide (or acoustic tube) simulations using standard lattice/ladder structures. Also, the necessary time corrections for the traveling waves, needed to output a simulated pressure or velocity, are achieved.
The waveguide structures of the present invention are useful for two distinct applications, namely, tone synthesis (the creation of a musical tone signal) and reverberation (the imparting of reverberation effects to an already existing audio signal). The present invention is directed to use of waveguide structures for tone synthesis. Use of such structures for reverberation is described in detail in U.S. Pat. No. 4,984,276, the disclosure of which is incorporated herein by reference.
Waveguide Networks with Non-Linear Junction--FIG. 14
In FIG. 14, a plurality of waveguides 53 are interconnected by a non-linear junction 52. In the particular embodiment of FIG. 14, the junction 52 has three ports, one for each of the waveguide networks 53-1, 53-2, and 53-3. However, junction 52 can be an N-port junction interconnecting N waveguides or waveguide networks 53. The control variable register 51 provides one or more control variables as inputs to the junction 52. In FIG. 14 when only a single waveguide is utilized, the single waveguide becomes a special case, single-port embodiment of FIG. 14. Single port examples of the FIG. 14 structure are described hereinafter in connection with reed instruments such as clarinets or saxophones. Multi-port embodiments of the FIG. 14 structure are described hereinafter in connection with stringed instruments such as violins. A multi-port variation of the FIG. 14 structure is also described hereinafter in connection with a reverberator. Many other instruments not described in detail can also be simulated in accordance with the present invention. For example, flutes, organs, recorders, basoons, oboes, all brasses, and percussion instruments can be simulated by single or multi-port, linear or non-linear junctions in combination with one or more waveguides or waveguide networks.
Waveguide with Non-Linear Terminating Junction--FIG. 15
In FIG. 15, a block diagram representation of a waveguide 53 driven by a non-linear junction 52 is shown. The non-linear junction 52 provides the input on the first rail 54 to the waveguide 53 and receives the waveguide output from the second rail on lines 55. A control variable unit 51 provides a control variable to the non-linear junction 52. The FIG. 15 structure can be used as a musical instrument for simulating a reed instrument in which case the control variable unit 51 simulates mouth pressure, that is the pressure drop across a reed. The non-linear junction 52 simulates the reed and the waveguide 53 simulates the bore of the reed instrument.
Non-Linear Junction--FIG. 16
FIG. 16 depicts further details of a non-linear junction useful in connection with the FIG. 15 instrument for simulating a reed. The control register input on lines 56 is a control variable, such as mouth pressure. The control variable forms one input (negative) to a subtractor 57 which receives another input (negative) directly from the most significant bits of the waveguide second rail on lines 55. The subtractor 56 subtracts the waveguide output on lines 55 and the control variable on lines 56 to provide a 9-bit address on lines 69 to the coefficient store 70 and specifically the address register 58. The address in register 58 provides the address on lines 68 to a table 59 and to a multiplier 62. The table 59 is addressed by the address, x, from address register 58 to provide the data, g(x), in a data register 61. The contents, g(x), in the data register 61 are multiplied by the address, x, from address register 58 in multiplier 62 to provide an output, x*g(x), in the multiplier register 63 which is equal to f(x). The output from the multiplier register 63 is added in adder 64 to the control variable to provide the first rail input on lines 54 to the waveguide 53 of FIG. 15.
In FIG. 16, table 59 in one embodiment stores 512 bytes of data and provides an 8-bit output to the data register 61. The multiplier 62 provides a 16-bit output to the register 63. The high order 8 bits in register 63 are added in saturating adder 64 to the 8 bits from the variable register 51' to provide a 16-bit output on lines 54. Similarly, the high order 8-bits from the 16-bit lines 55 are subtracted in subtractor 57.
The contents of the table 59 in FIG. 16 represent compressed data. If the coefficients required are f(x) from the compressed table 70, only a fewer number of values, g(x), are stored in the table 59. The values stored in table 59 are f(x)/x which are equal to g(x). If x is a 16-bit binary number, and each value of x represents one 8-bit byte of data for f(x), table 59 is materially reduced in size to 512 bytes when addressed by the high-order 9 bits of x. The output is then expanded to a full 16 bits by multiplication in the multiplier 62.
