A Review of DigitalTechniques for ModelingVacuum-Tube GuitarAmplifiers

Jyri Pakarinen∗ and David T. Yeh†

∗Department of Signal Processingand AcousticsHelsinki University of TechnologyP.O. Box 3000FI-02015 TKK Finlandjyri.pakarinen@tkk.fi†Center for Computer Researchin Music and AcousticsDepartment of MusicStanford UniversityStanford, California 94305-8180 USAdtyeh@ccrma.stanford.edu

Although semiconductor technologies have dis-placed vacuum-tube devices in nearly all fields ofelectronics, vacuum tubes are still widely used inprofessional guitar amplifiers. A major reason forthis is that electric-guitar amplifiers are typicallyoverdriven, that is, operated in such a way that theoutput saturates. Vacuum tubes distort the signal ina different manner compared to solid-state electron-ics, and human listeners tend to prefer this. Thismight be because the distinctive tone of tube am-plifiers was popularized in the 1950s and 1960s byearly rock and roll bands, so musicians and listenershave become accustomed to tube distortion. Somestudies on the perceptual aspects of vacuum-tubeand solid-state distortion have been published (e.g.,Hamm 1973; Bussey and Haigler 1981; Santo 1994).

Despite their acclaimed tone, vacuum-tubeamplifiers have certain shortcomings: large size andweight, poor durability, high power consumption,high price, and often poor availability of spare parts.Thus, it is not surprising that many attempts havebeen made to emulate guitar tube amplifiers usingsmaller and cheaper solid-state analog circuits (e.g.,Todokoro 1976; Sondermeyer 1984). The next stepin the evolution of tube-amplifier emulation hasbeen to simulate the amplifiers using computersand digital signal processors (DSP).

A primary advantage of digital emulation is thatthe same hardware can be used for modeling manydifferent tube amplifiers and effects. When a newmodel is to be added, new parameter values orprogram code are simply uploaded to the device.Furthermore, amplifier models can be implemented

Computer Music Journal, 33:2, pp. 85–100, Summer 2009c© 2009 Massachusetts Institute of Technology.

as software plug-ins so that the musician canconnect the guitar directly to the computer’s soundcard, record the input tracks, add effects and/orvirtual instruments, and then compile the song asa CD or upload it to the Internet. This is especiallyuseful for home studios and small ad hoc recordingsessions, because it eliminates several tedioustasks of acoustic recording, such as setting up theamplifier and recording equipment, selecting amicrophone position, finding a recording room, etc.

This article attempts to summarize real-timedigital techniques for modeling guitar tube ampli-fiers. Although a brief overview was presented inPakarinen (2008), to the authors’ knowledge, thereare no previous works that attempt a comprehensivesurvey of the topic. Because this topic is relativelynew and commercially active, most of the referencematerial can be found in patents rather than aca-demic publications. Judging from the large numberof amateur musicians and home-studio owners, aswell as the huge number of discussion threads onInternet forums, this topic is potentially interestingfor a wide spectrum of readers. Thus, a consciouschoice has been made to try to survey the modelingtechniques at an abstracted level, without delvinginto the underlying mathematics or electric circuitanalysis.

This review is organized into four sections. Wefirst describe the sources of the nonlinearities inguitar amplifier circuits. Then, we review publishedmethods for modeling the linear stages of guitaramplifiers. The heart of this survey is the reviewof methods for nonlinear modeling. Finally we

Pakarinen and Yeh 85

mention various other guitar-amplifier relatedtechnologies and present conclusions.

Vacuum-Tube Amplifiers

The purpose of this section is to present an overviewof the operation of vacuum-tube amplifiers and toillustrate the complex nature of their importantnonlinearities. An overview of vacuum tubes usedin audio applications can be found in Barbour(1998), and a detailed tutorial on classic vacuum-tube circuits is provided in Langford-Smith (1954).The physical principles governing the operation ofvacuum tubes are reviewed in Spangenberger (1948).Excellent Internet articles discussing the design ofguitar tube amplifiers can be found online (e.g., atwww.aikenamps.com and www.ax84.com).

A typical guitar tube amplifier consists of apreamplifier, a tone-control circuit (i.e., tone stack),a power amplifier, and a transformer that couplesto the loudspeaker load. The preamplifier magnifiesthe relatively weak signal from the magneticguitar pickups and provides buffering so that thepickup response is not altered by the amplifiercircuitry. The preamplifier is usually realized withtriode tubes. The tone stack provides a typical V-shaped equalization for compensating the pickup’sresonance at mid-frequencies, and it gives theuser additional tonal control. The power amplifierboosts the signal so that it is powerful enoughto drive a loudspeaker. In the so-called all-tubeguitar amplifiers, both the pre- and power-amplifiercircuits use tubes instead of transistors in amplifyingthe signal. Typically, these amplification circuitscontain one or more tube stages, namely, circuitblocks that consist of a tube connected to resistiveand capacitive (RC) components.

Vacuum Tubes

Vacuum tubes, or thermionic valves, were inventedin the early 1900s for amplifying low-level volt-age signals. Structurally, they consist of two ormore electrodes in a vacuum enclosed in a glassor metal shell. A two-terminal device is a diode,

Figure 1. Physicalconstruction (a) andelectrical representation(b) of a triode tube. (Figure(a) is adapted fromen.wikipedia.org/wiki/Vacuum tube.)

commonly used for signal rectification. Three-terminal devices are known as triodes and areprimarily used in preamplifier circuits. Four- andfive-terminal devices (tetrodes and pentodes, re-spectively) are used mainly for power amplificationpurposes to drive a loudspeaker, for example.

