Tech

John KreskovskyMusic and Design © November, 2008 |

In [1] Duelund presents a means of developing a crossover for s 3-way speaker system from a unique

approach with result that all the filter functions remain in phase at all frequencies. For details the reader is

referred to the web site where Steen Duelund’s article may be downloaded. Duelund expresses the

crossover transfer function for the complete system as

approach with result that all the filter functions remain in phase at all frequencies. For details the reader is

referred to the web site where Steen Duelund’s article may be downloaded. Duelund expresses the

crossover transfer function for the complete system as

where s is the complex frequency, j ω/ωo , and ωo is the center frequency of the bandpass response in

radians/sec. “**a**” is the inverse of the damping factor, a = 1/Qd. The crossover has the characteristic that

when separated into HP, BP and LP sections, all sections have the same phase response. While this looks

imposing it is actually very simple. Some clarity can be gained by retaining the denominator in factored

form:

radians/sec. “

when separated into HP, BP and LP sections, all sections have the same phase response. While this looks

imposing it is actually very simple. Some clarity can be gained by retaining the denominator in factored

form:

What we have here is a transfer function for which the denominator is the square on a 2nd order, all pole

low pass filter. We can then rewrite this transfer function as the sum of a high pass, band pass and low

pass filters:

**H(s) = HP(s) + BP(s) + LP(s)**

where

low pass filter. We can then rewrite this transfer function as the sum of a high pass, band pass and low

pass filters:

We see further that the HP and LP sections are just cascades of two 2nd order HP or LP filters with the

same Q and the bandpass is a cascade of one each of the 2nd order LP and HP sections with inverted

polarity and gain, G = (1/Qd)2 – 2.

As Duelund points out, when Qd = 1/√2, the 2nd order filters are of the Butterworth type and the HP and

LP section are Butterworth squared filters, and form the well known 4th order Linkwitz/Riley crossover.

Note that when Qd = 1/√2, the bandpass filter has zero gain. Thus, we have a 2-way LR4 crossover for Qd

= 1/√2. Duelund also looks at various other values for Qd and discusses then in sufficient detail that I will

not repeat it here.

However, it may be of interest to look at what happens when Qd is 0.5 or less. When Qd is 0.5 or less the

roots of P(s),

same Q and the bandpass is a cascade of one each of the 2nd order LP and HP sections with inverted

polarity and gain, G = (1/Qd)2 – 2.

As Duelund points out, when Qd = 1/√2, the 2nd order filters are of the Butterworth type and the HP and

LP section are Butterworth squared filters, and form the well known 4th order Linkwitz/Riley crossover.

Note that when Qd = 1/√2, the bandpass filter has zero gain. Thus, we have a 2-way LR4 crossover for Qd

= 1/√2. Duelund also looks at various other values for Qd and discusses then in sufficient detail that I will

not repeat it here.

However, it may be of interest to look at what happens when Qd is 0.5 or less. When Qd is 0.5 or less the

roots of P(s),

The significance of this is that the resulting 4th order transfer function has four poles which are given as

two set of two identical poles. As stated above, when a 2nd order transfer function has two identical poles

Q must equal 0.5. Thus we can rewrite the form of the LP filter as

two set of two identical poles. As stated above, when a 2nd order transfer function has two identical poles

Q must equal 0.5. Thus we can rewrite the form of the LP filter as

where

** γ = ω****1****/ω****o**** = ω****o****/ω****2**

ω1 being the greater that ωo. That is, for any value of Qd less than or equal to 0.5 the Duelund filter

sections can be expressed as two cascaded 2nd order filters with Q = 0.5 and staggered corner

frequencies ω1, ω2. In case you haven’t made the connection, since Q = 0.5 yields a 2nd order

Linkwitz/Riley filter, for Qd less that or equal to 0.5 the Duelund 3-way crossover HP, LP and BP section

reduce to cascaded 2nd order Linkwitz/Riley filters with staggered cut off frequencies, ω1 and ω2.

We can thus write the HP and BP sections in this form as well, and repeating the LP section for

completeness we have:

sections can be expressed as two cascaded 2nd order filters with Q = 0.5 and staggered corner

frequencies ω1, ω2. In case you haven’t made the connection, since Q = 0.5 yields a 2nd order

Linkwitz/Riley filter, for Qd less that or equal to 0.5 the Duelund 3-way crossover HP, LP and BP section

reduce to cascaded 2nd order Linkwitz/Riley filters with staggered cut off frequencies, ω1 and ω2.

