MODELING SALLEN-KEY AUDIO FILTERS IN THE WAVE DIGITAL DOMAIN

Mattia Verasani, Alberto Bernardini and Augusto Sarti

Dipartimento di Elettronica, Informazione e BioingegneriaPolitecnico di Milano, Piazza L. Da Vinci 32, 20133 Milano, Italy

Email: mattia.verasani@mail.polimi.it, alberto.bernardini@polimi.it, augusto.sarti@polimi.it

ABSTRACT

Sallen-Key filters are widespread in audio circuits. Therefore, ac-curate and efficient digital models of such filters are highly desir-able in audio Virtual Analog applications. In this paper, we will dis-cuss a possible strategy, based on Wave Digital Filters (WDFs), forimplementing all the analog filters described in the historical 1955manuscript by Sallen and Key. In particular, we will group the eigh-teen filter models presented by Sallen and Key into nine classes, ac-cording to their topological properties. For each class we will de-scribe the corresponding WDF structures. Finally, we will comparethe output signals of WDFs to the output signals of the same modelsimplemented in LTSpice.

Index Terms— Virtual Analog Modeling, Wave Digital Filters,Audio Filters

1. INTRODUCTION

Sallen-Key filters [1] are used pervasively in analog audio circuitry[2]. Therefore, accurate and efficient digital implementations of suchfilters are extremely useful in audio and music applications based onVirtual Analog (VA) modeling [3]. Some approaches exist in the lit-erature for digitally implementing Sallen-Key filters [4–6], but theyare mostly based on discretizing the transfer function, which charac-terizes the frequency response of the reference filter [4]. Moreover,using these kinds of digital filters, the information about the actualtopological structure of the reference analog filter is not preservedin the digital domain. Therefore, such approaches are probably notthe best solution for obtaining modular digital models characterizedby physically-meaningful parameters. Instead, Wave Digital Filters(WDFs) [7] are suitable for these purposes [8]. Moreover, WDFsare known to be an excellent tool for realizing efficient, accurate andstable digital implementations of linear and Non Linear (NL) refer-ence analog circuits [9–12], as they turn systems of implicit Kirch-hoff equations into a computable graph constituted of blocks, whichare connected through input-output relations [13]. For these rea-sons, they have been widely used in the literature on sound synthesisthrough physical modeling and interactive VA modeling [14]. How-ever WDFs present also some limitations. In fact, in many casesnon-computable Delay-Free-Loops (DFLs) arise also in WDF struc-tures. They may be caused by the presence of multiple non-adaptableNL elements or by complex topologies [15–17]. For accommodatingmultiple nonlinearities, many methods are under study: approachesbased on multi-port NL elements [18–21], iterative methods [8] ordynamic adaptation [22]. For facing topological problems, the ap-proach based on a graph decomposition of the reference circuit byFranken et al. [15] is quite effective, as the circuit can be representedas a computable structure constituted of series, parallel and R-type

junctions. R-type junctions can be implemented using one of thethree methods presented in [16, 17, 23].

In order to effectively implement Sallen-Key filters in the WaveDigital (WD) domain, besides having to accommodate complextopologies, it is also essential to rely on efficient and accurate op-erational amplifier (op-amp) models. In the literature on WDFsthere are only few works in which circuits containing op-amps arediscussed; [24], [25] and [26]. In particular, in [26] a ModifiedNodal Analysis (MNA)-based method is described for accommo-dating arbitrarily complex linear circuits containing op-amps, alongwith two possible op-amp models; one ideal model based on nullorsand one more complex linear macromodel containing some linearnonidealities. In the light of the approach described in [17, 26], inthis paper, we will analyze all the Sallen-Key filters and we willderive their WD implementations, using a simplified, though accu-rate, real op-amp macromodel. The paper is organized as follows.In Section 2 we will group the eighteen Sallen-Key filters presentedin [1] into nine classes, according to their topological properties.In Section 3, we will briefly describe how to proceed for modelinga generic circuit with complex topologies in the WD domain. InSection 4, for each one of the nine classes defined in Section 2, wewill show how to build the corresponding WDF model, followingthe approach presented in [17, 26]. For modeling op-amps, we willuse a static small-signal macromodel, which is simpler w.r.t. thelinear macromodel presented in [26]. In Section 5, we will testthe derived WDF models through some examples of applicationsand we will show the resulting output signals are nearly identicalto the corresponding LTSpice implementations. Finally, Section 6concludes this paper.

