THE INFLUENCE OF SMALL VARIATIONS IN A SIMPLIFIED GUITAR AMPLIFIER MODEL

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1 THE INFLUENCE OF SMALL VARIATIONS IN A SIMPLIFIED GUITAR AMPLIFIER MODEL Krstjan Dempwolf, Martn Holters, Stephan Möller and Udo Zölzer Dept. of Sgnal Processng and Communcatons, Helmut Schmdt Unversty Hamburg Hamburg, Germany dempwolf@hsu-hh.de ABSTRACT A strongly smplfed gutar amplfer model, consstng of four stages, s presented. The exponental sweep technque s used to measure the frequency dependent harmonc spectra. The nfluence of small varatons of the system parameters on the harmonc components s analyzed. The dfferences of the spectra are explaned and vsualzed. 1. INTRODUCTION Most gutar players love to experment wth dfferent sounds and use varyng equpment lke amplfers or effect unts. Thanks to amp modelng, the sgnal processor based smulaton of famous gutar amplfers, t has become easer for today s muscans to have a hgh number of sounds combned n one devce. The common way to do amp modelng s to dsassemble a real amplfer crcut, analyze the separated blocks accurately, and wrte a software that smulates the transfer behavor step-by-step. In dong so, a copy of the orgnal devce wth smlar sound propertes can be created, but qualty and smlarty of the result depend on the complexty and the analyss accuracy. Hence, good smulatons are stll computatonally ntensve and requre hgh-performance equpment. In ths paper gutar amplfers are understood as complex systems wth hghly nonlnear behavor. By measurng the frequency dependent harmonc spectrum, a unque "fnger prnt" s recorded, dentfyng the system s behavor for a gven snusodal nput. The nfluence of small varatons of the system parameters to the output spectrum can be characterzed by ths measurement technque. Ths study can help to analyze the nfluence of flters and transfer curves to the sound. Even though a strongly smplfed model s used, the tendences are n evdence. In addton to the recordng of these measurable parameters, a lstenng test s presented to statstcally dentfy the changes n tmbre for the same model. The assgnment between measurable values and tmbre attrbutes lke "aggressve" or "warm" s desred. 2. GUITAR AMPLIFIERS For the amplfcaton of electrc gutars, specal amplfer and loudspeaker combnatons are common, whch clearly dffer n ther transfer behavor from normal H-F unts. In use are combo amplfers combnng amplfer and speaker(s) n one enclosure or stacks, consstng of amplfer (top) and box separately. The amplfers show a strong nonlnear performance and enforce a voltonal dstorton of the gutar sgnal. The tone of the undstorted nstrument s rarely of nterest, the popular gutar sound s always assocated wth a more or less strong deformaton of the orgnal sgnal. Due to ther excellent sound behavor when drven n such crcuts, tubes are stll rulng todays gutar amplfers. The reproducton s done va specal loudspeaker cabnets, where the used speakers make a man contrbuton to the sound because of the non flat frequency response showng many resonances. The man stages of a common valve amplfer are [1]: The nput stage wth trode valves responsble for the preamplfcaton of the gutar sgnal. The tone stack suppressng the DC component from the sgnal and provdng smple equalzaton. The phase spltter provdng both orgnal and phase nverted sgnal as requred for the symmetrcal feedng of the followng power amplfer. The power amp stage wth pentode valves processng ndvdual amplfcaton of the orgnal and the phase nverted sgnal. The output transformer executng the subtracton of both sgnals delvered by the power amp stage to acheve the doublng of the sgnal ampltude. The transformer s also necessary for the mpedance matchng to the connected speaker (typcally 8 or 16 Ω). One or more speakers mounted n an open or closed cabnet. Commonly used speakers range from 10 to 12 nches. Each of these stages comes wth non-deal transfer characterstcs. When combnng all stages to the overall system the analyss gets very complex. The nonlnearty of a typcal phase spltter crcut s explaned n [2]. The analyss and synthess of the parastc nonlnear behavor of a gutar loudspeaker cabnet can be found n [3]. 