Testing Loudspeakers with Wavelets 5265

Marshall Buck Psychotechnology, Inc., Los Angeles, CA, USA

and

Audio Precision, Inc., Beaverton, OR, USA

Presented at the 109th Convention 2000 September 22-25 Los Angeles, California, USA

This preprint has been reproduced from the author’s advance manuscript, without editing, corrections or consideration by the Review Board. The AES takes no responsibility for the contents.

Additionalpreprints may be obtained by sending request and remittance to the Audio Engineering Society, 60 East 42nd St., New York, New York 10165-2520, USA.

All rights reserved. Reproduction of this preprint, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.

AN AUDIO ENGINEERING SOCIETY PREPRINT

Marshall Buck, Ph.D.

Los Angeles, CA 90034 Psychotechnology, Inc.

Audio Precision, Inc.

Testing Loudspeakers with Wavelets

Abstract

A shaped sine burst wavelet is very effective for revealing audible loudspeaker distortion. The same stimulus can be used to measure both frequency response and distortion, and lends itself to gated operation. A wavelet can be designed with a flat top spectrum one-half octave wide using an IFFT and windowing. A lower crest factor wavelet with attractive qualities can be constructed with multiple synchronous sine waves, and used in a measurement system particularly suitable for quality control testing. The synchronous wavelet has a flat spectrum over one-quarter to one-half octave, and is zero outside that region, leaving a wide dynamic range for distortion components to be detected. Comparisons with standard swept sine measurements are presented.

Introduction The standard stimulus for measuring frequency response is a slow sine wave sweep. This stimulus is also traditional for measuring harmonic distortion. A short impulse, or click, can also be used to measure frequency response with fast Fourier transform (FFT) analysis. The spectrum of the impulse is related to its duration and rise and fall times, and can be very wide in practice. This is the quickest method for measuring frequency response, or more precisely, the impulse response and thus the transfer function of a linear, time invariant system. However, it requires a very quiet environment, due to the high crest factor, and thus low energy, of the stimulus. The click impulse is not useful for measuring harmonic distortion. A noise burst, especially a pseudo-random maximum length sequence, has a low crest factor and can also be very fast in a frequency response measuring system. The MLS stimulus method is also unable to measure conventional distortion. It is mainly useful for response measurement of linear, time-invariant systems. A tone burst falls between a click and a slow sweep in the range of test stimuli, and it was decided to explore its possibilities for measuring both frequency response and distortion.

Background and History Tone burst stimuli have a long and valued history in the testing of loudspeakers. (Bunton & Small, Linkwitz, Keele) They are useful for time-gated measurements

in a non-anechoic environment. They are excellent for exploring the dynamic range of a speaker non-destructively. They are audibly very revealing of overload, resonance and buzzing problems because they stimulate the device under test and quickly go away, unmasking the ear so it can detect noises and distortions quite easily. Similar advantages are found in listening to room acoustic phenomena. The International Electrotechnical Commission standard IEC 60268-5 defines an Impulsive Signal suitable for measurement as “A short-duration pulse having a constant spectral power per unit bandwidth over at least the bandwidth of interest in the measurement.” Bunton & Small (1982 June JAES) found that in order to render Cumulative Spectral Decay plots in a manner that allowed for ease of visual analysis, it was necessary to apodize (literally “cut off the feet”) of a gated tone burst. By sending the burst through a low pass filter, the start and stop transients are smoothed, and the spectrum of the pulse is made narrower. Schoukens , et al (Broadband Versus Stepped Sine FRF Measurements, IEEE, April 2000) showed that in general, measurement time is significantly less when using broadband stimuli, although there is a strong dependence on signal to noise ratio. The spectrum of an apodized tone burst is related to the number of full cycles in the burst. The following graphics are from CoolEdit Pro, and show the appearance and spectrum of a five cycle shaped pulse.

The next three spectral graphs are from Sound Forge, and show the increasingly narrow spectrum of a seven eight, and nine cycle pulse.