The operation of vacuum tubes is analogous towater flow on a slope. First, the electrode termedthe cathode is heated, and the process known asthermionic emission acts like a pump that formsa pool of electrons at the top of a hill. A secondterminal called the plate (or anode) is at the bottomof a slope. Electrons will flow from the cathode tothe plate depending upon the relative height of theplate, which is controlled by the voltage appliedto it. Note that because a pump is at the cathode,electrodes can never flow backward from the plateto the cathode even though the plate may be raiseduphill of the cathode. This describes the rectificationbehavior of a diode tube.

The triode, illustrated in Figure 1, introducesa third terminal called the grid between the twoterminals. With the plate downhill of the cathode,the grid is like a raised barrier in the slope thatlimits the flow of electrons from the cathode tothe plate. If this barrier controlled by the grid ishigh enough, it stops the electron flow completely.This water-flow analogy motivates the British termreferring to vacuum tubes as “valves.”

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Nonlinear Amplification

The plate-to-cathode current is a nonlinear functionof both the grid-to-cathode and plate-to-cathodevoltages: Ipk = f (Vgk,Vpk). Note that a change involtage on the grid causes a change in currentflow between the cathode and plate. Amplificationoccurs when the change in current is converted toa change in voltage by a large-valued load resistor.Although amplification is nominally linear arounda central operating voltage known as the bias, atextreme signal levels, the amplified output willsaturate. When the grid-to-cathode voltage Vgk isvery small, current flow cuts off sharply. Very largeVgk causes the plate voltage to approach that of thecathode again, limiting the current and resulting ina nonlinearly saturating characteristic. To find thefull nonlinear transfer characteristic from input tooutput requires the solution of a nonlinear systemof implicit equations, because in a typical amplifiercircuit, Vpk depends on Ipk and vice versa.

In guitar-amplifier circuits, the operating point(bias), defined in terms of current through the tubedevice, is often set by a resistor connecting thecathode terminal to ground. The resistor introducesfeedback into the circuit, and its value influencesthe shape of the input-output curve and determinesthe offset about which the signal varies. Amplifierdesigns often include an AC bypass capacitor torecover gain in the passband lost to the feedback, butthis introduces memory effects into the nonlinearcharacteristic.

Dynamic Operation

Capacitive elements exist throughout the tube cir-cuit, preventing it from being accurately modeled asa static waveshaper (a memory-less nonlinearity). Iflarge transients are present in the input signal—as isoften the case with the electric guitar—the grid-to-cathode voltage could become positive, and currentIgk will flow from the grid to the cathode, eventu-ally causing the device to cut off, introducing anundesirable phenomenon called blocking distortion(Aiken 2006). Also, because a grid capacitor is oftenused to block the direct-current (DC) component

of the input signal, the grid current Igk charges thecapacitor and dynamically varies the bias point ofthe tube, leading to dynamically varying transientdistortion characteristics.

The cathode bypass capacitor retains memory ofthe tube bias and responds slowly to rapid changes insignal amplitude, causing signal history–dependentchanges in distortion characteristics. Furthermore,there exist parasitic capacitances in the tube itselfowing to the close proximity of its electrodes. Thedominant effect, Miller capacitance, is a low-passfilter resulting from the amplified capacitancebetween plate and grid; this is discussed morethoroughly in Aiken (1999a).

Amplifier Power Stage

The power amplifier can use either a single-endedor push–pull topology. In the single-ended topology,the signal is amplified in a single vacuum tube. Thistube conducts plate-to-cathode current during thewhole signal cycle (Class A biasing). Parallel tubestages can also be added if more output power isrequired.

The push–pull topology, perhaps more commonlyused, consists of two identical sets of output tubesdriven in opposite phases. The output of one setis inverted and combined with the other throughtransformer coupling. When a push-pull poweramplifier is operated in Class A biasing, bothtubes are actively amplifying during the entiresignal cycle. Alternatively, Class AB biasing canbe used, where one tube handles the signal forpositive signal excursions while the other tube isin a low current quiescent state, and vice versa fornegative excursions. Leaving the quiescent tubein a low-power state gives Class AB operationhigher power efficiency, but it may also introducecrossover distortion as the tubes transition betweenquiescent and amplifying states. Also, because ClassAB amplifiers draw current from the power supplyproportional to the signal amplitude, large input-voltage bursts can cause a momentary decreasein the supply voltage. This effect, called sagging,introduces further dynamic range compression(Aiken 1999b).

Pakarinen and Yeh 87

The power amplifier is coupled to a guitarloudspeaker through an output transformer, whichintroduces additional distortion and hysteresis (i.e.,an increasing signal is distorted differently than adecreasing signal). Furthermore, the loudspeakeritself can also contribute significant nonlinearbehavior both acoustically and electrically.

In conclusion, the complicated interdependen-cies and dynamic nonlinearities in vacuum-tubeamplifiers make their accurate physical modelingextremely demanding. As a result, approximatemodels simulating only some of the most noticeablephenomena have been developed by the amplifier-modeling community.

Modeling of Linear Filters in Amplifiers

To better understand nonlinear distortion modelinglater in this article, we will first consider thesimulation of the linear part of the amplifier, namely,the tone stack. The characteristics of linear filteringgreatly influence the tonal quality of electric-guitaramplifiers. Often, switches will be provided to allowa guitarist to choose between different componentvalues in a circuit to vary its frequency response.Certain frequency responses are associated withparticular genres or styles of music and are oftenassociated with specific guitar-amplifier models.

The unique quality of the tone stack of theelectric-guitar amplifier is significant enough towarrant several attempts in the patent literatureto invent methods to make a digital tone-stackmodel. The tone-stack configurations in guitaramplifiers are all very similar. Amplifiers are mainlydifferentiated by the component values of the circuitand the mapping from the controls to these values.The tone stack typically has up to three knobscontrolling the gains of three bands, loosely calledbass, middle, and treble. The middle band is a notchin the frequency response.