2. SALLEN-KEY FILTERS CLASSIFICATION

Sallen-Key filters are second order electronic active filters contain-ing op-amps, introduced in the 1955 manuscript [1] by Sallen andKey. They have been widely used, not only in audio circuits [27],for implementing low-pass, high-pass and band-pass frequency re-sponses. In the original work [1], such filters involved only resistorsand capacitors. In [1] a ‘‘catalog of second order active networks”with eighteen numbered filters is presented. Analyzing those filterswe identified nine recurring topological patterns, which are shown inFig. 1. The white rectangular blocks indicate generic impedances Zh(with h an integer), which could also be combinations of elements inseries or parallel. The numbered filters in the Sallen and Key catalogare classified in Fig. 1 as follows: (a) refers to filters 1, 3 and 9, (b)refers to 5 and 7, (c) refers to 10 and 13, (d) refers to 11 and 14,(e) refers to 16 and 17, (f) refers to 12 and 15, (g) refers to 6 and8, (h) refers to 2 and 4, while (i) refers to 18. As it is evident fromFig. 1, in the classification process, we considered also the op-amp

−

+

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1

(h)

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+

Z9

Z7

Z5Z8

Z4

Z6

−

+

Z9

Z7

Z5Z8

Z4

Z6

1

(i)

Fig. 1. Nine classes of Sallen-Key filters with different topologies.

polarity as a discriminative property. Therefore, the circuits pairs(a)–(b), (e)–(f) and (g)–(h) share the same topological structure, butthe positive and negative input terminals of the op-amp are swapped.

3. WDF MODELING OF CIRCUITS

WDFs are derived performing a lumped discretization of a referenceanalog circuit and applying a port-wise linear transformation. If thereference circuit is a N -port network, the vectors of port voltagesv = [v1, . . . , vN ]t and port currents i = [i1, . . . , iN ]t are mappedinto the vectors of incident waves a = [a1, . . . aN ]t and reflectedwaves b = [b1 . . . bN ]t, as follows

a = v + Ri , b = v −Ri , (1)

where R = diag[R01, . . . R0N ] is the diagonal matrix of theport free parameters, called reference port resistances. Often itis also useful to define the port admittance matrix G = R−1 =diag[G01, . . . G0N ]. In the WD domain, each element becomes ablock characterized by an input-output relation [7] and the topo-logical connections between the elements are implemented usingmulti-port junctions, called adaptors [28]. How to accommodatethe most spread linear circuit elements (e.g. real sources, resistors,capacitors) and traditional series and parallel adaptors is extensivelyexplained in [7, 28]. In order to solve more complex connectionsthan series and parallel, the method by Franken et al. [15] is usedfor turning the reference circuit graph into a SPQR-tree structure,in order to identify the S-, P-, Q- and R-type portions. Then, thecorresponding adaptors of the R-type nodes should be derived us-ing one of the approaches described in the three works [23], [16]and [17]. As pointed out in [25] in Fig. 8, where a circuit witha simple op-amp model is studied, in some cases a replacementgraph [15] must be added to the circuit graph representation inorder to embed multi-port active elements into R-type nodes andavoid DFLs. Therefore, especially in some particular cases wherecontrolled sources are involved, the implementation of hybrid activejunctions, which embody both topological connections and activemulti-port elements, seems unavoidable for ensuring computability.The MNA-based [29] method presented in [17, 25, 26, 30] allows usto derive the scattering matrix S of such active junctions as follows

S = I +[0 R 0

]X−1 [0 I 0

]t with b = Sa , (2)

where I is the identity matrix, 0 is a matrix of zeros with the appro-priate size and X is the so called MNA matrix. The MNA analysis isapplied to the circuit corresponding to the R-type components of theSPQR-tree to which ports are attached Thevenin equivalents. Theideal voltage sources of the Thevenin equivalents are ordered in thevector e = [e1, . . . , eN ]t , while the corresponding branch currentsare j1 = [j1, . . . , jN ]t. Applying the analogy, which is described in[17], using the concept of instantaneous Thevenin equivalents, andalso explained in [18], under the name of Kirchhoff Domain Analogy,we set e = a, j = [jt1, j

t2]t = [−it, jt2]t, where j2 are other branch

currents not passing through the instantaneous Thevenin equivalents(e.g. through Voltage Controlled Voltage Source (VCVS)), and weset the port resistances equal to the Thevenin equivalents series resis-tances. More precisely, X satisfies the following system of equations

X

[vn

j

]=

[Y AB D

] [vn

j

]=

[ise

](3)

where the X partitions Y, A, B, and D define the relation amonge, j, the node voltages vn and the current source values is, which inour context are always zero. Once all the needed WD elements andadaptors are derived, systematic methods for initializing and imple-menting generic WDF structures could be used, as explained in [13].