Note that as the spectrum becomes narrower, the level at the second harmonic of the lowest fundamental frequency gets lower, thus allowing for a better

.

floor for the measurement of distortion. A 12 cycle burst is about one sixth octave wide. D.B Keele, Jr. (Time-Frequency Display of Electro-Acoustic Data Using Cycle-Octave Wavelet Transform, 1995) showed that sixty-one such stimuli spaced on 1/6 octave centers could be used to measure both the frequency response and spectral decay of a loudspeaker over ten octaves. An advantage of such a wavelet scheme is that the desired logarithmic frequency scale is achieved with a sparse number of stimuli. My experimentation started with five cycle bursts, cosine windowed, with a one-third octave spectral width. With 33 such stimuli, the full ten octave audio band could be covered. Fast Fourier Transforms (FFT) on the resulting responses could be done with high resolution, such that the objective of 1/20 octave frequency resolution could be achieved. Due to the rounded nature of the spectral peak, it was necessary to apply a correction factor for frequencies away from the center of the 1/3 octave. This correction was about 2.2 dB. Experiments verified that this method was effective in measuring frequency response. An eight cycle pulse was created to provide a lower floor for distortion measurements, although it required 44 stimuli, since they were now ¼ octave wide.

Shaped Pulse Vs. Slow Sweep Measurements In order to explore the differences between the shaped sine pulse (wavelet) stimulus vs. the traditional slow sine sweep test, several measurements were performed. Electronic Equalizer A BSS VariCurve equalizer was set to exhibit a 15 dB dip 1/10 octave wide at 200 Hz, and a 15 dB peak 1/10 octave wide at 630 Hz. This measurement comparison uses an equalizer (BSS VariCurve). The major difference is that the pulse method underestimates the depth of the narrow dip by about 3 dB.

EQUALIZER RESPONSE - RED - PULSE, YELLOW - SW EEP. PEAK DIP BW = 1/10 OCTAVE, AMP=15 DB

Color L ine Style Thick Data Axis

Red Solid 1 Fft.Ch.1 Am pl!Norm alize LeftRed Solid 1 Anlr.Am pl!Norm alize Left

-14

+18

-12

-10

-8

-6

-4

-2

+0

+2

+4

+6

+8

+10

+12

+14

+16

dBV

80 3k90 100 200 300 400 500 600 700 800 900 1k 2k

Hz

Next the equalizer was set for 1/3 octave wide peak and dip of 15 dB amplitude. The following graph shows that the largest difference for a 1/3 octave dip is 1.4 dB and the frequency of the dip as measured by the pulse method is higher by 2.6%, which is less than 1/20 of an octave.

EQUALIZER MEASURED BY PULSE (RED) SW EEP (YELLOW )

Color L ine Style Thick Data Axis

Red Solid 1 Fft.Ch.1 Am pl!Norm alize LeftRed Solid 1 Anlr.Am pl!Norm alize Left

-14

+20

-12

-10

-8

-6

-4

-2

+0

+2

+4

+6

+8

+10

+12

+14

+16

+18

dBV

80 3k90 100 200 300 400 500 600 700 800 900 1k 2k

Hz

Acoustic Tests The acoustic tests of the shaped pulse method versus the slow sweep required an anechoic test environment. This is because the slow sweep method is not designed to gate out reflections, and delayed, reflected energy might be treated differently by the two techniques. It was decided to construct a plane wave tube, and use a compression driver as the transducer under test. The use of a plane wave tube (PWT) is standard practice for the testing of compression drivers, as the damped tube provides an acoustic impedance load for the driver that is similar to a large horn of the same throat diameter. When properly terminated, it is anechoic. Accordingly, a one inch inside diameter plastic tube three feet long was attached to a 1.25 inch thick 4 by 4 inch plastic plate that served as both the mounting surface for the driver and also a holder for the microphone, which was inserted through a hole perpendicular to the wall of the tube. The diaphragm of the microphone was tangent to the tube inner wall. The driver was a JBL Model 2426J with a one inch diameter throat (exit hole). The microphone was a Bruel & Kjaer Model 4133 ½ inch freefield mic and 2619 preamplifier, powered by a Bruel & Kjaer Model 2801 power supply. The absorbing wedge in the PWT was constructed with a number of varying lengths of long-haired wool.

Tests of the adequacy of the PWT were conducted using MLSSA measurements. The reference response is taken from the first 6 msec of energy, before the reflections from the open end of the three foot tube have a chance to interfere with the direct sound. The comparison is from an FFT of the full 200 msec of response. The error above 500 Hz is less than 1 dB, above 1200 Hz less than 0.1 dB, while the error at lower frequencies is 3 dB at points. We will concentrate on the data above 500 Hz, as that is the anechoic region with less than 1 dB error of this PWT. The error function is shown below:

Frequency Response Comparison Between Pulsed And Sweep Methods The following graph compares the fundamental response measured using the compensated shaped pulse technique and the standard slow sweep technique. Two drive levels were used for each method. The colors didn’t clipboard over correctly, but at the low frequencies, the higher amplitude drive response is

lower, while at the higher frequencies they are extremely close, within a small fraction of a decibel. In the frequency region above 500 Hz, where the PWT is anechoic, they are sufficiently close to indicate that the pulse technique works well. The following comparison shows that the swept vs pulse measurements are nearly identical at the same peak voltage, and that the effect of lowering the voltage gives similar results for both, especially in the anechoic region above 500 Hz.

