DSP Equalization of Electret Film Pickup for Acoustic Guitar 4907 (H6)

Matti Karjalainen, Vesa V_,lim_ki[1], Heikki R_is_nen, Harri Saastamoinen [2]

[1] HelsinkiUniversityof Technology,Espoo- [2] EMF AcousticsOy Ltd, Espoo,Finland

Presented at ^uD,othe 106th Convention1999 May 8-11Munich, Germany

Thispreprinthas beenreproducedfromthe author'sadvancemanuscript,withoutediting,correctionsorconsiderationby theReviewBoard. TheAES takesnoresponsibilityforthecontents.

Additionalpreprintsmay be obtainedby sendingrequestandremittanceto theAudioEngineeringSociety,60 East42nd St.,New York,New York10165-2520, USA.

All rightsreserved.Reproductionof thispreprint,or anyportionthereof,is not permittedwithoutdirectpermissionfromtheJournalof theAudio EngineeringSociety.

AN AUDIO ENGINEERING SOCIETY PREPRINT

DSP Equalization of Electret Film Pickupfor the Acoustic Guitar

Matti Karjalainen l, Vesa V/ilimSki l,

Heikki Rgis/inen 2, and Harri Saastamoinen 2

1Helsinki University of TechnologyLaboratory of Acoustics and Audio Signal Processing

Espoo, Finlandwww. acoustics, hut. fi

matti, karj alainen©hut, fi, vesa. valimaki_hut, fi

2EMF Acoustics Ltd, Espoo, Finlandwww. b-band, eom

ABSTRACT

The response of guitar bridge pickups does not correspond to the acousticradiation of the instrument. If authentic acoustic sound is desired, a mix-

ture of pickup and air microphone signals is often used. In this study we

apply digital signal processing to obtain high-quality acoustic sound usingonly an elastic electret film pickup and properly designed digital filtering

for equalization.

INTRODUCTION

Sound reinforcement of acoustic guitars is often needed since the sound

power radiated from the instrument is limited for example for large arena

or open-air concerts. Even when playing with a loud acoustic band ororchestra the sound level of the acoustic guitar may remain too low.

Different pickup and microphone arrangements have been used with

amplification. Normal microphones, external to the guitar, can yield nat-

ural sound but they are sensitive to feedback, they pick undesirable sounds,

and they cannot be used when the player moves around. Vibration pickups

attached to, e.g., the bridge or another vibrating part of the instrument

body may be less sensitive to disturbances and are better integrated with

the instrument. Microphones, such as small electrets, can also be used, for

example in the vicinity of the sound hole to pick the acoustic sound field.Both the vibration pickups and the electret microphones have problems

with sound coloration. The response of a pickup attached to the bridge (ortop-plate) is unavoidably different from the acoustic radiation as recorded

by a microphone in the near or far field of the guitar. This is causedby a more direct coupling of the bridge to the strings whereby the sound

transmission and radiation of other parts of the body are missing in the

bridge response. On the other hand, the response of a microphone close tothe sound hole is most sensitive to some air cavity resonances but not as

responsive to the body plate radiation.

The transfer function from the bridge vibration to the radiated sound

can be approximated as a linear and time-invariant (LTI) system. As faras this is true and the sound radiation is caused by the bridge vibration,

this transfer function can be simulated with signal processing [1, 2, 3, 4, 5,6, 7, 8, 9, 10]. This means that it is possible to take a signal from bridge

vibration pickup and to process it with a single digital filter, which is the

same for all notes (unlike in [9]), in order to approximate the radiated soundsignal. In this paper we study how well this principle works, especially

when using a novel pickup principle, an under-saddle elastic electret pickup

based on an electromechanical film transducer (EMFi l) [11], as producedin the B-Band 1 transducer.

The flexibility of digital signal processing (DSP) allows also for otherkind of processing. If the fidelity of acoustic reproduction is not the main

aim, the properties of the guitar sound can be adjusted by signal processing,

e.g., to make it sound like having a larger or smaller body. A solid bodyguitar may also be rendered to sound more like an acoustic guitar. A range

of effect-like features is also possible with DSP techniques.