3. MEASUREMENT OF NONLINEAR SYSTEMS The nonlnearty of a system leads to addtonal spectral content n the output sgnal. Ths means that the output spectrum wll show new parts at dscrete frequences kf 1 when t s caused wth a sne wave of frequency f 1, e.g. for a sne wave of 1 khz, the frst overtone occurs at 2 khz and the second at 3 khz. In musc, these nteger multples are called overtones. In physcs t s common to use the term harmonc wth a dfferent ndex relatng to the order: the exctaton frequency s called fundamental, the frst overtone k 2, the second overtone k 3 and so on. A common gage to descrbe nonlnear systems s the specfcaton of the total harmonc dstorton (THD) caused by the system, DAFX-1

2 defned as the rato of the sum of the powers of all harmonc components H to the power of all harmonc components H plus the fundamental H 1 [4]: H 2 THD = 2 + H H2 N H H2 2 + H (1) H2 N Dependng on the systems assembly, the value of the components n the output spectrum can vary extremely both over frequency and nput ampltude Exponental Sweep Technque The sne sweep technque s a well-known tool to measure mpulse responses (e.g. of rooms) featurng a hgh resoluton and robustness aganst tme varance. A useful varant of ths method, ntroduced by Farna [5], uses a real valued sweep x(t) = sn ( ω 1T ln( ω 2 ω 1 ) (e t T ln( ω 2 ω 1 ) 1) ) (2) Fgure 1: Measurement of a strongly nonlnear system, wth the lnear mpulse response and the harmonc responses. whose frequency ncreases exponentally from the start frequency ω 1 to the end frequency ω 2, wth sweep length T. In addton, an nverse sweep x 1 (t) has to be computed, fulfllng the condton x(t) x 1 (t) δ(t T ). (3) When x(t) s appled to a system, the convoluton of the measured output sgnal y(t) wth x 1 (t) leads to the mpulse response h(t) = y(t) x 1 (t). (4) For lnear systems, h(t T ) s zero and the lnear mpulse response s gven by h(t > T ). When the measured system s afflcted wth memoryless nonlnearty, h(t) wll show addtonal spkes, whch are located pror to the mpulse response of the lnear part of the system. For an exponentally swept sne, these spkes denote separated mpulse responses, whch belong to the harmonc components and are defned as harmonc mpulse responses (HIR) [6]. Begnnng wth the 2nd order component k 2 to the left of the man pulse the HIRs are separated neatly n tme one after another. The poston of the spke leadng to the th order HIR can be computed from t HIR() = t FUND log 2() sweep rate, (5) where s the order of the harmonc component and t FUND s the arrval tme of the fundamental response. Ths relaton s llustrated n Fgure 1. For hgh orders, the dstance between the spkes s gettng smaller and the responses wll overlap. A remedy s to record a longer sweep and/or to use a hgher samplng rate. By executng a FFT on each harmonc response, the frequency dependent harmonc dstorton s computed. Fgure 2 shows the frequency response for the fundamental wave and the frst harmoncs k 2 and k 3. In ths plot the harmoncs are dsplaced to the left so that the spectral content s plotted under the correspondng exctaton frequency. The components H, as used n (1), can be read out easly from the plot. The advantage of ths technque s, that even heavly dstorted systems can be measured accurately. The lnear response and the harmonc responses up to hgh orders are separated n tme leadng to segmented spectra for all orders. Fgure 2: The harmonc components k 2 (dashed) and k 3 (gray) plotted under the correspondng fundamental (black). 4. SYSTEM ANALYSIS 4.1. Smplfed Model of a Gutar Amplfer To analyze the behavor of gutar amplfers a software model had to be developed offerng the possblty of easy modfcatons. Dsregardng the lnear or nonlnear effects of transformer and phase spltter, the descrbed crcut can be smplfed to a four stage model as shown n Fgure 3. It conssts of the followng four stages: 1. a characterstc curve wth statc nonlnear behavor as found n preamplfers, 2. a lnear flter representng the tone stack, 3. a second characterstc curve as found n the power amplfer and 4. the loudspeaker smulaton performed by a FIR flter wth the mpulse response of a measured gutar cabnet. Ths model was mplemented on a professonal recordng PC usng the software Cubase 4. The characterstc curves were processed Fgure 3: The four stage amplfer model. DAFX-2

