Stiff Strings

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EURASIP Journal on Applied Signal Processing 2004:7, 964977c 2004 Hindawi Publishing Corporation

Physically Inspired Models for the Synthesis of Stiff Strings with Dispersive Waveguides

I. TestaDipartimento di Scienze Fisiche, Universit`a di Napoli Federico II, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy Email: [email protected]

G. EvangelistaDipartimento di Scienze Fisiche, Universit`a di Napoli Federico II, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy Email: [email protected]

S. CavaliereDipartimento di Scienze Fisiche, Universit a di Napoli Federico II, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy Email: [email protected]

Received 30 June 2003; Revised 17 November 2003

We review the derivation and design of digital waveguides from physical models of sti ff systems, useful for the synthesis of sounds from strings, rods, and similar objects. A transform method approach is proposed to solve the classic fourth-orderequations of sti ff systems in order to reduce it to two second-order equations. By introducing scattering boundary matrices,the eigenfrequencies are determined and their n2 dependency is discussed for the clamped, hinged, and intermediate cases.On the basis of the frequency-domain physical model, the numerical discretization is carried out, showing how the insertionof an all-pass delay line generalizes the Karplus-Strong algorithm for the synthesis of ideally exible vibrating strings. Know-ing the physical parameters, the synthesis can proceed using the generalized structure. Another point of view is o ff ered by Laguerre expansions and frequency warping, which are introduced in order to show that a sti ff system can be treated as anonsti ff one, provided that the solutions are warped. A method to compute the all-pass chain coe ffi cients and the optimumwarping curves from sound samples is discussed. Once the optimum warping characteristic is found, the length of the dis-persive delay line to be employed in the simulation is simply determined from the requirement of matching the desired fun-damental frequency. The regularization of the dispersion curves by means of optimum unwarping is experimentally evalu-ated.

Keywords and phrases: physical models, dispersive waveguides, frequency warping.

1. INTRODUCTIONInterest in digital audio synthesis techniques has been rein-forced by the possibility of transmitting signals to a wider au-dience within the structured audio paradigm, in which algo-rithms and restricted sets of data are exchanged [ 1]. Amongthese techniques, the physically inspired models play a privi-leged role since the data are directly related to physical quan-tities and can be easily and intuitively manipulated in orderto obtain realistic sounds in a exible framework. Applica-tions are, amongst the others, the simulation of a physicalsituation producing a class of sounds as, for example, a clos-ing door, a car crash, the hiss made bya crawling creature, thehuman-computer interaction and, of course, the simulationof musical instruments.

In the general physical models technique, continuous-time solutions of the equations describing the physical sys-

tem are sought. However, due to the complexity of the realphysical systemsfrom the classic design of musical in-struments to the molecular structure of extended objectssolutions of these equations cannot be generally found in ananalytic way and one should resort to numerical methods orapproximations. In many cases, the resulting approximationscheme only closely resembles the exact model. For this rea-son, one could better dene these methods as physically in-spired models, as rst proposed in [ 2], where the mathemat-ical equations or solutions of the physical problem serve asa solid base to inspire the actual synthesis scheme. One of the advantages of using physically inspired models for soundsynthesis is that they allow us to perform a selection of thephysical parameters actually inuencing the sound so that atrade-o ff between completeness and particular goals can beachieved.
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In the following, we will focus on stiff vibrating systems,including rods and sti ff strings as encountered in pianos.However, extensions to two- or three-dimensional systemscan be carried out with little e ff ort.

Vibrating physical systems have been extensively studiedover the last thirty years for their key role in many musi-

cal instruments. The wave equation can be directly approxi-mated by means of nite di ff erence equations [ 3, 4, 5, 6, 7],or by discretization of the wave functions as proposed by Jaff e and Smith [ 8, 9] who reinterpreted and generalized theKarplus-Strongalgorithm [ 10] in a wave propagation setting.The outcome of the approximation of the time domain solu-tion of the wave equation is the design of a digital waveg-uide simulating the string itself: the sound signal simulationis achieved by means of an appropriate excitation signal, suchas white noise. However, in order to achieve a more realisticand exible synthesis, the interaction of the excitation sys-tem with the vibrating element is, in turn, physically mod-eled. Digital waveguide methods for the simulation of physi-

cal models have been widely used [ 11, 12, 13, 14, 15, 16]. Oneof the reasons for their success is that they are appropriate forreal-time synthesis [ 17, 18, 19, 20]. This result allowed us tochange our approach to model musical instruments based onvibrating strings: waveguides can be designed for modelingthe core of the instruments, that is, the vibrating string, butthey are also suitable for the integration of interaction mod-els, for example, for the excitation due to a hammer [ 21] orto a bow [9], the radiation of sound due to the body of theinstrument [ 22, 23, 24, 25], and of diff erent side-e ff ects inplucked strings [ 26]. It must be pointed out that the interac-tions being highly nonlinear, their modeling and the deter-mination of the range of stability is not an easy task.

In this paper, we will review the design of a digital waveg-uide simulating a vibrating sti ff system, focusing on stiff strings and treating bars as a limit case where the tensionin negligible. The purpose is to derive a general framework inspiring the determination of a discrete numerical model.A frequency domain approach has been privileged, whichallows us to separate the fourth-order di ff erential equationof stiff systems into two second-order equations, as shownin Section 2. This approach is also useful for the simula-tion of two-dimensional (2D) systems such as thin plates.By enforcing proper boundary conditions, we obtain theeigenfrequencies and the eigenfunctions of the system asfound, for the case of strings, in the classic works by Fletcher[27, 28]. Once the exact solutions are completely charac-terized, their numerical approximation is discussed [ 29, 30]together with their justication based on physical reason-ing. The discretization of the continuous-time domain so-lutions is carried out in Section 3, which naturally leads todispersive waveguides based on a long chain of all-pass l-ters. From a diff erent point of view, the derived structure canbe described in terms of Laguerre expansions and frequency warping [ 31]. In this framework, a sti ff system can be shownto be equivalent to a nonsti ff (Karplus-Strong like) system,whose solutions are frequency warped, provided that the ini-tial and the possibly moving boundary conditions are prop-erly unwarped [ 32, 33]. As a side eff ect, this property can be

exploited in order to perform an analysis of piano soundsby means of pitch-synchronous frequency warped waveletsin which the excitation can be separated from the resonantsound components [ 34].

