Special Digital Filters for Audio Reproduction

Transcription

1 for Audio Reproduction Matti Karjalainen, Tuomas Paatero, Jyri Pakarinen, and Vesa Välimäki Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, P.O.Box 3, FI-25 TKK, Finland Correspondence should be addressed to Matti Karjalainen ABSTRACT Digital filtering is one of the key techniques in modern audio. In addition to using standard forms of digital filters, audio signal processing has required the development and deployment of special filters, often applied then also in other application fields. This paper presents an overview of some digital filter categories that have been found important in audio reproduction, covering particularly fractional delays, frequency warping techniques including Laguerre and Kautz filters, as well as filters used for physics-based modeling, including wave digital filters and digital waveguides. Applications of these filter categories to audio reproduction, such as loudspeaker and room response modeling and equalization, are discussed and demonstrated.. INTRODUCTION Digital signal processing is a fundamental part of modern audio. Many audio applications are built on generic DSP algorithms, such as the fast Fourier transform (FFT) and digital filters. The development of specific applications has required also special algorithms and signal processing principles that may then have been adopted in other application domains as well. Digital filtering is one of the most common building blocks in signal processing. The basic forms of digital filters can be designed and implemented in a routine way, and many DSP software tools and platforms support using them. Audio applications exhibit, however, certain commonly appearing requirements that need special consideration. Among such features are for example time-variance of filter parameters, compatibility to the properties of the human auditory perception, and ability to simulate the physical behavior of systems such as audio transducers and musical instruments. Carefully controlled nonlinearities are also needed in some audio processing tasks without producing audible artifacts. This paper discusses specific classes of digital filters that can meet the requirements mentioned above. Even in cases where the original formulations of the filters and algorithms may come from other problem domains, solving audio application problems has remarkably contributed to the development of these principles. The selection of the topics also reflects the research of the present authors. The paper starts with a brief reminder of standard form digital filters, as a reference for more specialized ones. The first special topic is a short discussion of fractional delay filters, which are already a commonly used practice in specific applications. The next topic is a relatively broad area of filters with controlled frequency resolution, including simple frequency warping, Laguerre, and Kautz filters, and their applications. The last category of special filters presented is related to modeling of physical systems, where two-way interaction is needed, such as in loudspeaker modeling or model-based sound synthesis. Realization of nonlinearities is also discussed. The paper is concluded by a general discussion and summary... Standard forms of digital filters In most cases of audio applications, the standard forms of digital filters [, 2], particularly FIR (finite impulse response) and direct form IIR (infinite impulse response) filters are useful enough. There are well established techniques to design these filters to meet given specifications. The most basic filter structures are presented here as a reference for special filters discussed in the following sections.... FIR filters A finite impulse response filter can be expressed by z-transform transfer function from input X(z) to output Y (z) as H(z) = Y (z) N X(z) = b n z n () n= and its transversal computational structure is depicted in Figure. FIR type filtering is often realized also through FFT, frequency domain multiplication, and inverse FFT, especially when extra latency due to blockwise processing is allowed. AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23

2 Karjalainen et al. in b b b 2 b N- out Fig. : FIR filter as a transversal computational structure. in + a b out b a 2 b 2 a M- (a) b N- in + a a 2 a M- (b) a M- out a M-2 a M-3 Fig. 2: Direct Form II filter structures: (a) IIR filter and (b) allpass filter...2. IIR filters Infinite impulse response filters can be expressed by rational form z-transform H(z) = Y (z) X(z) = N n= b nz n + m= M a (2) mz m One of the most popular and efficient implementations is the Direct Form II shown in Fig. 2(a). Other popular implementation structures are parallel and cascade forms of first and second order sections, as well as lattice and ladder formulations...3. Allpass filters A discrete-time allpass filter is a special kind of IIR filter. It has the following transfer function H(z) = D(z )z M+ (3) D(z) = M n= a M n z n+ + z M+ + n= M a (4) nz n where D(z) is the denominator polynomial of the form + a z + a 2 z 2 + :::+ a M z M+. The numerator of the transfer function is a reverse version of the denominator, i.e., it has the same coefficients but in opposite order. With this peculiar choice of the numerator polynomial, the poles and zeros cancel out each other s contribution to the filter gain. For this reason, the allpass filter has a gain of unity regardless of the values of its coefficients. The allpass filter s phase response, however, is nontrivial. It is twice the phase function of the denominator with an additional linear term, which corresponds to pure delay. Figure 2(b) shows the direct form allpass filter structure. It is seen that when all coefficients are set to zero, the allpass filter becomes a delay line of M - samples: The signal can then only propagate through all unit delays to the output, see Fig. 2(b). The allpass filter is a useful filter in audio signal processing, as it can be used as a phase equalizer or as a delay filter, and it is a building block in many special filter structures. 2. FRACTIONAL DELAY FILTERS A fractional delay refers to a delay that is not an integral multiple of sampling intervals. In discrete-time systems, the sample instants are uniformly spaced T seconds apart from each other, where T = = f s is the sampling interval and f s is the sampling rate. It is effortless to implement a time delay that is an integral multiple of T. This is called a digital delay line. However, it is not trivial to implement a digital delay of the form D = LT + d, where L is an integer and d is a decimal number ( < d < ). In this case, interpolation must be used. A fractional delay filter implements the interpolation required, and it produces a fractionally shifted version of the input signal at its output [3, 4, 5]. In audio, fractional delay filters can be used to adjust the time delay from the loudspeaker to a microphone or to a listener, for example. Time-varying fractional delay filters, in which the delay parameter is modulated, can be used to produce (or compensate for) the Doppler effect, for example. Sampling rate conversion can also be interpreted as fractional delay filtering, where the delay parameter is changed for every sample. In this section, we discuss FIR and allpass fractional delay filters. We select a simple design method from each category. 2.. FIR fractional delay filter based on Lagrange interpolation Lagrange interpolation refers to the classical polynomial interpolation method, in which a polynomial of order N isfit through N points of data, such as audio signal samples. Values between the points are then approximated by selecting them on the obtained polynomial AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page2of8