The content of this paper is as follows. In Section 1 the elastic electret

pickup and its use with the acoustic guitar is described. Section 2 includesa system analysis of the guitar with pickup from a signal processing point

of view to show what is possible and how it can be achieved. In Section 3

the measurements of the system behavior are presented and the estimation

o_'the equalizer filter is discussed. Section 4 includes a presentation of the

equalization filter realization techniques and discussion of the performanceof the equalizer designs, followed by conclusions in Section 5.

_B-BandTM and EMFiTM are trademarks of EMF Acoustics Ltd and its associates.

1 THE ELASTIC ELECTRET FILM PICKUP

There exist different kinds of pickups for mechanical vibration sensing of

the bridge or other parts of an acoustic musical instrument. Piezoelectric

sensors have been used typically but they often show level compression

and audible distortion with large signals. A new technology is based on

the EMFi (electromechanical) film [11, 12] which has been found very wellsuited to mechanical vibration sensing in general. In this study it was

decided to use such an under-saddle bridge pickup since it has an inherently

natural and distortion-free response.

Figure 1 illustrates the structure of the EMFi film. Permanently charged

dielectric material is filled with flat gas bubbles and placed between twoplastic sheets which are coated with electrodes and shields. Thus the struc-

ture which is cut into a strip form behaves like a condenser microphone.

Pressure variations are converted to corresponding changes of film thick-ness. The gas bubbles increase the sensitivity and linearity of EMFi as a

pressure sensor. EMFi has also a good impedance match to wooden parts

of an acoustic instrument. If a constant charge exists in the capacitance,

the change of thickness is converted to voltage change between the elec-

trodes. If the dielectric material has a permanent charge, an electret type

pressure transducer is obtained without the need for an external polariza-

tion voltage source.The EMFi material can be easily manufactured into a thin and small

strip that fits under the saddle of a string instrument. Figure 2 illustrates

the B-Band TM EMFi pickup manufactured by EMF Acoustics Ltd. It

consists of the transducer strip part and cable part that is a "passive"strip similar to the transducer part but without charged dielectric.

Compared with piezo transducers, EMFi-based electret transducers are

less sensitive and require more voltage gain in the first stage of the preamp-

lifier to achieve good signal-to-noise ratio. Also, the capacitance of the

practical B-Band transducers is relatively small, thus requiring a preamp-

lifier with a very high input impedance, located as close as possible to thetransducer.

2 ANALYSIS OF THE SYSTEM SIGNAL PATHS

Figure 3 illustrates the signal fiowgraph of an acoustic guitar with under-

saddle electret-type bridge pickup. The vibration of a string may be con-

3

ducted through the bridge in three ways: by two transversal (vertical andhorizontal) and a longitudinal vibration. Only the vertical movement of

the bridge (perpendicular to the top plate) is effectively transduced by thepickup. Fortunately this is the signal path with strongest coupling. The

longitudinal force due to tension variation is not important in the guitar

although it may be essential in some other string instruments [4].Based on this analysis we can see that the only signal path that can

be effectively equalized is the vertical vibration of the string through the

bridge to the body. Also any direct radiation from a string bypasses the

bridge pickup and cannot be simulated with the equalizer. Effects suchas 'friction hiss' when a left-hand finger moves on a string may introduce

such sounds. Furthermore, left-hand effects may generate vibrations that

radiate weakly through the vibration of the neck. As a conclusion, only

one of the signal paths in the guitar is captured by the bridge pickup 2, butfortunately this single path is the most important one and allows a usefultechnique of pickup acoustic equalization.

By assuming that the LTI transfer function Ha(co) from vertical bridge

excitation Vb(co) to acoustic radiation P(co) is

H (co)- P(co)vb(co)and the corresponding transfer function Hb(co) from bridge excitation tobridge pickup output X(co) is

x(co)Vb(co) (2)

then the equalizer transfer function Heq(co) yielding a good approximationof acoustic radiation from the guitar is

u (co) P(co)Ueq(co)-- U (co--X(co) (3)

Any bridge excitation with rich frequency content, such as hitting thebridge with an impulse hammer, can be used as excitation and the EQ

transfer function from Eq. (3) may be inverse Fourier transformed to yield

the corresponding impulse response heq(t). Different digital filter designtechniques may then be applied to realize or approximate it as discussedbelow.