The models presented in this paper provide at least twoentry points for the synthesis. If the physical parameters and

boundary conditions are completely known, or if it is de-sired to specify them to model arbitrary strings or rods, thenthe eigenfunctions, hence the dispersion curve, can be deter-mined. The problem is then reconducted to that of ndingthe best approximation of the continuous-time dispersioncurve with the phase response of a suitable all-pass chain us-ing the methods illustrated in Section 3. Another entry pointis off ered if sound samples of an instrument are available.In this case, the parameters of the synthesis model can bedetermined by nding the warping curve that best ts thedata given by the frequencies of the partials, together withthe length of the dispersive delay line. This is achieved by means of a regularization method of the experimental dis-

persion data, as reported in Section 4.The physical entry point is to be preferred in those sit-uations where sound samples are not available, for example,when we aremodelinga nonexisting instrumentby extensionof the physical model, such as a piano with unusual speak-ing length. The other entry level is best for approximatingreal instrument sounds. However, in this case, the synthesisis limited to existing sources, although some variations canbe obtained in terms of the warping parameters, which arerelated to, but do not directly represent, physical factors.

2. PHYSICAL STIFF SYSTEMS

In this section, we present a brief overview of the stiff string and rod equations of motion and of their solution.The purpose is twofold. On the one hand, these equationsgive the necessary background to the physical modeling of stiff strings. On the other hand, we show that their fre-quencydomain solution ultimately provides the link betweencontinuous-time and discrete-time models, useful for thederivation of the digital waveguide and suitable for their sim-ulation. This link naturally leads to Laguerre expansions forthe solution and to frequency warping equivalences. Further-more, enforcing proper boundary conditions determines theeigenfrequencies and eigenfunctions of the system, useful fortting experimentally measured resonant modes to the onesobtained by simulation. This t allows us to determine theparameters of the waveguide through optimization.

2.1. Stiff string and bar equationThe equation of motion for the sti ff string can be determinedby studying the equilibrium of a thin plate [ 35, 36]. One ob-tains the following 4th-order di ff erential equation for the de-formation of the string y (x , t ):

4 y (x , t )

x 4 +

2 y (x , t )x 2 =

1c2

2 y (x , t )t 2

,

= EI T

, c= T S

,(1)


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966 EURASIP Journal on Applied Signal Processing

featuring the Young modulus of the material E, the inertiamoment I with respect to the transversal axis of the cross-section of the string (for a circular section of radius r , I =r 4 / 4 as in [36]), the tension of the string T , and the massper unit length S. Note that for 0, (1) becomes the well-known equation of thevibrating string [ 35]. Otherwise, if the

applied tension T is negligible, we obtain

4 y (x , t )

x 4 = 2 y (x , t )

t 2 , =

EI S

, (2)

which is the equation for the transversal vibrations of rods.Solutionsof ( 1)and (2) are best found in termsof the Fouriertransform of y (x , t ) with respect to time:

Y (x , ) = +

y (x , t )exp(it )dt , (3)

where is the angular velocity related to frequency f by therelationship f =2.

By taking the Fourier transform of both members of ( 1)and ( 2), we obtain

4Y (x , )

x 4 2Y (x , )

x 2 2

c2 Y (x , ) =0 (4)

for the stiff string and

4Y (x , )

x 4 2Y (x , ) =0 (5)

for the rod.The second-order 2 /x 2 spatial diff erential operator isdened as a repeated application of the L2 space extension of

the i(/x ) operator [ 37]. To thepurpose, we seek solutionswhose spatial and frequency dependency can be factored, ac-cording to the separation of variables method, as follows:

Y (x , ) =W () X (x ). (6)Substituting ( 6) in (4) and (5) results in the elimination of the W () term, obtaining ordinary di ff erential equations,whose characteristic equations, respectively, are

4 2 2

c2 =0 (stiff string), 4 2 =0 (rod) .

(7)

The elementary solutions for the spatial part X (x ) have theform X (x ) = C exp( x ). It is important to note that in bothcases, the characteristics equations have the following form:

2 21 2 22 =0, (8)where 1 and 2 are, in general, complex numbers that de-pend on . Equation (8) allows us to factor both equationsin (4) and (5) as follows:

2

x 2 21

2

x 2 22 Y (x , ) =0. (9)

The operator 2 /x 2 is selfadjoint with respect to the L2scalar product [ 37]. Therefore, ( 9) can be separated into thefollowing two independent equations:

2

x 2 21 Y 1(x , ) =0,

2

x 2 22 Y 2(x , ) =0,

(10)

where

Y (x , ) =Y 1(x , ) + Y 2(x , ). (11)As we will see, (10) justies the use, with proper modica-tions, of a second-order generalized waveguide based on pro-gressive and regressive waves for the numerical simulation of stiff systems.

2.2. General solution of the stiff stringand bar equations

In this section, we will provide the general solution of ( 8).The particular eigenfunctions and eigenfrequencies of rodsand stiff strings are determined by proper boundary condi-tions and are treated in Section 2.3.From( 7), itcan beshownthat

1 = 1 + 42/c2 1

2

2 = 1 + 42/c2 + 1

2

(stiff string),

1 = 2 =

(rod) .