4. SALLEN-KEY FILTERS IN THE WD DOMAIN

In this Section, we describe how to apply the WDF approach re-sumed in Section 3, in order to implement all the Sallen-Key topolo-gies. The MNA method, described in Section 3, would work inde-pendently on the used op-amp macromodel. However, in this paper,we will describe op-amps with the linear small-signal static macro-model in Fig. 2. Rin is the finite, typically very high, input resis-tance. Rout is the non-zero, typically very low, output resistance.A0 is the finite open-loop gain. Rin, Rout and A0 are usually spec-ified in the reference op-amp datasheet. The VCVS is controlled by

Rin

+

−

RoutC

G

A0vid

A

B

+

−

vidRin

+

−

RoutC

G

A0vid

A

B

+

−

vid

1

Fig. 2. Op-amp linear static small signal macromodel.

the signal A0vid. The nodes A, B, C and G indicate the terminalsof the op-amp macromodel.

Let us now consider the nine models shown in Fig. 1. For eachfilter in Fig. 1 we connect a real voltage source Vin to the inputport and a load resistance Rload to the output port. In order to de-rive the corresponding WDF structures, we have firstly to build therelative graphs. Then we find the split components of the SPQR-tree structure, forcing the nodes A, B, C and G to be part of aR-type node, using a replacement graph, as explained in [15] andas shown also in Fig. 8 in [25]. A graph representation of the cir-cuits in Fig. 1 is shown in Fig. 3; the fictitious edges of the replace-ment graphs are colored in gray. We labeled the edges of the graphsusing the Zh impedances subscripts (with h an integer) shown inFig. 1, while we always assign the label 1 to the element Rin, 2 toRout and 3 to Rload. We named the different R-types using differ-ent subscripts. From the circuit graph representation, it is now easyto build the scheme of the corresponding WDF structures. As anexample, in Fig. 4 we show the WDF structure resulting from theSPQR-tree in Fig. 3.d. In order to find the scattering matrix of the R-node Rτ , we perform the MNA analysis of the circuit in Fig. 5, i.e.the R-node circuit with instantaneous Thevenin equivalents attachedat each port. The MNA matrix relative to Rτ is called Xτ and,according to Fig. 5, if we set vn = [vm, vn, vo, vt, vA, vC , vB ]t,j = [−i5,−i4,−i2,−i3, jV CV S ]t, e = [a5, a4, a2, a3]t, the corre-sponding Xτ partition matrices are: Dτ = 0,

Yτ =

G05 0 0 0 −G05 0 00 G04 0 0 −G04 0 00 0 G02 0 0 −G02 00 0 0 G03 0 0 −G03

−G05 −G04 0 0 (G05 + G04) 0 00 0 −G02 0 0 G02 00 0 0 −G03 0 0 G03

,

Aτ =

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 00 0 0 0 10 −1 −1 0 0

, Bτ =

1 0 0 0 0 0 00 1 0 0 0 0 −10 0 1 0 0 0 −10 0 0 1 0 0 00 0 0 0 −A0 1 A0

.

Finally we use equation (2) for finding Sτ . All the other scatteringmatrices of the R-type nodes in Fig. 3 can be derived similarly.

As in traditional adaptors, also one port of such R-type nodescan be made reflection free. For instance, in our example, we adaptthe left port of Rτ by setting G05 = G04G02+G04G03

G04+G02+G03+A0G02.

5. EXAMPLES OF APPLICATIONS

In this Section, we analyze the behavior of the derived WD Sallen-Key models, feeding them with a linear sinusoidal up-chirp signalwith f1 = 20 Hz and f2 = 2000 Hz as starting and ending fre-quencies. The amplitude of that signal is 3 V. A zoomed portion ofthe up-chirp is shown at the top of Fig. 6. The output signals arecompared to the outputs of the corresponding LTSpice simulations.

B

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Figure 1: Sallen-Key lowpass filter1

1

(c)

RτS1 P1

P2S2B

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Figure 1: Sallen-Key lowpass filter1

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Figure 1: Sallen-Key lowpass filter1

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Figure 1: Sallen-Key lowpass filter1

1

(i)

Fig. 3. Finding split components of the nine filters in Fig. 1.

Rτ

Z11

Vin Rload

Z9 Rout Rin Z8

Z7

Fig. 1: Sallen-Key lowpass filter with static small-signalmacromodel wave digital structureFig. 4. WDF structure relative to the graph in Fig. 3.d.

