Recovery of Low Frequency response by
the Matching Filter technique....
One of the major obstacles for the DIY speaker designed is obtaining reliable measurements of the low
frequency response of a speaker of driver. This obstacle is due to the usually limited environment available
for reflection free measurements. There are means of overcoming this obstacle such as combining far filed
measurements at higher frequency with near field measurement at low frequency. While this approach can
yield acceptable design data it is usually reliant on modeling of the low frequency baffle step effects, and
the procedure for obtained good data requires experience in knowing where and how to merge the far field
and near field data. Clearly a method which allowed a single full range measurement in a non or
quasi-anechoic environment would advantageous.

another article the recovery of the low frequency response of a speaker system in a room by editing the
impulse response was discussed. There it was shown that the in room response of a speaker system could
be separated into a series of impulse responses consisting of the direct sound impulse, a series of impulses
associated with baffle diffraction, and finally, a number of impulses associated with reflections from the
walls, floor and ceiling. Generically, a simplified version of such a response might appear as shown below in
Figure 1 for a 2 way speaker. For clarity the early time contribution from the baffle diffraction is shown in
blue. The direct impulse is a function of the high pass nature of the speaker woofer, the crossover, and the
low pass nature of the tweeter at high frequency. However, the long time behavior of the system is
fundamentally composed of the contribution from the woofer high pass response and the reflections.
Figure 1.
In this figure, the first reflection occurs at 6 msec. Thus, if we windowed the impulse to a length of less
that 6 msec we could obtain the response approximately 166 Hz through an FFt of the impulse with
reasonable accuracy. As discussed in the other article, if we knew the long time, anechoic behavior of
the impulse without reflection from the walls, etc. we could edit the impulse by deleting the response for t
greater than 6 msec and append the long time anechoic impulse response to it prior to performing the
FFt. This long time behavior could be estimated by examining the near field woofer response and
matching that response to a high pass target. The impulse of this high pass target would have a long
time response representative of the long time response of the speaker and could be appended to the
measured impulse to obtain the full range response of the speaker. The question becomes, how can we
effectively and easily edit such an impulse? There is, in fact, a means where by this can be
accomplished as was suggested by Benjamin [1] called the matching filter technique.

If we look at the in room impulse and compare it to the impulse associated with the high pass behavior of
the woofer on a vertically expanded scale the result is as shown in Figure 2. The blue trace is the same
system impulse as shown above. The red trace is that of the contribution from the woofer high pass
target in the absence of room reflections. The key here is that the woofer high pass impulse response
extends in time well past the time of the first reflection. We wish to remove this contribution from the
impulse response thus shortening the impulse as if, for example, the speaker was flat to DC.  A
hypothetical speaker which is flat to DC would have an impulse which approaches the ideal; a very
short, narrow initial spike of finite amplitude decaying to zero and remaining at zero to infinite time.
Figure 2.
Figure 3.
TIf we remove the high pass woofer response from the in room impulse the result is shown in Figure 3.
Here we see that the system impulse when flat to DC decays to zero well before the first reflection.
Editing the impulse in this case would be simple because it would only require that we curtail the
measured impulse just before the first reflection and pad the long time response with zeros. Of course,
real speakers are not flat to DC. The important point here is that if we can shorten the impulse so that it
decays to zero before the first reflection we can edit out the reflections without removing permanent
information about the speaker low frequency response because beyond the time of the first reflection
the reflection free impulse is zero.  

Since we are concerned with the long time behavior of the impulse we can restrict the discussion to the
high pass nature of the woofer response which is the component generating the nonzero, long time part
of the impulse. If the woofer high pass response is given by a transfer function Tw(f), then we could
ideally equalize the woofer flat to DC using equalization with a transfer function equal to the inverse of
the of the woofer response;

eq(f) = 1/Tw(f)

Unfortunately, we can not apply such equalization because the gain would go to infinity as the frequency
approached DC. However, we can compute T
eq(f) and form it compute the corresponding impulse
response, at least for a time duration of length suitable to cover the low frequency range of our
speaker, using an inverse FFt (IFFt).  

eq(t) = IFFt (Teq(f)) = FFt(1/Tw(f))

Now, if the in room measured impulse of our speaker (as shown in Figure 1), is expressed as h
then we can convolve** h
eq(t) with hsys to get h’sys(t)

sys(t) = heq(t) * hsys(t).

The consequence of this convolution is that the low frequency behavior of the measured impulse
associated with the woofer response is removed from the impulse and h’
sys(t) will now (hopefully) decay
to zero before the time of the first reflection. The degree to which this is accurate will depend on how
well the equalization function matches the inverse response of the speaker low frequency behavior. In
this manner we can then edit h’
sys(t) by curtailing it before the first reflection and padding it with zeros
as far out in time as we like without loosing information. Call the edited, reflection free impulse h”
Now, since the woofer high pass response, T
w(f) has an impulse,

w(t) = IFFt(Tw(f))

we can convolve the reflection free impulse,h"
sys(t), with hw(t) to restore the low behavior of the impulse
thus obtaining  a good representation of anechoic impulse response;

anec(t) = hw(t) * h”sys(t)

The FFt of h
anec(t) will yield an approximation to the full range, anechoic response of the speaker;

sys(f) = FFt (hanec(t)).