(12)

Note that in both cases, the eigenvalues 1 are complex num-bers, while 2 are real numbers. It is also worth noting that

21 + 22 = 1

(stiff string),

21 + 22 =0 (rod),(13)

where 1 corresponds to the positive choice of the sign infront of the square root in ( 12) and 2 = | 2 |. As expected, if we let T 0, then both sets of eigenvalues of the stiff stringtend to those found for the rod. Using the equations in ( 12),we then have for both strings and rods

Y 1(x , ) =c+1 exp 1x + c1 exp 1x ,Y 2(x , ) =c+2 exp 2x + c2 exp 2x ,

(14)

where c1 , c2 are, in general, functions of . Note thatY 1(x , ) is an oscillating term, while, since 2 is real, Y 2(x , )is nonoscillating. For nite-length strings, both positive andnegative real exponentials are to be retained.


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From ( 12), we see that the primary eff ect of stiff ness isthedependency on frequencyof the argument (from now on, phase) of the solutions of ( 4) and (5). Therefore, the propa-gation of the wave from one section of the string located at x to the adjacent section located at x + x is obtained by multi-plication of a frequency dependent factor exp( 1 x ). Conse-

quently, the group velocity u, dened as u (d 1 /d)1

, alsodepends on frequency. This results in a dispersion of thewavepacket, characterized by the function 1(), whose modulusis plotted in Figure 1 for the case of a brass string using thefollowing values of the physical parameters r , T , , and E:

r =1mm,T =9 107 dyne, =8.44g cm3,E =9 1011 dyne cm

2.

(15)

Clearly, the previous example is a very crude approximation

of a physical piano string (e.g., real-life piano strings in thelow register are built out of more than one material and acopper or brass wire is wrapped around a steel core). For thesake of completeness, we give the explicit expression of |u| inboth the cases we are studying. We have

|u| =2c c2 + 42

2c2 2c c2 + 42(stiff string),

|u| =2 (rod).(16)

If T 0, the two group velocities are equal. Moreover, if inthe rst line in ( 16), we let 0, then u c, which is thelimit case of the ideally exible vibrating string. These factsfurther justify the use of a dispersive waveguide in the nu-merical simulation. With respect to this point, a remark is inorder: the dispersion introduced by sti ff ness can be treated asa limiting nonphysical consequence of the Euler-Bernoullibeam equation:

d 2

dx 2 EI d 2 y dx 2 = p, (17)

where p is the distributed load acting on the beam. It is non-physical in the sense that u as . However, in thediscrete-time domain, this nonphysical situation is avoidedif we suppose all the signals be bandlimited.

2.3. Complete characterization of stiff stringand rod solution

Boundary conditions for real piano strings lie in between theconditions of clamped extrema:

Y L2

, =Y L2

, =0,Y (x , )

x L/ 2 =Y (x , )

x L/ 2 =0,(18)

0 0.5 1 1.5 2 2.5104Frequency (Hz)

0

0.5

1

1.5

2

2.5

3

3.5

1

( c m

1 )

Figure 1: Plot of the phase module of the sti ff model equation so-lution for =/ 4 cm2 and c2104 cm s1.

and of hinged extrema [ 5, 16, 31, 35, 36]:

Y L2

, =Y L2

, =0,2Y (x , )

x 2 L/ 2 =2Y (x , )

x 2 L/ 2 =0.(19)

Before determining the conditions for the eigenfrequenciesof the considered sti ff systems, we nd a more compact way of writing (18) and ( 19). Starting from the factorized form of the stiff systems equation (see ( 10)), and using the symbols

introduced in Section 2.2, we haveY 1(x , ) = +1 (x , ) + 1 (x , ),Y 2(x , ) = +2 (x , ) + 2 (x , ),

(20)

where we let

1 (x , ) =c1 exp 1 x , 2 (x , ) =c2 exp 2 x .

(21)

Conditions ( 18) can then be rewritten as follows:

Y 1 L2

, = Y 2 L2

, ,

Y 1L2

, = Y 2L2

, ,

Y 1(x , )x L/ 2 =

Y 2(x , )x L/ 2

,

Y 1(x , )x L/ 2 =

Y 2(x , )x L/ 2

.

(22)

At the terminations of the string or of the rod, we have

+1 + 1 = +2 + 2 , +1 +1 + 1 1 = +2 +2 + 2 2 ,

(23)


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which can be rewritten in matrix form:

1 1 +1 +2

+1 +2 =

1 1 1 2

1 2 .

(24)

By left-multiplying both members of ( 24) for the inverse of the 1 1

+

1 +

2matrix, we have

+1 +2 =Sc

1 2

, (25)

where we let

Sc

+2 + +1 +2 +1

2 +2

+2 +12

+1

+2 +1 +2 + +1 +2 +1

. (26)

The matrix Sc relates the incident wave with the reectedwave at the boundaries. Independently of the roots i, it hasthe following properties:

Sc = 1,S2c =

1 00 1 .

(27)

In the case of a hinged stiff system (see (19)) at both ends, wehave

+1 + 1 = +2 + 2 , +1

2 +1 + 12 1 = +2

2 +2 + 22 2

(28)

which, in matrix form, becomes 1 1 +1

2

+2

2 +1 +2 =

1 1 1

2

22

1 2 .

(29)

By taking the inverse of matrix 1 1( +1 )2 ( +2 )2 , we obtain

+1 +2 =Sh

1 2

, (30)

where

Sh = 1 00 1 . (31)

The Sh matrix for the hinged sti ff system is independent of roots i. The matrices Sh and Sc are related in the followingway:

Sh = Sc ,S2h =S2c.

(32)

In conclusion, the boundary conditions for sti ff systemscan be expressed in terms of matrices that can be used inthe numerical simulation of sti ff systems. Moreover, since thereal-life boundary conditions for sti ff strings in piano lie in

between the conditions given in ( 18) and (19), we can com-bine the two matrices Sc and Sh in order to enforce moregeneral conditions, as illustrated in Section 3. In the follow-ing, we will solve (4) and (5) applying separately these sets of boundary conditions.

