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ICTA Loudspeaker Development
By Music and Design TM....
By Music and Design TM....
Part I: Development
The transient perfect crossover presented for the ICTA woofer/midrange is not unique. In fact, the
development approach is straight forward. Small[1] briefly discussed related crossover in his 1971 paper on
constant voltage crossovers though he examined only the requirement that the summed (on axis) response
was unity and did not consider the polar radiation pattern arising form considerations of the system geometry.
Small developed asymmetric crossovers for which either the high pass or low pass section was 1st order, thus
of limited use. Small also looked only at symmetric higher order crossovers of 2nd and 3rd order. Lipshitz and
Vanderkooy [2] examined the use of overlap and equalization to develop transient perfect symmetric higher
order crossovers. They showed that it is possible to overlap standard high pass and low pass filters pairs of
the Butterworth, Bessel and Linkwitz/Riley families to obtain a minimum phase response which could then be
equalized to flat.
The approach presented here encompasses all of these preceding works, also including the B&O filler driver
type crossovers [3], and can be applied to develop transient perfect crossovers of arbitrary order for 2 or
more -way systems with symmetric or asymmetric filter characteristics. However, only examples of 2-way
crossovers are presented here.
We begin by noting that any system response that is flat and zero phase can be express as a transfer
function for which the numerator and denominator are identical polynomials.
The transient perfect crossover presented for the ICTA woofer/midrange is not unique. In fact, the
development approach is straight forward. Small[1] briefly discussed related crossover in his 1971 paper on
constant voltage crossovers though he examined only the requirement that the summed (on axis) response
was unity and did not consider the polar radiation pattern arising form considerations of the system geometry.
Small developed asymmetric crossovers for which either the high pass or low pass section was 1st order, thus
of limited use. Small also looked only at symmetric higher order crossovers of 2nd and 3rd order. Lipshitz and
Vanderkooy [2] examined the use of overlap and equalization to develop transient perfect symmetric higher
order crossovers. They showed that it is possible to overlap standard high pass and low pass filters pairs of
the Butterworth, Bessel and Linkwitz/Riley families to obtain a minimum phase response which could then be
equalized to flat.
The approach presented here encompasses all of these preceding works, also including the B&O filler driver
type crossovers [3], and can be applied to develop transient perfect crossovers of arbitrary order for 2 or
more -way systems with symmetric or asymmetric filter characteristics. However, only examples of 2-way
crossovers are presented here.
We begin by noting that any system response that is flat and zero phase can be express as a transfer
function for which the numerator and denominator are identical polynomials.
Here the coefficients, an, are real numbers and S is the complex frequency, S = jw/wo. The coefficients are
typically normalized such that ao = an = 1.0. Stability of the system response requires the roots of the
denominator (the poles of the system function) lie in the left half of the of the complex plane (the roots must
have negative real parts). Given that this is the case, since the numerator is the same function, the zeros of
the system function will also lie in the left half of the complex plan which renders the system function minimum
phase.
To obtain a transient perfect crossover we simply note that the system transfer function can be broken down
into the sum of two (or more) factors with the same denominator by grouping several terms of the numerator
for one section with the remaining terms assigned to the second (or additional) section(s). For example, a low
pass transfer function could be defined as,
typically normalized such that ao = an = 1.0. Stability of the system response requires the roots of the
denominator (the poles of the system function) lie in the left half of the of the complex plane (the roots must
have negative real parts). Given that this is the case, since the numerator is the same function, the zeros of
the system function will also lie in the left half of the complex plan which renders the system function minimum
phase.
To obtain a transient perfect crossover we simply note that the system transfer function can be broken down
into the sum of two (or more) factors with the same denominator by grouping several terms of the numerator
for one section with the remaining terms assigned to the second (or additional) section(s). For example, a low
pass transfer function could be defined as,
When the denominator is defined by a Butterworth, Bessel or Linkwitz/Riley polynomial a standard, run of the
mill n-th order low pass filter results. The corresponding high pass filter will have a transfer function given as
mill n-th order low pass filter results. The corresponding high pass filter will have a transfer function given as
This form yields the basic constant voltage crossovers discussed by Small[1] for which the low pass filter may
be defined by any of the standard polynomials or any other stable configuration. The high pass section so
derived will have a 1st order roll off since the first term in the numerator of the high pass transfer function is
first order.