3 Fgure 4: The nonlnear transfer curve of the preamp. octaves bandwdth was added to the model. The cases boost (setup 5) and cut (setup 6) wth gan = 20 db were appled. In setup 17 the cut-off frequency of the low cut flter was changed. Stage III: Smlar to stage I, the transfer curve was modfed both to a hard clpped and soft clpped curve (setup 3, 4) and to symmetrc / asymmetrc behavor (setup 20, 21). Stage IV: To smulate the nfluence of changes n the loudspeaker response, a peak flter was added to the orgnal speaker. The parameters are: f m = 800 Hz, bandwdth Q = 2 octaves, and gan = ±6 db (setup 7 and 8). In addton, further modfcatons n all stages were done ncludng some curves wth extreme or "crazy" characterstcs. An overvew over all setups s gven n Table 2. Fgures of all mportant curves are shown the appendx. 5. MEASUREMENT OF ALL SETUPS Fgure 5: Frequency response of the Blue Bulldog loudspeaker. wth the free plug-n Func Shaper 1, provdng the transfer curves by means of mathematcal expressons. The flter sectons were ntegrated wth the parametrc EQ of the Cubase plug-n Q. For the speaker smulaton the freeware plug-n kefir 2 was used. The sgnal processng was done wth help of the Powercore Frewre from TC ELECTRONIC and the Freface 400 Interface from RME Reference Setup The reference setup was desgned as a strongly smplfed amplfer model of the normal channel of a classc VOX AC30 gutar amplfer. The transfer curve for the stages I and III are shown n Fgure 4. In ths case the same soft clppng transfer curve s used both for pre- and power amplfer. The second stage performs the low cut flter caused by a couplng capactor plus the attenuaton of the hgh frequences caused by the phase spltter. The frequency response of the loudspeaker smulaton was taken from measurements wth the orgnal Blue Bulldog speaker and s shown n Fgure 5. All parameters are defned n Table 1. For stages I and III the functons are gven as pseudo code accordng to the syntax of the used plug-n Modfed Setups To analyze the presented four stage amplfer model, some modfcatons were done n all stages. In the followng 21 modfed setups only one stage was changed n the parameters whle all other stages reman as gven n the reference setup. Stage I: The characterstc curve of a 12AX7 trode was changed both to a hard clpped and soft clpped curve (setup 1, 2). The effect of a symmetrc / asymmetrc transfer curve was tested as well (setup 14, 15). Stage II: A peak flter wth md frequency f m = 800 Hz and two 1 Func Shaper Verson v0.5, 2 kefir, The reference setup and all modfcatons were analyzed wth the descrbed measurng technque. To ensure a suffcent resoluton at hgh orders, very long sweeps of 30 s were used at a samplng frequency of 44.1 khz. For each setup fve recordngs were done capturng the mportant range of the nput ampltude. Ths range was found to be 60 dbv to 20 dbv. Wth 1 reference and 21 modfcatons the number of sweep measurements s 22 5 at all, wthout countng addtonal ones used for ensurng accuracy. From each measurement the lnear response and all harmonc responses up to the order 9 were computed. The followng nomenclature s used to dentfy the sngular responses wth H j,v (k) (6) the harmonc order begnnng wth 1 for the fundamental, the ndex of the measured setup j = {1,..., 21, R} and the used nput ampltude V = { 60 dbv,..., 20 dbv} n steps of 10 dbv Interpretaton and Comparson For the comparson all measurements were arranged n a meanngful way one-to-one, leadng to a total of 15 pars. That way symmetrc and asymmetrc or hard- and soft clpped transfer curves were faced. The arrangements are shown on the left part of Table 3. Table 1: The parameters of the reference model. No. Parameters I SoftClp(0.5(x ) 2 1, a, b, c, d) wth a=-0.753, b=0.351, c=0.875 and d=0.625; gan=31.16 db and volume=5.89 db. II low-cut flter wth f c=56 Hz and a hgh shelvng flter wth f c=1973 Hz, gan 2.5 db. III SoftClp(0.5(x ) 2 1, a, b, c, d) wth a=-0.639, b=0.681, c=0.667 and d=0.771; gan=17.68 db and volume= db. VI A low-cut flter wth cut-off frequency f c=20 Hz and a 512 tap long FIR flter DAFX-3

