kautz filters and generalized frequency resolution: theory...

PAPERS

0 INTRODUCTION

In the context of signal processing, rational orthonor-mal filter structures were first introduced in the 1950s byKautz, Huggins, and Young [1]–[3]. Kautz showed that anorthogonalization process applied to a set of continuous-time exponential components produces orthonormal basisfunctions having particular frequency-domain expressions.Much earlier Wiener and Lee [4] proposed synthesis net-works based on some classical orthonormal polynomialexpansions [5]. The idea of representing functions in ortho-normal components is elementary in Fourier analysis, butthe essential observation in the aforementioned cases wasthat some time-domain basis functions have rationalLaplace transforms with a recurrent structure, defining anefficient transversal synthesis filter.

Discrete-time rational orthonormal filter structures can beattributed to Broome [6] as well as the baptizing of the dis-crete Kautz functions, consequently defining the discrete-time Kautz filter. The point of reference in the mathemati-cal literature is somewhat arbitrary, but reasonable choicesare the deductions made in the 1920s to prove intercon-nections between rational approximations and interpola-tions, and the least-square (LS) problem, which wereassembled and further developed by Walsh [7].

Over the last ten years there has been a renewed inter-

est toward rational orthonormal model structures, mainlyfrom the system identification point of view [8]–[11]. Theperspective has usually been to form generalizations to thewell-established Laguerre models in system identification[12]–[14] and control [15]. In this context the Kautz filteror model has often the meaning of a two-pole generaliza-tion of the Laguerre structure [9], whereupon further gen-eralizations restrict as well to structures with identicalblocks [16]. Another, and almost unrecognized, connectionto recently active topics in signal processing are the ortho-normal state-space models for adaptive IIR filtering [17],with some existing implications to Kautz filters [18], [19].

Kautz filters have found very little use in audio signalprocessing, to cite rarities [20]–[24]. One of the reasons iscertainly that the field is dominated by the system identifi-cation perspective. The more inherent reason is that there isan independent but related tradition of frequency-warpedstructures, which is already well grounded and sufficientfor many tasks in audio signal processing. Frequencywarping provides a (rough) approximation of the constant-Q resolution of modeling [25] as well as a good matchwith the Bark scale, which is used to describe the psy-choacoustical frequency scale of human hearing [26]. Fora recent overview of frequency warping, see [27].

In our opinion there is a certain void in generality, bothin the proposed utilizations of Kautz filters as well as inthe warping-based view of the frequency resolution ofmodeling. There are some implications [28], [29] thatKautz filters would also be well applicable to adaptivemodeling of audio systems, such as acoustic echo cancel-

J. Audio Eng. Soc., Vol. 51, No. 1/2, 2003 January/February 27

Kautz Filters and Generalized Frequency Resolution:Theory and Audio Applications*

TUOMAS PAATERO AND MATTI KARJALAINEN, AES Fellow

Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, Espoo, Finland

Frequency-warped filters have recently been applied successfully to a number of audioapplications. The idea of all-pass delay elements replacing unit delays in digital filters allowsfor focusing enhanced frequency resolution on the lowest (or highest) frequencies andenables a good match to the psychoacoustical Bark scale. Kautz filters can be seen as afurther generalization, where each all-pass element may be different, allowing also complex-conjugate poles. This enables an arbitrary allocation of frequency resolution to filter design,such as modeling and equalization (inverse modeling) of linear systems. Strategies for usingKautz filters in audio applications are formulated. Case studies of loudspeaker equalization,room response modeling, and guitar body modeling for sound synthesis are presented.

* An early version of this paper was presented in the 110thConvention of the Audio Engineering Society, Amsterdam, TheNetherlands, 2001 May 12–15; revised 2002 April 18.

PAATERO AND KARJALAINEN PAPERS

lation and channel equalization. In this paper we demon-strate the use of Kautz filters in pure filter synthesis, thatis, modeling of a given target response, which was actu-ally their original usage. Quite surprisingly, to our knowl-edge there have not actually been any proposals in thisdirection on the level of modern computational means indesign and implementation.

The objective of this paper is to introduce the conceptof Kautz filters in a practical manner and from the point ofview of audio signal processing. After a brief theoreticalpart we present various methods for the most essentialingredient in Kautz filter design, namely, choosing a par-ticular structure. Audio applications of Kautz filtering,including loudspeaker equalization, room response model-ing, and modeling of an acoustic guitar body, are demon-strated and compared with more traditional approaches.

1 KAUTZ FILTER STRUCTURES

For a given set of desired poles {zi} in the unit disk, thecorresponding set of rational orthonormal functions isuniquely defined in the sense that the lowest order rationalfunctions Gi(z), square-integrable and orthonormal on theunit circle, analytic for �z � > 1, are of the form [7]

, , ,G zz z

z z

z z

z zi

1

10 1�

�

�

�

��

*

* *

i

i

i i

j

j

j

i

1 1

1

0� �

�

�

f%^ h (1)

where z* denotes the complex conjugate of z. The mean-ing of orthonormality is most economically established inthe time domain: for the impulse responses gi(n) of Eq.(1), the time-domain inner products satisfy

!,

,

, .g g g n g n

i k

i k

0

1� �

�*

i k i kn 0�

3

!_ ^ ^i h h * (2)

For the remaining conditions of square integrability andanalyticity it is sufficient to presume stability and causal-ity of the rational transfer function, that is, that �z i� < 1 forall the poles.

Eq. (1) forms clearly a repetitive structure: up to a givenorder, that is, the number of poles, functions associatedwith the subsets {zj}

ij�0 of an ordered pole set {zj}

Nj�0 are

produced as intermediate substructures defining a tappedtransversal system. In agreement with the continuous-timecounterpart, a weighted sum of these functions is called aKautz filter and depicted in Fig. 1.

Defined in this manner, Kautz filters are merely a classof fixed-pole IIR filters, forced to produce orthonormaltap-output impulse responses.1 A particular Kautz filter isthus determined by a pole set, consequently defining thefilter order, the all-pass filter backbone, as well as the cor-responding tap-output filters. The Kautz filter response isthus defined by the linear-in-parameters {wi} formulas

andH z w G z h n w g n� �i ii

N

i ii

N

0 0� �

! !^ ^ ^ ^h h h h (3)

which despite their appearance as parallel systems are trans-versal in nature because of the particular form of Eq. (1).

A gentle approach to understanding Kautz filters from atraditional digital filtering point of view is to start from anFIR filter and replace the unit delays by first-order all-passsections, that is, by frequency-dependent delays. If thesections are equal, the resulting structure is a warped FIRfilter (WFIR) [27]. With an additional cascaded normaliza-tion term the structure is a Laguerre filter (normalizedwarped filter). Finally, when the all-pass sections have indi-vidual pole values, possibly complex ones, the result is aKautz filter, a generalized transversal filter. In analogy tothe FIR filter, Eq. (3) can be seen as a truncated generalizedz transform or as a generalization of the unit-delay decom-positions of a given system. Note that due to internal feed-backs it is a filter with an infinite impulse response.