3 curve. This idea can be directly used in interpolation of digital signals. The Lagrange fractional delay filter is also obtained as a maximally flat approximation of ideal bandlimited interpolation. In fact, it may be derived in many different ways [6, 7, 4]. The coefficients of the Lagrange fractional delay FIR filter are given by the following equation: Phase delay (samples) 8.5 (a) D = Frequency (khz) Magnitude (db) (b) Frequency (khz) N b(n)= k=;k6=n D k n k for n = ;;2;:::;N (5) where D is the desired total delay and N is the filter length. From this equation, it is easy to derive closeform polynomial formulas for FIR FD filters of different order. When N = 2, we obtain the linear interpolation coefficients b() =D and b() = D. This first-order Lagrange FD filter can be used for interpolation, when D is between. and.. Kootsookos and Williams [8] showed that Eq. (5) (for even order) can also be obtained by windowing the sinc function, where the window function is a scaled binomial window. This result has been generalized for odd-order Lagrange interpolators by Välimäki [4]. The following equation defines the windowing method for the design of a Lagrange interpolator for both odd and even N: b(n)=c bin (D;N)w bin (n)sinc(n D) (6) where sinc(x) =sin(πx)=(πx) and n = ;;2;:::;N, and the scaling coefficient C bin (D;N) and the binomial window w bin (n) are defined as C bin (D;N)=( ) N πn D and (7) sin(πd) N N w bin (n)= (8) n Figure 3 shows an example of a fractional delay filter design based on Lagrange interpolation. The sampling rate is 44 Hz. The dashed line in Fig. 3(a) shows the desired delay, which is constant over all frequencies. In this case it is 7.6 sampling intervals, which corresponds to.7 ms at 44. khz. The phase delay of the Lagrange FD filter of length 6 is also shown. It is seen that the design closely follows the specification at low and mid frequencies, but at very high frequencies the filter s phase delay approaches 8 samples. Thus, the performance of this filter is very good at low frequencies, but it fails in the vicinity of 2 khz. Figure 3(b) shows the magnitude Coefficient value.5 (c) Coefficient index Imaginary part 5 (d) 5 5 Real part Fig. 3: Example design of a Lagrange fractional delay FIR filter of length 6 for fractional delay of 7.6 samples (fractional part.6 samples): (a) the desired (dashed line) and realized (solid line) phase delay response, (b) the desired (dashed line) and realized (solid line) magnitude response, (c) the FIR filter coefficients (i.e., impulse response), and (d) the roots of the filter (circles). The sampling rate used is 44. khz. response of the same filter (solid line) and the desired filter gain, which is always db. A similar behavior as with the phase delay response is observed: the magnitude response is nearly constant below 5 khz or so, but at high frequencies it deviates from the specification. In this example the signal is attenuated about 6 db at 2 khz. The approximation error of Lagrange FD filters depends on the value of the delay parameter D so that when D is close to an integer, the filter gets better. When the decimal part of D is close to.5 (such as D = 7:5), the filter gets worse. The results become generally better, when the filter order is increased. Unfortunately the increase is slow as function of filter order, which can be frustrating. In Figure 3 also shows the impulse response of the filter and its root constellation. The thick vertical line in Fig. 3(c) illustrates a virtual unit impulse delayed by 7.6 samples, i.e., it is located between the sample points - the impulse response of the fractional delay filter is approximating a sinc function centered at this point. The root constellation in Fig. 3(d) demonstrates that the fractional delay FIR filter is a non-minimum phase filter, since some of its zeros are outside the unit circle. AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page3of8