Digital Filtering

A system that introduces no new frequenciesto the signal is linear and can be characterizedcompletely by its impulse response. The impulse

response describes how the system reacts to aunit impulse. The frequency representation of thisimpulse response is known as the frequency responseand describes the gain or attenuation applied to theinput signal at various frequencies. Once the impulseresponse is known, e.g., on the computer in digitalform, convolution with this impulse response willrecreate the effect of this filter.

There are two general methodologies of modelinglinear systems in guitar circuits. The black-boxsystem identification approach views the systemas an abstract linear system and determines coeffi-cients replicating the system. A white-box approachderives a discretized frequency response transferfunction for the system based upon knowledge of itslinear, constant-coefficient differential equations.Because the linear systems in guitar amplificationare often parametrically controlled (e.g., by poten-tiometers in tone or volume controls), the modelingapproach must be parametric.

Black-Box Approach

In the black-box approach, the linear system isexcited with a test signal that covers all frequenciesof interest. This signal is usually a frequency sweepof a low-amplitude sinusoidal input or broadbandwhite noise. A set of measurements is obtainedfor various settings of the parameters, which maybe multivariate as for the low, mid, and high toneknobs of the guitar tone stack. Various techniquesare well known for extracting a frequency responsefrom these measurements (Foster 1986; Abel andBerners 2006).

Once the impulse response is found, it can beused directly as a finite impulse response (FIR)filter to simulate the measured system. Because theoriginal systems are typically low-order infinite im-pulse response (IIR) systems, it is computationallyadvantageous to identify IIR filters correspondingto the measured response. The digital filter systemidentification process optimizes either the error inimpulse response (time-domain identification) orfrequency response (frequency-domain identifica-tion) over the set of digital filter coefficients, givena desired filter order. Preferably, optimizing over the

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impulse response captures phase information and isa simpler, more robust formulation.

Because the parameterized filter coefficients areusually implemented as lookup tables, the patentscovering linear modeling of amplifier componentsgenerally concern methods to reduce table size andstorage costs in a practical implementation. TheFender tone-stack patent (Curtis, Chapman, andAdams 2001) covers an active filter topology thatreplicates the range of frequency responses of atone stack. Assuming this filter structure, systemidentification comprises obtaining coefficients forvarious knob settings by manual tuning to match theresulting frequency responses. The mapping from pa-rameters to coefficients is compressed for implemen-tation by sparse sampling (a suggested five points perknob) and 3D linear interpolation of the coefficients.

The Gustafsson et al. (2004) patent also de-scribes multidimensional linear interpolation forthe compression of mapping from parameters tofilter coefficients. This approach improves uponthe accuracy of classical linear interpolation andreduces the number of entries needed in the tableby warping each parameter dimension using non-linear mapping functions prior to interpolated tablelookup. The patent also describes the decompositionof the resulting filter into a linear combination ofKautz basis filters, a particular form of second-orderdigital filter, for stability in implementation. Thisis a special case of the general technique in digitalsignal processing to ensure numerically stable filterimplementations by decomposition into second-order sections. More information concerning Kautzfilters in audio applications can be found in Paateroand Karjalainen (2003).

A gray-box approach incorporating some insightinto the structure of the circuit, described in apatent application by Gallien and Robertson (2007),divides the tone stack into a parallel bank of twofirst-order filters, one high-pass and one low-pass,which are weighted and added. The filters arecleverly devised approximate equivalent circuitscomprising resistors and capacitors that allow forimplementation of the parameter mapping. Theequivalent circuits are simulated and compared toa simulation of the full circuit to derive componentvalues for the equivalent circuits and the filter

weights so that the resulting response matches thatof the actual circuit. The circuits, which are definedusing capacitors and resistors, are taken into thediscrete time domain by the bilinear transform fordigital implementation.

In summary, black-box approaches decide on aparticular filter structure, and then they decide oncoefficients for that structure to match the responseof the target system. Ad hoc mappings from parame-ter space to coefficient space parameterize the filter.

White-Box Approach

Yeh and Smith (2006) propose an analytical approachto the full tone-stack circuit and suggest that theresulting parameter update equations are not pro-hibitively complicated. This approach derives thefull third-order transfer function with no approxi-mations for the filter by symbolic circuit analysis.Because the coefficients are described as algebraicfunctions of the parameters, this method is fullyparametric. Yeh, Abel, and Smith (2007) applied thisapproach to filters based upon operational amplifiers.The tone stack for the Boss DS-1 distortion pedalwas implemented by interpreting the analog filter asa weighted sum of high-pass and low-pass functionsand implementing the analogous structure digitally.

Nonlinear Modeling

Nonlinear signal processing is at the heart oftube-amplifier modeling. Here, we review staticwaveshaping with memoryless nonlinearities,which is a fundamental technique in digital-distortion implementations, and several categoriesof methods to reintroduce memory into thenonlinearity: ad hoc nonlinear filters based uponthe circuit signal path, analytical approaches,and nonlinear filters derived from solving circuitequations using numerical methods.

Static Waveshaping

The most straightforward method for obtainingsignal distortion with digital devices is to apply an

Pakarinen and Yeh 89

Figure 2. Construction ofthe digital effects devicedescribed in Araya andSuyama (1996). Thedistortion block consists ofthree identical

nonlinearities and suitablescaling coefficients. Theamount of distortion canbe varied by changing thescaling coefficients.