Fig. 2 illustrates time- and frequency-domain responsesof an example Kautz filter with a real pole set {0.1 0.2 0.30.4 0.5 0.6 0.7 0.8 0.9}. In the time domain the tap outputshave excessively increasing delays, similar to FIR filtertaps but with infinitely long responses. In the frequencydomain each tap has a magnitude response that is inde-pendent of the tap position, in this sense resembling a par-allel filter bank.

1.1 Kautz Modeling of Signals and SystemsDefined by any set of points {zi}

∞i�0 in the unit disk, Eq.

(1) forms an orthonormal set that is complete, or a base,with a moderate restriction on the poles {zi} [7]. The cor-responding time-domain basis functions {gi(n)}∞i�0 areimpulse responses or inverse z transforms of Eq. (1). Thisimplies that a basis representation of any causal and finite-

28 J. Audio Eng. Soc., Vol. 51, No. 1/2, 2003 January/February

1Kautz filters are clearly related to other orthogonal filter for-mulations, but here we just exploit this connection by referringto favorable numerical properties implied by the orthonormality[30]–[33].

Fig. 1. Kautz filter. For zi � 0 in Eq. (1) it degenerates to an FIR filter and for zi � a, �1 < a < 1, it is a Laguerre filter, where tap fil-ters are replaced by a common prefilter.

PAPERS KAUTZ FILTERS AND FREQUENCY RESOLUTION

energy discrete-time signal is obtained as its generalizedFourier series, that is, as infinite sums of the form of Eqs.(3) with corresponding Fourier coefficients wi � (h, gi) �(H, Gi) with respect to the time- or frequency-domain ortho-normal basis functions and definitions for the inner products.The bounded-input–bounded-output (BIBO) stability condi-tion for a causal system implies square summability,

� �n nnn

2

00 ��

33

< <� �h h&3 3!! ^ ^h h (4)

that is, impulse responses of causal and stable (CS) lineartime-invariant (LTI) systems form a subset of finite energysignals, which makes Eqs. (3) valid model structures forinput-output data identification.

Kautz filters provide models for many types of systemidentification and approximation schemes, includingadaptive filtering, for both fixed and nonfixed pole struc-tures. Also in the Kautz model case there are various inter-pretations, criteria, and methods for the parameterizationof the model. Here we address only the “prototype” LSapproach implied by the signal space description: tap-output signals of a Kautz filter xi(n) � Gi[x(n)] to inputx(n) span a finite-dimensional approximation space, pro-viding an LS optimal approximation to any CSLTI systemwith respect to the basis. The parameterization, that is, thefilter weights are solutions of the normal equations assem-bled from correlation terms of the tap outputs and thedesired output y(n). For example, in matrix form, definingthe correlation matrix R, [r]i j � (xi, xj), the weight vectorw is the solution of the matrix equation

, ,p y xRw p� �i i_ i7 A (5)

where p is the correlation vector.2 This simply utilizes thepreviously defined time-domain inner product Eq. (2) ofthe signals, so there are also definitions for the frequency-domain correlation terms, and separately for deterministicand stochastic signal descriptions, but this notion is just

included to indicate similarities with the FIR modelingcase.

Actually we further limit our attention to the approxi-mation of a given target response since the input–outputsystem identification framework is a bit pompous andimpractical for most audio signal processing tasks.Although typical operations such as inverse modeling andequalization are basically identification schemes, they areusually convertible to the approximation problem.Furthermore we are not going to address here the interest-ing question of the invertibility of a Kautz filter, whichwould require elimination of delay-free loops, or an imple-mentation method proposed in [34].

For a given system h(n) or H(z), Fourier coefficientsprovide LS optimal parameterizations for the correspon-ding Kautz model, or synthesis filter, with respect to thepole set. Evaluation of the Fourier coefficients

, ,c h g H G� �i i i_ _i i (6)

can be implemented by feeding the time-reversed signalh(�n) to the Kautz filter and reading the tap outputs xi attime n � 0 ,

, .c x x n h n g n0� � �i i i i*^ ^ ^ ^h h h h (7)

This implements inner products by filtering (convolution[*]), and it can be seen as a generalization of the FIRdesign by truncation. It should be noted that here the LScriterion is applied on the infinite time horizon and not, forexample, in the time window defined by h(n). We use thesetrue orthonormal expansion coefficients because they areeasy to obtain, providing implicitly simultaneous time- andfrequency-domain design and powerful means to the Kautzfilter structure (that is, pole position) optimization. More-


2The term correlation is used to point out that this is a genuinegeneralization of the Wiener filter setup.

Fig. 2. (a) Time-domain Kautz functions (tap outputs of Kautz filter backbone) gi(n), i � 0, … , 8, for real-valued Kautz poles {0.1 0.20.3 0.4 0.5 0.6 0.7 0.8 0.9}. Repetitive offset of �0.5 applied for improved readability. Compare gi(n) with shifted impulses in an FIRfilter. (b) Frequency-domain Kautz magnitude functions 20 log�Gi(ω)� for same case.

(a) (b)0 5 10 15 20 25 30 n/samples

-4

-3

-2

-1

0

1

g0(n)

g1(n)

g2(n)

g3(n)

g4(n)

g5(n)

g6(n)

g7(n)

g8(n)

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10

dB

G0(²)

G8(²)

Normalized frequency f/fs


over, the coefficients are independent of ordering andapproximation order,3 which makes choosing poles, approx-imation error evaluation, and model reduction efficient.

1.2 Real-Valued Kautz Functions for Complex-Conjugate Poles

A Kautz filter produces real tap-out signals only in thecase of real poles. In principle this does not in any waylimit its potential capabilities of approximation a real sig-nal or system. However, we may want the processing, thatis, signals internal to a filter, coefficients, and arithmeticoperations to be real-valued. A restriction to real linear-in-parameter models can also be seen as a categoric step inthe optimization of the structure.4

From a sequence of real or complex-conjugate poles it isalways possible to form real orthonormal structures.Symbolically this is done by applying a block diagonal uni-tary transformation to the outputs, consisting of 1’s corre-sponding to real poles and 2 � 2 rotation elements corre-sponding to pole pairs. From the infinite variety of unitarilyequivalent possible solutions5 it is sufficient to use the intu-itively simple structure of Fig. 3, proposed by Broome [6].

The second-order section outputs of the backbone struc-ture in Fig. 3 are already orthogonal, from which tap-output pairs are produced as orthogonal components ofdifference and sum, x(n) � x(n � 1) and x(n) � x(n � 1),respectively. The orthonormalizing terms are then

andp z q z1 1� �i i1 1� �

` `j j (8)

and they are determined by the corresponding pole pair{z i, z*

i} so that

/

/

ρ ρ γ

ρ ρ γ

γ

ρ

Re

p

q

z

z

1 1 2

1 1 2

2

� � � �

� � � �

� �

�

i i i i

i i i i

i i

i i2

_ _

_ _

i i

i i

# -

(9)

where γi and ρi can be recognized as corresponding second-order polynomial coefficients. The construction worksalso for real poles, producing a tap-output pair correspon-ding to a real double pole, but we use a mixture of first-

and second-order sections in the case of both real andcomplex-conjugate poles.

