Recovery of Low Frequency response by

the Matching Filter technique....

(preliminary)

the Matching Filter technique....

(preliminary)

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.

In 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.

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.

In 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.

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;

Teq(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 Teq(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).

heq(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 hsys(t),

then we can convolve** heq(t) with hsys to get h’sys(t)

h’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”sys(t).

Now, since the woofer high pass response, Tw(f) has an impulse,

hw(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;

hanec(t) = hw(t) * h”sys(t)

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

Tsys(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.

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;

Teq(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 Teq(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).

heq(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 hsys(t),

then we can convolve** heq(t) with hsys to get h’sys(t)

h’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”sys(t).

Now, since the woofer high pass response, Tw(f) has an impulse,

hw(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;

hanec(t) = hw(t) * h”sys(t)

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

Tsys(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 Tcr

(f) then we must convolve the matching filter impulse, heq(t) with the impulse of the inverse of Tcr(f)*

h’cr(t) = Ifft(1/Tcr(f))

so that

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

for which

hanec(t) = [hw(t) * hcr(t)] * h”sys(t)

and the system response is obtained by taking the FFt of hanec(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).

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 Tcr

(f) then we must convolve the matching filter impulse, heq(t) with the impulse of the inverse of Tcr(f)*

h’cr(t) = Ifft(1/Tcr(f))

so that

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

for which

hanec(t) = [hw(t) * hcr(t)] * h”sys(t)

and the system response is obtained by taking the FFt of hanec(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.

Convention paper 6218, 2004.

Figure 6.

** The convolution can be

performed in either the time or

frequency

domain.

performed in either the time or

frequency

domain.

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

response.

of a 2 way speaker with 40 Hz, Q = 1.5 high pass

cut off along with the individual woofer and tweeter

response.

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.

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.

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.

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.

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.

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.