4 Phase delay (samples) Impulse response (a) D = Frequency (khz).5 (c) Sample index Phase (rad/pi) Imaginary part (b) Frequency (khz) 5 (d) Real part Fig. 4: Example design of a Thiran fractional delay allpass filter of order 8 for fractional delay of 7.6 samples (fractional part.6 samples): (a) the desired (dashed line) and realized (solid line) phase delay response, (b) the desired (dashed line) and realized (solid line) phase response, (c) the beginning of the impulse response, and (d) the pole-zero plot of the filter (o = zero, x = pole) Allpass fractional delay filter based on Thiran s method Allpass filters are in general more complicated to design than FIR filters. The reason is that the relationship between the allpass filter s properties and its coefficients is nonlinear. By contrast, FIR filters are simpler, because their frequency response is directly the Fourier transform of the coefficients. Luckily, there is also a simple method to design allpass FD filters: the Thiran allpass filter that has closed-form coefficient formulas [9,, 3, 4]. M M a(m)=( ) m m n= M n D M m n D (9) for m = ;;2;:::;M. An example of a fractional delay filter design using the Thiran method is given in Fig. 4. The dashed line in Fig. 4(a) shows the desired delay, which is constant (D = 7.6 samples) at all frequencies. The phase delay of the Thiran allpass filter of order 8 is also shown. Note that the computational complexity (i.e., number of multiplications and additions) is almost the same as that of the Lagrange filter of length 6 studied in Fig. 3. Again, it is seen that the design closely follows the specification at low and mid frequencies, but at very high frequencies the filter s phase delay is heading towards an integer number of 8 samples. Thus, the filter performs well at low frequencies, but poorly at frequencies around 2 khz. Figure 4(b) shows the phase response of the same filter and the desired phase response, which is a decreasing straight line. The deviation is at its largest near the Nyquist limit. We do not show the magnitude response of the allpass filter, because it is always flat. Figure 4(c) shows the first 7 impulse response samples of the allpass filter. As explained above, the thick vertical line illustrates a virtual delayed impulse. It is interesting to see that while the allpass filter s impulse response is infinitely long in theory (as it is an IIR filter), in practice the impulse response of a stable allpass filter, such as this one, decays fast. The shape of the impulse response is quite similar to the one in Figure 3(c), which is known to be a windowed shifted sinc function centered at the virtual delayed impulse. The poles and zeros of the allpass filter are shows in Figure 4(d). The allpass filter is stable, so the poles are inside the unit circle. All zeros are outside the unit circle and their radii are the inverse of the pole radii. A MATLAB toolbox for fractional delay filter design is available at /software/fdtools/ 3. FREQUENCY-WARPED FILTERS The inherent frequency resolution of digital filters (particularly of FIR filters) is uniform over the entire range from zero to Nyquist frequency. Non-uniform resolutions and related frequency scales are found to be more useful, however, in particular cases. Firstly, a system to be simulated by a digital filter may exhibit such a nonuniform character. Secondly, and often more importantly, the human auditory system follows a nearly logarithmic frequency scale, approximated by the Bark scale [] or the ERB (equivalent rectangular bandwidth) scale [2]. These properties can be obtained by frequency warping, overviewed in this section and discussed thoroughly in [3], or by transversal pole-zero filters based on orthonormal basis functions, as discussed in Section 4. The basic frequency warping in Subsections 3. and 3.2 is well matched to the psychoacoustic Bark scale of critical bands [4], while Kautz filters in Section 4. allow for more flexibility to have control over frequency resolution. AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page4of8

5 Normalized warped frequency λ =.8 λ =.6 λ =.4 λ =.2 λ =. λ = -.2 λ = -.4 λ = -.6 λ = Normalized original frequency Fig. 5: Frequency warping of the first-order allpass section D (z) for different values of λ. Frequencies are normalized to the Nyquist rate. 3.. Warped FIR filters The idea of warped filters is best understood by starting from the transversal FIR structure in Fig.. If each unit delay z of an FIR filter is replaced by a new frequencydependent (dispersive) delay element z, the filter can be designed and realized on a non-uniform warped frequency scale. This is possible by using the first order allpass filter D (z) = z λ λz () where λ, < λ <, is a warping parameter and D (z) = z is the dispersive delay element. Figure 5 shows how the frequency warping depends on the warping parameter λ. The bilinear mappings between z- domain and z-domain [3] are s(n)z n z = k λ w(k) λz () n= k= w(k) z k = k= n= s(n) z n + λ + λ z (2) These equations define all necessary mappings between a traditional impulse response s(n) and its warped counterpart w(k). The right-hand side of Eq. () can be considered as warped filter realization (synthesis) formula. while the right-hand side of (2) is for inverse warping (i.e., analysis). Notice the similarity of () and (2), where negation of λ leads from warping to inverse warping or the way around. Mappings between s(n) and w(k) in x D (z) x D (z) x 2 β β β 2 etc. (a) out in + + λ λ (b) z - β β z - z - β 2 etc. out Fig. 6: Warped FIR filter structure: a) with allpass delay elements and b) as a computationally efficient version. are linear but not shift-invariant, so they are easily applicable to system impulse responses with unique time origin but not as easily to arbitrary signals that don t have an inherent zero time moment. Figure 6 shows the basic principle of a WFIR filter and a computationally efficient structure for its implementation. Using the notations of the figure, the z-domain reponse can be written as M H WFIR (z) = h(n) z n M = n= n= β n fd (z)g n (3) Notice also that both mappings () and (2) yield responses of infinite length even if one of the sequences to be mapped is of finite length 2. Since the coefficient sequence β i in (3) must in practice be of finite length, we have to limit M, for example by truncation or windowing. By expanding the bilinear mapping, inherent in the delays D (z) of warped filters, we may at least in theory transform any warped filter into an equivalent traditional structure, such as direct form I or II. Warped filters have, however, advantages that may compensate for the extra complexity of implementation: ffl Filters can be designed directly on a warped frequency scale such as the psychoacoustically motivated Bark scale [4]. ffl Warped structures are more robust and require less precision if the poles and zeros are mapped so that 2 Warped FIR filters have infinite impulse responses since allpass delay elements are internally recursive. Thus the term WFIR is somewhat contradictory but describes well the structural analogy to transversal FIR filters. AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page5of8