2.3.1. The clamped stiff string and rod In order to characterize the eigenfunctions in the case of con-ditions ( 18), in (12) we let

1 =i 1 (33)for both the sti ff string and the rod solution. By denition, 1 is a real number. Moreover, for the rod, we have 1 = 2.With this position, it can be shown that conditions ( 18) forthe stiff string lead to the equations [ 35, 38]

tan 1L2

tanh 2L2

tanh 2L2 tan 1

L2

1

2 =

0

0, (34)

while, for the rod, we have

cos 1L cosh 2L =1. (35)Equations ( 34) and ( 35) can be solved numerically. In partic-ular, taking into account the second line in ( 12), solutions of (35) are [35]

n = 2

43.0112, 52, 72, . . . , (2n + 1)2 2,

=

4 L .

(36)

A similar trend can be obtained for the sti ff string. In view of their historical and practical relevance, we here report thenumerical approximation for the allowed eigenfrequencies of the stiff string given by Fletcher [27]:

n n cL

1 + n2 22 1 + 2 + 42 ,

= L .

(37)

If we expand the above expression in a series of powers of truncated to second order, we have the following approxi-mate formula valid for small values of sti ff ness:

n n cL

1 + 2 + 1 + 18

n2 2 (2)2 . (38)

The last approximation does not apply to bars. For =0, wehave =0 and the eigenfrequencies tend to the well-knownformula for the vibrating string [ 35]:n =n1. (39)

Typical curves of the relative spacing n n /1, where n n+1 n, of eigenfrequencies for the sti ff string are


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r =3 mm

r =1 mm

0 10 20 30 40

Partial number

0

1

2

3

4

5

6

7

8

9

R e l a t i v e s p a c i n g

Figure 2: Typical eigenfrequencies relative spacing curves of theclamped stiff string for diff erent values of the radius r of the sec-tion S.

shown in Figure 2 with variable r , where values of the otherphysical parameters are the same as in ( 15).

Due to the dependency on the frequency of the phase of the solution, the eigenfrequencies of the sti ff string are notequally spaced. For a small radius r , hence for low degreeof the stiff ness of the string (see (1)), the relative spacing isalmost constant for all the considered order of eigenfrequen-cies. However, for higher stiff ness, the spacing of the eigen-frequencies increases, in rst approximation, as a linear func-tion of the order of the eigenfrequency. The above results are

summarized by the typical warping curves of the system,shown in Figure 3, in which the quantity n n, which rep-resents the deviation from the linearity, is plotted in terms of spacing n between consecutive eigenfrequencies.

In the stiff string case, we have two sets of eigenfunctions,one having even parity and the other one having odd parity,whose analytical expressions are respectively given by [38]

Y (x , )=C ()cos 1L2

cos 1x cos 1(L/ 2)

cosh 2x cosh 2(L/ 2)

,

Y (x , )=C ()sin 1L2

sin 1x sin 1(L/ 2)

sinh 2x sinh 2(L/ 2)

,

(40)

where C () is a constant that can be calculated imposing theinitial conditions.

2.3.2. Hinged stiff string and rod

Conditions ( 19) lead to the following sets of equations forthe stiff string:

sin 1L sinh 2L =0, 1

2 + 22 =0,(41)

r =3 mm

r =1 mm

0 500 1000 1500 2000

Frequency (Hz)

0

500

1000

1500

2000

2500

3000

3500

D e v i a t i o n f r o m l i n e a r i t y ( H z )

Figure 3: Typical warping curves of the clamped sti ff string for dif-ferent values of the radius r of the section S.

and for the rod:

sin 1L sinh 2L =0. (42)The second line in ( 41) has no solutions since both 1

2 and 22 are real functions. It follows that hinged sti ff systems areonly described by ( 42). In this equation, sinh( 2L) = 0 hasno solution, hence the eigenfrequencies are determined by the condition

1 =

n

L . (43)

Using the parameters and respectively dened in ( 36)and ( 37), the eigenfrequencies for the hinged sti ff string areexactly expressed as follows:

n = n cL n

2 22 + 1 , (44)

while for the rod, we have

n =n2 2 2. (45)As the tension T 0, (44) tends to ( 45). Figure 4 showsthe relative spacing of the eigenfrequencies in the case of thehinged stiff string.

Relative eigenfrequencies spacing curves are very similarto the ones of the clamped string and so are the warpingcurves of the system, as shown in Figure 5.

Using (45), we can give an analytic expression for the rel-ative spacing of the eigenfrequencies of the hinged rod. Wehave

2 2(2n + 1). (46)

Equation ( 43) leads to the following set of odd and even


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r =3 mm

r =1 mm

0 10 20 30 40

Partial number

0

2

4

6

8

10

R e l a t i v e s p a c i n g

Figure 4: Typical eigenfrequencies relative spacing curves of thehinged stiff string for diff erent values of the radius r of the sectionS.

r =3 mm

r =1 mm

0 500 1000 1500 2000

Frequency (Hz)

0

500

1000

1500

2000

2500

3000

3500

4000


Figure 5: Typical warping curves of the hinged sti ff string for dif-ferent values of the radius r of the section S.

eigenfunctions for the sti ff string [38]:

Y n(x , ) =2D()sin2n

L x ,

Y n(x , ) =2D()cos(2n + 1)

L x ,

(47)

where D() must be determined by enforcing the initial con-ditions. It is worth noting that both functions in ( 47) are in-dependent of the sti ff ness parameter . In Section 3, we willuse the obtained results in order to implement the dispersivewaveguides digitally simulating the solutions of ( 4) and ( 5).