A Kautz filter example with complex-conjugate pole pairsis characterized by frequency-domain responses in Fig. 4(b)and (c). Notice the complementary orthogonal behavior ofresonances for odd versus even tap outputs. Kautz pole posi-tions in the z plane are shown by circles in Fig. 4(a).

2 METHODS FOR POLE DETERMINATION

Contrary to all-pole or all-zero modeling and filterdesign there are in general no analytical methods to solvefor optimal pole–zero filter coefficients. Kautz filterdesign can be seen as a two-step procedure, involving thechoosing of a particular Kautz filter pole set and the eval-uation of the corresponding filter weights. The fact thatthe latter task is much easier, better defined, and inher-ently LS optimal makes it tempting to use sophisticatedguesses and random or iterative search in the pole opti-mization. For a more analytic approach, the whole idea inthe Kautz concept is how to incorporate desired a prioriinformation to the Kautz filter. This may mean knowledgeon system poles or resonant frequencies and the corre-sponding time constants, or indirect means, such as anyall-pole or pole–zero modeling method to find potentialKautz filter poles.

A practical way to limit the degrees of “freedom” in fil-ter design is to restrict to structures with identical blocks,


3According to Eq. (5), coefficients ci depend only on the cor-responding basis function. This means that for a fixed (ordered)base {gi}

∞i�0 we obtain the same coefficients for any chosen

approximation order. Moreover, the energy of the approximationis distributed orthogonally in the coefficients, which makes theapproximation error E � (h, h) � ΣN

i�0 �ci �2, independent of the

ordering of the basis functions. For various permutations of afixed basis function set, {gi}

Ni�0 the tap-output magnitude

responses are indeed independent of ordering, but the phaseresponses differ. However, the phase response of the approxima-tion h(n) � ΣN

i�0 cigi(n) is independent of ordering.4Assuming that we are approximating a real-valued response,

we know that the “true” poles of the system are real or occur incomplex-conjugate pairs, which makes all other choices suboptimal.

5Some of the Kautz filter deductions are made directly on thereal function assumption [6]. Moreover, the state-space approachto orthonormal structures with identical blocks, the generalizedorthonormal basis functions of Heuberger, is based on balancedrealizations of real rational all-pass functions [8].

Fig. 3. One realization for producing real Kautz filter tap-output responses defined by a sequence of complex-conjugate pole pairs.Transversal all-pass backbone of Fig. 1 is restored by moving denominator terms one step to the right and compensating for change intap-output filters.


that is, to use the same smaller set of poles repeatedly. Thepole optimization and the model order selection problemsare then essentially separated, and various optimizationmethods can be applied to the substructure. In addition,for the structure of identical blocks, a relation betweenoptimal model parameters and error energy surface sta-tionary points with respect to the poles may be utilized[35] as well as a classification of systems to associate sys-tems and basis functions [36]. However, in the followingwe mainly focus on practical methods for choosing a dis-tributed pole set.

2.1 Generalized Descriptions of FrequencyResolution

As mentioned earlier, frequency-warped configurationsin audio signal processing [27] constitute a self-contained

tradition originating from warping effects observed inanalog-to-digital mappings and digital filter transforma-tions [37]. The concept of a warped signal was introducedto compute nonuniform resolution Fourier transforms usingthe fast Fourier transform (FFT) [25] and in a slightly dif-ferent form to compute warped autocorrelation terms forwarped linear predictions [38]. The original idea of replac-ing a unit delay element with a first-order all-pass opera-tor in a transfer function, that is,

λ

λz

z

z

1 �

�1

1

1�

�

�

! (10)

was restated and applied to general linear filter structures[39]–[41].

The warping effect or resolution description introducedby first-order all-pass warping is defined by the all-pass


-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real part

Imag

inar

y pa

rt

(a)

Fig. 4. Example of real-valued Kautz filter for complex-conjugate pole pairs. (a) Poles (only one per pair shown) positioned logarith-mically in frequency (� pole angle) with pole radius of 0.97. (b) Magnitude responses for odd tap outputs. (c) Magnitude responsesfor even tap outputs. (See Fig. 3.)

(c)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-30

-20

-10

0

10

20( ) g p p p


Mag

nitu

de /

dB

(b)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-30

-20

-10

0

10

20


Mag

nitu

de /

dB


phase function of Eq. (10). The warping parameter λ canbe chosen to approximate desired frequency-scale map-pings, such as the perceptually motivated Bark scale, withrespect to different error criteria and sampling rates [26].This is complete in the sense that the first-order all-passelement is the only (rational, stable, and causal) filter hav-ing a one-to-one phase function mapping. Parallel struc-tures can be constructed to approximate any kind of warp-ing [42], including the logarithmic scale [43], but the aimof this chapter is to broaden the concept of frequency res-olution to account for the resolution allocation introducedby a Kautz filter.

There is a certain deep-rooted conceptual difficulty inplacing Kautz filter techniques in the general frameworkof LTI filtering and its warped counterparts. Warped filtersprovide an efficient way to incorporate a desired fre-quency resolution in traditional FIR and IIR filter design,if the bilinear mapping in Eq. (10) is considered sufficientand flexible enough. However, a more fine-structured orcase-specific frequency resolution allocation would oftenbe preferable. On the other hand, it can be theoreticallyproven that FIR filters are the optimal choice if (exponen-tial) stability is the only thing that is known of the systemto be modeled. Actually the opposite of this statement isused in motivation of the Kautz model for approximatingresonant systems [36]. Furthermore, in our cases of typicalaudio-related responses it can be readily shown that theKautz filter techniques proposed in Section 2.3 usuallyprovide in the LS sense the best IIR filter designs, whichmay seem surprising regarding the vast number of methodsavailable. Warped counterparts of FIR and IIR filters mightbe the optimal choices in producing a desired simple globalallocation of frequency resolution, but for a specific filterdesign problem this is probably not the case. Combiningthese arguments, our hypothesis is that if a target responsehas clearly distinguishable resonant features, it is alwayspossible to design a Kautz filter with at least the same tech-nical and perceptual quality, but with better efficiency.