Figure 2

instantaneous nonlinear mapping from the inputvariable to the output variable. This type of timbrealteration is called waveshaping (Arfib 1979; LeBrun 1979). If the mapping does not change in time,this method is called static waveshaping. An earlyYamaha patent (Araya and Suyama 1996) describesa digital guitar effects device using this technique.This is illustrated in Figure 2.

In Figure 2, the signal from the instrumentis first fed to the distortion block through ananalog-to-digital (A/D) converter (including ananalog amplifier for setting a suitable input level).The distortion effect is obtained by feeding thesignal into a nonlinear function through a scalingcoefficient. The nonlinear function used in Arayaand Suyama (1996) is of the form

y = 3x2

(1 − x2

3

)(1)

where x is the input (bounded between [−1, 1]) and yis the output signal. The nonlinear curve produced byEquation 1 is illustrated in Figure 3 with a solid line.Because the curve is fairly linear in the operationrange of the device, the scaling and nonlinearity isapplied three times in cascade (i.e., sequentially) forobtaining more distortion. After leaving the distor-tion block in Figure 2, the signal is fed to a collectionof linear effects (e.g., chorus or reverberation) andfinally to a digital-to-analog (D/A) converter. Arayaand Suyama also suggest adding a digital equalizerbetween the A/D converter and the distortion.

More nonlinear functions are suggested in Doidicet al. (1998), including a symmetric function of theform

f (x) = (|2x| − x2) sign(x) (2)

where sign(x) = 1 if x > 0, and sign(x) = −1 other-wise. Alternatively, a hard-clipping function or apiecewise-defined asymmetric static nonlinearity of

Figure 3. Solid line:input–output plot of thenonlinear function ofEquation 1 used in Arayaand Suyama (1996); dottedline: the input-output plotof the symmetricnonlinearity in Equation 2,

used in Doidic et al.(1998). Dash-dotted line:the asymmetricnonlinearity in Equation 3,also used in Doidic et al.(1998). The allowedoperation range is denotedwith dashed lines.

Figure 3

the form

f (x) = −34

{1 − [1 − (|x| − 0.032847)]12

+13

(|x| − 0.032847)}

+ 0.01,

for − 1 ≤ x < −0.08905

f (x) = −6.153x2 + 3.9375x,

for − 0.08905 ≤ x < 0.320018, and

f (x) = 0.630035, for 0.320018 ≤ x ≤ 1 (3)

can be used. Figure 3 illustrates the input-outputcurve defined by Equation 2 using a dotted line andthe curve defined by Equation 3 using a dash-dottedline. It must be noted that the original patent (Doidicet al. 1998) has some typographical errors in theequation of the asymmetric nonlinearity, and thusit does not produce the input–output relationshipillustrated in Figure 3.

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As displayed in Figure 3, all the input–outputcurves are fairly linear for small-amplitude signals,that is, signal values near the origin. This obviouslymeans that the smaller the signal is, the less itis distorted. A patent by Toyama (1996) uses asignal-dependent scaling procedure with a nonlinearfunction to also distort small-amplitude signals.This technique can add harmonic content to varioussignals regardless of their amplitude levels, althoughit does not resemble the behavior of vacuum-tubedistortion. A further Yamaha patent (Shibutani1996) describes a computationally simple methodfor creating piecewise-linear distortion functions bybranching the signal via various scaling coefficientsand adding the output. Graphically, this means thateach of the scaling coefficients determines a slopefor a linear segment in the input–output plot.

Another simple digital distortion circuit, “man-tissa fuzz,” is described in Massie (1996). This exoticalgorithm uses a simple bitshifing operation in dis-torting the input signal. Although the mantissa-fuzztechnique is computationally extremely efficient, itseems virtually impossible to match the distortioncurve to a desired nonlinearity.

Moller, Gromowski, and Zolzer (2002) describea technique to measure static, nonlinear transfercurves from all stages of a guitar amplifier. Theirgoal is to mimic the nonlinearities and filters inthe signal path of the amplifier, approximating thenonlinearities as static, the filters as linear, andneglecting loading between stages. Santagata, Sarti,and Tubaro (2007) introduce a model of the triodepreamplifier with an added hard-clipping feature.This model uses an iterative technique for evaluatingthe nonlinear tube equations, but it does notincorporate the capacitive effects of the triode stage;therefore, it can be considered as computing theimplicitly defined waveshaping curve “on the fly,”based on parameters measured from an actual tube.

Lookup-Table Nonlinearity

Preceding the patent by Araya and Suyama (1996),there had already been some studies on how to obtaindigital distortion effects. Kramer (1991) introduceda simple method for obtaining arbitrary nonlineardistortion in real time using a lookup table. This

means that instead of applying a nonlinear algebraicfunction, such as the one in Equation 1, the systemreads the input–output relation from a pre-storedtable, for example, a digitized version of Figure 3.The advantage of this technique is that it is easierto obtain a desired type of input-output relation,because the designer can freely draw the input–output curve for the lookup table.

On the other hand, a high-resolution lookuptable would consume an excessive amount ofmemory, so low-resolution lookup tables andinterpolation algorithms must be used. Also, run-time modification of the nonlinearity becomesdifficult. Digidesign implemented this type oflookup-table waveshaping in their early softwaresynthesizer Turbosynth in 1989.

In an early study by Sullivan (1990), a simple non-linear function or a lookup table is used in distortingthe output of a synthesized guitar string. In fact,the nonlinear function in Equation 1 can be seenas a scaled version of the one suggested in Sullivan(1990). Sullivan’s article also introduces a system forsimulating the acoustic feedback between synthe-sized guitar strings, amplifier, and a loudspeaker.