PULSE (RED 8.8 VOLTS GRN .88 V) VS. SW EEP (MGNTA 8.8 Y .88 V) RESPONSE PW T. NORM @ 1KHZ

Color L ine Style Thick Data Axis

Red Solid 1 Fft.Ch.1 Am pl!Norm alize LeftRed Solid 1 Fft.Ch.1 Am pl!Norm alize LeftRed Solid 1 Fft.Ch.1 Am pl!Norm alize LeftRed Solid 1 Fft.Ch.1 Am pl!Norm alize LeftRed Solid 1 Anlr.Am pl!Norm alize LeftRed Solid 1 Anlr.Am pl!Norm alize Left

-24

+4

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

+0

+2

dBV

80 3k90 100 200 300 400 500 600 700 800 900 1k 2k

Hz

Harmonic Distortion Comparison Between Pulsed And Sweep Methods The following graph shows the fundamental, second (dashed) and third harmonics as measured by a slow sweep, at 8.8 volts continuous. Compare with the next graph.

The graph above is a measure of the fundamental, second (dashed) and third order harmonic distortion plus difference distortion components using the shaped pulse technique. The scalloped appearance of the distortion data suggests that the difference distortion components add distortion in a non-uniform manner, as a function of frequency. Nonetheless, the overall shape and level of the pulse distortion data is very similar to that from the slow sweep, although somewhat lower in amplitude. This result is adequate for a Proof-of-Concept validation.

The following graphs compare the fourth (dashed) and fifth harmonic with sweep vs. pulse measurements

The graph above shows the fourth (dashed) and fifth order distortion with the pulsed stimulus. As with the second and third harmonic measurement comparisons, the pulse data is scalloped in appearance and somewhat lower in level than the swept data. A careful look at the numbers at 500 Hz show that the fourth order harmonic plus difference distortion is lower by eight dB and the fifth order distortion is lower by seven dB when measured by the pulse method, and using the tops of the scallops as the data points to join. As before, the general shape of the distortion curves is similar when measured by the pulse and the sweep methods. There does not seem to be a systematic increase in scallop depth with increasing harmonic number, although there is a tendency in that direction. Certainly this tendency is not as strong as to require multiplying the compensation factor by the harmonic number. In fact, no compensation was applied to the distortion data.

Precompensation Of The Pulse

Next, we explore a modification of the shaped pulse: apply the frequency response compensation to the pulse, rather than to the FFT of the response. This may be most easily researched by performing an inverse FFT on an ideally shaped spectral clump. This ideal shape would have a flat top and a steep, deep skirt at the upper band edge. This might reduce computation time during the measurement, as the correct pulse shape could be simply read out from a look-up table, and the flat energy within the band would pre-compensate the fundamental.

Designer Wavelet Increases Measurement Speed A designer wavelet that stimulates over ½ octave instead of ¼ octave requires only half as many stimuli to be delivered to cover the entire audio band; twenty two will cover 20 to 30kHz. Compared to a conventional approach that requires 220 stimuli to measure this band with 1/20 octave resolution, the new wavelet is ten times as efficient. The new wavelet requires no equalization.

Details of Designer Wavelet I next experimented with designer wavelets, and I found one that satisfies a one half octave criterion. I started with a frequency spectrum, edited in MLSSA, that had the desired characteristics, i.e. a flat top along the fundamental frequencies, and a fast and deep –90 dB rolloff so that second order distortion could be measured accurately. I then performed an IFFT on this 4096 point spectrum to generate an impulse response. Some further manipulation of the resulting impulse was applied, specifically truncation and windowing. Following are the results for the one-half octave wavelet.

Designer Wavelets The following graph shows the spectrum desired. It is centered at 1000 Hertz, and the flat top extends ½ octave from 840 Hz to 1180 Hz. The second harmonic of 840 Hz is 1680 Hz, and the skirt is down 92 dB at this point, dropping further to –95 dB.