6 in + g=/σ σ σ 2 σ 3 λ λ etc. + + β β β2 out Fig. 7: A realizable WIIR structure with first-order allpass delays and a single unit delay. they are more uniformly distributed over the warped frequency scale. ffl The order of the warped filter may in some cases be considerably lower (e.g., 5 times in Bark scale modeling) than a filter designed on a uniform Hz scale. ffl The warping parameter λ may be used as a control parameter for filters where the resonances and cutoff frequencies have to be controlled by a single factor Warped IIR filters A general form for the transfer function of a warped IIR (WIIR) filter is H WIIR (z) = M i= β Λ i i D (z) + R i= α Λ (4) i i D (z) A realization problem appears since delay elements D (z) contain a delay-free component as is seen from the modified form λ 2 z D (z) = λz λ (5) Implementation of Eq. (4) is not possible in a regular way 3 because of delay-free feedback paths when λ 6=. There exist several solutions that make WIIR filters realizable, such as early proposals by Strube [6] and Steiglitz [7]. Imai [8] and later Karjalainen et al [9, 2] 3 In fact, delay-free loops can be computed using the approach presented in [5]. introduced a robust WIIR structure where the delay chain has full all-pass characteristics. The first delay is a unit delay and the other ones are first-order allpass sections. Recursive feedbacks a i (see Fig. 2) are mapped to coefficients σ i which feed back from the outputs of the unit delays of the allpass sections in order to avoid delay-free loops. A gain term g = =σ is also needed. The feedforward part of the WIIR filter can be implemented directly, without modifications, using β i coefficients. The resulting WIIR structure is shown in Fig. 7. The β i coefficients are directly obtained from Eq. (4), while the coefficients σ i, k = ; :::;R+, may be computed from α i using iteration: σ R+ = λα R ; S R = α R ; for i = R;R ; :::;2 S i = α i λs i ; σ i = λ S i + S i ; end σ = S ; =g = σ = λs ; 3.3. Frequency-warped equalizer design One of the most successful applications of frequencywarped filters has been the correction of frequency responses in audio reproduction [2, 5]. Minimum-phase equalization (no phase equalization) using WFIRs and WIIRs is straightforward and results in computationally efficient low-order filters. Figure 8 shows a comparison of three different equalization filter designs. The FIR equalizer is based on ARmodeling (linear prediction) to yield a filter of order 5. It is clearly seen that it results in almost perfect flattening of magnitude response at high frequencies, but low frequencies remain prolematic. The WFIR equalizer in Fig. 8 shows a much better balanced correction on the logarithmic frequency scale than the FIR design. The filter has been designed by prewarping the measured minimum-phase response and then applying linear prediction of order 35 [2]. The WIIR design of order 24 has been obtained by first applying minimum-phase inversion of the measured response and then modeling the inverted response using Prony s method (ARMA-modeling) [2]. A MATLAB toolbox for warped DSP is available at AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page6of8

7 2 25 FIR5 Magnitude (db) WFIR35 WIIR24 ORIGINAL Frequency (Hz) 4 Fig. 8: Comparison of three equalization filter designs applied to measured (ORIGINAL) loudspeaker response: FIR design of order 5, WFIR of order 35, and WIIR of order GENERALIZED TRANSVERSAL FILTERS As shown earlier, frequency-warped configurations in audio signal processing [3] constitute a self-contained tradition that can be traced back to warping effects observed in analog-to-digital signal mappings and digital filter transformations [22]. The concept of a warped signal was introduced to compute non-uniform resolution Fourier transforms using the Fast Fourier Transform (FFT) [23], and in a slightly different form, to compute warped autocorrelation terms for warped linear prediction [6]. The idea of replacing a unit delay element with a first-order allpass filter was subsequently restated and generalized to include warped counterparts of arbitrary linear filter structures [7, 8, 9]. The inherent nonorthogonality of the underlying signal transformation is treated in these considerations in a pragmatic way, if and when needed. It is nevertheless somewhat surprising that the warping tradition did not emerge from the network synthesis approach to rational orthonormal filter structures, pioneered in the 95 s by Kautz, Huggins and Young [24, 25, 26], which in turn were inspired by the work of Lee and Wiener on Laguerre filter synthesis [27]. In the following, basic forms of orthonormal transversal filters are introduced and related to warped filters from the viewpoint of potential applicability to audio signal processing. The Laguerre and Kautz filters to be discussed may look complicated at the first sight. Conceptual complexity is much removed when considering them just as a generalization Fig. 9: The Kautz filter with (possibly complex) poles fa ;a 2 ;:::g, normalization coefficients n i = p ja i j 2, and tap output weights c i. An FIR filter is attained with a i =, whereas a Laguerre filter results for a i =a;jaj <, where tap filters can be replaced by a common pre-filter. of the FIR transversal filter structure. 4.. Laguerre and Kautz filters An orthogonalization process applied to a set of complex exponentials, defined in the frequency domain by a set of stable complex poles fa ;a 2 ;:::g, results in the synthesis filter structure of Fig. 9 [24, 28]. The filter structure is in fact quite intuitive: the original set of parallel blocks generating the exponentials appear as tap-output filters of the transversal structure, where the role of the allpass filter chain is to introduce pole-zero cancelations that ensure orthogonality of the tap-outputs. From the perspective of a particular tap-output, the corresponding input to tapoutput transfer function has zeros that are reciprocal to all poles of the previous blocks. The orthogonality is reflected in the fact that the ordering of the previous blocks is irrelevant and that the choice of subsequent poles is arbitrary. Filter configurations of the form in Fig. 9 are generally called Kautz filters and the transfer function related to a particular choice of poles fa ;a 2 ;:::;a N g is given by Ĥ(z) = N i= c i G i (z) = N i= c i ψp ai a Λ i a i z i j=! z a Λ j a j z ; (6) where c i, i = ;:::;N, are somehow assigned tap-output weights. The time-domain counterpart of (6), the Kautz filter impulse response, is given by N ĥ(n) = c i g i (n); (7) i= AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page7of8