Oversampling

Nonlinear signal processing blocks are known toexpand the bandwidth of the incoming signal,which in a DSP system can cause aliasing if thebandwidth of the output exceeds the Nyquistfrequency (i.e., half the sampling rate). An amplifiermodel can distort harmonic signals such as aguitar tone and produce many new harmonics inthe output that, through aliasing into the audiorange, are no longer harmonically related to theoriginal tone. The resulting noisy, “dissonant”sound owing to aliasing is characteristic of low-cost digital implementations of strong distortionsand is typically mitigated through running thedistortion algorithm at an oversampled rate, whichis computationally expensive.

In the late 1990s, the Line 6 Company patented adigital guitar amplifier, i.e., an amplifier and effectsemulator combined with a loudspeaker (Doidic et al.1998). This device used a sampling rate of 31.2 kHzfor most of the signal processing, but it included

Pakarinen and Yeh 91

Figure 4. Tube-amplifiermodeling scheme, assuggested in the Line 6TubeTone patent (Doidicet al. 1998). Thenonlinearity is evaluated

at a higher sampling rateto avoid aliasing.Multichannel output canbe used, for example, inconjunction with stereoeffects.

an eight-times oversampling circuit for evaluatinga static nonlinearity at 249.6 kHz, thus attenuatingthe aliased distortion components. This straightfor-ward technique, named TubeTone Modeling, wasused in several commercially successful Line 6digital guitar-amplifier emulators.

Figure 4 illustrates the system described in Doidicet al. (1998). Here, the digital signal is first fed toa collection of preamplifier effects—that is, effectsthat are typically located between the guitar andamplifier, such as a noise gate, compressor, or awah-wah. Next, eight-times oversampling withlinear interpolation is applied to the signal, and itis fed to a nonlinearity. After the nonlinearity, thesignal is lowpass-filtered using an antialiasing FIRfilter, and it is downsampled back to the samplingrate of 31.2 kHz.

Figure 5 visualizes what happens to the waveformand spectrum of a sinusoidal input signal whendistorted by the nonlinear Equations 2 and 3.The top row illustrates the waveform (left) andspectrum (right) of a 1.2-kHz sinusoidal signal withan amplitude of 0.8. The middle row shows thesignal after the symmetric distortion defined byEquation 2. As expected, the symmetric distortioncreates a “tail” of odd harmonics in the outputsignal spectrum. For frequencies above the Nyquistlimit (a sampling frequency of 44.1 kHz was usedhere), the harmonics fold back to the audio band,resulting in frequency components that are not inany simple harmonic relation with the input tone.The bottom row shows the input signal after theheavy-clipping asymmetric distortion defined byEquation 3. As can be seen in the lower right graph,the asymmetric distortion creates even and oddharmonic components. The upper components areagain aliased back to the audio band, resulting in aninharmonic spectrum.

In Doidic et al. (1998), the output signal from thedistortion is fed to a collection of linear effects, such

as tremolo, chorus, or delay. If headphones or lineoutput are used, a simple low-pass filter can also beapplied for simulating the effect of the loudspeakercabinet. Finally, the signal drives a loudspeaker (orseveral loudspeakers, if for example stereo effectsare used) after a D/A conversion and amplification.

Customized Waveshaping

An interesting method for obtaining a highlycustomized type of distortion has been introducedin Fernandez-Cid and Quiros (2001). This technique,illustrated on the left of Figure 6, decomposes theinput signal into frequency bands using a filterbank,and it then applies a different static nonlinearityfor each band separately. Thus, only narrow-bandsignals are inserted to the nonlinear waveshapers,and the perceptually disturbing intermodulationdistortion is minimized. The authors call this tech-nique multiband waveshaping. The delay imposedon the direct signal in Figure 6 equals the delaycaused by the filterbank, so that the signal phase iscorrectly preserved after the final summation.

Fernandez-Cid and Quiros (2001) suggest usingChebychev polynomials as the nonlinearities. Thesepolynomials are a special type of function allowingthe designer to individually set the amplitude ofeach harmonic distortion component, providedthat the input signal is purely sinusoidal withunity amplitude. Furthermore, using this type ofpolynomial approximation, aliasing can be avoidedfor sinusoidal input signals, because the designercan simply choose not to synthesize the highestharmonics. The right part of Figure 6 illustratesthe construction of a single Chebychev-basedwaveshaper used in Fernandez-Cid and Quiros,where the overall signal level is set between [–1, 1]prior to the evaluation of the nonlinearity. Dynamicnonlinearities can be imitated by using two differentpolynomials ( fA(x) and fB(x) in the right part of

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Figure 5. Signal waveforms(left pane) and thecorresponding frequencyspectra (right pane) forsinusoidal input-outputsignals. Top row: a

sinusoidal input signalwith a frequency of 1.2kHz; middle row: the inputsignal after the symmetricdistortion defined byEquation 2; bottom row:

the input signal after theheavy-clipping asymmetricdistortion defined byEquation 3.

Figure 6) in parallel, and varying their mix ratioaccording to the signal level of the correspondingband. Finally, the original dynamics of the signalare restored by multiplying the polynomial outputwith the signal level, as shown in the right part ofFigure 6. The authors claim that the waveshapersperform well, even though their input is not asinusoid but rather a narrowband signal.

Patents by Jackson (2003) and Amels (2003)present trigonometric functions for creating staticwaveshapers where the distortion component levelscan be set by the designer. Schimmel and Misurec(2007) implemented and analyzed static nonlinear-ities using piecewise-linear approximations of thenonlinear input-output curves. These three meth-ods use oversampling to suppress aliasing. Also,a polynomial approximation of a static nonlinear-ity without aliasing suppression is presented inSchimmel (2003).