Obviously, any number of terms form the numerator can be associated with the low pass filter and the
remaining assigned to the high pass section for a two way system. It is even possible to separate the system
transfer function into a multi-way crossover, with up to n separate pass bands, while maintaining transient
perfect response of the system. However, special cases are obtained when assignments are as follows:
be defined by any of the standard polynomials or any other stable configuration. The high pass section so
derived will have a 1st order roll off since the first term in the numerator of the high pass transfer function is
first order.
Obviously, any number of terms form the numerator can be associated with the low pass filter and the
remaining assigned to the high pass section for a two way system. It is even possible to separate the system
transfer function into a multi-way crossover, with up to n separate pass bands, while maintaining transient
perfect response of the system. However, special cases are obtained when assignments are as follows:
As an example, when n = 5 the highest order term in the low pass numerator is 2nd order which yields a 3rd
order low pass roll off. The lowest order term in the high pass numerator is 3rd order yielding a 3rd order roll
off of the high pass section. Thus, when n is odd we have symmetric high and low pass roll offs of order
(n+1)/2. When n is even the result is an asymmetric filter pair with a high pass roll off of order n/2+1 and a
low pass roll off of oder n/2. This asymmetry can be reversed by moving the highest order term form the
numerator of the low pass function to the numerator of the high pass function. However, since it is usually
more important to protect drivers form excessive low frequency signals the asymmetry as presented above
would usually be preferable. Obviously, the task now remains to choose the values for the coefficients ao
through an. These must be chosen such that the polynomial defining the system is stable and so that the
resulting filters remain minimum phase. (A stable system polynomial results in a minimum phase system but
not necessarily minimum phase filter sections. For example, if n is even and all the odd coefficients (a1, a3, ...,
an-1) are zero then while the system will have a flat, minimum phase response the filter sections will be linear
(zero) phase.)
A design example for n = 4 was presented on the ICTA development page. Below an example for n=7,
yielding a symmetric 4th order crossover is shown. As the order becomes higher while the final roll off will be
higher order as well, there is more overlap in the crossover region making the initial roll off more gradual. The
initial slope of the roll off and degree of peaking can be controlled somewhat by the choice of the coefficients
ao through an in the system transfer function. However, steep initial roll off leads to greater peaking in the
crossover region. While some peaking is acceptable, it would seem best to minimize it. It must be realized that
in all cases these response functions must be applied as acoustic targets with the actual filter networks
designed to shape the native driver response to these targets. This is most effectively performed using active
circuits.
When applies as a woofer to midrange crossover the polar plot shows with appropriate driver spacing (see
ICTA page) a symmetric WM-MW format will provide excellent polar response (red) while the WM format will
yield an asymmetric lobing pattern. Due to the requirement that the driver separation be much less then a
wave length at the crossover frequency such crossovers are not generally recommended for the midrange
/tweeter.
order low pass roll off. The lowest order term in the high pass numerator is 3rd order yielding a 3rd order roll
off of the high pass section. Thus, when n is odd we have symmetric high and low pass roll offs of order
(n+1)/2. When n is even the result is an asymmetric filter pair with a high pass roll off of order n/2+1 and a
low pass roll off of oder n/2. This asymmetry can be reversed by moving the highest order term form the
numerator of the low pass function to the numerator of the high pass function. However, since it is usually
more important to protect drivers form excessive low frequency signals the asymmetry as presented above
would usually be preferable. Obviously, the task now remains to choose the values for the coefficients ao
through an. These must be chosen such that the polynomial defining the system is stable and so that the
resulting filters remain minimum phase. (A stable system polynomial results in a minimum phase system but
not necessarily minimum phase filter sections. For example, if n is even and all the odd coefficients (a1, a3, ...,
an-1) are zero then while the system will have a flat, minimum phase response the filter sections will be linear
(zero) phase.)