Further compression is possible by interpolating values in the table 59. Many table interpolation techniques are well known. For example, linear interpolation could be used. Interpolation can also be used to compress a table of f(x) values directly, thus saving a multiply while increasing the needed table size, for a given level of relative error.
Other examples include a double look-up, address normalization, root-power factorization, address and value quantization, address mapping to histogram. Other compression techniques can be employed.
The manner in which the data values for a reed instrument are generated is set forth in APPENDIX A.
In FIG. 17, further details of a schematic representation of the waveguide 53 are shown. The waveguide 53 includes a first rail receiving the input on lines 54 and comprising a delay 65. A terminator 67 connects the delay 65 to the second rail delay 66 which in turn provides the second rail output on lines 55.
In an embodiment where the FIG. 16 signal processor of FIGS. 16 and 17 simulates a reed instrument, the terminator 67 is typically a single pole low-pass filter. Various details of a clarinet reed instrument in accordance with the signal processor of FIGS. 16 and 17 appear in APPENDIX B.
To simulate clarinet tone holes, a three-port scattering junction is introduced into the waveguide. Typically, the first three or four adjacent open tone holes participate in the termination of the bore.
In FIG. 17, the terminator 67 includes a multiplier 74, an inverting low-pass filter 72 and a DC blocking circuit 73. The multiplier 74 multiplies the signal on line 75 from the delay 65 by a loss factor g.sub.1 where g.sub.1 is typically 1-2.sup.-4 =0.9375 for a clarinet. The output from the multiplier 74 is designated y.sub.1 (n) where n is the sampled time index. The output from the low-pass filter 72 is designated y.sub.2 (n), and the output from the DC blocking unit 73 is designated y.sub.3 (n).
For a clarinet, the low-pass filter 72 has a transfer function H.sub.12 (Z) as follows:
Therefore the signal y.sub.2 (n) output from the low-pass filter 72 is given as follows:
In the above equations, g is a coefficient which is typically determined as equal to 1-2.sup.-k where k can be any selected value. For example, if k is 3, g is equal to 0.875 and g equal to 0.9 is a typical value. As another example, 1-2.sup.-3 +2.sup.-5 =0.90625.
In FIG. 17, the transfer function, H.sub.23 (Z), of the DC blocking circuit 73 is given as follows:
With such a transfer function, the output signal y.sub.3 (n) is given as follows:
In simulations, the value of r has been set to zero. In actual instruments, DC drift can cause unwanted numerical overflow which can be blocked by using the DC block unit 73. Furthermore, when using the compressed table 70 of FIG. 16, the error terms which are produced are relative and therefore are desirably DC centered. If a DC drift occurs, the drift has the effect of emphasizing unwanted error components. Relative signal error means that the ratio of the signal error to signal amplitude tends to remain constant. Therefore, small signal values tend to have small errors which do not significantly disturb the intended operation.
In FIG. 17, for a clarinet, the delays 65 and 66 are typically selected in the following manner. One half the desired pitch period less the delay of the low-pass filter 72, less the delay of the DC block in unit 73, less the delay encountered in the non-linear junction 52 of FIG. 16.
When a saxophone is the reed instrument to be simulated by the FIG. 16 and FIG. 17 devices, a number of changes are made. The non-linear junction of FIG. 16 remains the same as for a clarinet. However, the waveguide network 53 of FIG. 15 becomes a series of cascaded waveguide sections, for example, of the FIG. 4 type. Each waveguide section represents a portion of the bore of the saxophone. Since the bore of a saxophone has a linearly increasing diameter, each waveguide section simulates a cylindrical section of the saxophone bore, with the waveguide sections representing linearly increasing diameters.
For a saxophone and other instruments, it is Useful to have a non-linear bore simulation. Non-linearity results in excess absorption and pressure-dependent phase velocity. In order to achieve such non-linear simulation in accordance with the present invention, one method is to modify the delays in the waveguide structure of FIG. 8. In FIG. 8, each of the delays, Z.sup.-2T, includes two units of delay. In order to introduce a non-linearity, one of the two units of delay is replaced by an all-pass filter so that the delay D changes from Z.sup.-2T to the following:
With such a delay, the output signal, y.sub.2 (n) is given in terms of the input signal, y.sub.1 (n) as follows:
In the above equations, in order to introduce the non-linearity, the term h is calculated as a function of the instantaneous pressure in the waveguide, which is the sum of the travelling-wave components in the first rail and the second rail. For example, the first rail signal input to the delay, y.sub.1.sup.+ (n) is added to second rail signal y.sub.1.sup.- (n) and then utilized by table look up or otherwise to generate some function for representing h as follows:
The delay of the first-order all-pass as a function of h can be approximated by (1-h)/(1+h) at low frequencies relative to the sampling rate. Typically, h is between 1-.epsilon. and 0 for some small positive .epsilon. (the stability margin).