8 Imaginary part (a) phase /rad (b) phase derivative.5.5 Real part log angle 2 (c) Fig. : An example of a real Kautz filter for the choice of poles fa ;a 2 ;a 3 ;a 4 ;a 5 ;a 6 ;:::g, where a is real, a 2 =, and where a 4 = a Λ 3 and a 6 = aλ 5 are complex conjugate pairs. Magnitude / db 2 3 where functions fg i (n)g are impulse responses or inverse z-transforms of functions fg i (z)g. A natural choice of tap-output parametrization in the case of modeling a given target response h(n) or H(z) is to use (Kautz- Fourier) expansion coefficients given by c i =(h;g i ) or c i =(H;G i ), where ( ; ) is an appropriately defined inner product. The use of orthogonal expansion coefficients corresponds to least-square (LS) approximation in the time- or frequency-domain, respectively. Similarly, a LS identification setup is attained as a straightforward generalization of the Wiener filtering (normal equation) configuration. It is also noteworthy that the tap-output weights affect only the zeros of the overall transfer function of Eq. (6). The filter structure in Fig. 9 is in general complex-valued, meaning that the tap-output signals as well as the identified tap-output weights are strictly real only in the case of real-valued poles. The Laguerre filter is an example of a real Kautz filter of the form in Fig. 9, where all the poles are constrained to the same real value, a i = a, < a <, in which case the identical tap-output filters can be replaced by a common pre-filter. The Laguerre filter is nothing but a warped FIR filter equipped with an all-pole pre-filter. As expected, the frequency resolution allocation induced to modeling is in both cases (Laguerre and WFIR) characterized by the allpass phase mapping. The spectral emphasis caused by omitting the pre-filter in WFIRs can be compensated or utilized as a desirable feature [6]. It is interesting to notice that both filter structures degenerate to an FIR filter when a!. Also in the more general Kautz case it is presumed that the model is real-rational, that is, the poles are real or occur 4 Normalized log frequency Fig. : Allpass filter characteristics for varying pole radius damping: (a) five pole sets with logarithmic distribution, (b) phase functions and phase derivatives on logarithmic pole angle scale, and (c) magnitude responses of real Kautz filter tap-outputs for a single set of poles. in complex conjugate pairs. A modified real Kautz filter structure is displayed in Fig., where the block related to a complex conjugate pair of poles is implemented by a second-order section configuration [28]. The desire of utilizing warping more freely than what the first-order allpass mapping offers has been there from the beginning [23]. In particular, a better approximation of the logarithmic frequency scale would be very tempting from the audio engineering point of view. One way to look at the matter is to use the orthogonal structure of Fig. and a distribution of (complex conjugate) poles that reflects the desired frequency resolution allocation, such as a logarithmic spacing of pole angles, complemented with a suitable mapping of pole radii [29, 3]. In contrast to basic warping, where the frequency resolution depends on the single warping parameter λ, in a Kautz filter the resolution should be considered from the perspective of the whole allpass filter chain [3]. Figure characterizes the behavior of the Kautz filter allpass backbone with complex conjugate pole pair angles and radii logarithmically distributed in the z-plain [3]. Pane (a) shows spiral-like pole distributions for five AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page8of8