In summary, the procedure is as follows: 1) measure the in room impulse of the system. 2) Process the
impulse using the inverse filter representative of the speaker low frequency roll off. 3) Window the
processed impulse to remove room reflections. 4) Apply the low frequency matching filter to the
windowed impulse to restore the long time low frequency behavior. 5) Preform the FFt of the processed
impulse to obtain the full range response of the speaker.

This approach to extending quasi-anechoic measurement to low frequency was presented by Benjamin
[1] in 2004 and has been incorporated into SoundEasy with release 16 and is referred to as the
Matched filter approach since the filter upon which the equalization is matched to the speaker’s low
frequency cut off. Further details are to be provided in the manual for V16. For sealed box system it is
fairly easy to apply since a simple measurement of the box alignment will yield the corner frequency and
Q of the required matching filter. As an example of how this works I measured a small 2-way speaker
system using the combined near filed/far filed technique and the matching filter technique in a beta
version of SoundEasy V16. The result is shown in Figure 4.
Figure 4.
In Figure 4 we see 3 traces. The green trace is the response obtained using and MLS generated
impulse with a 50 msec window. The red trace is a merged far field/near field measurement. A 5 msec
widow was used for the far field and the near field response was corrected for the baffle step. The
blue trace is the response obtained with the matching filter technique. It should be observed that
below 300 Hz the matching filter approach appears to yield a better representation than the near
field/far field approach. (It should be noted that the raggedness of the tweeter response is due to the
presence of grill cloth covering the speaker.)

In the case presented above the speaker had a crossover frequency of about 3k Hz. Thus, the part of
the impulse associated with the crossover decayed to zero well before the first reflection. But what
about the case of a 3 way speaker system? Figure 5 shows results for a 3 way speaker system with a
woofer response corresponding to a 30 Hz, Q = 0.5 high pass response. The red trace is the in room
response with at 120 msec window. The green trace is the matching filter result when a 5 msec window
is used to eliminate the reflections. Obviously the result is not at all satisfactory. But what is wrong?

The problem is that the speaker has a crossover with all pass response between woofer and midrange
at about 100 Hz. If we look at the impulse for the 100 Hz all pass crossover it appears as shown in
Figure 6. It is apparent that this crossover contributes to the impulse beyond the 5 msec window used
to remove the early reflections. The fix which I developed, and which was subsequently incorporated in
SoundEasy, is straight forward. If the transfer function of the woofer/midrange all pass crossover is T
(f) then we must convolve the matching filter impulse, heq(t) with the impulse of the inverse of Tcr(f)*

cr(t) = Ifft(1/Tcr(f))

so that

sys(t) = [heq(t) * h’cr(t)] * hsys(t).

for which

anec(t) = [hw(t) * hcr(t)] * h”sys(t)

and the system response is obtained by taking the FFt of h
anec(t). When the contribution to the long
time impulse form the woofer/midrange crossover is included the technique yields the result shown in
blue in Figure 5. The ability to construct such matching filters, including the midrange/woofer
crossover all pass response, has been implemented in
SoundEasy V16 (to be released).
Figure 5,
1. Benjamin, E., Extending Quasi-Anechoic Electroacoustic Measurements to Low Frequencies, AES
Convention paper 6218, 2004.
Figure 6.
** The convolution can be
performed       in either the time or
March, 2009
An illustrative example:
Figure 7.
Figure 8.
Figure 9.
Figure 10.
Figure 11.
Figure 7 shows a hypothetical anechoic response
of a 2 way speaker with 40 Hz, Q = 1.5 high pass
cut off along with the individual woofer and tweeter
The anechoic impulse response of this system is shown
in Figure 8. The impulse consists of a short impulse
which deviates from the ideal impulse of a perfectly flat
system due to the crossover and the high pass cut off of
the woofer. Room related reflection have been omitted
so that we may observe how the application of the
matching filter removes the long time behavior of the
system impulse.
Figure 9 shows the same impulse as in Figure 8 but with
expanded vertical scale. The area shaded in yellow would
potentially be reflection free if measured in a reflective
environment where the first reflection came after 4 msec.
The portion of the impulse extending past 4 msec is due to
the 40 Hz, 2nd order high pass nature of the system. The
distortion of the impulse from the ideal due to the
crossover is limited to times less than 4 msec. If room
reflection were present they would appear in the impulse
response after 4 msec adding additional distortion to
behavior of the impulse for times greater than 4 msec.
The inverse of the matching filter for this hypothetical
system is shown in Figure 10 and is the inverse of the high
pass system response. This response would be obtained
either by measurement of the system T/S parameters or,
perhaps through simulation of the enclosure alignment.
When the anechoic impulse of the system is convolved
with the impulse of the inverse of the matching filter the
result is as shown in Figure 11. As can be seen, the
component of the impulse from the high pass nature of
the system response has been eliminated. The anechoic
impulse decays to zero well before the time where
potential room reflections would contaminate the
impulse. Thus, if room reflections were present for times
greater than 4 msec they would be the only long time
components in the impulse and can overwritten by zeros,
thus eliminating them.
Figure 12.
When the edited impulse is convolved with the impulse
of the matching filter the long time anechoic behavior
associated with the system low frequency cut off is
restored and an FFt of this impulse will yield the
anechoic system response.