The IFFT of this spectrum created the following wavelet:

The total length of this wavelet is 54.2587 milliseconds. The original eight cycle cosine shaped 1 kHz 1/4 octave wavelet had a total length of eight milliseconds. We would like to truncate the wavelet as much as possible while still retaining the desired spectral shape. We truncate to 20 milliseconds to get the following wavelet:

When we perform a 4096 point FFT on a Cosine windowed version of this wavelet, we get a spectrum with the following useful features: The amplitude deviation of all frequencies within the ½ octave near 1 kHz is 0.07 dB. This means that no equalization of the fundamentals will be required. See Graph below:

Further, the skirt is down 92 dB at the second harmonic of the lowest fundamental, 840 Hz. At the second harmonic of all frequencies above 900 Hz, it is 95 dB down. This allows distortion measurements to –72 dB (0.25%) with 1 dB accuracy, which is adequate for loudspeaker evaluation. See Graph below:

.

Measurement Time with the Designer Wavelet The implications for lessening total measurement time using this wavelet are substantial. The measurement time for the 905 Hz stimulus is currently 22 msec, using the ¼ octave pulse. The truncation of the wavelet to 20 msec is practical, and will result in no additional measurement time per stimulus, assuming that the measurement epoch is simultaneous with the stimulus epoch. Now the stimulus duration is equal to the measurement duration. Thus the total measurement time for a 20 Hz to 30 kHz measurement would drop to about 3.2 seconds, as compared to 6.3 seconds for the ¼ octave scheme, because there are half as many stimuli.

Generating the Designer Wavelet A Look-up-Table scheme is indicated to generate the designer wavelet, as it is not an analytic function. Thus, we did no further experiments with the Designer Wavelet. The difficulty in generating such a non-analytic wavelet led to the next scheme, which was to create synchronous sine waves spaced by the lowest frequency

resolution of the analysis. For example, the sines would be spaced every two Hz when a 32768 point FFT is to be performed on data sampled at 65536 Hz. This generated a wavelet very similar in appearance to the “designer wavelet” in the time domain, but with improved qualities. By making the sine waves synchronous with the sampling rate, no windowing is required for the FFT, and the spectral signature is perfectly flat over the 1/2 octave and zero outside that region. This allows distortion components to be seen that are over 100 dB down. Although these distortion measures include difference frequency as well as harmonic distortion, it is probable that they can be very useful for quality control of loudspeakers.

Improved PWT Before proceeding with the next round of testing, the anechoic properties of the PWT were improved by adding additional absorption to the wedge. The error above 65 Hz is less than 1 dB, between 500 Hz and 8KHz less than 0.1 dB, while the error at lower frequencies is 1.5 dB at points. We will concentrate on the data above 65 Hz, as that is the anechoic region with less than 1 dB error of this PWT. The new error function is shown below:

Above is a graph showing the swept sine response4 and second and third harmonic distortion over the full audio band, of the JBL 2426J compression driver on the PWT.

Above is a graph of the synchronous sine wavelet response of the compression driver on the PWT including second and third order distortion. In this case, the distortion levels are higher than the swept single tone measures, especially in the higher frequencies. This is due to the fact that difference tones are also produced in the analysis band, adding to the levels caused by simple harmonic distortion. This effect could be a real advantage, because it simulates musical signals much more closely than do single sine waves.

Conclusion The use of wavelet stimuli is very promising for use in testing of transducers, because they can quickly and simultaneously measure both frequency response

and distortion, and they are especially well suited psychoacoustically for revealing audible problems. The correlation between measurement and audibility is less elusive than with other, more traditional stimuli. Appreciation is given to Tom Kite, Joe Rayhawk, and Rich Cabot of Audio Precision for practical and theoretical discussions, and to Jim Williams and Bill Rich for AP System Two and System Two Cascade programming.

References Bunton, John D., and Small, Richard H.; “Cumulative Spectra, Tone Bursts, and Apodization”. Journal of the Audio Engineering Society, 1980. In: Loudspeakers, Vol. 2, an AES Anthology. Keele, Donald B., Jr. “Time-Frequency Display of Electro-Acoustic Data Using Cycle-Octave Wavelet Transform”. AES Preprint 4136, October 1995. Schoukens, Johan, Pintelon, Rik M., and Rolain, Yves J.;“Broadband Versus Stepped Sine FRF Measurements”. IEEE Transactions on Measurement, April 2000. pp 275-278. MATLAB Version 5.2. MathWorks, Inc. www.mathworks.com MLSSA, DRA Laboratories. www.mlssa.com CoolEdit Pro, David Johnston, www.syntrillium.com Sound Forge, www.sonicfoundry.com