Tech Design.....
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A Discussion of Stored Energy/Linear Distortion, Part III.
Figure 1. Amplitude response of the subject test unit.


This response was captured and then used as the target response for a digital filter with the assumption of
minimum phase. The impulse response of this filter was then measured and used to determine the response of the
driver to 5 cycle bursts at 1.2 and 1.58 k Hz by convolution of the burst signal with the measured impulse
response. These frequencies were used originally to test the driver in [1]. We note that the impulse response of the
driver could have been determined directly from the reported frequency response through an IFFt, however it was
more convenient to proceed by measurement.

Having obtained the impulse response consistent with the amplitude response of Figure 1 the driver’s response to
a 1.2k Hz burst is shown in Figure 2. The black line is the originally measured [1] response and the red line is the
response found by convolution of the measured impulse response of the DSP filter with the 1.2K Hz burst. The
agreement is quite good all the way out to 6 msec which is 2 msec after the burst terminates. The vertical scale is
10dB/division. This result indicates that generating the impulse response from the captured amplitude response
Figure 2. 1200 Hz burst comparison, base frequency response.
provides a reasonable representation of the driver’s  impulse response, at least in this frequency range.

The next figure, Figure 3, shows a similar result for a 1.58k Hz burst. Note the expanded time scale. Again we
note the excellent comparison between the measured data (black trace) and the result of the convolution (red)
with regard to the rise and initial fall of the response, to about 2.5 msec. However, beyond that time the result of
the convolution shows a much more rapid decay compared to the original result. It is the characteristic of the
original result that supported the conclusion that such burst testing can reveal resonances that are not apparent
in the basic frequency response. The current result also  tends to support this conclusion since when the
impulse is derived from the frequency response obtained using burst testing, as in Figure 1, no significant
resonant behavior is observed in the convoluted burst response. However, a resonance was originally observed
in the measured burst that was used to generate the frequency response data. This discrepancy leads to
several possibilities:

1) The impulse response determined form the captured frequency response is in error.

2) The impulse response obtained from the captured data is correct but the  frequency response of Figure 1
does not accurately represent the actual steady state driver response. This could be the case if, as suggested in
the original work, the resonance behavior observed in Figure 3 arises from a high Q, narrow bandwidth
resonance that could be stepped over when steady state amplitude response measurement were made, or
because the time required for the resonance to reach its full amplitude is longer than the burst period when the
amplitude is inferred from the burst response.

3) Some other, unknown inconsistency exists in the original data.
Figure 3. 1580 Hz burst comparison, base frequency response.
The first issue is easily addressed by taking the FFt of the impulse response of the DSP filter based on the base
frequency response shown in Figure 1. This was done and indicated that the impulse response was indeed
consistent with the captured frequency response. Item 2 leads to the conclusion that the currently measured
impulse response, while being consistent with the frequency response shown in Figure 1, somehow does not
adequately represent the true impulse response of the driver actually tested. Since there is a 1 to 1
correspondence between the impulse and steady state frequency responses this leads further to the observation
that the frequency response shown in Figure 1 does not accurately represent the true steady state frequency
response of the driver.  We shall turn to examining this consideration next. It is of interest to note that the steady
state frequency response of the driver under test was not presented at all in [1].

By examining the original burst response shown in Figure 3 we can determine several things about the 1.58k Hz
resonance. Since the input burst lasts 5 cycles we know that the burst ends at 3.165 msec. The vertical white lines
in the figure have been spaced at 1 cycle for to aid in interpretation. Thus we can see that the amplitude envelope
of the original response at the time the burst ends is about -15dB. Additionally, the decay of the response after the
burst ends is fairly linear at a rate of approximately -4.5dB/cycle. This rate of decay indicates that the Q of the
resonance is on the order of 6.0. The spacing of the vertical white lines also appears a little closer in the decay
region. This may indicate that the resonance is actually slightly higher than 1.58k Hz, but we shall assume that it is
at 1.58k for the present. Thus we have some knowledge of the resonance.