2In principle it is possible to add sensors to the system to capture also other vibrationsthan the vertical string vibration.

4

3 MEASUREMENTS AND EQUALIZER TARGET

RESPONSE ESTIMATION

In this section we discuss the DSP equalization of a fiat-top acoustic guitar

with metal strings and equipped with an electret-type bridge pickup. The

measurement and EQ target filter estimation was carried out in two ways:

1. Impulse hammer excitation to the bridge

2. String plucking (music playing) method

In both cases the response was registered at the sample rate of 44100 Hz.

First the method with impulse (impact) hammer technique was applied.The top of the bridge was excited in an anechoic chamber by a vertical hit.

The strings were damped by absorptive material between them and the

fretboard. Signals from the bridge pickup and from a microphone 1 meter

in front of the sound hole of the guitar were registered. Figure 4 shows the

time responses. It is easy to see that the pickup signal (bottom curve) is

impulse-like in time while the radiation response (top curve) is temporallyspread and resembles reverberation.

The target response for the acoustic equalization filter may be computed

by frequency-domain deconvolution, based on Equations (1)-(3), as

(FFT{p(n))heq(r/) _- FFT-i_ FFT{x(n)}' (4)

where discrete-time signals p(n) and x(n) are the acoustic response and

the bridge pickup response, respectively, n is the discrete time variable,

and FFT is the fast Fourier transform. Figure 5 (top) shows the mag-nitude response of the target EQ filter, as obtained from impact hammer

measurement and deconvolution by Eq. (4). In general the envelope of the

spectrum is quite fiat, although resonances and antiresonances are found.

In Fig. 5 (bottom) the low-frequency part of the response is zoomed into.

It shows that the resonance peak around 160 Hz is strong, meaning thata sharp boosting of about 20 dB at this frequency is needed. The low-

est resonance (modified Helmholtz resonance) at about 80 Hz needs also

remarkable emphasis.

As discussed in section 4.1, the impact hammer measurement mW not

yield the best estimate for the EQ filter. An alternative method of obtain-

ing the desired EQ transfer function was also based on Eq. (4) but insteadof a mechanical impulse the estimation was computed from responses to

5

playing the instrument. The requirements for useful estimation of heq(t)are that the excitation from the string is spectrally rich, not too much

noise is injected to the registered signals, and that the signal path of Fig.3 where the pickup exists is strong enough compared to side paths.

We applied several seconds of playing, including pluckings of all strings

and various pitches (frets), in the excitation signal. Fig. 6 shows therecorded response envelopes from the pickup and a microphone 1 m in

front of the guitar sound hole, respectively, for 7.5 s of playing (pickingwith a plectrum). Notice that the details of the signal envelopes fromthese two sources look quite different.

Deconvolution of the playing response signals of Fig. 6 yields the im-

pulse response curve of Fig. 7a that looks very noisy, even for negative

(noncausal) time values. However, when windowing it properly (Fig. 7b),the result turns out to be more useful than the impact hammer measure-

merit described above. Fig. 7c shows the magnitude response of 7b and 7d

zooms to the low-frequency region of the spectrum. Notice that a boost of

about 15 dB around 160 Hz is needed in this equalizer, but at 80 Hz thereis no need to amplify the resonance much.

4 EQ FILTER DESIGN AND SOUND QUALITY

Assuming that the bridge pickup signal can be equalized to sound more

acoustic by an LTI response function heq(n), as defined above, this responsecan be approximated with various digital filter structures. Several filter

types will be discussed below, some of them studied in more detail.

The problem of the guitar body modeling using DSP techniques has

earlier been studied for sound synthesis and demonstrated for example in

[2] and [7]. The latter reference presents several digital filter solutions that

can yield good results. Among these are conventional FIR and IIR filters,warped FIR and IIR structures, and solutions where the filter is partitioned

into subfilters, for example for individual resonance modes (see also [8] and[10]). These cases are discussed in this paper from the viewpoint of bridge

pickup equalization.

4.1 EQ target response quality

The most straightforward way to realize an equalizer is to approximate the

given target response heq(n) by an FIR filter

N

H(z) = Y] w(i) heq(n) z -i (5)i=0

where N is the order of the FIR filter and w(i) is a windowing function,in the simplest case just zeroing samples of target response outside of a

desired span. We started first from this approach since it is flexible in

experimenting with the equalization quality achievable using a measured

target response.

