Jiffer Harriman

Filtering Techniques for Piezoelectric Transducers

(or How To Make Your Guitar Sound Like A Banjo)

Abstract
By extracting both the impulse response of an acoustic instrument piezoelectric pickup and body response to that same impulse, a filter can be designed to accurately make up the difference. The piezo output of an instrument can then be run through the resulting filter to achieve a sound more closely resembling the natural sound output by the acoustic body as captured by a microphone. Additionally interesting combinations of new instruments can be made by applying the body response of one instrument to the piezo output of another.


Introduction
F
or live applications acoustic instruments typically use piezoelectric pickups mounted to the bridge. This style of pickup captures the vibrations imparted on the bridge by the strings and transducers the vibrations into an electrical signal. This transduction imparts some frequency response on the signal which is characteristic of the piezoelectric crystal used. Additionally the placement at the bridge allows it to only pickup energy which is present at the bridge, whereas an acoustic guitar produces its sound from the vibration of the entire body which will have characteristics and coloration not sufficiently delivered by the bridge piezo-transducer.

By using modeling of an acoustic body, a more realistic characteristic of the instrument can be realized by combination of the piezo-transducer output with a filter representing the acoustic body which are not fully captured by the pickup alone.

Prior work
For acoustic guitars t
here are various approaches to capturing the sound which fall short of a nice microphone placed far enough away to capture the entire characteristic of the instrument. In practice, these type of microphone techniques are only practical in the recording studio. Even still, in this setting there may be use in being able to record directly from the guitar's pickup (perhaps in the presence of additional instruments).

Taylor Guitars has attempted to reconcile this issue by placing additional contact transducers on the body (not on the bridge) and combining it with an electromagnetic pickup in the neck. Line-6 is another company which has made their own attempt at this solution. The Variax line of guitars uses modeling to produce various instrument sounds including acoustic guitars.

Modeling acoustic instruments is a fairly well worn path although I'm not aware of any instances where a combination of a real string pickup are used in conjunction with a model. (It's unclear from the documentation if the Line-6 Variax does this). The technique described in the paper “Techniques for Digital Filter Design and System Identification with Application to the Violin” is particularly applicable to extracting the body response and formulating usable filters for what is known as the body loop filter.

Project Description
My goal was to capture the body response of an acoustic guitar in order to find a way to combine it with the input response of a piezo pickup. By capturing both the piezo output and the output captured by a microphone simultaneously the two responses can be compared to identify the differences. With this information a filter could be fit to apply to the piezo output and get something more resembling the whole acoustic guitar response.

The strings were heavily wrapped in a soft material in an attempt to dampen any string harmonics from becoming activated. The goal being to capture the body response only. I used a thick pick to pluck the strings in an attempt to capture a good impulse response. I also tried tapping on the bridge with the end of a large screwdriver.



Results
I was able to get some good impulse responses with the plucked string. The recordings with the taps from the screwdriver end tended to saturate the piezo pickup.


Body response of an acoustic guitar (Gibson J45):


To filter the piezo output with a body response filter I tried using the microphone impulse recording as the coefficients of a large FIR filter. The resulting filter was used to filter a short sample recorded with the piezo output. The same sample was also captured with a microphone simultaneously, this could be considered the golden data that we are aiming at.

The result is a very dark and muddy sound since portions of the body response are found in the piezo output which essentially doubles these responses.


Next I wanted to remove the piezo response from the mic response. The resulting frequency response could then be applied to a recording with the piezo pickup. To do this I divided the complex frequency data from the microphone impulse piecewise by the complex frequency data from the piezo impulse response. The resulting coefficients and frequency response are shown below.


Guitar Mic impulse response divided by Piezo impulse response

Here's how that sounds:


This sounds better but still is a little too “boomy” but it's a good start and sounds more natural than the dry piezo output. More careful and controlled recordings may result in better results, although the Gibson J45 is known for it's deep, full sound, so this is representative of the actual guitar body response.


The next goal would be to find a more efficient filter implementation which captures the body response (less the piezo response) which can be implemented at a lower computational cost but still captures the essence of the instrument. An initial attempt was made using the MATLAB function invfreqz with emphasis on the low end of the spectrum where the first several modes reside.


Here's how that sounds:



Other instruments:

Similar measurements were taken with a banjo and a square neck resonator guitar (sometimes called a Dobro after the original brand). The piezo-pickup on these instruments are mounted differently. For the dobro, a piezo disk is mounted below the resonator cone which is attached to the bridge via a screw. The banjo uses an adhesive to affix the piezo disc to the back side of the banjo's resonant head just below the bridge.