The Laguerre filter differs structurally from the warpedFIR filter only in the orthonormalizing all-pole prefilter,

, , ,λ

λ

λ

λH z

zw

z

zi

1

1

10 1�

�

�

�

��i

i

N i

1

2

01

1

��

�

�

f!J

L

KK^

N

P

OOh

(11)

that is, the term before the weighted tap summation (seealso Fig. 1). The effect of this distinction can be seen as amere technicality, which can be compensated if needed, oreven as a favorable frequency emphasis in the modeling[38]. Actually we have already referred to three variationsof warping, the Laguerre and the all-pass warping and theOppenheim et al. construction [25], which is a biorthogo-nal analysis–synthesis structure. In practice the questionwhich warping to choose may not be of importance, butsignal and system transformations defined by the Laguerreanalysis–synthesis structure are the only truly orthonor-mal transform pairs related to mapping (10).

The orthonormal Laguerre transformation of signalsand systems has a generalization to Kautz structures withidentical blocks defined by any rational all-pass transfor-

mation [44], [10], so actually it is just a question of howto interpret the well-defined phase characteristics as a fre-quency resolution mapping. The same applies to the gen-eral Kautz model. The phase function of the transversalall-pass backbone is completely determined by the poleset (as a whole) and we should “just” find a way to decodethis information as a generalized frequency resolutionallocation. Audio application cases presented in Section 3help with understanding intuitively how to focus resolu-tion on the frequency axis in a desirable way.

Fig. 5 also characterizes this from a couple of view-points for the same case of complex-conjugate poles withlogarithmically distributed pole angles and constant poleradius, as was used in Fig. 4. Fig. 5(b) shows the phasefunctions for each individual pole-pair section and Fig.5(c) plots the accumulated phases at the tap outputs of theall-pass sections (from high-frequency to low-frequencypole angles). Fig. 5(d) is the group delay τg � �dφ/dω ofthe all-pass backbone. The accumulated phase in Fig. 5(c)is equivalent to the effect of frequency warping in theLaguerre case, although the concept of warping is not asclear and should be used with caution. Similarly, the groupdelay can be interpreted as a measure of frequency resolu-tion. It illustrates well how the resolution in this case is high-est at low frequencies and then decreases, except locally atfrequencies corresponding to the resonating pole pairs.

A trivial way to attain a desired frequency resolutionallocation in Kautz modeling is to use suitable pole distri-butions. We may place (complex-conjugate) poles withpole angle spacings corresponding to any frequency reso-lution mapping. The choosing of pole radii is also to betaken into account so that we can use more sophisticatedchoices than a constant radius. Motivated from our exper-iments on producing warping on a logarithmic scale withparallel all-pass (and generalized parallel orthonormal)structures [43] we have, for example, used pole radiiinversely proportional to the pole angles.

2.2 Manual Fitting to a Given ResponseIt is always possible to simply adjust the Kautz filter

poles manually through trial and error in order to producea Kautz filter matching well with a given target response.A Kautz filter impulse response is a weighted superposi-tion of damped sinusoids which provide for direct tuningof a set of resonant frequencies and the correspondingdecay time constants. In principle this allows for veryflexible time-domain design, especially if some otherweighting than the LS parameterization is chosen.

By direct inspection of the time and frequency responsesit is relatively easy to find useful pole sets by selecting a setof prominent resonances and proper pole radius tuning.Choosing the complex-conjugate pole angles is more crit-ical than the pole radius selection in the sense that the fil-ter coefficients perform automatic weighting of the sinu-soidal components. In practice, however, when designinglow-order models for highly resonant systems, poles mustbe fine-tuned very close to the unit circle.

An obvious way to improve the overall modeling with astructure based on a fixed set of resonances is to use thecorresponding generating substructure repetitively, produc-



ing a Kautz filter with identical blocks. There is no obliga-tion to use the same multiplicity for all poles, but it makesmodel reduction easier. If a set of poles is determined for asubstructure that is used repetitively, some kind of damp-ing by reduced pole radii would be advisable.

2.3 Methods Implied by OrthogonalityWe have chosen our model to a given target response

h(n) to be the truncated series expansion

, ,h n c g n c h g� �i i i ii

N

0�

!t^ ^ _h h i (12)

which makes approximation error evaluation and controleasy. For any (orthonormal) {gi} and approximation orderN, the approximation error energy satisfies

E c H c� � �i ii

N

i N

2 2

01 ��

3

!! (13)

where H � (h, h) is the energy of h(n). Hence we get theenergy of an infinite-duration error signal as a by-productfrom finite filtering operations, previously described forthe evaluation of the filter weights.

As a more profound consequence of the orthogonality,the all-pass operator defined by the (transversal part ofthe) chosen Kautz filter induces a complementary divisionof the energy of the signal h(�n), n � 0, … , M [3]. Anall-pass filter A(z) is lossless by definition, and (from thepreceding) it can be deduced that the portion of theresponse a(n) � A[h(�n)] in the time interval [�M, 0]corresponds to the approximation error E, that is, the

Kautz filter optimization problem reduces to the mini-mization of the energy of a finite-duration signal.

We have encountered three attempts to utilize the con-cept of complementary signals in the pole position opti-mization, at least implicitly [45]–[47]. McDonough andHuggins replace the all-pass numerator with a polynomialapproximating the denominator mirror polynomial to pro-duce linear equations for the polynomial coefficients in aniteration scheme [45]. Friedman has constructed a networkstructure for parallel calculations of all partial derivativesof the approximation error with respect to the real second-order polynomial coefficient associated to the complexpole pairs, to be used in a gradient algorithm [46].6

Brandenstein and Unbehauen have proposed an iterativemethod resembling the Steiglitz–McBride method of pole –zero modeling to pure FIR-to-IIR filter conversion [47],which we have adapted to the context of Kautz filter opti-mization. We have implemented and experimented with allthe aforementioned algorithms, but in general just themethod of Brandenstein and Unbehauen (BU method) isfound reliable. It genuinely optimizes in the LS sense thepole positions of a real Kautz filter, producing uncondition-ally stable and (theoretically globally) optimal pole sets fora desired filter order. Furthermore the BU method works onrelatively high filter orders, for example, providing sets of300 distributed and accurate poles, which would beunachievable with standard pole–zero modeling methods.


6An iterative solution based on a modified error criterion ispresented in [48] as well as a connection to the generalizedSteiglitz–McBride method in adaptive IIR filtering in [17].

Fig. 5. Phase and resolution behavior of complex-conjugate pole Kautz filter characterized in Fig. 4. (a) Pole positions. (b) Phase func-tions of individual all-pass sections. (c) Accumulated phase in all-pass tap outputs. (d) Group delay (phase derivative).

(c) (d)

0 1 2 30

50

100

angle / rad

phas

e / r

ad

1 0 1

1

0.5

0

0.5

1

Real part

Imag

inar

y pa

rt

(a)

0 1 2 3

103

102

angle / rad

0 1 2 30

2

4

6

angle / rad

phas

e / r

ad

(b)


An outline of our Kautz filter pole-generation method,modified from Brandenstein and Unbehauen [47] follows.