A design example for n = 4 was presented on the ICTA development page. Below an example for n=7,
yielding a symmetric 4th order crossover is shown. As the order becomes higher while the final roll off will be
higher order as well, there is more overlap in the crossover region making the initial roll off more gradual. The
initial slope of the roll off and degree of peaking can be controlled somewhat by the choice of the coefficients
ao through an in the system transfer function. However, steep initial roll off leads to greater peaking in the
crossover region. While some peaking is acceptable, it would seem best to minimize it. It must be realized that
in all cases these response functions must be applied as acoustic targets with the actual filter networks
designed to shape the native driver response to these targets. This is most effectively performed using active
circuits.
When applies as a woofer to midrange crossover the polar plot shows with appropriate driver spacing (see
ICTA page) a symmetric WM-MW format will provide excellent polar response (red) while the WM format will
yield an asymmetric lobing pattern. Due to the requirement that the driver separation be much less then a
wave length at the crossover frequency such crossovers are not generally recommended for the midrange
/tweeter.
References:
1. Small, Constant-Voltage Crossover Network Design, J. AES,Vol. 19, No. 1,
January 1971.
2. Liptshitz, S. P. and Vanderkooy, J., Use of Frequency Overlap and
Equalization to Produce High-Slope Linear-Phase Loudspeaker Crossover
Networks, J. AES, Vol. 35, No. 3, March, 1985.
3. Bekgaard, E., A Novel Approach to Linear Phase Loudspeakers Using
Passive Crossover Networks, J.AES, Vol. 24, No. 5, May, 1977.
1. Small, Constant-Voltage Crossover Network Design, J. AES,Vol. 19, No. 1,
January 1971.
2. Liptshitz, S. P. and Vanderkooy, J., Use of Frequency Overlap and
Equalization to Produce High-Slope Linear-Phase Loudspeaker Crossover
Networks, J. AES, Vol. 35, No. 3, March, 1985.
3. Bekgaard, E., A Novel Approach to Linear Phase Loudspeakers Using
Passive Crossover Networks, J.AES, Vol. 24, No. 5, May, 1977.
The mixed order crossover with 2nd order low pass section and 3rd order high pass section presented on the
ICTA page was chosen because at this time it was found to provide the steepest initial high pass roll off with the
required, minimum 3rd order slope as well as minimizing "overlap" and overshoot in the crossover region.
Investigation into a symmetric 3rd order crossover is continuing to determine if it may provide additional
advantages.
ICTA page was chosen because at this time it was found to provide the steepest initial high pass roll off with the
required, minimum 3rd order slope as well as minimizing "overlap" and overshoot in the crossover region.
Investigation into a symmetric 3rd order crossover is continuing to determine if it may provide additional
advantages.
Part II: Active Filter Implementation with Model Drivers
In this part the implementation of a symmetric, 3rd order crossover is demonstrated. In doing so it is
necessary to recognize that the filter transfer functions represent the acoustic targets to be imposed
on the individual drivers. Additionally, the phase response of the drivers will also be superimposed on
the filters as well. To begin we examine the woofer and midrange native response as shown below. For
this example the woofer model is representative of a Peerless 10" XXLS and that of the midrange a
Scan Speak 21W8545. The SPL amplitude is shown at the left and the phase at the right.
In this part the implementation of a symmetric, 3rd order crossover is demonstrated. In doing so it is
necessary to recognize that the filter transfer functions represent the acoustic targets to be imposed
on the individual drivers. Additionally, the phase response of the drivers will also be superimposed on
the filters as well. To begin we examine the woofer and midrange native response as shown below. For
this example the woofer model is representative of a Peerless 10" XXLS and that of the midrange a
Scan Speak 21W8545. The SPL amplitude is shown at the left and the phase at the right.
As shown in the figure to the upper right, the phase response of the drivers through the anticipated
150 Hz crossover region is significantly different. The woofer response has a 25 Hz, Q = 0.5, 2nd
order high pass characteristic while the midrange has a 2nd order, Q = 0.22, 23 Hz characteristic.