Ad Hoc Nonlinear Filters

Because the assumption that the nonlinearitiesare memory-less does not hold for describing thebehavior of real tube amplifiers, researchers haveproposed various dynamic waveshapers, namely,

nonlinearities that change their shape according tothe input signal or some system-state variables.

An early digital system for emulating a tubeamplifier was outlined by Pritchard (1991). Hesuggested using two nonlinear distortion blockswith a digital equalization unit in between. Ideally,the first distortion block would have a high-passfilter with the cutoff frequency controlled by theinput-signal polarity, and an asymmetric staticnonlinearity for producing mainly even harmonics.The second distortion block would generate botheven and odd harmonics and emulate the saggingeffect of the power amplifier using a dynamicnonlinearity. Aliasing problems, however, are notaddressed by Pritchard.

A more detailed description of a dynamic tube-amplifier model has been discussed in a Yamahapatent (Kuroki and Ito 1998). There, a single tubestage is again modeled using a lookup table, butthe DC offset of the input is varied according tothe input-signal envelope. The authors give theimpression that this bias variation would be causedby grid capacitor charging owing to grid current,although a more realistic explanation would be thevariation of the cathode voltage owing to a change inplate current. A tube preamplifier can be simulatedby connecting several tube-stage models in cascade.

Pakarinen and Yeh 93

Figure 6. Construction ofthe multiband waveshaperdistortion, described inFernandez-Cid and Quiros(2001). The overallstructure is illustrated inthe left half of the figure,

and the signaling inside anindividual waveshaper isdepicted in the right half.In the right half, theoutput of the averager canbe seen as a measure of theoverall signal level.

Sign inversion is applied between tube stages formodeling the phase-inverting behavior of a real tubestage. Note that, owing to the dynamic nonlineari-ties (i.e., signal history-dependent DC offsets), thepreamplifier stages cannot be combined as a singleequivalent lookup table. A push–pull power ampli-fier can be simulated by connecting two tube-stagemodels in parallel and reversing the sign of the otherbranch. With suitable DC-offset values, crossoverdistortion can be emulated, if desired. The systemproposed in Kuroki and Ito is illustrated in Figure 7.

Another dynamic model of a guitar preamplifierhas been presented in Karjalainen et al. (2006). Thismodel assumes that the plate load of the tube stage isconstant and resistive, so that the tube nonlinearitysimplifies to a mapping from the grid voltage Vgkto plate voltage Vp. This curve is measured from thetube by shorting the cathode to ground and varyingthe grid voltage. Grid current is also measured asa function of the grid voltage. These curves arecombined in a single precomputed Vgk-to-Vp table.Bias variation is simulated using a feedback loop,as in Kuroki and Ito (1998). The filtering effectcaused by the grid resistor and Miller capacitanceis modeled with a low-pass filter at grid input,while a high-pass filter emulates the interstageDC-blocking filter. Three tube-stage models areused in series and connected to a loudspeaker modelvia an equalizer. A minimum-phase FIR filter isused as a loudspeaker model.

An interesting system-identification-based ap-proach has been presented by Gustafsson et al.(2004), the founders of the Swedish company Soft-ube AB (producers of Amp Room software). Here,the dynamic nonlinearity is simulated by feedingthe signal through a nonlinear polynomial functionand varying the polynomial coefficients accordingto the input signal. Figure 8 illustrates this. The

signal-analysis block estimates the signal energyfor the last few milliseconds, and it checks whetherthe input signal is increasing or decreasing. Next,the polynomial coefficients are interpolated froma set of pre-stored coefficient values according tothe signal energy. The pre-stored coefficients areobtained from measured tube data using system-identification techniques (see, e.g., Nelles 2000).The hysteresis effect can be simulated by usinga different set of polynomial coefficients for in-creasing and decreasing input signals. The authorssuggest implementing the static nonlinearities withChebychev polynomials to avoid aliasing, and alsobecause the accuracy of the Chebychev polynomialapproximation is highest near the signal extrema(i.e., around ±1, near saturation).

Analytical Methods

Several methods exist for analyzing a nonlinearitywith memory. These are based upon Volterra seriestheory and can be used to implement nonlinearaudio effects.

Volterra Series

The Volterra series expansion (Boyd 1985) is arepresentation of systems based upon a nonlinearexpansion of linear systems theory. Analogous toconvolution with the impulse response vector of alinear system, the Volterra series is a multidimen-sional convolution with nonlinear system-responsematrices. Whereas in linear systems the impulse re-sponse fully characterizes the system and allows itsoutput to be predicted given an input, Volterra sys-tems are characterized by special functions, calledkernels, that correspond to the multidimensionalimpulse response of the nonlinear terms. It can also

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Figure 7. A dynamic tubeamplifier model asdescribed in a Yamahapatent (Kuroki and Ito1998). The model of asingle tube stage consistsof a lookup table, added

with a signal-dependentDC-offset. An entirepreamplifier can besimulated by connectingtube-stage models incascade with a phaseinversion in between. A

push–pull power amplifieris simulated by connectingthe tube stage models inparallel and in oppositephase.

be regarded as a Taylor series expansion with thepolynomial terms replaced by multidimensionalconvolution, accounting for the memory associatedwith different orders of nonlinearity.

Volterra series have been used extensively tomodel nonlinear acoustic systems including loud-speakers. In particular, they can linearize low-orderdistortion circuits and loudspeakers in real time (e.g.,Katayama and Serikawa 1997). Farina, Bellini, andArmelloni (2001) and Abel and Berners (2006) useda technique to identify parameters for a subclassof Volterra systems based upon a frequency-sweepexcitation of the system. A similar technique is usedin the Nebula effects sampler by Acustica Audio(www.acusticaudio.net), which allows the user tocreate soft-saturating models of several audio effectsbased on the system response. Helie (2006) applieda specific Volterra series expansion to create a real-time effect that includes the third-order nonlinear-ities of the Moog ladder filter. Schattschneider andZolzer (1999) report an efficient implementation ofa type of Volterra series and a system-identificationtechnique to derive parameters for theirmodel.