• The algorithm is based on approximating an all-passoperator A(z) of a given order N with

A zD z

z D z�( )

( )

( )k

k

N k

1

1

�

� �t

^^

`

hh

j(14)

where {1, D(1)(z), D(2)(z), …} are iteratively generatedpolynomials, restricted to the form

,D z d z D z1 1� � � �( ) ( )kik i k

i

N

11

�

�

!^ ^h h

, , . ( )k 1 2 15� f

• Eq. (14) converges to an all-pass function if ��D(k)(z) �D(k�1)(z)�� → 0.

• The objective is to minimize the output of Eq. (14) tothe input X(z) � z�L H(z�1) [the z transform of h(�n)].Denote U(k)(z) � A(k)[X(z)].

• Define Y(k)(z) � X(z)/D(k�1)(z) [all-pole filtered h(�n)].• Now U(k)(z) � z�N D(k)(z�1)Y(k)(z), and by substituting

Eq. (15) and rearranging,

.Y z z D z U z z Y z� �( ) ( ) ( ) ( ) ( )k N k k N k11

1� � � �^ ` ^ ^h j h h

(16)

• Collecting common polynomial terms into a matrixequation produces A(k) d(k) � u(k) � b(k), where d(k) andu(k) are unknown. The solution of A(k) d(k) � b(k) mini-mizes the square norm of u(k) � A(k) d(k) � b(k).

• The BU algorithm proceeds as follows:Step 1: For k � 1, 2, … , filter h(�n) by 1/D(k�1)(z) toproduce the elements of A(k) and b(k).Step 2: Solve A(k) d(k) � b(k), d(k) � A(k) \b(k), the (mir-ror) polynomial coefficients of D1

(k)(z). Go to step 1.Step 3: From a sufficient number of iterations, chooseD(k)(z) that minimizes the true LS error. The Kautz fil-ter poles are the roots of D(k)(z).

2.4 Hybrid MethodsBy trying to categorize various methods for choosing a

particular Kautz filter we do not in any way intend to becomplete or exclusive. There are certainly many other pos-sibilities as well as modifications and mixtures of the onespresented. For example, we have not addressed pole posi-tion optimization methods based on input–output descrip-tions of the system, which could in some cases be usefuleven if the target response is available.

An obvious way to modify the optimization is to manip-ulate the target response. Time-domain windowing orfrequency-domain weighting by suitable filtering may beapplied, or alternatively, the optimization can be parti-tioned using selective filters. For example, the BU methodis based on solving at every iteration a matrix equationwith the dimensions implied by the duration of the targetresponse and the chosen model order, and therefore patho-logical behavior can be decreased by truncation. By divid-

ing the frequency range into two (or more) parts, separatemethods and allocation of the modeling resolution on thesubbands can be applied.

In the case where a substructure is assigned and used ina Kautz filter with identical blocks there is always thequestion of compensating for the repetitive appearance ofthe poles. We have used constant pole radius damping,individual tuning of the pole radii, and an ad hoc methodwhere we simply raise the pole radius to the power of thenumber of blocks used in the Kautz filter. For poles withdissimilar radii this approach is certainly better justifiedthan using constant damping.

The most efficient strategy, in our experience, is tocombine various methods with the BU method. At theleast this may mean examining the produced pole set andpossibly omitting some of the poles. For a sharply reso-nant system, poles very close to the unit circle areobtained, and although the BU method produces (at leasttheoretically) unconditionally stable poles, the resultingpole set should be checked for numerical oddities. Theremay also appear weak poles that make practically no con-tribution to the model. Furthermore one does not usuallycare for real poles because by definition they are eitherweak or products of the anomalies in the target response atfrequency band edges. In addition sometimes a cluster ofpoles may be represented by a single conjugate pole pair.

A more systematic pruning of the BU pole set may alsobe performed, for example, by simply omitting the polesoutside a specific frequency (pole angle) region, or byspacing out the pole set according to some rule. The factthat a high-order distributed BU pole set is a good repre-sentation of the whole resonance structure can be used toassociate poles directly with different choices of promi-nent resonances. Typically a sharp resonance is repre-sented by a single pole pair, with pole angles correspon-ding (almost) exactly to the peak frequency, andappropriately tuned pole radii. Because of the latter, thishas also proven to be a useful approach in selecting a sub-structure for the Kautz filter with identical blocks. As aninverse to the pruning, one may also add and manually orby rule tune poles to the BU pole set. This is demonstratedin the loudspeaker case study that follows.

We have also developed and applied successfully acombination of frequency warping and the BU method.By using the warped target response in the BU method wemay optimize the pole positions corresponding to Kautzmodeling on a warped frequency scale. The poles in theoriginal domain are obtained simply by mapping themaccording to the inverse all-pass transformation. At pres-ent this may be the best way to incorporate a perceptuallyjustified allocation of the modeling resolution to the Kautzfilter,7 but warping can also be utilized to focus the reso-lution in a more technical fashion. It should also be notedthat our interpretation of warping has a definite effect onthe poles we obtain. However, the possible “tilt” in the


7From a perceptual viewpoint, loudness scaling of magnitudeand the masking effect are also important in optimizing for audioapplications, but these tend to make the optimization a morenonlinear problem.


Kautz model magnitude response is due to the choice ofpoles, and it is not a property of the model, in contrast tosome warped designs. A simplified version of the pro-posed method is to down-sample the response prior tothe BU method and to map the produced poles back tothe original frequency domain. This approach is used incase 2.

3 AUDIO APPLICATION CASES

We have tested the applicability of Kautz filter design inthree audio-oriented applications. The first is a loud-speaker equalization task where frequency resolution isdistributed both globally and locally. The second case, aroom impulse response, is modeled at low frequencies tocapture the modal behavior as a robust filter structure. Inthe third case we use Kautz filters to model the bodyresponse of an acoustic guitar where, similarly, the lowestfrequencies are of primary interest.

3.1 Case 1: Loudspeaker Response EqualizationAn ideal loudspeaker has a flat magnitude response and

a constant group delay. Simultaneous magnitude andphase equalization of a nonperfect loudspeaker would beachieved by modeling the response and inverting themodel, or by identifying the overall system of the responseand the Kautz equalizer, but here we demonstrate the useof Kautz filters in pure magnitude equalization8,9 based onan inverted target response.

A small two-way active loudspeaker was selected as aKautz equalization experiment due to its clear deviationsfrom ideal magnitude response. The measured responseand a derived equalizer target response are included inFig. 6. The sample rate is 48 kHz.

Magnitude response equalization consists typically ofcompensating for three different types of phenomena:1) slow trends in the response, 2) sharp and local devia-tions,10 and 3) correction of rolloffs at the band edges.This makes “blind equalization” methods, which do not

utilize audio-specific knowledge, ineffective. We proposethat Kautz filters provide a useful alternative betweenblind and hand-tuned parametric equalization.