4 Table 2: Overvew over the settngs for the modfed setups. No. Stage Descrpton Parameters 1 I hard clpped preamp curve c=0, d=0 2 I soft clpped preamp curve c=1, d=1 3 III hard clpped power amp curve c=0, d=0 4 III soft clpped power amp curve c=1, d=1 5 II added peak flter boost f m=800 Hz, Q=2 octaves, gan=20 db 6 II added notch flter cut f m=800 Hz, Q=2 octaves, attenuaton=20 db 7 IV peak flter added to loudspeaker (boost) f m=800 Hz, Q=2 octaves, gan=6 db 8 IV notch flter added to loudspeaker (cut) f m=800 Hz, Q=2 octaves, attenuaton=6 db 9 I unrealstc freeform curve z=x+a sn(2 π b x);e=tanh(c z) wth a=0.56, b=2.28, c=1.84, drve=20.2 db and volume=0 db 10 I rectfer half-abs abs(x)/2 +x/2 wth drve=31.16 db and volume=0 db 11 I rectfer abs abs(x) wth drve=31.16 db and volume=0 db 12 I squared curve x 2 /2 + x/2 wth drve=31.16 db and volume=0 db 13 I tlted curve e=below(-a,x); f=above(b,x); f(e,x,-1-2x)+f(f,x,1-2x); wth a=0.33, b=0.33, drve=26.11 db and volume=0 db 14 I symmetrc soft softclp(select(x, c tanh(a x), d tanh(a x))); wth a=3.125, c=1, d=1, drve=31.16 db and volume=5.89 db 15 I asymmetrc soft same as No.14, but wth d=0.5, drve=31.16 db and volume=5.89 db 16 IV added hgh-shelvng flter f s=1428 Hz, gan=8 db 17 II low-cut frequency moved f c=140 Hz 18 I reduced gan drve=8 db 19 III reduced gan drve=0 db 20 III symmetrc soft softclp(select(x, c tanh(a x), d tanh(a x))); wth a=3.125, c=1, d=1, drve=17.68 db and volume= db 21 III asymmetrc soft same as No. 20, but wth d=0.5, drve=17.68 db and volume= db As a metrc for the smlarty the averaged dfference between two transfer curves H j1 (k) and H j2 (k) s ntroduced, computed by (j1,j2),v = 1 N N H j1 k=0 (k) H j2 (k) (7) wth the mean over all frequency bns k. A small value for denotes a hgh smlarty between two curves. Pror to the computaton of ths value, t makes sense to prepare the measurements. Frst a smoothng has to be appled to the frequency responses, as explaned n [7], to ensure nsenstvty aganst addtonal nose. Wth 1/3 octave smoothng good results were acheved. The second step s to algn the magntude plots n a meanngful way. Therefor the fundamental responses H 1 were normalzed to a mean value of 0 db and the harmonc responses were algned accordngly. The smlarty s then calculated for the measurements done wth the nput levels V= 30 dbv through 50 dbv. Table 3 gves an overvew over the smlarty between all measured curves. Specfed s the maxmum value calculated by ( max (j1,j2),v ), V = { 50 dbv,..., 30 dbv}, (8) for all pars j 1, j 2 and the harmoncs ndex. The mappng s ++ for very hgh smlarty and - - for no observable smlarty. Although ths representaton s just a rough estmaton, some patterns are vsble. Because of the hgh number of pars only a selecton of the results can be presented n ths paper. Some observatons reflect the well-known behavor of nonlnear transfer curves, e.g. that symmetrcal nonlneartes (setup 14, 20) wll cause odd order harmoncs whle asymmetrcal nonlneartes cause even order dstorton products. Further on t can be seen that the characterstc of the fundamental (row f 1) conssts manly of the loudspeakers response. Snce ths LTI system has ts poston at the end of the chan, all harmoncs show more or less the same form Selected Results Asymmetrc vs. symmetrc soft transfer curves: A symmetrc transfer curve drven n the preamplfer, as smulated n setup 14, causes hgh levels for all odd harmoncs, but mnor levels for the even harmoncs. Partally s the dfference bgger than 40 db, see Fgure 6(a). The thrd harmonc s the strongest component at all nput levels. A dfferent trend can be observed for the asymmetrc preamp curve, as depcted n Fgure 6(b). Agan, all odd and all even harmoncs run n groups parallel, but the dfference s much smaller. For small nput ampltudes the second harmonc can be stronger than the domnatng thrd component. No dfference can be found for the fundamental response. The same tendency can be found comparng the measurements from the second transfer curve, setup 20 and 21. Ths condton can be seen clearly n Table 3, too. Soft clpped vs. hard clpped transfer curves: The dfferences n the measurements of the hard clpped preamp curve (setup 1) and the soft clpped one (setup 2) are not as dstnctve. Agan, the DAFX-4