We are now in a position to examine item 2 in our list of possibilities. We shall assume first that the resonance is of
such magnitude that it might not be obviously visible from examination of frequency response data, or that the Q is
sufficiently high that the resonance did not reach its full amplitude during the burst cycle. To test these hypotheses
the base frequency response was modified by adding a 1.58k Hz resonance to it; the new impulse response was
measured; and the response to the burst, for the system with modified frequency response, was obtained by
convolution with the new impulse response.  Two cases were considered: the first with Q = 5.0 and 0.5 dB
amplitude, the second with Q=10 and 0.5 dB amplitude. The reasoning here is that a 0.5 dB blip in the response
may not have been observable in the original measurements and the range of Q spans the valued expected for the
actual resonance.

Figure 4 show the result for the case where a Q=5.0, 0.5 dB resonance at 1.58k Hz was added to the base
frequency response. Here we see that the initial response still shows the same excellent agreement with the
original data and, while the decay takes longer than that shown in Figure 3, it still does not match the decay of the
original result. The slope of the decay from the convolution result does indicate that the Q of the resonance is
indeed close to 5 but the amplitude is too low.

When the test was repeated with a Q = 10, 0.5 dB amplitude resonance in the response the result was as shown in
Figure 5. The decay of the resonance from the convolution show in Figure 5 is much slower that that of the original
result indicating that Q =10 is far too high. And once again the amplitude at the onset of the decay is too low
indicating that amplitude of the resonance peak amended to the base frequency response was also too low. To
obtain the original burst response the amplitude of the resonance must be higher.
Figure 4. Comparison with Q=5, 0.5dB 1580 Hz resonance added to base frequency response.
Figure 5. Comparison with Q=10, 0.5 dB 1580 resonance added to base response.
With this information in hand we set about to determine what value of Q and what peak amplitude would yield a
result that matched the original result. In Figure 6 the result for the case where a 1.58k Hz resonance with Q = 5.5
and a 3 dB peak was amended to the base frequency response. As shown, the match is now excellent. Since the
burst response is a known and the agreement with the original burst response is now in good agreement it can be
concluded that the impulse response used in the convolution must be close to that of the driver originally used
in the test, at least with respect to the 1.58k resonance.
Figure 6. Comparison with Q= 5.5, 3dB 1580 resonance added to base response.
Performing an FFt of the impulse response yields the result of Figure 7 where the originally reported frequency
response from burst testing is shown in blue and the amended response is shown in dark red. The amended
response is a reasonable representation the steady state response, at least in the area of the resonance, which the
driver would necessarily have to have to exhibit the observed burst behavior at 1.58k Hz. It is obviously very
different than the original amplitude response obtained through burst testing and the resonance is clearly
discernable in the steady state frequency response.
Figure 7. Dark blue = base response as reported. Dark red = base
response with Q=5.5, 3dB 1580 resonance added. The modified
response agrees with the originally reported measurement as shown
above.
To further verify that the amended frequency response shown in Figure 7 is reasonable the result of convolution of a
1.2k Hz burst with the impulse response for the amended system is shown in Figure 8. The agreement is still very
good and indicates that the addition of the 1.58k Hz resonance has minimal effect on the 1.2k Hz burst response.
Fine tuning of the 1.58k Hz resonance characteristics could be performed to obtain even better agreement.
Figure 8. 1.2k Hz burst result based on amended frequency response with
1.58k Hz, Q=5.5, 3.0dB resonance added.
Since there is a 1 to 1 correspondence between the system impulse and steady state frequency responses, and since the
burst response is unequivocally determined by the convolution of the input burst signal with the system impulse response,
it is unarguable that such resonance behavior, as that observed in the 1.58k Hz burst response, must be reflected in the
steady state frequency response. As demonstrated here, it must also be of such magnitude that it is clearly observable.
The results also show that while the transient burst response reveals the character of such resonances the amplitude
response inferred from the burst masks them. This is demonstrated in Figure 9 where the transient response of a Q=5,
+3dB, 1k Hz resonance to a 1k Hz burst is shown. In this case only the rise of the burst is windowed. The output is allowed
to reach its steady state amplitude and remain there. The black line is the envelope of the input signal. The red trace is the
output. Note that after 2 ½ cycles the amplitude of the output is still almost 3 dB down relative to its steady state level, 0dB.
Thus any amplitude response inferred from a short, windowed burst of 4 or 5 cycles would reflect this lower amplitude
completely masking the 3dB resonance peak; however it is not masked in the steady state frequency response.
Figure 9. Response of a1k Hz Q=5, 3dB resonance to a 1K input
signal with windowed rise.
It is possible that other significant resonances in the response of the driver used in the original test are also
hidden in the response of Figure 1 due to the use of burst tests to determine the amplitude data. Such resonances
would, no doubt, be better revealed using steady state frequency response measurements where they would be
clearly obvious as demonstrated by the result of Figure 7. The single piece of missing evidence is the true steady
state frequency response of the driver used in the original test which was not reported in [1].

