Preserving the LTI system topology in s- to z-plane transforms

Vadim Zavalishin∗

May 5, 2008

Abstract

A method for preserving the LTI system topologyduring the application of an s- to z-plane trans-form is described. The produced discrete-time sys-tems keep the important characteristics of the orig-inal continuous-time systems in cases of modulatedparameters. A further extension to the case of con-tained nonlinearities is made.

1 Introduction

In music DSP applications, two approaches for con-versions of continuous-time (s-plane) LTI systemsinto the discrete-time (z-plane) case are widely used.

The first approach is a general DSP one. It as-sumes the application of an s- to z-plane transformsubstitution, usually the bilinear one, to the system’stransfer function H(s). The resulting discrete-timetransfer function H(z) is subsequently decomposedinto biquad sections, with optional 1st-order sections,which are then implemented in one of the classicdiscrete-time forms such as direct or canonic one, in-cluding transposed versions of those. The transferfunction H(z) is obviously in the desired relation toH(s), which in case of the bilinear transform can beconsidered ‘reasonably perfect’. However, the specificset of state variables and the topology of the origi-nal continuous-time system are completely ignored inthis approach. As a result, the obtained discrete-timesystems usually exhibit strong undesired artifacts ifthe system parameters are modulated, especially ata high rate, which is not unlikely in the music DSP

∗Native Instruments GmbH

area. Besides, the completely different system topol-ogy makes it impossible to model the nonlinearitiescontained in the original system.

The second approach is rather music DSP specific.It starts with decomposing the s-plane system intolower-order elements such as integrators or one-polefilters, which are then converted to the z-plane us-ing the transform approach. For those lower-ordersystems, one can often construct topologies, whichfeature nice modulation behaviors. These lower-order discrete-time blocks are then connected to eachother in exactly the same way as their continuous-time counterparts. Therefore, the state variables andthe system topology are preserved to a good extent,which usually ensures good modulated-case behav-ior and also makes it possible to clone nonlinear ele-ments present in the original system. Unfortunately,the resulting systems often turn out to have delaylessfeedback loops, which means they are not directly im-plementable. A trivial approach is commonly in usehere, to simply put additional z−1 delays into the of-fending feedback paths. This often results in accept-able transfer functions, except for the high frequencyareas and/or certain values of system parameters,where the transfer function gets distorted beyond rea-sonable, often leading to unstable systems. The usualways to counteract these effects are to clip the pa-rameters to safe ranges, to design specific methodsfor the parameter prewarping ([1]), and/or to simplyincrease the sampling rate.

Apparently, a third possiblity exists, which com-bines the benefits of the first two. Surprisingly, it ismuch less known. Particularly, the important workson music DSP filter design, such as [1] and [2], makeno mention of it. It seems, as if only brief hints to this

1

MMqq// + '!&"%#$// •//

z−1

��

•��

OO + '!&"%#$// //OOωc/2

Figure 1: Bilinear integrator.

approach, such as [3] exist.12 This article is supposedto fill the gap.

2 The basics

We begin by mentioning the fact, that any differ-ential LTI system can be ‘implemented’ in a struc-ture diagram consisting only of adders, gain elementsand integrators, with a possibility to extend the spec-trum of such systems by allowing delay elements. Ofthese, adders, gain elements, and delays are natu-rally present in discrete-time systems as well. Theintegrators need to be converted by means of somes- to z-plane transform, the bilinear one being prob-ably the best choice. A good possible topology for abilinear-transformed integrator H(s) = ωc/s, whereωc is the cutoff, is shown in Fig. 1 (in some cases,the opposite order of the IIR and FIR parts could beused).

Thus, all basic elements are available for thediscrete-time case as well. So, for a given continuous-time structure we simply need to replace the integra-tors by their digital models. Notice, that the trans-form used to convert the integrators thereby becomesthe transform applied to the entire system.

Let’s take a basic 1-pole lowpass RC-filter as anexample. Such filter can be completely equiva-lently represented in the state-variable form (Fig. 2).Replacing the integrator with its digital model, suchas the one in Fig. 1, we obtain a bilinear-transformed

1While [3] is the first known to the author reference to thediscussed approach, the author would like to mention that hehas come up with a similar idea independently.

2Due to the limited author’s acquaintance with the respec-tive literature, it might as well be that the approach is alreadydocumented elsewhere. Anyway, the author hopes that this ar-ticle will not be useless.

+ '!&"%#$//∫

// •// //

oo−

OO

Figure 2: Equivalent state-variable representation ofa 1-pole lowpass RC-filter. The cutoff of the integra-tor is set to ωc = 1/RC.