Finally, we need to stress the fact that the eigenfrequen-cies of the hinged stiff string are similar to the ones for theclamped case except for the factor (1 + 2 + 42). Therefore,for small values of stiff ness, they do not di ff er too much. This

X (z ) Y (z )2P delays

z 2P

G(z )(low-pass)

Figure 6: Basic Karplus-Strong delays cascade.

can also be seen from the similarity of the warping curves ob-tained with the two types of boundary conditions.

Taking into account the fact that real-piano stringsboundary conditions lie in between these two cases, we canconclude that the eigenfrequencies of real-piano strings canbe calculated by means of theapproximated formula[ 27, 28]:

n An Bn2 + 1, (48)where Aand B canbe obtained from measurements. Approx-imation ( 48) is useful in order to match measured vibratingmodes against the model eigenfrequencies.

3. NUMERICAL APPROXIMATIONS OF STIFF SYSTEMS

Most of the problems encountered when dealing with thecontinuous-time equation of the sti ff string consist in de-termining the general solution and in relating the initialand boundary conditions to the integrating constants of theequation. In this section, we will show that we can use a sim-

ilar technique also in discrete-time, which yields a numericaltransform method for the computation of the solution.

In Section 2, we noted that ( 1) becomes the equation of vibrating string in the case of negligible stiff ness coefficient .It is well known that the technique known as Karplus-Strongalgorithm implements the discrete-time domain solution of the vibrating string equation [ 8], allowing us to reach goodquality acoustic results. The block diagram of the adoptedloop circuit is shown in Figure 6.

The transfer function of the digital loop chain can bewritten as follows:

H (z ) = 1

1 z 2P

G(z ), (49)

where the loop lter G(z ) takes into account losses due tononrigid terminations and to internal friction, and P is thenumber of sections in which the string is subdivided, as ob-tained from time and space sampling. Loop lters designcan be based on measured partial amplitude and frequency trajectories [ 18], or on linear predictive coding (LPC)-typemethods [ 9]. The lter G(z ) can be modelled as IIR or FIR and it must be estimated from samples of the sound or froma model of the string losses, where, for stability, we need

|G(e j)| < 1. Clearly, in the IIR case or in the nonlinear phaseFIR case, the phase response of the loop lter introduces a


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u =0.9

u = 0.90 1 2 3

Discrete frequency (Hz)

0

0.5

1

1.5

2

2.5

3

D i s c r e t e f r e q u e n c y ( H z )

Figure 7: First-order all-pass phase plotted for various values of u.

limited amount of dispersion. Additional phase terms in theform of all-pass lters can be added in order to tune thestring model to the required pitch [ 13] and contribute to fur-ther dispersion.

Since the group velocity for a traveling wave for a sti ff sys-tem depends on frequency (see ( 16)), it is natural to substi-tute, in discrete time, the cascade of unit delays with a chainof circuital elements whose phase responses do depend onfrequency. One can show that the only choice that leads torational transfer functions is given by a chain of rst-orderall-pass lters [39, 40]. More complex physical systems, forexample, as in the simulation of a monaural room, call forsubstituting the delays chain with a more general lter as il-lustrated in [ 41]:

A(z , u) = z 1 u1 uz 1

(50)

whose phase characteristic is

( )= + 2 arctan usin( )1 u cos( )

. (51)

The phase characteristics in ( 51) are plotted in Figure 7 forvarious values of u.

A comparison between the curve in Figure 1 and the onesin Figure 7 gives more elements of plausibility for theapprox-imation of the solution phase of the sti ff model equations,given in (12), with the all-pass lter phase ( 51). Adopting asimilar circuital scheme as in the Karplus-String algorithm[10] in which the unit delays are replaced by rst-order all-pass lters, the approximation is given by

1 f s P L

( ), (52)

X (z ) Y (z )2P all-pass cascade

A(z )2P

G(z )(low-pass)

Figure 8: Dispersive waveguide used to simulate dispersive systems.

where f s is the sampling frequency. Note that, by denition,both members of ( 52) are real numbers. Therefore, in the z -domain, a nonsti ff system can be mapped into a sti ff systemby means of the frequency warping map

z 1 A(z ). (53)The resultingcircuit is shown in Figure 8. Note, that the feed-back all-pass chain results in delay-free loops. Computation-ally, these loops can be resolved by the methods illustrated in[34, 42, 43]. Moreover, the phase response of the loop lterG(z ) contributes to the dispersion and it must be taken intoaccount in the global model.

The circuit in Figure 8 can be optimized in order to takeinto account the losses and the coupling amongst strings(e.g., as in piano). In the framework of this paper, we con-ned our interest to the design of the sti ff system lter. For areview of the design of lossy lters and coupling models, see[17].

3.1. Stiff system lter parameters determination

Within the framework of the approximation ( 52) in the caseof dispersive waveguide, the integer parameter P can be ob-tained by constraining the two functions to attain the samevalues at the extrema of the bandwidth. Since ( ) = , wehave

P = 1 f s L

. (54)

As we will see, condition (54) is not the only one that can beobtained for the parameter P . The deviation from linearity

introduced by the warping (

) can be written as follows:

( ) ( ) =2 arctan u sin( )1 u cos( )

. (55)

The function ( ) is plotted, for di ff erent values of u, inFigure 9.

One can see that the absolute value of ( ) has a max-imum which corresponds to the maximum deviation fromthe linearity of ( ). It can be shown that this maximum oc-curs for

= M =arccos(u) (56)


9/15


u =0.9

u = 0.90 1 2 3

Discrete frequency (Hz)

3

2

1

0

1

2

3


Figure 9: Plot of the deviation from linearity of the all-pass lterphase for diff erent values of parameter u.

for which the maximum deviation is

M , u =2 arcsin(u). (57)Substituting ( 56) in (51), we have

M = 2

+ arcsin( u). (58)

Since the solution phase 1 is approximated by ( ), it has tosatisfy the condition

1 M

T LP

2

+ arcsin( u) (59)

and therefore, we have the following bound on P :

P L 1 f s arccos(u)/ 2 + arcsin( u) .