As is well known, FIR modeling and equalization hasan inherent emphasis on high frequencies on the auditorilymotivated logarithmic frequency scale. Warped FIR (orLaguerre) filters [27] shift some of the resolution to thelower frequencies, providing a competitive performancewith 5 to 10 times lower filter orders than with FIR filters[49]. However, the filter order required to flatten the peaksat 1 kHz in our example is still high, on the order of 200,and in practice Laguerre models up to the order of about50 are able to model only slow trends in the response. In arecent publication [50] general real-pole all-pass cascadeswere proposed for low-frequency equalization, but theywere found difficult to design. Here we demonstrate effi-cient design methods for the orthonormal and complex-pole counterpart.

The simplest choice of Kautz filter poles in this case isto focus on the 1-kHz region with a single (or multiple)pole pair. By tuning the pole radius we trade off betweenthe 1-kHz region and the overall modeling. Quite interest-ingly, as a good compromise, we end up with a radiusclose to a typical warping parameter at this sampling rate(for example, λ � 0.76) [26] and we get surprisingly sim-ilar results for the Laguerre and the two-pole Kautz equal-izers for filter orders 50–200. Actually this simply meansthat a perceptually motivated warping is also technically agood choice for the flattening of the 1-kHz region. This isdemonstrated in Fig. 7.

The obvious way to proceed would be to add anotherpole pair corresponding to the 7-kHz region. In search forconsiderably lower Kautz filter orders, compared to theLaguerre equalizer, we however utilize the BU methoddirectly. It provides stable and reasonable pole sets fororders at least up to 40. In Fig. 6 we have presented Kautzequalizers and equalization results for orders 9, 15, 30,and 38. These straightforward Kautz filter constructionsare already comparable to the FIR and Laguerre counter-parts, but we may further lower the filter orders by omit-ting some of the poles. For example, for orders above 15,the BU method produces poles really close to z � 1because of the low-frequency boost in the target, andomitting some of these poles actually tranquilizes the low-frequency region. We obtain, for example, quite similarequalization results for orders 28 and 34 from the sets with30 and 38 original poles, respectively (Fig. 8).

To improve the modeling at 1 kHz, we added three tofour manually tuned pole pairs to the BU pole sets, corre-


8For a relatively high-quality loudspeaker the deviations ingroup-delay response are often within 2–3 ms, except for thelowest frequencies, which means that phase equalization ishardly needed.

9Kautz filters are inherently well suited for overall equaliza-tion of magnitude and phase, for example, by applying LS opti-mization as discussed.

10 In this study we have presented the equalization case as anillustrative example of the controllability of frequency resolutionrather than the practicality of results. It may not even be desir-able to flatten sharp resonances in the main-axis free-fieldresponse of a loudspeaker, since off-axis responses can becomeworse and degrade the overall quality of sound reproduction.

Fig. 6. Measured magnitude spectrum of loudspeaker understudy, equalization results for Kautz orders 9, 15, 30, and 38, tar-get of equalization filter, and Kautz equalizer responses (bottomto top). Kautz equalizers are designed using BU method.


sponding to the resonances in the problematic area. This isactually not too hard since the 1-kHz region is isolatedfrom the dominant pole region, which allows for undis-turbed tuning. As starting points we used the 15th- and30th-order Kautz equalizers of Fig. 6, omitting three polepairs in the latter case. Three pole pairs were tuneddirectly to the three prominent resonances, and one polepair was assigned to improve the modeling below the 1-kHz region. Results for final filter orders 23, 32, and 34are displayed in Fig. 9, where the last two differ only inthe optional compensating pole pair.

Finally we abandon the pole sets proposed by the BU

method and try to tune 10 pole pairs manually to the tar-get response resonances. The design is based on 10selected resonances, represented with 10 distinct polepairs, chosen and tuned to fit the magnitude response. Thisis of course somewhat arbitrary, but it seems to work. InFig. 10, along with the equalizer and target responses,there are vertical lines indicating pole-pair positions. Thisis clearly one form of “parametric equalization” withsecond-order blocks since each resonance is representedwith a single pole pair. However, with Kautz filters wehave, at least to some extent, separated the choice of theresonance structure and the fine-tuning produced by the


Fig. 9. Kautz equalizers and equalization results for orders 23, 32, and 34, with combinations of manually tuned and BU-generated poles.

Fig. 8. Kautz equalization results for orders 28 and 34, with pruned BU poles. Compare with orders 30 and 38 in Fig. 6.

Fig. 7. Comparison of 100th-order Laguerre and Kautz equalization results. Kautz filters with 50 complex-conjugate pole pairs corre-spond to 1-kHz pole angles, and pole radius varied from 0.5 to 0.9 in steps of 0.05.


(linear-in-parameter) model parameterization.Fig. 11 compares some of the Kautz equalization results

to those achieved with FIR and Laguerre equalizers oforders 200 and 100, respectively. If we are only concernedwith the number of arithmetic operations at run time, weshould compare Laguerre and Kautz equalizers at aboutthe same filter orders. The actual complexity depends onmany details, but in any case we have achieved very low-order and accurate Kautz equalizers. Furthermore, the lowfilter orders enable in principle filter transformations toother structures, possibly more efficient ones, such asdirect-form filters or otherwise preferable structures.

3.2 Case 2: Room Response ModelingModels for a room response, that is, a transfer function

or an impulse response from a sound source to an obser-vation location in a room, are used for different purposesin audio signal processing, typically as part of a larger sys-tem. Room response modeling may constitute a majorcomputational burden, because of both the complexity oftarget responses and the difficulty of incorporating properperceptual criteria in those models.

An obvious difficulty in modeling a room response isthat the duration of the target response is usually long andthat the time-frequency structure is very complicated.Physically speaking there are low-frequency modes deter-mined essentially by the room dimensions and, on theother hand, a reverberation structure produced by the mul-titude of reflections. There are methods proposed to takeinto account various time- and frequency-domain model-

ing aspects as LTI digital filter models [51], [52], includ-ing also reverberation designs that approximate reflectionsand reverberation by complicated parallel and feedbackstructures [53]. Thus it is interesting to see how a Kautzfilter, a generalized transversal filter, can perform similartasks.

We have chosen as an example a measured room11

impulse response (re)sampled at 22 050 Hz. The targetresponse for modeling, shown in Fig. 12, is composed ofthe original signal by omitting the early delay and by trun-cation to 8192 samples, which yields a duration of 372 ms.Such a target is clearly not an ideal Kautz modeling tasksince the early response is not a pure superposition ofcoincident damped exponentials. However, if the Kautzfilter is long enough, it should in theory be able to modelall the temporal details, though with a potential ineffi-ciency, for example, in producing pure delays. After all,functions in Eq. (1) define an exact representation of any(finite-energy) signal, so this is a brute FIR-type design,but with a more delicate choice of basis functions.

It turns out that the proposed iterative pole optimizationmethod, the BU algorithm, is in trouble in accurate mod-eling of the full audio frequency range and temporal decayin the form of a single Kautz filter. Therefore we firstfocus on the low-frequency range where each prominentmode may have a noticeable perceptual effect.