Using the principles described, simulation of a non-linear waveguide medium (such as air in a clarinet bore) is achieved. For clarinet and other instruments, the bore which is modeled by the waveguides of the present invention, includes tone holes that are blocked and unblocked to change the pitch of the tone being played. In order to create the equivalent of such tone holes in the instruments using waveguides in accordance with the present invention, a three-port junction can be inserted between cascaded waveguide sections. One port connects to one waveguide section, another port connects to another waveguide section, and the third port is unconnected and hence acts as a hole. The signal into the third port is represented as P.sub.3.sup.+ and this signal is equal to zero. The radiated signal from the third port, that is the radiated pressure, is denoted by P.sub.3.sup.-. The three-port structure for the tone hole simulator is essentially that of FIG. 14 without the waveguide 53-3 and without any control variable 51 input as indicated by junction 52 in FIG. 14. The junction 52 is placed as one of the junctions, such as junction 26-i in FIG. 4. With such a configuration, the junctions pressure, P.sub.J, is given as follows: ##EQU4## where,
.alpha..sub.i =2.GAMMA..sub.i /(.GAMMA..sub.1 +.GAMMA..sub.2 +.GAMMA..sub.3),
.GAMMA..sub.i =characteristic admittance in i.sup.th waveguide
P.sub.i.sup.- =P.sub.J -P.sub.i.sup.+
P.sub.J =.alpha..sub.1 P.sub.1.sup.+ +.alpha..sub.2 P.sub.2.sup.+ =.alpha..sub.1 P.sub.1.sup.+ +(2-.alpha..sub.1 -.alpha..sub.3)P.sub.2.sup.+
P.sub.1.sup.- =P.sub.J -P.sub.1.sup.+ =(.alpha..sub.1 -1)P.sub.1.sup.+ +.alpha..sub.2 P.sub.2.sup.+
P.sub.2.sup.- =P.sub.J -P.sub.2.sup.+ =.alpha..sub.1 P.sub.1.sup.+ +(.alpha..sub.2 -1)P.sub.2.sup.+
P.sub.3.sup.- =P.sub.J -P.sub.3.sup.+ =P.sub.J (tone hole output) ##EQU5##
Then, with P.sub..DELTA..sup.+ =P.sub.1.sup.+ -P.sub.2.sup.+, we obtain the one multiply tone-hole simulation:
In a smooth bore, .GAMMA..sub.1 =.GAMMA..sub.2 =.GAMMA. and .GAMMA..sub.3 =.beta..GAMMA., where .beta. is the cross-sectional area of the tone hole divided by the cross-sectional area of the bore. For a clarinet, .beta.=0.102 and for a saxophone, .beta.=0.436, typically. So we have: ##EQU6## Then,
There is now a single parameter ##EQU7## So, the tone hole simulation is given by
Summary: ##EQU8##
P.sub.J is radiated away spherically from the open hole with a (1/R) amplitude attenuation.
Reed Simulation
In FIG. 20, a graph is shown representing the data that is typically stored in the table 59 of FIG. 16 for a reed instrument. The output signal R.sup.- (n) on line 54 is as follows:
The control variable input on line 56 is P.sub.m (n)/2 and the input on line 68 to the table 59 is
where R.sup.+ (n) is the signal sample on line 55 of FIG. 16.
The table 59 is loaded with values which, when graphed, appear as in FIG. 20. The curve 92 in FIG. 20 has a maximum value of one and then trails off to a minimum value of zero. The maximum value of one occurs between (P.sub..DELTA.,min.sup.+)/2 and (P.sub..DELTA.,c.sup.+)/2. The value (P.sub..DELTA.,c.sup.+)/2 corresponds to the closure of the reed. From (P.sub..DELTA.,c.sup.+)/2 to (P.sub..DELTA.,max.sup.+)/2 the curve 92 decays gradually to zero. The equation for the curve 92 is given as follows,
where l=1, 2, 3, . . .