We compared the quality of the two response estimates: the impulse

hammer and the guitar playing cases. The EQ response estimated from

the impulse hammer measurement (as illustrated by Figures 4 and 5) andfurther windowed to FIR filter hih(n) of tap length 1000-2000, was used

to convolve a played music signal $dry(n) recorded from the bridge pickup,into an equalized signal

8eq(n) _ bib(n) * 8dry(n) (6)

where, denotes discrete convolution. This was then compared with the

corresponding acoustic recording $wet(n) by informal listening.

The equalized signal Seq(n) sounded "more acoustic" than the "dry"

pickup output Sdry(n) but in this case the low-frequency resonances tendedto be too emphasized and the result was too "boomy". Also the high-

frequency sounds were a little too damped. The exact reasons to thesedeficiences were not found. One possible cause may be that the strings

were fully damped in impulse hammer measurements which is not a natural

condition during normal playing.

Next the same experiment was repeated using the target response es-

timated from playing the guitar, hmus(n). Since the response looks very

noisy (Fig. 7a), a careful windowing was applied to cut away signal beforethe main attack and after the length of a rectangular window (1000-2000

samples). The equalization was carried out in a similar way as above by

8eq(n) = hmus(n)*Sdry(n) (7)

In informal listening this was found to yield a clearly better acoustic timbrethan when using hih(n). The overall balance of the equalized sound was

good and the smoothly "reverberant" quality was more natural than the

"crispy" sound of the bridge pickup alone 3. The only feature that was

noticeably missing was the friction "hiss" or "squeak" of left-hand finger

sliding over a string. Since this radiates directly or through longitudinal

string vibration, the bridge pickup EQ filter cannot generate it.

4.2 Performance of FIR equalizers

After finding that equalization target response hmus(n) was the most usefulone the next step was to study the effect of FIR filter length on sound

quality. The target response was windowed to lengths

N = {4000, 2000, 1000, 500,300,150}

by fading the response in and out with a half-Hanning function of length

100 samples (50 samples for N = 150). The sampling rate was 44100 Hz.The FIR equalizer lengths of 1000 4000 yield very good acoustic timbre.

Actually longer lengths start sounding like playing in a room since the long

noise-like response generates extra reverberation to the equalized sound.

When shortening the FIR length below 1000 the results start soundingmore dry and electric. Below order 500 both high and low fi'equencies start

to decrease in acoustic quality. At high frequencies the attack becomessharp and at low frequencies the body resources come to an end too fast.

The experiments show that the impulse-response length of EQ is a some-what noncritical but still important factor in achieving natural acoustic

sound. The need for a relatively long response can be understood by con-

sidering the instrument body as a reverberator -- comparable to a smallroom -- whereby both temporal and spectral properties of the impulse

response must be natural. At low f,'equences the modes are relatively sep-

arated and the frequency response of the equalized system must follow themodal structure of the acoustic guitar. Also the exponential decay of these

modes has to be similar enough.

At mid to high frequencies the mode density is higher and therefore

individual modes are not as important anymore. From the frequency-domain point of view the human hearing is integrating frequency compon-

ents within a critical band so that only the spectral envelope is important.

aNotice that the desired sound characteristics is a matter of taste and style wherebysuch "crispy" and relatively "electric" sound of the pickup itself may sometimes be highlypreferred. In this study, however, we concentrate on making it to resemble the inherentacoustic characteristics as much as possible.

8

In the time domain, however, the temporal spreading of transients, espe-

cially the attacks of plucked sounds, make the body sound acoustic. Thistemporal spreading is necessary in the equalizer, and thus relatively high-

order digital filters are needed.

4.3 IIR equalizer filters

The body response of the guitar, especially at low frequencies, incorpor-ates modal resonances that can be described by exponentially decaying

sinusoids in the time domain and by complex conjugate pole pairs in the

frequency domain. Thus it sounds reasonable to model a body response

using a recursive, i.e., IIR filter that has this property. The same canbe done for equalization filter designs to approximate the target response

heq(t) described above. In this study we tried two different IIR filter designtechniques: all-pole design with linear prediction (LP) and pole-zero design

with Prony's method.