+ '!&"%#$// G(z)// •// //

qqMM oo−

OO

k

Figure 3: A generic discrete-time system with feed-back.

digital implementation of the original filter. Noticethat the transfer function H(z) of the resulting filteris exactly the result of bilinear-transform substitutioninto H(s). However, since the integrator in Fig. 1contains a delayless path from input to the output,our digital filter is not directly implementable. Thisis pretty much the intermediate result we would haveobtained from the second approach mentioned in theintroduction.

3 The method

Now, instead of inserting a z−1 delay into the feed-back, we are going to claim that the digital filter cor-responding to Fig. 2 is implementable. We can evenextend it to a somewhat more general case, picturedin Fig. 3.

Let’s consider a fixed discrete time moment n, atwhich we consider the output signal value y = y[n]as a function of the input signal value x = x[n]. Forany difference LTI system, and for G(z) in Fig. 3 inparticular, this function will have the linear form:

y = f(x) = gx + s

where x is the system input, y is the system output,the offset s is defined by the state of the system andby the system parameters, and the instant gain g is

2

defined by the system parameters. Both s and g arefixed at a given n. Typically, g > 0. Often, g ≤ 1.

Therefore, for the entire system in Fig. 3 one couldwrite:

y = f(x− ky) = g(x− ky) + s (1)

where x and y are this time the input and outputof the entire system. Instead of rushing to solve theabove equation, we first examine its meaning.

In the analog world, the signals cannot abruptlychange their value. Each element of the circuit,including wires, would have a (possibly negligiblysmall, but still nonzero) complex impedance, all suchimpedances working together to smooth the suddenchanges.

It is suggested to treat the digital structures witha similar approach. E.g. in the case of (1) we wouldnot assume that y instantly takes the value equal tothe solution of the equation. Initially y would retainits value from the previous sample y0 = y[n − 1].Then f(x − ky0) would be computed according to(1) producing a new desired value for y. Now thevalue of y will start changing in a continuous waytowards this new value. Simultaneously f(x − ky)will be changing further, correcting the ‘destination’value for y until both of them coincide, in which casethe system has reached the state corresponding tothe solution of (1), all of that happening within theduration of the current sample.

Solving (1) according to the above spirit we get

y =

gx + s

1 + gkif 1 + gk > 0

+∞ · sgn[(gx + s)− (1 + gk)y0] otherwise

(2)

Notice that for a typical G(z), the system would beunstable for k ≤ −1 anyway, although if 1 + gk > 0such instability would take more than one sample toreach the infinity.

Thus we have just described a way to treat the digi-tal system in Fig. 3 as implementable. If the feedbackpath doesn’t contain any nonlinearities, then, by in-corporating the explicit computation of (2) into thesystem structure, the latter can be converted into anequivalent structure without delayless feedback. Of-

+ '!&"%#$//∫

// •//∫

// •// //

��

+ '!&"%#$oooo

−OO

11 ��

��

55kkkkk1/Q

y

Figure 4: 2-pole state-variable lowpass filter.

ten however, one might simply use (2) during theimplementation of the system in program code.

An extension of the method to the cases of multiplefeedback paths is simple, if all paths share at leastone common point. E.g. for a 2-pole state-variablefilter (Fig. 4) we simply can (temporarily) choose thefeedback signal as y, again resulting in an equationof the form (1) (with k = 1).

The case of a system, where some interdependentdelayless feedback loops do not have a common point,is somewhat more complicated. Still, it is always pos-sible to write a linear equation or a system of linearequations analogous to (1). The extension of the ap-proach to this case is clear.

Also, often it is possible to separate the ‘smaller’loops into LTI subsystems, which can be convertedto equivalent structures without delayless feedbackas mentioned earlier. E.g. the ladder filter structurecan be decomposed into a number of serially con-nected 1-pole filters, where each one of those wouldbe converted independently first, thereafter the entiresystem having only a single delayless feedback loop.

4 Nonlinearities

The described method can be extended to the case ofsystems containing nonlinear elements. We use theladder filter emulation ([1], [2]) as an example. Toreduce the problem complexity, including the com-putational one, we choose to use a single saturationstage at the feedback mix-in point (Fig. 5). Thisshould be the best point, if only a single saturator isto be used. Notice that the structure in Fig. 5 has ageneric character, not restricted to the ladder filter.

Let the saturator in Fig. 5 be a memoryless wave-

3

+ '!&"%#$// S// G(z)// •// //

qqMM oo−

OO

k

Figure 5: Saturation in feedback (‘S’ is a memorylesssaturator).

shaper defined by the function

xout = S(xin) (3)

In this case (1) becomes

y = gS(x− ky) + s (4)

If S′(x) ≥ 0 ∀x and if k ≥ 0 and g ≥ 0 then (4) al-ways has a unique solution, this solution also being inagreement with the ‘nonzero-impedance’ paradigm.Therefore, we can simply solve (4).