9 Magnitude / db Pole positions Equalizer Equalized response Measured loudspeaker Frequency / Hz (a).5 Measured loudspeaker response (early part) Samples.5 5 (c).5 Equalized response: poles from inv target Kautz filter order 38 EQ delay=2 samples (b).5 Equalized response: poles from inv target Kautz filter order 38 "minphase equalizer" Samples.5 5 (d).5 Equalized response: including 2 poles at z= (order 5) EQ delay=2 samples Fig. 2: Magnitude response equalization of a loudspeaker response. The 38th order Kautz equalizer filter is constructed with respect to a minimum-phase inverse of the loudspeaker response. different pole radius scalings, each exhibiting decreasing radius with increasing pole angle. Pane (b) depicts the phase function and phase derivative as a function of logarithmic pole angle, i.e., logarithmic pole frequency. The phase function that represents the frequency mapping is almost logarithmic as desired, quite independent of the radius scaling. The phase derivative characterizes the frequency resolution, which is higher at low and lower at high frequencies, also as desired. The pole radii scaling has a quite strong effect on the resolution locally so that a pole closer to the unit circle results in increased resolution in the vicinity of the corresponding frequency. Thus the frequency resolution can be controlled both locally and globally by proper positioning of the pole set. Figure (c) shows the magnitude responses of individual orthonormal tap outputs in a real-valued Kautz filter (right-hand side structure in Fig. ) for a logarithmically distributed pole set. It shows an almost constant Q-value behavior with pairs of responses sharing the same resonance frequency. Such logarithmic distribution of poles is a good first choice in audio applications if nothing more is known about the allocation of required frequency resolution Equalizer design using Kautz filters The present authors have demonstrated the potential applicability of Kautz filters for modeling audio-related measured responses such as instrument body responses [32, 33, 34] and various room responses [35, 36], as well as loudspeaker equalizer design based on inverted minimum phase target responses [32, 3]. A generalization of Samples.5 5 Samples.5 5 Fig. 3: Loudspeaker response equalization in the timedomain using different versions of the direct method of Kautz equalizer design, as described in the text. the FIR LS equalizer configuration [37] was proposed in [29]. This approach can be considered as direct LS Kautz equalizer design, in contrast to an equalizer that approximately models an indirectly constructed inverse of the system [3]. Examples of both approaches are given in the following. In the first example of Kautz filter equalizer design, a DFT-based inverse of a measured loudspeaker response is used to identify the equalizer (indirect method). Instead of simple logaritmic spacing, the Kautz filter poles are generated with respect to a warped response utilizing a specialized iterative method [33]. The equalizer filter order is chosen to be 38 (8 complex conjugate pole pairs and two real poles) and the results in terms of magnitude responses are depicted in Fig. 2. It has also been shown in [29] that the direct method of LS Kautz equalizer configuration, where the cascade of loudspeaker and equalizer identifies a unit impulse, provides a very similar parametrization of the equalizer for the same set of equalizer poles. Possible deviations are due to differences in the treatment of necessary band-edge compensation in the DFT-inverse and correlation analysis, respectively. The early part of the measured loudspeaker impulse response and the LS equalized response are displayed in Fig. 3 (a) and (b), respectively. The minimum-phase equalization obtained removes some of the ripple in response, but the main part is not much corrected towards an ideal impulse. AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page9of8

10 In Fig. 3(c), an LS equalizer is designed with respect to a delay of 2 samples in the target impulse. The pole set that is generated from a minimum-phase target response is not particularly good at producing pure delay components, which results also in inefficiency in magnitude equalization (not shown). A somewhat trivial way to attain better equalization in the time-domain response is to include zeros in the Kautz filter pole set: in Fig. 3(d) the equalizer is equipped with 2 additional poles at the origin, that is, part of the Kautz filter is implemented as an FIR filter substructure as was described in Fig.. This mixed-phase equalization configuration introduces phase correction in addtion to magnituide correction, although the delay required to attain an actual causal inverse is in this case about 2 samples. As yet another example, a loudspeaker plus room response is corrected using Kautz LS equalization. The loudspeaker used had a lower roll-off frequency of about 8 Hz that was compensated in the construction of the Kautz LS equalizer. The room was a listening room of 33 m 2 with fairly well controlled acoustics. A Kautz equalizer of order 24 (2 pole pairs) was designed with logarithmically positioned pole frequencies between 5 Hz and 2 khz. This procedure is described in more detail in [29]. The resulting impulse response and magnitude response are plotted in Fig. 4. The magnitude response is flattened as desired and some of the low-frequency oscillations are damped in the time-domain response, but the peaks corresponding to reflections from surfaces cannot naturally be canceled out by such a low-order equalizer. However, even equalizer orders of 8 2 (4 6 pole pairs) provide useful equalization results in this particular case. A MATLAB toolbox for Kautz filter demos and design is available at demos/kautzeq/. 5. PHYSICS-BASED MODELING TECHNIQUES Regular DSP is based on one-directional interaction between processing elements. When modeling and simulating physical systems, such as loudspeakers and amplifiers, the interaction between elements is (especially in passive systems) two-directional. While computation of transfer functions between signal inputs and outputs in any system can be done by basic DSP, true physical modeling requires different approaches. There exist a number of time-domain modeling paradigms with a physics-based approach, such as finitedifference time-domain (FDTD) modeling, wave digital Amplitude level [db] level [db].5 Room response.5 Time [ms] Amplitude Magnitude response and /3 octave smoothed Equalized response 2 3 Frequency [Hz] 4 Time [ms] Equalized response and /3 octave smoothed Fig. 4: Room response correction using a 24th order Kautz LS equalizer with logarithmically spaced poles (2 complex conjugate pairs of poles). Top panes: early part of original and equalized impulse response. Middle pane: original magnitude response, Bottom: Equalized magnitude response. filters (WDFs), digital waveguides (DWGs), and modal decomposition techniques, just to mention a few. For an overview of these techniques, as applied to musical acoustics and sound synthesis, see [38]. The methods are applicable generically to different physical domains, such as the electrical, mechanical, and acoustical domains. Here we discuss the wave digital filtering approach and briefly also digital waveguides from the point of view of audio reproduction. 5.. Wave digital filters Wave digital filters are a special class of digital signal processing blocks with physically meaningful formulation and parameters. WDFs are best suited for modeling lumped element systems, although they can be extended to distributed systems, for example for simulating multidimensional systems [39]. The WDF technique was formulated by Alfred Fettweis in the late 96s [4] for discrete-time modeling of analog electric circuits. For a tutorial on WDFs, see [4]. Only recently wave digital filters have been systematically applied to modeling of various physical systems. This section gives a short overview on WDF modeling, while a more thorough study using the same approach can be found in [38] Analog elements with wave ports As discussed in [38], there are two different ways of rep- AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page of 8