Although Volterra series are a theoreticallyvalid black-box method for simulating variousnonlinearities, real-time emulation of stronglysaturating distortion poses a problem. This isbecause Volterra series involve a convolution of adimension equal to the order of the nonlinearityfor each nonlinear term in the model, making thenumber of coefficients and computational costgrow rapidly with increasing order of nonlinearity.Because guitar distortion often involves very strong,clipping-type nonlinearities, Volterra series are notthe preferred technology for this application.

Dynamic Convolution

Kemp (2006) has patented a black-box method,dynamic convolution, for nonlinear system analysisand emulation. The basic idea of this technique issimple: several impulses with different amplitudesare inserted into the distorting system during theanalysis, and the resulting impulse responses arerecorded. System emulation is carried out using con-volution, so that the amplitude of each input sampleis detected and compared to the set of impulseamplitudes used in the analysis. Once the near-est measured impulse is found, the correspondingimpulse response is used in evaluating the convo-lution. Because this procedure is applied for eachinput sample, the convolution coefficients changeaccording to the input signal level during run-time.

Although a promising technique, dynamic con-volution has some limitations. First, the amountof stored data can be prohibitively large if a high-amplitude resolution is used. Secondly, dynamicconvolution can be used for modeling static nonlin-earities, but it fails to model dynamic nonlinearities,namely, systems for which the shape of the nonlin-earity changes due to the input signal (Berners andAbel 2004). Note that the nonlinear convolutionintroduced by Farina, Bellini, and Armelloni (2001)can be seen as the Volterra representation of thedynamic convolution method.

Circuit Simulation-Based Techniques

The preceding techniques have all treated thedistortion device as a nonlinear black box, possiblywith memory. Techniques based upon solving the

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Figure 8. A dynamicamplifier stage model,described in Gustafsson etal. (2004). The nonlinearfunction f(x) is varied eachtime sample according tothe input signalcharacteristics. Chebychev

polynomials are suggestedfor implementing thenonlinearities. A completeamplifier can be simulatedby connecting severalamplifier stage models incascade.

ordinary differential equations (ODEs) that describethe behavior of the circuit have also been attempted.

Transient Modified Nodal Analysis

Integrated circuit design involves the engineer-ing of analog and digital systems based uponhighly nonlinear integrated circuit devices such asmetal-oxide-semiconductor field-effect transistors(MOSFETs) and bipolar transistors. Verification ofthe designs depends critically on the accuracy ofnumerical circuit simulators, e.g., the SimulationProgram with Integrated Circuit Emphasis (SPICE;Vladimirescu 1994). SPICE uses transient modifiednodal analysis (MNA) with nonlinear components inaudio circuit simulation. MNA solves the equationsdescribing circuit behavior in matrix form, GV = I,where V is a vector containing the node voltages;I is a vector containing the current contributed bythe nonlinear devices, capacitors, and sources; andG is the conductance matrix representing the linearcurrent-to-voltage relation of each component in thecircuit. MNA is particularly convenient, becausethe computer can easily derive the circuit equationsgiven a circuit schematic.

The matrix G is typically sparse, because itencodes the connections between the componentsof the circuit, which are typically connected to justa few neighbors. MNA requires the solution of thisequation, usually by LU decomposition. Althoughthe complexity of a general matrix solve is O(N3),

Figure 9. A single stage ofthe nonlinear digital Moogfilter (Huovilainen 2004).The nonlinearity isembedded within thedigital filter feedback loop.

Equivalently, this is anonlinearity withembedded memory,derived by discretizing thecircuit equations.

where N is the number of rows or columns of thesquare matrix G, it has been found empirically thatfor typical circuits a sparse LU solve is O(N1.4),owing to the sparse nature of the matrix equations(White and Sangiovanni-Vincentelli 1987). As com-putational power increases and researchers modelmore complex circuits, MNA offers a simple way toconstruct circuit schematic-based audio effects.

Custom, Simplified Ordinary DifferentialEquation Solvers

For commercial digital audio effects, the simplestacceptable implementation is desired, becausecompanies boast of their capability to providea multitude of real-time effects simultaneously.To this end, several researchers have developedeffects based on simplifying the ODE model of thecircuit and trading off accuracy for efficiency in thenumerical ODE solvers.

Huovilainen reported nonlinear models of theMoog ladder filter (2004), as well as operationaltransconductance amplifier (OTA)-based all-passfilters (2005), by deriving a minimal ODE from thecircuit equations and solving it using Forward-Eulernumerical integration. The result is a nonlinearrecursive filter structure with a nonlinearity embed-ded in the filter loop. Huovilainen’s nonlinear Moogfilter model is illustrated in Figure 9. A simplifiedversion of this model has been presented in Valimakiand Huovilainen (2006).

Yeh et al. (2008) extended this approach tostrongly clipping diode-based distortion circuitsand found that for circuits in general, implicit ODEmethods such as Backward Euler or TrapezoidalRule are needed to avoid numerical instability attypical sampling rates. Implicit methods require the

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numerical solution of an implicit nonlinear equationby iterative fixed-point methods, a general subclassof which are the Newton–Raphson methods. Yehand Smith (2008) also extended this approach tothe triode preamplifier using a state-space approachwith a memory-less nonlinearity (the vacuum-tubeIpk expression itself), demonstrating that implicitmethods transform the ODEs for audio circuits intoa recursive state-space structure with a multidimen-sional static nonlinearity embedded in the feedbackloop. This approach accounts for both the implicitnonlinearity of the circuit and the memory intro-duced by bypass, coupling, and Miller capacitancesin the circuit. It can be considered a brute-force,fixed-sampling-rate simulation of the circuit.