The output from the table 59 is the variable k as given in FIG. 20, that is,
Bowed-String Simulation
In FIG. 21, a graph is shown representing the data that is typically stored in the coefficient table 59 of the signal table 70 (see FIG. 16) of FIG. 18. The output signals V.sub.s,1.sup.- on line 54 and V.sub.s,r.sup.- on line 49 are as follows:
The control variable input on line 56 is bow velocity, V.sub.b, and the input on line 68 to the table 59 is
where V.sub.s,1.sup.+ is the signal sample on line 55 and V.sub.s,r.sup.+ is signal sample on line 50 of FIG. 18.
The table 59 is loaded with values which, when graphed, appear as in FIG. 21. The curve 93 in FIG. 24 has a maximum value of one and then trails off to a minimum value of zero to the left and right symmetrically. The maximum value of one occurs between -V.sub..DELTA.,c.sup.+ and +V.sub..DELTA.,c.sup.+. From (V.sub..DELTA.,c.sup.+) to (V.sub..DELTA.,max.sup.+) the curve 93 decays gradually to zero The equation for the curve 93 is given as follows,
where l=1, 2, 3, . . .
The output from the table 59 is the reflection coefficient k as given in FIG. 21, that is,
Compressed Table Variations
The compressed table 59 of FIG. 16 containing g(x)=f(x)/x is preferable in that quantization errors are relative. However, alternatives are possible. The entire table compressor 70 of FIG. 16 can be replaced with a simple table. In such an embodiment, the round off error is linear and not relative. For linear errors, the error-to-signal ratio tends not to be constant. Therefore, for small signal amplitudes, the error tends to be significant so that the error may interfere with the intended operation. In either the table compressor embodiment 70 of FIG. 16 or a simple table previously described, the tables can employ compression techniques such as linear, Lagrange and quadratic interpolation with satisfactory results. In a linear interpolation example, the curve 92 of FIG. 20 would be replaced by a series of straight line segments thereby reducing the amount of data required to be maintained in the table.
Also table 59, address register 58 and data register 61 of FIG. 16 each have inputs 94, 95 and 96 from processor 85 (FIG. 22).
The inputs from processor 85 function to control the data or the access of data from the table 59. Modifications to the data in the table can be employed, for example, for embouchure control for reed synthesis. Similarly, articulation control for bowed-string synthesis is possible. In one example, the address register 58 has high order address bits, bits 10 and 11, which are supplied by lines 95 from the processor. In this manner, the high order bits can be used to switch effectively to different subtables within the table 59. This switching among subtables is one form of table modification which can be used to achieve the embouchure and articulation modifications.
Non-Linear Junction with Plural Waveguides--FIG. 18
In FIG. 18, further details of another embodiment of a non-linear junction is shown connected between a first waveguide 76 and a second waveguide 77. The non-linear junction 78 receives an input from the control variable register 51' and provides inputs to the waveguide 76 on lines 54 and receives an output on lines 55. Also the non-linear junction 78 provides an output to the waveguide 77 on lines 49 and receives an input on lines 50.
In FIG. 18, the non-linear junction 78 includes an adder 57 receiving as one input the control variable from the control variable register 51' on lines 56. The other input to the subtractor 57 is from the difference register 79 which in turn receives an output from an adder 80. The adder 80 adds the inputs on lines 55 from the waveguide 76 and lines 50 from the waveguide 77.
The output from the subtractor 57 on lines 68 is input to the table compressor 70. The table compressor 70 of FIG. 12 is like the table compressor 70 of FIG. 10 and provides an output on lines 69. The output on lines 69 connects as one input to each of the adders 81 and 82. The adder 81 receives as the other input the input from lines 50 from the waveguide 77 to form the input on lines 54 to the first waveguide 76. The second adder 82 receives the table compressor signal on lines 69 and adds it to the input from the first waveguide 76 on lines 55. The output from adder 82 connects on lines 49 as the input to the second waveguide 77.