11 resenting a physical system: by using Kirchhoff or wave quantities. The former uses actual physical quantities such as voltage and current, or pressure and velocity, as system variables, while the latter uses wave components of these variables. The wave formalism is based on the traveling-wave solution of the D-wave equation, as introduced by d Alembert in 747. The physical Kirchhoff quantities are obtained as a superposition of the wave variables, if desired. The behavior in a Kirchhoff-based representation is usually defined by impedance or admittance, i.e., the ratio between related Kirchhoff variables in the frequency domain. The operation in a wave-based representation, however, is characterized by port reflectance, i.e., the frequency-domain ratio between the wave leaving the port of the element and the incoming wave. For linear and time-invariant (LTI) systems, the reflectance is represented by the elements s port transfer function. In practice, the transform between actual physical quantities and wave variables is carried out using the Kirchhoffto-wave transform» a(t) b(t) = R p i(t)» Rp» u(t) ; (8) where u(t) and i(t) represent a Kirchhoff pair related to the port, voltage and current in this case. Variables a(t) and b(t) denote the incoming and outgoing waves, respectively (voltage waves in this case). Variable R p is a computational port resistance that can have an arbitrary nonnegative value. It must be noted that the port resistance does not represent the actual physical impedance of the system, but is rather an additional degree of freedom which will be used in making the model computable, as will be shown later. As discussed above, the system s reflectance can now be given as S(s) = L fb(t)g L fa(a)g = Z(s) R p ; (9) Z(s)+R p where L denotes the Laplace transform, s is the complex frequency variable, and Z(s) is the actual physical impedance of the system. For an inductor, for example, the impedance is Z L (s) =Ls, where L is the inductance in Henrys, and so the reflectance is S L (s) = s R p=l s + R p =L : (2) Wave digital elements For digital simulation, the analog reflectance of Eq. (9) must be discretized. Wave digital filters use the bilinear transform for obtaining the discrete-time reflectance. For the inductor example in Eq. (2), the discrete-time reflectance becomes S L (z) = (2=T R p=l) (R p =L + 2=T)z ; (2) (2=T + R p =L)+(R p =L 2=T)z where T is the temporal sampling interval. The transfer function in Eq. (2) now shows that there is an instantaneous dependency between the input and output due to the first term of the numerator. By selecting a port resistance value R p = 2L=T for the inductor, the digital reflectance reduces to S L (z) = z : (22) Similarly for a capacitor, selecting the port resistance as R p = T =(2C), where C is the capacitance in Farads, yields S C (z) =z (23) for the discrete reflectance. For a resistor, a port resistance value R p = R (Ohms) gives the discrete reflectance S R (z)=: (24) In other words, the input wave to a WDF resistor does not generate any output wave. It can be seen above that by properly selecting the port resistance values, the WDF realizations of the analog circuit elements become extremely simple and the instantaneous input-output dependency that generally causes delay-free loops is avoided. Table summarizes the element impedance values and the resulting WDF elements for the basic one-port electric components and for the ideal transformer as an example of two-port elements. The wave digital formalism is known to have many advantageous features, such as element-wise localization of computation, numerical robustness, and stability of passive networks [4]. It must be noted, however, that since the bilinear transform maps the analog frequency axis in the s-domain onto the unit circle in the z-domain, the infinite analog frequency is mapped to the Nyquist frequency, which causes frequency warping at high frequencies. This warping takes place in the system transfer functions (not in signal frequencies) so that for example the resonances of the system to be modeled are shifted AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page of 8

12 Table : Table of the basic electric components (column 2) and their WDF realizations (column 5). Columns 3 and 4 contain the element impedances and the corresponding reflectances, respectively. For the source components, e denotes the voltage generated by the source. For ideal transformers g equals the turns ratio. (After [38]). Element type Analog element Impedance Reflectance Wave digital element Inductor Z L = 2L=T S L (z) = z Capacitor Z C = T =(2C) S C (z) =z Resistor Z R = R S R (z) = Open circuit Z oc = S oc (z) = Short circuit Z sc = S sc (z) = Voltage source Z u = S u (z) =2e Current source Z i = S i (z) = 2e i Transformer Z 2 = g 2 Z lower than desired. The warping effect can be compensated by oversampling or by prewarping the desired system frequency response Adaptors and wave digital networks A convenient feature of WDFs it their modularity, where circuits and networks can be easily built from elements. The one-port WDF elements listed in Table can be AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page 2 of 8