4.3.1 All-pole filter design

A simple IIR filter estimation technique, especially in speech processing, is

to apply linear prediction [13] of order M to the target response in order toobtain the denominator polynomial coefficients ai of the transfer function

bo bo

HLp(z) - A(z_ - 1 -.}-_Mi=iaiz-i (8)

where bo is a gain term.It can be expected that a low-order LP model can do overall spectral

envelope equalization, but since the body response contains a large num-

ber of modes, a high-order all-pole filter is needed to implement a detailed

spectral structure. As a time-domain interpretation this means that a high-

order filter is needed to make the exponentially decaying modal resonances

to decay slowly enough for a reverberant sound quality. Note that LP mod-

eling yields a mininmm-phase transfer function where temporal spreadingof the response is minimized.

In LP modeling experiments, we tried filter orders of M -- 1000, 500,

and 300, and filtered the "dry" bridge pickup recording with the corres-

ponding all-pole filters HLp(z). Figure 8 shows the impulse and magnitude

response of an all-pole filter of order 500 designed using linear prediction.

The minimum-phase character of the filter appears as a fast decay of the

9

response. The magnitude response contains a large number of peaks, some

of which are very sharp. It was verified that the spectral properties of

the target impulse response of length 2000 samples (which was used in the

design) have been well preserved.Informal listening showed that filter orders M = 500 and M = 1000

yield a sound quality almost comparable to FIR filter orders of 1000-2000

but it was disappointing to notice that the timbre is still more "electric"

than with FIR equalizers. Lowering the IIR filter order degraded the sound

gradually towards more "dry" quality of the pickup itself. It seems thatthe all-pole IIR solution does not give an advantage since the minimum-

phase behavior is against the rule of temporal spreading that is needed for

a good reverberant characteristic.

4.3.2 Pole-zero filter design

Pole-zero modeling can realize the mixed-phase (non-minimum phase) be-

havior of reverberant systems. We tried Prony's method [14, 15] where

the FIR part B(z) of filter Hpz(z) replicates the beginning of the target

impulse response and the all-pole part A(z) tries to capture the modal be-havior with exponentially decaying components. The transfer function of

the pole-zero filter is

B(z) N -iEi=0 biz (9)

HPZ(Z) ----- A(z) = 1 "Jr- EMi ai Z-i

We tried filter orders of N around 100 and M from 100 up. The impulse

and magnitude responses of an IIR filter designed using Prony's methodare presented in Fig. 9. The orders of the numerator and denominator

of the transfer function are M = 50 and N = 150. Now the beginning of

the impulse response agrees with the target response, but the tail decays

too fast. The spectral properties are captured rather inaccurately by thisfilter. It turns out that no relatively low-order pole-zero filter yields as

good result as, e.g., an FIR filter with N = 1000. Since the pole-zero

filter structures are more complex to compute than the straightforwardFIR structure, not much motivation is found for using high-order pole-zero

equalization filters.In conclusion, the IIR filter design methods tested in this study seem

not to be superior over a simple FIR design by truncation of the target

impulse response. Minimum-phase equalization is not an option, since it

degrades the reverberant character of the response.

l0

4.4 Warped equalizer filters

The formulation of warped digital filters was presented in [7] to model the

body of the acoustic guitar. The idea of warped filter structures is to use

allpass sections instead of unit delays in order to realize a desired digital

filter on a warped (nonuniform) frequency scale. Any traditional digitalfilter may have an equivalent warped filter although it mw not always

be realized directly. It was shown in [7] that, in spite of more complexfilter structures, the order of a warped filter can be reduced even more

radically without compromising too much the sound quality, thus resulting

in computationally efficient implementations.

Warped filters mw be found an attractive solution also to bridge pickup

equalization. Based on experience from the study in [7], however, they mw

not be optimal if the highest acoustic quality is needed at high frequences.

Warped filter equalization was not experimented in this study.

4.5 Separation of isolated modes of the body

Many other possible digital filter principles exist that, while being equival-ent in transfer function, can be advantageous from some specific point of

view. It is possible to separate the modes (pole pairs) of a transfer func-

tion and use a cascade structure (product of partial transfer functions) or

a parallel structure (sum of partial transfer functions). In [7], [8], and [10]some of these techniques have been discussed.