Now that we have developed a clear understanding that resonances, hence linear distortion and energy storage
are clearly indicated in the steady state frequency response lets look at the relative importance of such testing in
choosing drivers for speaker systems. Figure 10 shows a hypothetical driver’s steady state frequency response in
blue. The response is typical of many modern day drivers and show a 2nd order 60 Hz high pass characteristics at
the low frequency limit, a rising response with a 9dB peak at 756 Hz, a dip to a minimum at 2.4k Hz and a 13dB
peak associated with cone breakup at approximately 5k Hz. Beyond 5 k Hz the response drops off steeply at
24dB/octave. From inspection of this response data we would expect linear distortion in the 750, 2.4K and 5K
frequency ranges.
Figure 10. Steady state frequency response (blue) and filtered              
                   acoustic  response (brown) of a hypothetical driver.
The impulse response of this hypothetical driver was measured and then computationally convoluted with 5 cycle
bursts at 756, 2.4k and 5k Hz to ascertain the level of linear distortion at these frequencies. The results are shown in
Figure 11. In this figure the input (black) and output (red) have been presented in the form of their envelopes.
Furthermore, the output signal has not been scaled to the 0dB level and is representative of the output level the actual
signal would obtain during the playback of a burst. As can be seen in the top plot, for 756 Hz, the output shows an
overshoot of the input, but does not obtain the level reflected in the steady state frequency response of Figure 10. A
considerable “tail” after the input signal has ended is also observed. Surely this is an indication that the driver would
perform poorly as a midrange? At 2.4k Hz the burst response is as appears in the center plot. There doesn’t appear to
be a great deal of amplitude distortion, but there are obvious problems in the decay of the output. Finally, the lower
plot shows the burst response at 5k Hz. Clearly this is a serious problem. However, since it is outside the intended
bandwidth, hopefully the crossover filter would remove this problem from further consideration.
Figure 11. Burst response for the hypothetical  driver at three           
                   problem  frequencies.
Figure 12. Burst response of the hypothetical driver/crossover filter
combination.
But what do these linear distortion results mean? Would this really be a poor driver selection for a midrange? The
problem here is that these burst tests of the driver have little or no impact on how the driver will actually perform
when then input signal is passed through a correctly designed crossover filter. After all, this is linear distortion
associated with a linear system and that means it can be corrected by a crossover filter and/or equalization net
work which is also a linear system. In fact, this is probably one of, if not the single most important issue in
designing loudspeakers: Drivers are fundamentally linear systems whose responses can be shaped to conform to
the desired response by other linear systems (crossover and equalization networks). And since once again, there
is a 1 to 1 correspondence between the systems frequency response and impulse response, and the relationship
between the impulse and burst response is simply one of convolution, then obviously what is relevant is the burst
response of the driver connected to the intended crossover filter. To demonstrate this, the burst response of the
hypothetical driver was measured when driven by a crossover which yielded an approximate 4th order, 2.5 k Hz,
Linkwitz/Riley low pass response. This is shown as the dark red curve in Figure 10. The response matches the
target to 8k Hz and then roll off at 48dB/octave above that point. As shown in Figure 12, the burst response at
756 Hz is now basically perfect. There is no over/undershoot and no significant tail compared to the response of
the raw driver shown in Figure 11. The only artifact apparent is the slight delay associated with the group delay of
the LP response. Similarly, at 2.4k Hz what you are seeing is indicative of the burst response of a 4th order, 2.5k
Hz, Linkwitz/Riley LP filter. The reduced amplitude is associated with the roll off of the response and the time shift
to the right is, again, a result of the group delay of an LR4 LP response. Finally the burst response at 5k Hz is
shown. Again this primarily shows the characteristic of the LP filter with the burst amplitude reflecting the roll off.
Clearly there is nothing here above the -40dB level that is of any concern.