Since ultimately we desire to have the system phase response and related group delay defined by the
system band pass nature the first step in designing the woofer/midrange crossover is to shift the
poles of the midrange response such that the midrange roll off at low frequency is identical to the
woofers. This can be performed using a pair of 1st order shelving filters. Upon application of the
appropriate filters the woofer and midrange response are as shown in the figure directly below.
150 Hz crossover region is significantly different. The woofer response has a 25 Hz, Q = 0.5, 2nd
order high pass characteristic while the midrange has a 2nd order, Q = 0.22, 23 Hz characteristic.
Since ultimately we desire to have the system phase response and related group delay defined by the
system band pass nature the first step in designing the woofer/midrange crossover is to shift the
poles of the midrange response such that the midrange roll off at low frequency is identical to the
woofers. This can be performed using a pair of 1st order shelving filters. Upon application of the
appropriate filters the woofer and midrange response are as shown in the figure directly below.
The result shows that the low frequency response of the woofer and midrange as now the same
(within the accuracy of the equalization). However, there is still a significant discrepancy in the
phase in the 150 Hz region owing primarily to the 1st order high frequency roll off of the woofer.
This must also be compensated for using a shelving filter with result shown below.
(within the accuracy of the equalization). However, there is still a significant discrepancy in the
phase in the 150 Hz region owing primarily to the 1st order high frequency roll off of the woofer.
This must also be compensated for using a shelving filter with result shown below.
The amplitude and phase of both drivers has now been corrected such that they are identical through the
crossover region. As such, any text book crossover filter response could be applied with essential perfect
results. Application of the current 3rd order, symmetric, transient perfect filters yields the response shown
in the next series of figures.
crossover region. As such, any text book crossover filter response could be applied with essential perfect
results. Application of the current 3rd order, symmetric, transient perfect filters yields the response shown
in the next series of figures.
The figure above to the left shows the woofer (red), midrange
(blue) and summed amplitude response (green) with the
crossover in place. The figure directly above show the phase of
the woofer (red), midrange (blue) and summed response (green).
Directly to the left is the group delay using the same color scheme.
The three figure below show a comparison to the summed
woofer/midrange response with the system 25 Hz, Q = 0.5, 2nd
order high target. The target is in red. As shown, the agreement
with the target is very good. At low frequency the phase and
group delay are those of the 25 Hz high pass target. The
crossover introduces no additional group delay. )The noise in the
GD result is because of the limited SPL data for the drivers.) The
divergence at higher frequencies is due to the midrange roll off
and will be corrected once the tweeter/midrange filter is in place.
(blue) and summed amplitude response (green) with the
crossover in place. The figure directly above show the phase of
the woofer (red), midrange (blue) and summed response (green).
Directly to the left is the group delay using the same color scheme.
The three figure below show a comparison to the summed
woofer/midrange response with the system 25 Hz, Q = 0.5, 2nd
order high target. The target is in red. As shown, the agreement
with the target is very good. At low frequency the phase and
group delay are those of the 25 Hz high pass target. The
crossover introduces no additional group delay. )The noise in the
GD result is because of the limited SPL data for the drivers.) The
divergence at higher frequencies is due to the midrange roll off
and will be corrected once the tweeter/midrange filter is in place.
Finally, the figure below shows a comparison of the net
electrical filter functions, including the required driver
equalization, to the acoustic targets. The woofer acoustic
target is shown in violet and the required electrical filter in red.
The midrange target is presented in blue with the required
electrical filters in green. The electrical filter response are both
of the minimum phase type and can be implemented using the
most efficient topology (fewest active stages) which will
produce the desired response.
electrical filter functions, including the required driver
equalization, to the acoustic targets. The woofer acoustic
target is shown in violet and the required electrical filter in red.
The midrange target is presented in blue with the required
electrical filters in green. The electrical filter response are both
of the minimum phase type and can be implemented using the
most efficient topology (fewest active stages) which will
produce the desired response.