A recent patent by Gallo (2008), the founder ofGallo Engineering (producers of Studio Devil soft-ware), introduces a tube-stage emulation algorithmusing a parametric nonlinear function. The biasvariation is modeled by evaluating the cathodevoltage ODE using a numerical solver, such asthe fourth-order Runge–Kutta algorithm. The platevoltage variation is neglected here, as in Karjalainenet al. (2006).

Wave Digital Filters

Wave digital filters (WDFs; Fettweis 1986) are aspecial class of digital filters with parameters thatdirectly map to physical quantities. Each of thebasic electrical circuit elements has a simple WDFrepresentation, and, through the use of “adaptors,”the resulting filters connect to each other as realelectric components do. Thus, the user can build theWDF circuit model by connecting elementary blocks(resistors, capacitors, etc.) to each other like a realamplifier builder. A real-time model of a WDF tube-amplifier stage has been presented in Karjalainenand Pakarinen (2006). Here, the tube is modeledusing a two-dimensional lookup table for simulatingthe bias variation, while the effect of the gridcurrent is neglected. Sound examples are availableat www.acoustics.hut.fi/publications/papers/icassp-wdftube. Yeh and Smith (2008) demonstrated thatthe WDF can efficiently represent certain guitarcircuits, such as the bright switch and the two-capacitor diode clipper.

Although WDFs are a computationally efficient,modular physical-modeling technique—and thusa promising method for flexible real-time audiocircuit simulation—some barriers to widespreadapplication of WDFs remain. Finding a generalmethodology in the WDF framework to modelinstantaneous feedback loops between differentparts of the amplifier circuitry presents a significantchallenge. Also, certain circuit topologies, such asbridges, do not easily map to connections of theadaptors commonly used for WDFs.

Other Models

A hybrid DSP/tube amplifier has been patented byKorg (Suruga, Suzuki, and Matsumoto 2002). Theirsystem uses an upsampled nonlinear function inmodeling the preamplifier, while the power amplifieris emulated using two push-pull triodes, connectedto a solid-state power circuit via a transformer. Acentral processing unit (CPU) controls the biasing ofthe tubes and the filtering of the feedback from theoutput to the input. The power amplifier state canbe switched between class A and class AB biasing bythe CPU. Furthermore, the solid-state power circuitcouples the output transformer to the loudspeaker sothat the output power rating can be varied withoutaltering the interaction between the tubes and theloudspeaker. Vox Amplification, a subsidiary ofKorg, manufactures a hybrid DSP/tube amplifiermodeling system called Valvetronix.

A recently introduced exotic sound effect(Pekonen 2008) uses a time-varying allpass filterin adding phase distortion to the input signal. Al-though various types of distortion could be emulatedby suitably modulating the filter coefficients, thecurrent usage of this effect does not allow convincingemulation of vacuum-tube distortion.

Summary and Discussion

Digital emulation of guitar tube amplifiers is avibrant area of research with many existing com-mercial products. Linear parts of the amplifier,such as the tone stack, are modeled using digital

Pakarinen and Yeh 97

filters, for which the parameters are found withsystem-identification methods or by using a prioriknowledge of the underlying circuitry. In the sim-plest case, the distortion introduced by the tubestages is modeled using static waveshaping. Aliasingproblems can be avoided using oversampling. More-sophisticated methods can be used for the simulationof dynamic nonlinearities. Most of these methodscan be classified as being inspired by circuit signalpaths, which try to model the signal path from theamplifier’s input to the output. There are also somemethods that attempt to simulate the operation ofthe underlying electric circuit, but these are ofteneither greatly simplified or still too demanding com-putationally for real-time modeling of complex cir-cuits. Alternatively, some analytical methods, suchas Volterra series or dynamical convolution, havealso been suggested. Owing to the complex dynam-ical nonlinearities of the tube-amplifier circuit, truephysics-based models for accurate real-time simula-tion of the tube amplifier have yet to be discovered.

It must be noted that owing to the essentiallynonlinear, complex nature of tube amplifiers, objec-tive evaluation of their sound quality—and hencethe sound quality of tube emulators—is extremelydifficult. Thus, the best way to rate different emu-lation schemes is by listening. Marui and Martens(2002) have presented some studies discussing per-ceptual aspects of amplifier modeling. As a resultof the subjectivity of human listeners, one shouldbe careful not to underestimate certain amplifier-modeling schemes just because the method used issimple or physically inaccurate. Careful tuning ofthe emulation parameters can make a tremendousimprovement in the resulting sound.

Existing emulation techniques are improving inboth physical accuracy and sound quality. Owingto the easy distribution of digital media, softwareamplifier emulators are also constantly gaining newusers. Although some tube-amplifier enthusiastsmight feel that digital emulation is a threat to thetube-amplifier industry, the authors believe that itshould rather be viewed as an homage. It can also beseen as a form of conservation, because the quantityand quality of available tube-amplifier componentscontinues to dwindle. After all, the ultimate goalof amplifier emulation is to convincingly reproduce

all the fine details and nuances of the vacuum-tubesound, and to make it widely available for use inartistic expression.

Acknowledgments

Jyri Pakarinen’s research is funded by Helsinki Uni-versity of Technology. David Yeh was supported by aNational Science Foundation Graduate Fellowship.The authors wish to thank Prof. Matti Karjalainen,Prof. Vesa Valimaki, Miikka Tikander, and JonteKnif for helpful comments.

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