We use a LTSpice model of the LM324 op-amp, which is very com-mon in some audio synthesizers, such as the Korg Monotron [31].The needed parameters are taken from the LM324 datasheet [32].The LM324 LTSpice model is employed in all the circuit simula-tions. According to the LM324 datasheet, we set the small-signalstatic macromodel parameters to Rin = 106 Ω, Rout = 10 Ω andA0 = 105. We implement variants of the filters in Fig. 1. Here weshow the results of five significant Sallen-Key configurations. Thesampling rate is Fs = 48 kHz. For the sake of brevity, we define

+−a′

5

R′5

+ −

a′4

B

R′4

A

−

+ +

−

R2

G

C

+ −

a2

+−a3

R3

Fig. 1: Sallen-Key lowpassfilter

Fig. 5. Kirchhoff Domain Analogy Circuit for MNA Analysis (re-ferred to Fig. 3.d).

the capacitance parameter C = 0.1 nF and the resistance parameterR = 100 kΩ. In the corresponding figures captions, we will specifythe actual filter impedances values used in each simulation. As faras the following plots are concerned, we define the Vload LTSpicesignal as Vload and the same signal taken from the WDF structureas Vload. In each plot we compare Vload to Vload and we show thesquared error signal V 2

error = (Vload − Vload)2.

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

Vin

[V]

-2

0

2 WDS

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

Vlo

ad

[V]

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2WDSSPICE

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

V2 erro

r[V

2] #10-3

2

6

10 Squared Error

Fig. 6. Top: Up-chirp signal Vin used for all the simulations. Center:Vload signal WDF model vs LTSpice. The reference model is theone in Fig. 1.a; in particular Z4 = 1/sC, Z5 = R, Z6 = 1/sC andZ8 = R. Bottom: Squared Error signal WDF model vs LTSpice.

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

Vlo

ad

[V]

-2

0

2 WDSSPICE

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

V2 erro

r[V

2]

0.02

0.04Squared Error

Fig. 7. Vload WDF model vs LTSpice. The reference Sallen-Keymodel is the one in Fig. 1.c; in particular Z4 = 1/sC, Z5 = R,Z6 = 1/sC and Z8 = R.

6. CONCLUSION AND FUTURE WORK

In this paper, we analyzed the eighteen RC active filters proposedby Sallen and Key in their original manuscript and we divided them

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

Vlo

ad

[V]

-2

0

2 WDSSPICE

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

V2 erro

r[V

2] #10-3

51015 Squared Error

Fig. 8. Vload WDF model vs LTSpice. The reference Sallen-Keymodel is the one in Fig. 1.d; in particular Z7 = 1/sC, Z8 = R,Z9 = 1/sC and Z11 = R.

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

Vlo

ad

[V]

-2

0

2 WDSSPICE

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

V2 erro

r[V

2] #10-5

1

2

3Squared Error

Fig. 9. Vload WDF model vs LTSpice. The reference Sallen-Keymodel is the one in Fig. 1.e; in particular Z4 = 1/sC, Z5 = R,Z6 = 1/sC and Z7 = R.

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

Vlo

ad

[V]

-0.5

0

0.5 WDSSPICE

T ime[sec]0.02 0.04 0.06 0.08 0.1 0.12

V2 erro

r[V

2] #10-4

5

10

15Squared Error

Fig. 10. Vload WDF model vs LTSpice. The reference Sallen-Keymodel is the one in Fig. 1.g; in particular Z4 = R/3, Z5 = R/3,Z6 = 1/sC, Z7 = 1/sC, Z8 = R and Z10 = R/3.

into nine possible classes, according to their topological properties.Then we showed how to develop the parametric WDF model foreach class. Finally, we showed that the WDF output signals greatlymatch the corresponding LTSpice simulation results, even thoughsome small errors are present. Such errors are not necessarily pro-duced entirely by WDFs. In fact, they might be partially due tothe numerical inaccuracies of the iterative solvers employed by LT-Spice and/or the re-sampling process (from non-uniform sampling touniform sampling) performed on LTSpice output signals. The pre-sented models are highly modular, independent from the actual filterimpedances and they preserve the topological properties of the ref-erence analog circuits. Moreover, as all traditional WDFs, they aresuitable to be used for interactive VA applications [3, 14].

As far as future work is concerned, we are planning to apply thesame methodology in order to derive WD models of more complexfilter topologies involving op-amps, such as biquad filters.

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