13 Fig. 5: A simple electric circuit (a) and its WDF implementation (b). In (b), the RLC elements are illustrated as one-port blocks, which are connected using three-port adaptors. The lower adaptor (marked with ffij ) denotes a series connection. The upper adaptor (marked with k ) stands for a parallel connection. interconnected using special adaptor elements. These adaptors come in two types, serial and parallel, and they implement the wave scattering between the elements for serial and parallel connections, respectively, fulfilling the Kirchhoff s energy conservation laws. A straightforward implementation method is to connect the one-port elements using three-port adaptors, so that the WDF circuit becomes a binary tree [42]. This approach is explained further in the following. Denoting the port index as m, the reflectance rules [4] give b m = a m 2R N m N k= R a j (25) k j= for the series connection and b m = 2 N j= a j (=R j ) N k= (=R k ) a m (26) for the parallel connection. In the above equations, R k denotes the resistance of port k, which must equal the resistance of the one-port element connected to it, and the total number of the ports is denoted by N. Obviously, N = 3 for three-port adaptors. Figure 5 illustrates a simple electrical circuit and its WDF realization. Since the signal flow between each port is bidirectional, special scheduling is needed in order for the WDF network to be computable. This can be implemented using reflection-free ports, where each adaptor will have one port whose output does not depend on its input. Figure 6 shows symbols for a series and a parallel tree-port adaptor with a reflection-free port denoted by j. Formally, for a port k of a series adaptor to be reflectionfree requires R k = N j=; j6=k R j and port input-output relations become Fig. 6: (a) Series and (b) parallel tree-port adaptor with reflection-free port marked by j. N b k = a j ; (27) j=; j6=k for the series reflection-free port and b = a n;n6=k n R n (a R k b k ) (28) k for other ports. For the parallel adaptor, port k is reflection-free if =R k = N j=; j6=k (=R j ), and the port inputoutput relations become N a j b k = R k ; b j=; j6=n R = b n;n6=k k + a k a n: (29) j In practice, scheduling a WDF network as a binary tree means that one of the one-port elements is chosen as a root node (without reflection-free port), while other oneports are the leaves and adaptors are the nodes of the binary tree. In Fig. 5 the WDF diagram is not yet made a directed binary tree, while the WDF models in Figs. 7 and 8 are binary trees with a root node. At the first step of computation, the wave components traveling towards the root are evaluated. This can be implemented using reflection-free ports, where each adaptor will have one port whose output does not depend on its input. When all the adaptors have a reflection-free port on the signal path towards the root node, all waves traveling towards the root can be evaluated without calculating any other waves. After the wave has reached the root node, a reflecting wave is calculated and propagated towards the leaves Modeling of nonlinearities If the system to be modeled by WDFs is LTI, the port resistances remain constant throughout the simulation, and the control flow of computation is as described above. For a nonlinear WDF element, the port resistance changes as a function of the incoming wave. This is more problematic in the implementational point of view, since all port resistances through the adaptor tree structure towards the root element need to be updated. This requires AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page 3 of 8

14 V + C i R p V p R g C o Vg Vgk b + N.L. + a triode z - Rk Vk V i R i (a) V g V k R k C k V o R o V+ = 25 V Rp = kω Ro = MΩ Co = nf Rk = kω Ck = µf (b) V+ E/Rp Ck Co Ro Vo Fig. 7: A triode vacuum-tube amplifier stage (a) and its WDF realization (b). In (b), the vacuum-tube is simulated as a nonlinear resistor (root node marked N.L.) using a lookup table, and the input circuit of the tube stage is omitted. (Figure adopted from [43]). extra processing, but more challenging is the implicit equations (i.e., delay-free loops) to be computed. This is because the wave variables can be computed only when port impedances are known, which on the other hand are dependent on wave variable values. An elegant special case solution is available in the binary connection tree approach [42], where a single nonlinear resistive element is located at the root node of a binary tree network. Then no other parts of the network need to change port resistances. The nonlinear wave reflection a = f (b) can be computed either by a function approximating the desired behavior (often not easy to find), by table lookup and interpolation (fast and robust), or by iteration (may have converge problems). If the nonlinear root element is reactive (capacitive or inductive), special mutator elements can be used [44]. In a general case of more than one nonlinearity 4 the new values of port resistances and wave variables need to be solved iteratively, and the energy preservation and stability requirements need to be met if the system to be modeled is passive. The change of port resistance in a reactive element will change its stored energy, unless so called power-normalized waves [45] or controllable ideal transformers [46] are used. Iteration of the parameters and variables can become computationally expensive and in complex cases it is difficult to guarantee the convergence of iteration. Another 4 More than one nonlinearity is no problem if they are separated by two-port delay-lines. possibility is to approximate new signal or parameter values by prediction if the variables change smoothly and regularly. A simple solution is to add an unphysical delay, for example to use the previous value of port resistance instead of a new one. This should be used with caution from a stability point of view. Oversampling is also useful in such case, because a high enough oversampling factor approximates full iteration. Oversampling is often needed in nonlinear simulation anyhow, because it may be the only way to reduce audible distortion due to aliasing WDF modeling of vacuum-tube amplifier A WDF model for a triode tube stage, commonly found for example in guitar amplifiers, has been introduced in [43]. A circuit diagram of the triode stage is illustrated in Fig. 7(a). The WDF model is obtained by treating the vacuum-tube as a nonlinear resistor, whose current depends on the grid-to-cathode and plate-to-cathode voltages V gk and V pk, respectively. The input circuit, containing components C i, R i, and R g is omitted for simplicity, and the input signal to the tube stage is inserted directly as the grid voltage V g. The resulting WDF network is a binary tree, containing the tube element as its root node, as illustrated in Fig. 7(b). There has been need for one unphysical delay from cathode voltage v k to to the grid circuit, on top of Fig. 7(b), in order to avoid a delay-free dependency when computing the grid to cathode voltage. This makes little harm because the cathode voltage changes only slowly due to capacitor C k. The WDF tube stage has been implemented as real-time AES 32 ND INTERNATIONAL CONFERENCE, Hillerød, Denmark, 27 September 2 23 Page 4 of 8