(60)

For higher-order Q all-pass lters, (60) can be written as fol-lows:

P 1QQ

i=1 1 f s arccos ui L/ 2 + arcsin ui

. (61)

An optimization algorithm can be used to obtain the vectorparameter u. Based on our experiments, we estimated thatan optimal order Q is 4 for the piano string. Therefore, us-ing the values in (15) for the 58Hz tone of an L = 200cmbrass string, we obtain P =209. Although this is not a modelfor a real-life wound inhomogeneous piano string, this ex-ample gives a rough idea of the typical number of the re-quired all-pass sections. The computation of this long all-pass chain canbe tooheavy for real-time applications. There-

fore, an approximation of the chain by means of a cascade of an all-pass of order much smaller than 2 P with unit delays isusually sought [ 13, 29, 30]. A simple and accurate approachis to model the all-pass as a cascade of rst-order sectionswith variable real parameter u [38]. However, a more gen-eral approach calls for including in the design second-order

all-pass sections, equivalent to a pair of complex conjugatedrst-order sections [ 29]. In Section 4, we will bypass this esti-mation procedure based on the theoretical eigenfunctions of the string to estimate theall-pass parameters and the numberof sections from samples of the piano.

3.2. Laguerre sequences

An invertible and orthogonal transform, which is related tothe all-pass chain included in the sti ff string model, is givenby the Laguerre transform [ 44, 45]. The Laguerre sequencesl i[m, u] are best dened in the z -domain as follows:

Li(z , u)

=

1 u21 uz

1 z 1 u1 uz

1

i. (62)

Thus, the Laguerre sequences can be obtained from the z -domain recurrence

L0(z , u) = 1 u21 uz 1

,

Li+1(z , u) = A(z )Li(z , u),(63)

where A(z ) is dened as in (50). Comparison of ( 62) with(50) shows that the phase of the z transform of the Laguerresequences is suitable for approximating the phase of the solu-tion of the stiff model equation. A biorthogonal generaliza-tion of the Laguerre sequences calling for a variable u fromsection to section is illustrated in [ 46]. This is linked to therened approximation of the solution previously shown.

3.3. Initial conditions

Putting together the results obtained in Section 1, we canwrite the solution phase of the sti ff model Y ( , x ) as follows(see (11) and (14)):

Y (, x ) =c+1 ()exp i 1x + c1 ()exp i 1x . (64)We are now disregarding the transient term due to 2 since itdoes not inuence the acoustic frequencies of the system. Indiscrete time and space, we let x = m(L/P ) as in [10]. Withthe approximation ( 52), (64) becomes

Y (m, ) c+1 ( )exp im ( ) + c1 ( )exp im ( ) .(65)

Substituting ( 63) in (65), we have

Y ( , m) c+1 ( )Lm( , u)L0(z , u)

+ c1 ( )Lm( , u)

L0(z , u) , (66)

where we have used the fact that

A ei , u = ei u1 uei =

exp i ( ) . (67)


10/15


By dening

V + ( ) c+1 ( )L0(z , u)

, V ( ) c1 ( )L0(z , u)

, (68)

(66) can be written as follows:

Y (m, ) V + ( )Lm( , u) + V ( )Lm( , u). (69)

Taking the inverse discrete-time Fourier transform (IDTFT)on both sides of ( 69), we obtain

y [m, n] y + [m, n] + y [m, n], (70)

where

y + [m, n] =

k=v + [n k]l m[k, u],

y [m, n] =

k=

v [n k]l m[k, u],(71)

and the sequences v (n) are the IDTFT of V ( ). For thesake of conciseness, we do not report here the expression of v [n] in terms of constants c1 . For further details, see [31,38]. The expression of the numerical solution y [m, n] can bewritten in terms of a generic initial condition

y [m, 0] = y + [m, 0] + y [m, 0]. (72)In order to do this, we resort to the extension of Laguerresequences to negative arguments:

l m[n, u] =l m[n, u], n

0,

l m[n, u], n < 0, (73)and to the property

l m[n, u] =l n[m,u]. (74)If we introduce the quantity

y k [u] =

m=0 y [m, 0]l k [m, u],

l k [m, u] =l m[k, u],(75)

with a simple mathematical manipulation, ( 71) can be writ-ten as follows:

y + [m, n] =

k= y +k [u]l m[k + n, u],

y [m, n] =

k= y k [u]l m[k + n, u].

(76)

Therefore, the numeric solution becomes

y [m, n] =

k= y +k l m[k + n, u] +

k= y k l m[k + n, u]. (77)

We have just shown that the solution of the discrete-timestiff model equation can be written as a Laguerre expansionof the initial condition. At the same time, this shows that thestiff string model is equivalent to a nonsti ff string model cas-caded by frequencywarping obtained by Laguerre expansion.

3.4. Boundary conditions

In Section 1, we discussed the stiff model equation bound-ary conditions in continuous time (see ( 18) and (19)). Inthis section, we will discuss the homogenous boundary con-ditions (i.e., the rst line in both ( 18) and (19)) in thediscrete-time domain. Using approximation ( 52) and lettingthe number of sectionsof the stiff system P be an even integer,we can write the homogenous conditions as follows (see also(69)):

Y P 2

, =0

=V + ( )LP/ 2( , u) + V ( )LP/ 2( , u) =0,Y + P 2 ,

=0

=V + ( )LP/ 2( , u) + V ( )LP/ 2( , u) =0.