In FIG. 18, the waveguide 76 includes the top rail delay 65-1 and the bottom rail delay 66-1 and a terminator 67-1.
Similarly, the second waveguide 77 includes a top rail delay 65-2 and a bottom rail delay 66-2 and a terminator 67-2.
In the case of a violin in which the long string portion is approximately one foot and the short string portion is one-fifth of a foot, the waveguides of FIG. 18 are as follows. The terminator 67-1 is merely an inverter which changes the sign of the first rail value from delay 65-1 going into the delay 66-1. For example, the changing the sign is a 2's complement operation in digital arithmetic. Each of the delays 65-1 and 66-1 is the equivalent of about fifty samples in length for samples at a 50 KHz frequency. The terminator 67-2 in the waveguide 77 is typically ten samples of delay at the 50 KHz sampling rate. The terminator 67-2 can be a single pole low-pass filter. Alternatively, the terminator can be a filter having the empirically measured bridge reflectance cascaded with all sources of attenuation and dispersions for one round trip on the string. Various details of a violin appear in APPENDIX C.
Musical Instrument--FIG. 19
In FIG. 19, a typical musical instrument, that is signal processor, employing the waveguide units of the present invention is shown. In FIG. 19, a processor 85, such as a special purpose or general purpose computer, generates a digital signal representing the sound to be produced or a control variable for a synthesizer. Typically, the processor 85 provides an address for a random access memory such as memory 86. Memory 86 is addressed and periodically provides a digital output representing the sound or control variable to be generated. The digital sample from the memory 86, typically at a sampling rate T.sub.s (usually near 50 KHz), is connected to the waveguide unit 87. Waveguide unit 87 processes the digital signal in accordance with the present invention and provides an output to the digital-to-analog (D/A) converter 88. The converter 88 in turn provides an analog output signal through a filter 89 which connects to a speaker 90 and produces the desired sound.
When the signal processor of FIG. 19 is a reed instrument, the structure of FIGS. 15, 16 and 17 is typically employed for waveguide unit 87. In FIG. 15, the control variable 51 is derived from the processor 85 and the memory 86 of FIG. 19. The structure of FIGS. 15, 16 and 17 for a clarinet uses the FIG. 17 structure for waveguide 53 with a simple inverter (-1) for terminator 67. For a saxophone, the waveguide 53 is more complex, like FIG. 4.
When the signal processor of FIG. 19 is a bowed-string instrument, the waveguide unit 87 in FIG. 19 typically employs the structure of FIG. 18. The control variable input to register 51' of FIG. 18 comes from the memory 86 of FIG. 19. The output from the waveguide unit of FIG. 18 is derived from a number of different points, for example, from the terminals 54 and 55 for the waveguide 76 or from the terminals 49 and 50 from the waveguide 77 of FIG. 18. In one typical output operation, an adder 71 adds the signals appearing at terminals 49 and 50 to provide an input at terminal 20 to the D/A converter 88 of FIG. 19. The sum of the signals in adder 71 corresponds to the string velocity at the location of the bow on the instrument.
When reed and other instruments are employed, it has been found useful to introduce white noise summed with the control variable input to register 51' of FIG. 16. Additionally, the introduction of tremolo and other musical effects into the control variable enhances the quality of the sound produced.
Claims
What is claimed is:
1. An electronic musical instrument for synthesizing musical tones comprising: control signal generating means for generating a control signal for controlling the generation of a musical tone; a signal processing unit including wave transmission means and converting means for processing a musical tone signal, said wave transmission means for conducting said tone signal, including a first rail in one direction, a second rail in the opposite direction of said first direction, and a plurality of delays connected to at least either of said rails and selectively interconnectable, said converting means for determining a conversion characteristic on the basis of said control signal and for converting said signal transmitted from said wave transmission means in accordance with said conversion characteristic and transmitting the converted signal to the wave transmission means; controlling means for controlling the interconnection of said delays to be connected in response to a selected tone pitch of said musical tone, and output means for extracting a signal from the signal processing unit as a synthesized musical tone.