If (3) is a second-order polynomial equation rel-ative to xin and xout, then (4) can be easily solvedanalytically. The same obviously holds, if (3) definesa curve which can be broken into multiple second-order segments (although a search for the applicablesegment will be necessary). This is particularly thecase for parabolic:

S(x) =

{x

4· (4− |x|) if |x| < 2

sgn x otherwise

and hyperbolic:

S(x) =x

1 + |x|

saturation curves.A very nice curve to use for such saturator is a hy-

perbolic tangent. It particularly occurs in differentialamplifiers, such as ones used in the transistor ladderfilter ([2]). In this case (4) takes the form

y = g tanh(x− ky) + s (5)

and is obviously not solvable analytically. Thereforeone needs to resort to numeric methods.

y

h(y)g

−g

s− g s + g

x/ks

Figure 6: Graphical illustration of (5).

One possibility would be to break tanh x intosecond-order approximation segments. Another ap-proach is to build an approximation of a functionF (u, v) defined as:

F (u, v) = tanh(u− vF (u, v)) (6)

where u = x − ks, v = gk. Otherwise the Newton-Raphson method looks like a good candidate, exceptthat some convergence problems may occur for poorlychosen initial values. To come up with a reasonablygood initial value choice we consider (5) as an inter-section of a hyperbolic tangent curve g tanh(x− ky)with a straight line y − s (Fig. 6).

Looking at the picture, we come up with the fol-lowing rule. If s − g < x/k < s + g, then the initialpoint is y = x/k. Otherwise one of the points s − gand s+g (the one lying between s and x/k) is chosen.This choice ensures the monotonous3 convergence ofthe algorithm. For 0 < g ≤ 1 and 0 ≤ k ≤ 8, whichare reasonable ranges (particularly for the ladder fil-ter), it was found that about 5 iterations are generallynecessary for the convergence to 10−5 precision. Yetanother iteration will generally double the number ofdigits.

A number of tricks can be further employed to re-duce the computation times. E.g. one could use anapproximate computation with reduced precision atearlier iterations. A second-order segmented approxi-mation of tanh x or an approximation of (6) might be

3Because the initial point is lying between y = x/k and thetrue solution.

4

used improve the initial value choice. Once could alsofirst check whether the previous output value y[n−1]doesn’t provide a better initial guess, although theconvergence speed may significantly vary in this case.

Returning to the generic nonlinearity S(x), we’dlike to mention a cheap and dirty but computation-ally efficient approach, which is to simply solve (1)against x− ky, allowing the possibility of infinite re-sults, and then compute S(x− ky) just once.

The discussed nonlinear-case method can be alsogeneralized to a case of multiple contained nonlinear-ities, However, the solution of the equation similar to(4) will become more computationally intensive, andin certain cases more challenging.

If k < 0 and/or if the saturation curve is havinga complex shape, such that (4) can have multiplesolutions, the ‘nonzero-impedance’ approach can beused to pick up the right one.

5 Conclusion

The described method allows the conversion ofcontinuous-time LTI system prototypes to discrete-time implementations with the transfer functionmapped according to the desired transform formula,while the system topology and state variables are pre-served. Thus one can obtain implementations whichhave high quality amplitude and phase responses andsimultaneously have nice behavior in the modulated-parameter case and (if desired) certain nonlinear ele-ments of the original prototype. Obviously the samemethod also can be applied to the direct constructionof discrete-time systems.

The practical cases of application of the methodinclude filters, phasers, flangers, etc. Since evenbilinear-transformed systems feature significant high-frequency warping in their responses compared tothe original prototypes, sampling rates higher than44kHz might still be desired.

The nonlinear case, often requiring iterative nu-meric methods, is significantly more computationallyintensive. Taking the aliasing produced by the non-linearities into account, an oversampling at a higherrate instead of solving (4) might often provide a rea-sonable compromise at better performance.

Acknowledgements

The author would like to thank NI, particularlyDaniel Haver, Tom Kurth, Nicolas Gross, and An-dreas Gloggengiesser for their support and for help-ing him to release this article. Very special thanks toStephan Schmitt.

References

[1] Stilson T., Smith J.O. III “Analysing the MoogVCF with Considerations for Digital Implemen-tation.”http://ccrma.stanford.edu/˜stilti/papers

[2] Huovilainen A. “Non-linear Digital Implmenta-tion of the Moog Ladder Filter.” Proc. of the7th Int. Conference on Digital Audio Effects(DAFx’04), Naples, Italy, October 5-8, 2004http://dafx04.na.infn.it

[3] Landeholm E. “Zero delay feedback using sampleprediction.” Posted Mar 8, 2002 to the music-dspmailing listhttp://music.columbia.edu/pipermail/music-dsp

c© Vadim Zavalishin. The right is hereby granted to freelydistribute this article further, provided no profit is made fromsuch distribution.

5