We noticed that there is a strong resonance at 160 Hz in the target

equalizer response (see Fig. 7d). We decided to extract this resonance,

and model it using a second-order digital resonator. When removing thismode it is possible to model the residual with lower-order filters and then,

by realizing this single mode in series or parallel with the main filter, tohave a full resolution behavior for this resonance. The extraction is realized

with a notch filter of the form

A(z) 1 4- al z-1 4- a2z-2 (10)Hr(z) - A(z/r_ - 1 4- alcz -1 4- a2c2z-2

where al = -2r cos(_o) and a2 = r2, where r is the pole radius and v:o

is the center frequency (in radians), and c is a coefficient slightly smallerthan 1 used to flatten the magnitude response of the filter far away from

the notch frequency.

11

The estimated parameter values used in this study were coo -- 0.0228056,

ri -- 0.999970, and c = 0.995000. The lower part of Fig. 10 shows the mag-

nitude response of the equalizer impulse response after notch filtering with

filter Hr(z) using the above coefficient values. It is seen that the resonance

has disappeared while other parts of the spectrum remain unchanged.The resonance can be synthesized with a second-order resonator that is

connected in parallel with an equalizer designed based on the above notch-

filtered response. We chose to use the following second-order resonator,

which has a sharp notch at zero and the Nyquist frequencies in additionto a single tuneable resonance peak:

i - z -2

R(z) = (1 - b)l _ 2boos(co0)Z -1 -{-(2b - 1)z -2 (11)

with 1b= (12)

1 - tan(Aw/2)

where coois the center frequency and Aco _ 2(r - 1) is the 3-dB bandwidthof the resonance. This digital resonator has been discussed in [16] (see pp.

248-253 and pp. 583-590).

Figure 11 illustrates how the resonator is capable of synthesizing theextracted resonance. The upper part of Fig. 11 shows the extracted 160-

Hz resonance computed by subtracting the notch-filtered equalizer response

from the original one. The lower part of Fig. 11 presents the impulse

response of the resonator R(z) of Eq. (11).It is obvious that such isolation of a single or a few modes does not help

in achieving a top-quality equalizer where modes at all frequencies should

have high spectral resolution, proper temporal decay, and temporal spread-

ing of the attack part. There is one advantage of such partitioning of thetransfer function, however, the parametric controllability of the equalizer.

The frequency, the Q value, and the relative magnitude level of each suchresonator can be adjusted with an equivalent change in the timbre of the

equalized sound. Since the lowest modes have known interpretations interms of the physical behavior of the body, this leads to controllability of

the body response in such respects.We conclude that an effective solution for equalizing the film pickup

signal is an equalizer implemented as a parallel connection of resonator

R(z) and an FIR filter H(z) of order 500 to 1000.

12

5 CONCLUSIONS AND FUTURE WORK

In this study we have shown that it is possible to improve the sound quality

of the acoustic guitar equipped with a bridge pickup using digital signalprocessing. We focused on the use of an elastic electret-type under-saddle

pickup (B-Band TM pickup from EMF Acoustics Ltd) that itself has a nat-

ural and clear sound. It was found that a digital equalizer filter needs to

improve the low-frequency modes as well as mid to high-frequency tem-poral diffusion (reverberation). This means that a relatively high-order

digital filter is needed. Digital signal processors available can, however,

accomplish this task in real time with good results. An excellent choice is

to model the lowest top-plate mode with a digital resonator and the restof the equalization with an FIR filter of order 500 or higher.

Some remaining problems to achieve acoustic sound of very high fidelity

are related to picking such vibrations that are not well exhibited by thevertical bridge vibration. At least in principle these features can be added

by using extra sensors, such as longitudinal bridge vibration sensing or neck

and fretboard vibration sensing. Such sounds that radiate only directly

from the string -- if any -- cannot be captured and equalized with thetechniques described above.

The utilization of DSP opens up also possibilities to create new sounds

by acoustic and non-acoustic effects, e.g., characteristics of the guitar or

other instruments can be changed to sound different. The generation ofsuch effects can be integrated within a single guitar DSP processor.

ACKNOWLEDGEMENT

The work of Dr. V_limSki has been financed by the Academy of Finland.