2. An electronic musical instrument as in claim 1 wherein the conversion characteristic is a non-linear characteristic.
3. A tone generation system comprising: control means for providing a control signal, the value of which is variable within a range including plural non-zero values in accordance with performance variation, for initiating and thereafter controlling generation of a tone; wave transmission means for initially creating a wave signal in response to the control signal and thereafter sustaining and varying the wave signals by interacting the control signal with the created wave signal, said wave transmission means including a signal path and means for delaying signals propagating therethrough; and means for providing at least one wave signal from the wave transmission means as an output tone signal, wherein the pitch of the output tone signal is a function of the amount of delay imparted by the wave transmission means.
4. An electronic musical instrument for synthesizing musical tones comprising: control signal generating means for generating a control signal for initiating and thereafter controlling generation of a musical tone; a signal processing unit, including a wave transmission section and converting means, for processing a musical tone signal, said wave transmission section for conducting said tone signal and including a first rail in one direction, a second rail in the opposite direction of said first direction, and at least one delay element contained in a rail, said converting means for determining a non-linear conversion characteristic on the basis of said control signal and for converting signals transmitted from said wave transmission section in accordance with said non-linear conversion characteristic and transmitting the converted signals to the wave transmission section, wherein the amount of said delay is controlled on the basis of a tone pitch of said musical tone; and output means for extracting a signal from the processing unit as a synthesized musical tone.
5. A real time tone generation system comprising: control means for providing a control signal, the value of which is variable within a range including plural non-zero values in accordance with performance variation, for initiating and thereafter controlling generation of a tone; wave generation means, having a closed signal propagating path including a delay therein, for creating and circulating a periodic wave by mathematically combining the control signal with a signal propagating in the signal propagating path, wherein said wave generation means is capable of creating a periodic wave in response to a control signal containing no periodic components; and output means for extracting a signal from the wave generation means as a musical tone signal.
6. A real time tone generation system comprising: control signal generating means for generating a variable control signal for initiating and thereafter controlling generation of a tone; wave transmission means, having an input terminal and output terminal and at least one signal path including a delay and coupling the input terminal to the output terminal, for propagating wave signals therein; converting means for receiving a first wave signal from the output terminal, converting it to a second wave signal in accordance with a non-linear conversion characteristic based on the value of the control signal, and supplying the second wave signal to the input terminal; and output means for outputting a wave signal from at least one of the wave transmission means and converting means as a tone signal, wherein transmission characteristics of the wave transmission means determine the pitch of the tone signal.
7. A tone generation system as in claim 6 wherein said wave transmission means and converting means forms a loop for propagating wave signals, wherein the total amount of delay of a wave signal propagating in the loop is controlled to provide a desired pitch.
8. A method of generating tones in real time, comprising the steps of: providing a signal propagation network having an input for receiving signals, an output for providing signals and a signal propagation path coupling the input and output and including a delay therein; introducing a control signal to the input of the network to initiate signal propagation in the network and varying the value of the control signal within a range including plural non-zero values in accordance with performance variation; repeatedly performing processing of the control signal and a signal from the output of the network and providing the result to the input to cause a periodic signal to be created which is propagated through the network; and extracting a signal from the network as a musical tone signal which has a pitch determined by signal transmission characteristics of the propagation network.
9. The method of claim 8 wherein the step of introducing a control signal includes the step of varying the value of the control signal over time.
10. The method of claim 8 wherein the step of performing processing includes the steps of receiving the control signal and the signal from the output of the network and determining the value of an interaction signal to be provided to the network in accordance with a predetermined relationship, the interaction signal forming one instantaneous part of the periodic signal.
11. The method of claim 10 wherein the step of determining includes the step of mathematically combining the control signal and the signal from the output of the network.
12. The method of claim 11 wherein the step of determining includes the steps of calculating the difference in value between the control signal and the signal from the output of the network and deriving a value of the interaction signal in accordance with the difference.
13. The method of claim 12 wherein the step of deriving includes the step of determining the value of a coefficient based on said difference and determining the value of the interaction signal as a function of the coefficient and the control signal.
14. The method of claim 13 wherein the reflection coefficient is determined by looking up a value stored in a table.
15. The method of claim 8 wherein the step of performing processing includes the step of converting the signal from the output of the network to a result signal in accordance with a conversion characteristic and further comprising the step of controlling the conversion characteristic in response to the control signal.
16. The method of claim 15 wherein the conversion characteristics include non-linear characteristics.
17. The method of claim 16 wherein the conversion characteristics are stored in a table.
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