In conclusion, while understanding the burst response can give insight into the behavior of a driver it would
appear that driver characteristics are better determined by starting with standard steady state frequency
response measurements through either direct measurement or through measurement of the impulse response
and then applying FFt techniques. This will allow identification of the areas where resonance problems may exist.
Then burst tests, or convolution of a burst signal with the driver's impulse response can be used to reveal the
characteristics of such resonances as desired and guide in the development of suitable crossover filters.
However, the burst response of the raw, unfiltered driver can lead to undue concern about the suitability of a
driver for a given application any may lead to the discarding of a driver that otherwise has excellent
characteristics in such areas as dispersion, nonlinear distortion, power handling, etc. Rather than judge the driver
on it’s apparent undesirable linear distortion characteristics it is far more relevant to first examine the steady state
frequency response of the driver and assess the difficulty involved in shaping the driver’s response to match the
desired target. With current DSP crossovers and design software that can emulate filters to match a desired
target response in the click of a mouse this assessment can readily be performed with little effort. If such an
assessment shows that the required filter is of greater complexity than desired, then this may provide justification
for discarding the driver. If not, burst response testing can then be applied to verify the results. However, since
the system impulse response can be obtained directly (or by MLS) or by an IFFt of the steady state frequency
response (swept sine) of the driver/filter combination and the burst response computed easily by convolution of
the impulse with the burst signal, it would seem that such a computational approach would provide an efficient
alternative to actually sweeping through the frequency range of interest with burst testing.

The primary thing to remember is that given the impulse response or frequency response of a linear system, then
things like burst response, waterfall or CSD plots, etc, are all just different ways of displaying the same
information. The FFt conveniently allows us to move from the frequency and time domain.  

1.) Linkwitz, S, L. Shaped Tone-Burst Testing, Presented at the 60th AES Convention, May, 1978.
Interpreting Shaped Tone-Burst Responses
                     John Kreskovsky
                         © April, 2004

Shaped tone burst testing can be useful in examining the transient response of a loudspeaker system or driver.
The shaped burst consists of a single frequency burst windowed using a raised cosine or Hamming window. The
burst may contain any number of cycles. Typical 4 or 5 cycles is sufficient to stimulate the test unit and observe its
response. Shaped tone-burst testing has been applied to test drivers for linear distortion or stored energy. Linear
distortion, or stored energy, is associated with irregularities or resonances in the frequency response of the test
unit. It has also been suggested [1] that such tests can reveal significant, high Q resonances that are not observed
or are difficult to pinpoint in the frequency response. This seems inconsistent with the physics of such resonances
and this premise is examined here by going back and analyzing the data of [1] using current signal processing
techniques.

We start by recognizing that all the characteristics of the unit under test can be determined from the
unit’s impulse response under the assumption that the system is linear and time invariant. This is a reasonable
assumption for loudspeakers and drivers, at least over their intended operating range. The frequency dependent
amplitude and phase responses are obtained by taking the Fourier transformation of the impulse response.
Likewise, the impulse response can be determined by taking the inverse Fourier transform of the amplitude
response. Once we have the impulse response of the driver its response to any input signal can be obtained
through convolution of the input signal with the driver’s impulse response. Since the impulse response is at the
heart of both the steady state frequency response and the response of the driver to a any signal it seems unlikely
that a significant resonance observed in the burst response would not be observable in the driver’s steady state
frequency response as well, provided the frequency response is measured with even moderate resolution. In fact,
choosing a burst frequency to examine the behavior of a resonance is dependent on knowing the frequency of the
resonance before hand. If the resonance is not shown in the frequency response then there is no guidance with
respect to what frequencies should be used in burst testing. Additionally, a frequency response curve generated
by plotting the maximum amplitude of the response to discrete bursts at various frequencies will not  necessarily
represent the true frequency response of the driver.

To examine the idea that shaped burst testing can expose resonances otherwise unseen in the amplitude
response we shall consider and examine the data used to support this conclusion in [1]. The frequency response
for the driver in question, as reported in [1], is shown in Figure 1. This is the response as obtained through burst
testing and does not necessarily represent the steady state frequency response. The response is not particularly
smooth, but does
not visually show any serious resonances. In fact, the response looks rather benign.