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13

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14

[12] http://valo, uts. fi/emf/.

[13] J. D. Markel and J. A. H. Gray, Linear Prediction of Speech. Berlin,Germany: Springer-Verlag, 1976.

[14] T. W. Parks and C. S. Burrus, Digital Filter Design. Englewood

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15

Fig. 1: Computer model of EMFi film structure. Electret film material

contains fiat gas bubbles. The film is coated with metal electrodes on both

sides. The thickness of the film is about 50/am.

Fig. 2: B-Band TM EMFi-type under-saddle pickup element with cable. Thethickness of the element is about 0.3 mm. The active sensor area is about

80 nlm in length and 3 nlm in width.

16

PICKUPSIGNAL EQ OUT

x(t) _ Heq(°3) _"

STRING L'O''N'aAT"'L_ .....

p(t)

IBDIRECTSTRINGRADIATION

Fig. 3: Signal paths in an acoustic guitar with bridge pickup.

iili0.02 0.03 0.04 sec

10000 t

OOOoV0.02 0.03 0.04 sec

Fig. 4: Responses to impulse hammer excitation of the guitar bridge, re-

corded from the electret pickup (bottom) and from a microphone 1 m in

front of the sound hole (top).

17

20i__....

0 5000 10000 freq/ Hz

203oo0 500 1000 1500 freq / Hz

Fig. 5: Equalizer target response when measured with impulse hammer

techniques: audio frequency range (top) and low-frequency range (bottom).

Fig. 6: Signal envelopes from 7.5 s playing of the guitar: EMFi pickup

recording (top) and microphone recording (bottom).

18

0.2

0.1

0

-0.1

0 0.05 0.1 0.15 time /

0.2

o.10 b-o.1

0.01 0.02 time / sdB

1o · c)

0

-10

5000 10000 freq / Hz

500 1000 1500 freq / Hz

Fig. 7: Equalizer target response estimated from playing by deconvolution

of the acoustic response by the pickup response. The top pane shows about

0.25 s of the deconvolution result and the second from top shows a shorter

windowed portion (about 30 ms) that has been used as a target response

for equalizer design. The third from top is the magnitude response and

the bottom curve zooms into the low-frequency region of the magnitude

response.

19

Impulse response, linear prediction

0.5

0

-0.50.01 0_.02 time/s

0.2

Target impulse response

0.1

0

-0.1

11.89 11.9 11.91 time / sdB :.......................................

20

10 i

-1

5000 10000 freq/HzdB

20 A Magnituderesponse,lowfrequenciest_

l0 ·

o I

Fig. 8: Equalizer IIR filter design with linear prediction. All-pole filter

impulse response for M = 500 (top), compared with original target re-

sponse (second from Cop). The maginitude response of the filter designedwith linear prediction is shown second from bottom and the low-frequency

portion of it in the bottom curve.

2O

0.2 Impulse resp0nse, Prony'smethod

0.1

0 0.01 0.02 time / s

0'21 , Targetimpulseresponse0.1 ·

0t i-0.1 ·

1189 119 119

dB _ _ ·10 Magnitude response_I_rony'smethod

0

-10

-200 5000 · 10000 freq / Hz

dB _ I

-10

-200 500 1000 1500 freq / Hz

Fig. 9: Equalizer design with Prony's method. Pole-zero filter impulse

response for N = 50 and M = 150 (top), compared with original target

response (second from top). The magnitude response of the filter designed

with Prony's method is shown second from bottom and the low-frequency

portion of it in the bottom curve.

21

i i , i , i i

920

10

:_-lo ,t_/_' ,' .....

9 20 160 Hz mode removed10

L°00 50 100 150 200 250 300 350 400

Frequency (Hz)

Fig. 10: Equalizer target impulse response originally (top) and after the

removal of the 160-Hz mode (bottom).

i i i i i

0.01

0.005

0

-0.005

-0.01i i i i i i i I

i i

0.01

0.005

0 --

-0.005

-0.01I I i I I I I i

-50 0 50 100 150 200 250 300 350 400Time (ms)

Fig. 11: Response of the isolated 160-Hz mode as estimated from the ori-

ginal target impulse response (top) and the impulse response of the digital

resonator R(z) (bottom).

22