11The room has approximate dimensions of 5.5 � 6.5 � 2.7m3 and shows relatively strong modes with long decay times atlow frequencies.

102

103

104

10

20

30

Frequency / Hz

Mag

nitu

de /

dB

Target response

Equalizer response

Fig. 10. Manually tuned 20th-order Kautz equalizer and target responses, with lines indicating pole-pair positions.

Fig. 11. Comparison of FIR, Laguerre, and Kautz equalization results.

102

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-20

-10

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dB

FIR filter, order 200

Laguerre filter, order 100

Partly manually tuned Kautz, order 34 (26+8)

Partly manually tuned Kautz, order 23 (15+8)

Manually tuned 20th order Kautz

Measured response


Fig. 13 plots the magnitude spectrum of the roomresponse up to 220 Hz at the top. Below it a pole anglebifurcation plot is shown as a function of the Kautz filterorder, produced by the BU method, applied to the down-sampled target response of 512-sample duration, and formodel orders 1–120.

Fig. 14. illustrates the accuracy of the magnituderesponse compared to the target response (top curve) forKautz model orders 20–100 in steps of 5. For orders 100 –120 the match is practically perfect.

The quality of time–frequency modeling can bechecked by a waterfall plot (cumulative spectral decay), asshown in Fig. 15 for the target response and for a Kautzmodel of order 80. In careful comparison it can be noticedthat less prominent modal resonances are weaker andshorter in the model response. Increasing the filter order to120 makes the model practically perfect in a time–fre-quency plot too.

As mentioned, full-bandwidth12 Kautz modeling is notpossible by directly utilizing the BU method because ofnumerical limitations in the iterative algorithm and in rootsolving of the poles for filter orders above 200–300. If themodeling accuracy can be compromised at some frequen-cies, or only a spectral envelope model is needed, severalKautz pole determination techniques can be applied. Fig.16 illustrates magnitude responses of a 320-order model(top curve) and the target response (bottom curve) whenKautz poles are positioned logarithmically in the fre-

quency and with a constant pole radius of 0.98. At lowfrequencies below 200–300 Hz the magnitude responsefit is quite complete, but the poor performance in model-ing the dense resonance structure at higher frequencies isevident.

Note that by using a large set of weak poles, the Kautzfilter acts like a “slightly recursive FIR filter,” and there isa clear transition from this kind of FIR-type fit of the earlyresponse to representing resonances by correspondingpole pairs. In responses where the high-frequency compo-nents are short in time we could actually force some of thepoles to zi � 0, that is, use a mixture of Kautz and FIR fil-ter blocks. It is noteworthy that this may be done in anypermutations and that the same chosen filter parameteri-zation applies also to the FIR blocks.

The application of the BU method to the room responseis studied in Fig. 17. A 318th-order Kautz model is pre-sented in Fig. 17(a). It can be seen that the modeling res-olution is worse than the target resolution at all frequen-cies. By utilizing an intermediate warping step in thedesign phase,13 better modeling of the lower frequencies isachieved. In Fig. 17(b) Kautz model poles of the 240th


12 In the examples that follow we have used the sampling rateof 22 050 Hz and a bandwidth of approximately 10 kHz.

13The method was outlined in Section 2.4. The target signal iswarped prior to the pole optimization, and then the producedpoles are mapped back to the original frequency domain accord-ing to the inverse all-pass mapping.

Fig. 12. Measured room impulse response.

Fig. 13. Frequencies corresponding to pole angles produced byBU method for filter orders 1–120 for frequency range 0–220Hz of room response (bottom) compared with magnituderesponse (top).

Fig. 14. Magnitude responses of Kautz models of order 20–100,in steps of 5 from bottom up, compared with target magnituderesponse (top).



0

0.5

10 20 40 60 80 100 120 140 160 180 200 220

-40

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time/s

ORIGINAL

frequency/Hz

leve

l/dB

(a)

0

0.5

10 20 40 60 80 100 120 140 160 180 200 220

-40

-20

0

time/s

MODELED

frequency/Hz

leve

l/dB

(b)

Fig. 15. Cumulative spectral decay plot in frequency range 0–220 Hz. (a) Target. (b) Order 80 modeled room response.

102

103

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ĭ40

ĭ20

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Frequency / Hz

Mag

nitu

de /

dB

Fig. 16. Magnitude responses for room. 320th-order Kautz model (top) and target response (bottom) plotted with vertical offset. Polesare placed with logarithmically spaced angles and constant radius 0.98. Pole-pair positions are indicated by vertical lines.

Fig. 17. Kautz model magnitude responses produced using BU method with upward offset to target response. (a) Filter order 318.(b) Warping and back-mapping at filter order 240.

(b)

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(a)


order are produced using the appropriate Bark-warpingparameter value λ � 0.646 [26]. As a tradeoff, poorermodeling of the higher frequencies is evident.

It should be noted that this was not a detailed case studyon room response modeling and that especially the tem-poral evolution of the full-band model should be investi-gated more carefully. In principle it is possible to applydifferent kinds of partitioning, such as modulation anddecimation techniques, to get a partial set of poles, andthen decompose a full model. One such approach isthrough subband modeling, for example, in critical bands[54]. In addition Kautz models for the lower part of thefrequency scale can be supplemented with differentapproaches for the high-frequency modeling.

3.3 Case 3: Guitar Body ModelingAs another example of high-order distributed-pole

Kautz modeling we approximate a measured acoustic gui-tar body response sampled at 22 kHz (Fig. 18). Theimpulse response was obtained by tapping the bridge of anacoustic guitar with an impulse hammer [55], [56], withstrings damped.14 A guitar body response is typically likethe response of a small room, but since the density of themodes is proportional to the volume, the body response isactually a better justified Kautz modeling task than theroom response modeling case discussed in Section 3.2.

The obvious disadvantage of a straightforward FIR fil-ter implementation for the body response is that modelingthe slowly decaying lowest resonances requires a veryhigh filter order (1000–3000). All-pole and pole–zeromodeling are the traditional choices to improve the flexi-bility of spectral representation. However, model ordersremain problematically high, and the basic design meth-ods seem to work poorly. A significantly better approachis to use separate IIR modeling for the slowly decayinglowest resonances combined with the FIR modeling [57].Perceptually motivated warped counterparts of all-poleand pole–zero modeling pay off, even in technical terms[55], but in this study we want to focus on the modelingresolution more freely.

Direct all-pole and pole–zero modeling were found toproduce unsatisfactory pole sets {zi} for the Kautz filter,even in searching for a low-order substructure. On theother hand, it is relatively easy to find good pole sets bydirect selection of prominent resonances and proper poleradius tuning. We will demonstrate next some Kautz mod-eling cases, with unwarped and warped pole determina-tion based on the BU method. In all cases the poles havebeen determined from a minimum-phase version of a

given body response. Also the model weight coefficientshave been obtained by fitting to the minimum-phase ver-sion. If the perceptually best body response model isdesired, it is better to apply the final fitting to the originalversion (but with the initial delay removed). This mayrequire a slight increase in filter order, although Kautz fil-ters are inherently well suited to this due to good modelfitting in the time domain.