(78)

Like (34), (78) can be expressed in matrix form:

LP/ 2( , u) LP/ 2( , u)LP/ 2( , u) LP/ 2( , u)

V + ( )V ( ) =

00 . (79)

As shown in Section 3.3, the functions V ( ) aredeterminedby means of Laguerre expansion of the initial conditions se-quences through ( 71) and ( 76). For any choice of these initialconditions, the determinant of the coe ffi cients matrix in ( 79)

must be zero, obtaining the following condition:

LP/ 2( , u)2

LP/ 2( , u)] 2 =0. (80)Recalling the z -transform expression for the Laguerre se-quences, we have

sin ( )P =0, ( ) = k

P , k =1,2,3, . . . . (81)

In the stiff string case, the eigenfrequencies of the system arenot harmonically related. In our approximation of the phaseof the solution with the digital all-pass phase, the harmonic-ity is reobtained at a di ff erent level: the displacement of theall-pass phase values is harmonic according to the law writ-ten in ( 81). The distance between two consecutive values of this phase is /P . Due to the nonrigid terminations, the real-life boundary conditions can be given in terms of frequency dependent functions, which are included in the loop lter.In mapping the sti ff structure to a nonsti ff one, care must betaken into unwarping the loop lter as well.

4. SYNTHESIS OF SOUND

In order to implement a piano simulation via the physicalmodel, we need to determine the design parameters of the


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0 5 10 15 20 25 30

Partial number

0.2

0.3

0.4

0.5

0.6

0.7

W a r p i n g p a r a m e t e r

Figure 10: Computed all-pass optimized parameters u.

dispersive waveguide, that is, the number of all-pass sectionsand the coe ffi cients ui of the all-pass lters. This taskcould beperformed by means of lengthy measurements or estimationof the physical variables, such as tension, Youngs module,density, and so forth. However, as we already remarked, dueto the constitutive complexity of the real-life piano stringsand terminations, this task seems to be quite di ffi cult and tolead to inaccurate results. In fact, the given physical modelonly approximately matches the real situation. Indeed, inorder to model and justify the measured eigenfrequencies,we resorted to Fletchers experimental model described by (48). However, in that case, we ignore the exact form of theeigenfunctions, which is required in order to determine thenumber of sections of the waveguide and the other param-eters. A more pragmatic and e ff ective approach is to esti-mate the waveguide parameters directly from the measuredeigenfrequencies n. These can be extracted, for example,from recorded samples of notes played by the piano underexam. Fletchers parameters A and B can be calculated asfollows:

A = 12n

162n 22n3

,

B = 1n2

42

1

1 162 , = n2n .

(82)

In practice, in the model where the all-pass parametersui are equal throughout the delay line, one does not evenneed to estimate Fletchers parameters. In fact, in view of theequivalence of the stiff string model with the warped non-stiff model, one can directly determine, through optimiza-tion, the parameter u that makes the dispersion curve of theeigenfrequencies the closest to a straight line, using a suitabledistance. A result of this optimization is shown in Figure 10.It must be pointed out that our point of view di ff ers fromthe one proposed in [ 29, 30], where the objective is the min-

0 20 40 60 80

Partial number

20

40

60

80

100

120

140

160

180

S p a c i n g o f t h e p a r t i a l s ( H z )

Figure 11: Warped deviation from linearity.

0 10 20 30 40 50

Partial number

0.016

0.017

0.018

0.019

0.02

0.021

0.022

N o r m a l i z e d f r e q u e n c y

Figure 12: Optimized all-pass parameters u for A#3 tone.

imization of the number of nontrivial all-pass sections in thecascade.

Given the optimum warping curve, the number of sec-tions is then determined by forcing the pitch of the cascadeof the nonsti ff model (Karplus-Strong like) with warping tomatch the required fundamental frequency of the recordedtone. An example of this method is shown in Figure 11,where the measured warping curves pertaining to several pi-ano keys in the low register, as estimated from the resonanteigenfrequencies, are shown. In Figure 12, the optimum se-quence of all-pass parameters u for the examined tones isshown. Finally, in Figure 13, the plot of the regularized dis-persion curves by means of optimum unwarping is shown.For further details about this method, see [ 47, 48, 49]. Fre-quency warping has also been employed in conjunction with2D waveguide meshes in the e ff ort of reducing the articial


12/15


0 10 20 30 40Partial number

0

50

100

150

200

250

F r e q u e n c y ( H z )

Figure 13: Optimum unwarped regularized dispersion curves.

dispersion introduced by the nonisotropic spatial sampling[50]. Since the required warping curves do not match therst-order all-pass phase characteristic, in order to overcomethis diffi culty, a technique including resampling operatorshas been used in [ 50, 51] according to a scheme rst in-troduced in [ 33] and further developed in [ 52] for thewavelet transforms. However, the downsampling operatorsinevitably introduce aliasing. While in the context of wavelettransforms, this problem is tackled with multichannel lterbanks, this is not the case of 2D waveguide meshes.

5. CONCLUSIONSIn order to support the design and use of digital dispersivewaveguides, we reviewed the physical model of stiff systems,using a frequency domain approach in both continuous anddiscrete time. We showed that, for dispersive propagation inthe discrete-time, the Laguerre transform allows us to writethe solution of the sti ff model equation in terms of an or-thogonal expansion of the initial conditions and to reob-tain harmonicity at the level of the displacement of the all-pass phase values. Consequently, we showed that the sti ff string model is equivalent to a nonsti ff string model cas-caded with frequency warping, in turn obtained by Laguerreexpansion. Finally, we showed that due to this equivalence,the all-pass coeffi cients can be computed by means of opti-mization algorithms of the sti ff model with a warped nonsti ff one.

The exploration of physical models of musical instru-ments requires mathematical or physical approximations inorder to make the problem treatable. When available, thesolutions will only partially reect the ensemble of mechan-ical and acoustic phenomena involved. However, the phys-ical models serve as a solid background for the construc-tion of physically inspired models, which are exible nu-merical approximations of the solutions. Per se, these ap-proximations are interesting for the synthesis of virtual in-

struments. However, in order to ne tune the physically in-spired models to real instruments, one needs methods forthe estimation of the parameters from samples of the instru-ment. In this paper, we showed that dispersion from sti ff -ness is a simple case in which the solution of the raw phys-ical model suggests a discrete-time model, which is exible

enough to be used in the synthesis and which provides real-istic results when the characteristics are estimated from thesamples.