5 (a) Symmetrc soft (setup 14), (b) asymmetrc soft (setup 15), (c) hard clpped (setup 1) (d) and soft clpped (setup 2). Fgure 6: Fundamental (black), odd (dashed gray) and even order harmonc responses (sold gray), measured wth nput level 40 dbv. Table 3: The smlarty (max) for all arrangements, as ntroduced n equaton (8). changes n the odd order components are smaller than n the even harmoncs. The man devaton s gven for the nput ampltude 40 dbv, where the hard clpped system shows a dp for all even harmoncs, contrary to the soft clpped system, see Fgures 6(c) and 6(d). par harmonc component j 1 j 2 f 1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k R R R R R R R R R Lstenng Test Modfcatons n the setup wll of course lead to changes n the tmbre of the perceved sound. Some changes wll result n a complete dfferent sound, others may be almost naudble. For the analyss whch modfcatons result n an audble change, and how those changes can be descrbed, an onlne lstenng test has been developed. Three short sound clps of an electrc gutar played wth dfferent technque were processed wth the dfferent setups. The same arrangement of setups as presented n Table 3 s compared and the user has to rate dfferent attrbutes descrbng the sounds (e.g. "warm", "aggressve" or "transparent"). The lstenng test can be found on the webste The purpose of ths s to jon and compare the results of the measurements and the lstenng test and fnd relatons between perceved sound and measurable parameters. DAFX-5

6 6. DISCUSSION The harmonc spectrum of a gutar amplfer s wde. For the comparson made n secton 5, only the harmonc components up to k 9 have been taken nto account. Dsregardng hgher harmoncs of course leads to an naccuracy. But the dea s to brng out the tendences of the changes, for ths reason the lmtaton to only 9 harmoncs s suffcent. The reference model sounds good, but due to the smplfcatons t works statc. Ths means, the model can not react to the dynamc of the playng, lke real amplfers and (extensve) smulatons wll do. But t was not the task to develop an excellent model and the cognton of ths study holds for real amplfers models as well. For the comparson of asymmetrc and symmetrc preamplfer curves, t has to be consdered that the output level of the asymmetrc curve s lower. Therefor the followng second transfer curve s not drven n the same ampltude range. But the same tendency of the harmoncs can be found n setup 20 and 21, where the second transfer curve s modfed n the same way. A. APPENDIX: CHARACTERISTIC CURVES OF THE MODIFIED SETUPS (a) Hard clpped, (b) soft clpped, 7. CONCLUSIONS In ths paper, the nfluence of small varatons n a gutar amplfer model s analyzed. A strongly smplfed model s presented, consstng of two nonlnear transfer functons, a lnear flter n between, and a loudspeaker smulaton. By usng the exponental sweep technque, the frequency dependent harmonc spectra of all systems are measured, gvng an dentfcaton of the system s behavor for a gven nput ampltude. In addton, the system parameters are vared and the nfluence to the output spectra are characterzed and compared. It s shown, that the spectral components can vary n a wde range, dependng on the performed modfcatons. The dfferences are explaned and vsualzed for selected results. (c) freeform, (e) abs, (d) half-abs, (f) squared, 8. REFERENCES [1] U. Zölzer, Ed., DAFX: Dgtal Audo Effects, John Wley & Sons, Inc., New York, NY, USA, [2] S. Möller, M. Gromowsk, and U. Zölzer, A measurement technque for hghly nonlnear transfer functons, n Proc. Dgtal Audo Effects (DAFx-02), Hamburg, Germany, Sept [3] D. Yeh, B. Bank, and M. Karjalanen, Nonlnear modelng of a gutar loudspeaker cabnet, n Proc. Dgtal Audo Effects (DAFx-08), Espoo, Fnland, Sept [4] S. Temme, Audo dstorton measurements, Tech. Rep., Brüel & Kjaer, [5] A. Farna, Smultaneous measurement of mpulse response and dstorton wth a swept-sne technque, n 108th AES Conventon, Pars, France, Feb [6] S. Müller, Measurng transfer-functons and mpulse responses, n Sgnal Processng n Acoustcs. 2008, vol. 1, pp , D. Havelock, S. Kuwano and M. Vorländer (Eds.), Sprnger. [7] P. Hatzantonou and J. Mourjopoulos, Generalzed fractonal-octave smoothng of audo and acoustc responses, J. Audo Eng. Soc., vol. 48, no. 4, pp , Aprl (g) tlted, (h) symmetrc and () asymmetrc soft. Fgure 7: The modfcatons of the transfer curves. DAFX-6