Fig. 19 demonstrates that the proposed pole positionoptimization scheme, the BU method, is able to captureessentially the whole resonance structure. The Kautz filterorder is 262, and the poles are obtained from a 300th-orderBU pole set, omitting some poles close to z � �1. In gen-eral, the BU method works quite well, at least up to anorder of 300, and the lower limit for finding the chosenprominent resonances is about 100.

The relatively high order of the model shown in Fig. 19is required to obtain a good match of the lowest(strongest) modal resonances. The direct application ofthe BU method pays on average too much attention to thehigh frequencies compared to the importance of the low-est modes, both physically and from a perceptual point ofview. A better overall balance of frequency resolution canbe achieved with a lower order by applying the BUmethod of the Kautz pole determination to a frequency-warped version of the measured response. Fig. 20 depictsthe magnitude response of a 120th-order Kautz modelobtained this way. A warping coefficient value of λ � 0.66is applied, corresponding approximately to the Bark scalewarping [26].

Fig. 21 gives further information about the modelingpower of the warped BU method by depicting magnituderesponses for a set of low-order models (order � 10, 18,39, 60, and 90), and comparing with the original response.As with any reverberant system, the magnitude responsedoes not tell the full story of how we perceive the response.Comparing time–frequency plots may be needed to evalu-ate both the temporal and the spectral evolution of mod-eled responses.

4 DISCUSSION AND CONCLUSIONS

Kautz filter design can be seen as a particular IIR filterdesign technique. The motivation from an audio-processing


14A more “natural” impulse response would be achieved byextracting it by deconvolution from an identification setup usingspectrally rich real playing of the acoustic guitar as excitation[57], although the signal-to-noise ratio is better through animpact hammer measurement.

Fig. 18. Measured impulse response of acoustic guitar body.

0 0.05 0.1 0.15 0.2 0.25 0.3

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perspective is twofold. Many audio-related target responsescan be modeled well by a combination of distributeddecaying exponential components, which by definition iswhat a Kautz filter does, but in an orthonormalized form,providing many favorable properties. On the other hand,Kautz filtering techniques provide many ways to incorpo-rate an auditorily meaningful allocation of frequency res-olution to the modeling.

The cases of modeling and equalization were selectedas audio examples to show the applicability of Kautz fil-ters. Many specific questions, such as the audio engineer-ing relevance of the modeling details, perceptual aspectsof the designs, as well as computational robustness andexpense have been addressed only briefly or not at all.These details call for further investigations.

The aim of this study was to show that it is possible toachieve good modeling or equalization results with lowerKautz filter orders than, for example, with warped(Laguerre) or traditional FIR and IIR filters. In the loud-speaker equalization case Kautz filter orders of 20–30 canachieve similar results of flatness as warped IIR equaliz-

ers of order 100–200, or much higher orders with FIRequalizers. This reduction is due to a well-controlledfocusing of frequency resolution on both the global shapeand in particular on local resonant behaviors.

The room response modeling case should be taken as amere methodological study, except possibly regarding theusefulness of obtained low-frequency models. In the caseof guitar body response modeling the low-frequencymodes are important perceptually, and low-order Kautzfilters can focus sharply on them, showing an advantageover warped, IIR, and FIR designs, especially when thefocus is on separate low-frequency modes of the bodyresponse.

In this study we have successfully applied the BUmethod of Kautz pole determination, with and without fre-quency warping. There are numerous other possible tech-niques and strategies to search for an optimal model for agiven problem, including exhaustive search for very low-order models and, for example, genetic algorithms forsomewhat higher orders. Different tasks can be solvedbest with different approaches. The cases investigated herejust hint on general guidelines, and a fully automaticsearch for optimal solutions, even in the cases presented,requires further work. We, however, have demonstratedthe potential applicability of Kautz filters. They are foundto be flexible generalizations of FIR and Laguerre filters,providing IIR-like spectral modeling capabilities withwell-known favorable properties resulting from the ortho-normality. The competitiveness compared to Laguerremodeling is based on the fact that the generalization stepimposes little or no extra computation load at runtime,even if the design phase may become more complicated.

MATLAB scripts and demos related to Kautz filterdesign can be found at http://www.acoustics.hut.fi/software/kautz.

5 ACKNOWLEDGMENT

This work has been supported by the Academy ofFinland as part of the projects “Sound Source Modeling”and “Technology for Audio and Speech Processing.”


Fig. 19. 262th-order Kautz model and original response. Verticallines indicate pole frequencies, obtained by direct application ofBU method.

Fig. 20. Magnitude response for 120th-order Kautz model ofacoustic guitar body, using Bark-warped BU poles, and originalresponse. Vertical lines indicate pole frequencies.

Fig. 21. Kautz models of orders 10, 18, 39, 60, and 90 of guitarbody response and target magnitude response.


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THE AUTHORS

T. Paatero M. Karjalainen



Tuomas Paatero was born in Helsinki, Finland, in 1969.He studied mathematics and physics at the University ofHelsinki, from which he received a Master of Sciencedegree in 1994. He has been a postgraduate student in theLaboratory of Acoustics and Audio Signal Processing atthe Helsinki University of Technology, where he receiveda Licentiate in Technology in 2002. As a researcher in theLaboratory his current interests include rational orthonor-mal model structures and their audio signal processingapplications. He is in the process of completing his doc-toral dissertation.

●

Matti Karjalainen was born in Hankasalmi, Finland, in1946. He received M.Sc. and Dr. Tech. degrees in electri-cal engineering from the Tampere University ofTechnology in 1970 and 1978, respectively. His doctoralthesis dealt with speech synthesis by rule in Finnish.

From 1980 he was an associate professor and since1986 he has been a full professor in acoustics and audiosignal processing at the Helsinki University of Technology

on the faculty of electrical engineering. In audio technol-ogy his interest is in audio signal processing, such as DSPfor sound reproduction, perceptually based signal pro-cessing, as well as music DSP and sound synthesis. Inaddition to audio DSP his research activities cover speechsynthesis, analysis, and recognition, perceptual auditorymodeling, spatial hearing, DSP hardware, software, andprogramming environments, as well as various branchesof acoustics, including musical acoustics and modeling ofmusical instruments. He has written 250 scientific andengineering articles and contributed to organizing severalconferences and workshops. In 1999 he served as thepapers chair of the AES 16th International Conference onSpatial Sound Reproduction.

Dr. Karjalainen is an AES fellow and a member of theInstitute of Electrical and Electronics Engineers, theAcoustical Society of America, the European AcousticsAssociation, the International Computer Music Assoc-iation, the European Speech Communication Assoc-iation, and several Finnish scientific and engineeringsocieties.

kautz filters and generalized frequency resolution: theory...

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