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I. Testa was born in Napoli, Italy, onSeptember 21, 1973. He received the Lau-rea in Physics from University of NapoliFederico II in 1997 with a dissertation onphysical modeling of vibrating strings. Inthe following years, he has been engagedin the didactics of physics research, in theeld of secondary school teacher trainingon the use of computer-based activities andin teaching computer architecture for theinformation sciences course. He is currently teaching electronicsand telecommunications at the Vocational School, Galileo Fer-raris, Napoli.

G. Evangelista received the Laurea inphysics (with the highest honors) from theUniversity of Napoli, Napoli, Italy, in 1984and the M.S. and Ph.D. degrees in electri-cal engineering from the University of Cal-ifornia, Irvine, in 1987 and 1990, respec-tively. Since 1995, he has been an Assistant

Professor with the Department of PhysicalSciences, University of Napoli Federico II.From 1998 to 2002, he was a Scientic Ad- junct with the Laboratory for Audiovisual Communications, SwissFederal Institute of Technology, Lausanne, Switzerland. From 1985to 1986, he worked at the Centre dEtudes de Math ematique etAcoustique Musicale (CEMAMu/CNET), Paris, France, where hecontributed to the development of a DSP-based sound synthesissystem, and from 1991 to 1994, he was a Research Engineer atthe Microgravity Advanced Research and Support Center, Napoli,where he was engaged in research in image processing appliedto uid motion analysis and material science. His interests in-clude digital audio, speech, music, and image processing; coding;wavelets and multirate signal processing. Dr. Evangelista was a re-cipient of the Fulbright Fellowship.

S. Cavaliere received the Laurea in elec-tronic engineering (with the highest hon-ers) from the University of Napoli FedericoII, Napoli, Italy, in 1971. Since 1974, he hasbeen with the Department of Physical Sci-ences, University of Napoli, rst as a Re-search Associate and then as an AssociateProfessor. From 1972 to 1973, he was withCNR at the University of Siena. In 1986, hespent an academic year at the Media Lab-oratory, Massachusetts Institute of Technology, Cambridge. From1987 to 1991, he received a research grant for a project devoted tothe design of VLSI chips for real-time sound processing and for the

realization of the Musical Audio Research Station, workstation forsound manipulation, IRIS, Rome, Italy. He hasalso been a ResearchAssociate with INFN for the realization of very-large systems fordata acquisition from nuclear physics experiments (KLOE in Fras-cati and ARGO in Tibet) and for the development of techniques forthe detection of signals in high-level noise in the Virgo experiment.His interests include sound and music signal processing, in partic-ular for the Web, signal transforms and representations, VLSI, andspecialized computers for sound manipulation.


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Photograph Turisme de Barcelona / J. Trulls

Preliminary call for papers

The 2011 European Signal Processing Conference (EUSIPCO 2011) is thenineteenth in a series of conferences promoted by the European Association for

Signal Processing (EURASIP, www.eurasip.org ). This year edition will take placein Barcelona, capital city of Catalonia (Spain), and will be jointly organized by theCentre Tecnolgic de Telecomunicacions de Catalunya (CTTC) and theUniversitat Politcnica de Catalunya (UPC).EUSIPCO 2011 will focus on key aspects of signal processing theory and

Organizing Committee

Honorary Chair Miguel A. Lagunas (CTTC)

General Chair

Ana I.

Prez Neira

(UPC)

General Vice Chair Carles Antn Haro (CTTC)

Technical Program Chair Xavier Mestre (CTTC)

Technical Pro ram Co Chairsapp ca ons as s e e ow. ccep ance o su m ss ons w e ase on qua y,relevance and originality. Accepted papers will be published in the EUSIPCOproceedings and presented during the conference. Paper submissions, proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.

Areas of Interest

Audio and electro acoustics.

Design, implementation, and applications of signal processing systems.

Javier Hernando (UPC)Montserrat Pards (UPC)

Plenary TalksFerran Marqus (UPC)Yonina Eldar (Technion)

Special SessionsIgnacio Santamara (Unversidad de Cantabria)Mats Bengtsson (KTH)

Finances

Mu ltimedia signal processing and coding. Image and multidimensional signal processing. Signal detection and estimation. Sensor array and multi channel signal processing. Sensor fusion in networked systems. Signal processing for communications. Medical imaging and image analysis. Non stationary, non linear and non Gaussian signal processing .

TutorialsDaniel P. Palomar (Hong Kong UST)Beatrice Pesquet Popescu (ENST)

Publicity Stephan Pfletschinger (CTTC)Mnica Navarro (CTTC)

PublicationsAntonio Pascual (UPC)Carles Fernndez (CTTC)

Procedures to submit a paper and proposals for special sessions and tutorials will be detailed at www.eusipco2011.org . Submitted papers must be camera ready, nomore than 5 pages long, and conforming to the standard specified on theEUSIPCO 2011 web site. First authors who are registered students can participatein the best student paper competition.

Important Deadlines:

Industrial Liaison & ExhibitsAngeliki Alexiou (University of Piraeus)Albert Sitj (CTTC)

International Liaison Ju Liu (Shandong University China) Jinhong Yuan (UNSW Australia)Tamas Sziranyi (SZTAKI Hungary)Rich Stern (CMU USA)Ricardo L. de Queiroz (UNB Brazil)

Webpage: www.eusipco2011.org

roposa s or spec a sess ons ec

Proposals for tutorials 18 Feb 2011Electronic submission of full papers 21 Feb 2011Notification of acceptance 23 May 2011Submission of camera ready papers 6 Jun 2011

Documents

Stiff Strings