The results of applying the same techniques were mixed in these cases. I believe a significant contributing factor is the quality of recordings. These were done in a different room than the guitar recordings and I believe the room response is prevalent in these recordings as a slapback echo effect is heard when running the piezo recordings through the mic filter and mic./piezo filter. This is a result of a bad impulse response measurement in a small square room without acoustical treatment. Essentially you are hearing a convolution reverb of a instrument body in this room.


The resulting recordings can be found in the appendix section at the end of this page.


Other interesting combinations (a.k.a. cross breading instruments):

Finally I wanted to see if I could impart some of the sonic characteristics of one instrument onto another. To achieve this I used a similar technique as described above for the guitar but in varying combinations. For instance I would take the spectrum of the impulse from a microphone on a banjo (representing the body response) and divide by the spectrum of the impulse response the guitar piezoelectric pickup. Then apply the resulting filter to the output of the guitar recording from the piezoelectric pickup. The resulting “guitjo” recording does contain some of the signature of a banjo.



Some of the responses also contain some of the slapback echo mentioned above, which should not be there.


Full results of combining various instruments through different body filters formulations can be found below.


Conclusions

This method of extracting the body response and subtracting the piezo response proved to be a promising approach. This technique can allow for anyone with an instrument with a piezoelectric pickup to be run through the body filter of any number of different body filters from highly desired instruments, or of their own.


Careful measurements of other instruments provide a unique opportunity to experiment with the sound of one instrument bodies as applied to another. This could be done with two similar instruments or to vastly different ones to discover new sounds. This has the potential to make available the sounds of instruments not owned (or not played) to a player of another instrument. The typical attack, decay, sustain, release curve of each of the combined instruments will impact the realistic impression of the results. These curves are directly related to the body filter, or how the bridge reflects the energy back into the string but in these cases the body loop filter is not actually impacting the reflection of energy back into the string since it is only being applied on the output of the instrument's piezoelectric pickup output.


One interesting approach to this I'd like to explore is if a body filter could be simulated mechanically on a stringed instrument by having an active mechanical system that would take energy out of the string in the same way that a particular acoustic resonator would. This would be best on a very rigid termination so that the predominant energy decay is a result of the body loop filter being implemented.


I wonder if an acoustic shaker as used in measurement systems could be used in such an application where the input signal were a signal from a piezoelectric crystal mounted on the bridge along with the shaker and the vibrations on the shaker were the output of the input signal applied to a particular body filter. In this way the body response could be physically imprinted on the string as though it were on the body of the modeled instrument. In this way, an accurate ADSR curve would also accompany the frequency response curve.


Finally, more efficient, psychoacoustically equivalent filters can be realized than those presented here. The filters presented are formulated as very large FIR filters which would be expensive in memory. Introducing some feedback coefficients would be another logical step to reduce the filter order.




Appendix

Recordings and Filtered Results

Impulse response recordings:

Guitar

Dobro

Banjo

Piezo Impulse

Piezo Impulse

Piezo Impulse

Mic Impulse

Mic Impulse

Mic Impulse



Playing samples:

Guitar

Dobro

Banjo

Piezo recording

Piezo recording

Piezo recording

Mic recording

Mic recording

Mic recording




Filtered recordings:

Filtered through:

Guitar

Dobro

Banjo

Guitar mic

filtered_through_mic_response.wav

dobro_guitar_mic.wav

banjo_guitar_mic.wav

Guitar mic ./ piezo

mic_minus_piezo.wav

dobro_through_guitar_minus_piezo.wav

banjo_through_guitar_minus_piezo.wav

Guit mic./piezo invfreqz

guitar_piezo_filt_through_invfz.wav

dobro_through_guitar_invfz.wav

banjo_through_guitar_invfz.wav

Dobro mic

guitar_dobro_mic.wav

dobro_filtered_through_mic_response.wav

banjo_dobro_mic.wav

Dobro mic./piezo

guitar_through_dobro_minus_piezo.wav

dobro_mic_minus_piezo.wav


banjo_through_dobro_minus_piezo.wav

Dobro mic./piezo invfrqz

guitar_through_dobro_invfz.wav

dobro_piezo_filt_through_invfz.wav

banjo_through_dobro_invfz.wav

Banjo mic

guitar_banjo_mic.wav

dobro_banjo_mic.wav

banjo_filtered_through_mic_response.wav

Banjo mic./piezo

guitar_through_banjo_minus_piezo.wav

dobro_through_banjo_minus_piezo.wav

banjo_mic_minus_piezo.wav

Banjo mic./piezo invfreqz

guitar_through_banjo_invfz.wav

dobro_through_banjo_invfz.wav

banjo_piezo_filt_through_invfz.wav



Bibliography

Techniques for Digital Filter Design and System Identification with Application to the Violin

Julius O. Smith

https://ccrma.stanford.edu/STANM/stanms/stanm14/



Physical Audio Signal Processing For Virtual Musical Instruments and Digital Audio Effects

Julius O. Smith