We can thus write the HP and BP sections in this form as well, and repeating the LP section for

completeness we have:

All that remains is to relate G and γ to the Duelund parameters, Qd, and ωo. To that end we find the

following relationships useful:

** ω****o**** = √( ω****1****× ω****2****)**

γ = √( ω**1****/ ω****2****)**

G = 1 + 1/γ4

Q**d**** = γ / (γ2 + 1)**

This form is appropriate for all γ ≥ 1.0. The benefit of this cascaded 2nd order Linkwitz/Riley form should be

obvious; we can choose the woofer/midrange, midrange/tweeter crossover points, ω1, ω2, directly and then

construct the filters, setting the band pass gain factor from knowledge of γ. The band pass gain factor and

Qd as a function of γ is shown below.

following relationships useful:

γ = √( ω

G = 1 + 1/γ4

Q

obvious; we can choose the woofer/midrange, midrange/tweeter crossover points, ω1, ω2, directly and then

construct the filters, setting the band pass gain factor from knowledge of γ. The band pass gain factor and

Qd as a function of γ is shown below.

are both real numbers. These roots of the denominator of the basic 2nd order transfer function form the

transfer functions poles. These roots correspond to frequencies given as

** ω1,2 = ω****o**** [1/(2Q****d****) +/– Sqrt((1/2Q****d****)2 -1)]**

where the plus sign is used for ω1 and the minus for ω2. These poles characterize the behavior of the filter

response. We note that for Qd = 0.5, ω1,2 are real and equal, and when Qd is less than 0.5 there are two

unique, real values for ω1,2, one above ωo and one below ωo.

This leads to an interesting observation. Looking at the LP transfer function we see that there are a total

of four poles, 2 each form each 2nd order section, as indicated below. Each section as the same poles so

that of the four, at most only two are unique. When Qd is less than 0.5, the poles are all real as was stated

previously.

transfer functions poles. These roots correspond to frequencies given as

response. We note that for Qd = 0.5, ω1,2 are real and equal, and when Qd is less than 0.5 there are two

unique, real values for ω1,2, one above ωo and one below ωo.

This leads to an interesting observation. Looking at the LP transfer function we see that there are a total

of four poles, 2 each form each 2nd order section, as indicated below. Each section as the same poles so

that of the four, at most only two are unique. When Qd is less than 0.5, the poles are all real as was stated

previously.

Band pass gain factor and Qd vs. γ for reformulated Duelund crossover.

It should be noted that while Duelund goes to the trouble of defining formulas for passive crossover

components for the filter sections, if the characteristic of the Duelund crossover that all drivers being in

phase at all frequencies, then there is little point of these ideal filter component values. Rather, the transfer

functions should be used as the acoustic targets for LP, BP and HP drivers and the filters developed to

achieve these results. In one considers the midrange driver we see further that given ω2, the midrange to

woofer crossover frequency, then what is really needed is a sealed midrange enclosure which yields a Qtc =

0.5 at ω2. In this regard, the T/S parameters of the midrange driver will dictate the mid/woofer crossover

point. Additionally, the acoustic output of a tweeter is basically that of a 2nd order HP response. Thus to

achieve the Duelund HP acoustic response the poles of the tweeter response must be shifted to that of

either HP section of the cascaded Duelund filter with an additional electrical filter performing the function of

the second section. Similar consideration must be given to the LP response of the woofer and midrange,

taking the acoustic roll off of the midrange driver and woofer into consideration. One important note should

be emphasized. Since the midrange should, in theory, run without a HP filter in a Q = 0.5 sealed enclosure,

there is a danger of over driving the midrange leading to higher distortion and even failure.

Without passing judgment one must wonder as to the benefits of theoretically having all drivers in phase at

all frequencies. The figure below shows an overlay of a 3-way using the Duelund crossover with x-o points at

300 and 2k Hz (γ = 2.582) compared with a 3-way using all LR2 filters. The vertical scale is 5dB/dividion.

Below that are the theoretical 50 Hz square wave response for each, Red = woofer, Green = midrange, Blue

= tweeter and violet = system. Neither shows ringing in the system response while both show some

overshoot in the response of the midrange and tweeter. Additionally, the early time response is not a

monotonic rise in either case.

It should be noted that while Duelund goes to the trouble of defining formulas for passive crossover

components for the filter sections, if the characteristic of the Duelund crossover that all drivers being in

phase at all frequencies, then there is little point of these ideal filter component values. Rather, the transfer

functions should be used as the acoustic targets for LP, BP and HP drivers and the filters developed to

achieve these results. In one considers the midrange driver we see further that given ω2, the midrange to

woofer crossover frequency, then what is really needed is a sealed midrange enclosure which yields a Qtc =

0.5 at ω2. In this regard, the T/S parameters of the midrange driver will dictate the mid/woofer crossover

point. Additionally, the acoustic output of a tweeter is basically that of a 2nd order HP response. Thus to

achieve the Duelund HP acoustic response the poles of the tweeter response must be shifted to that of

either HP section of the cascaded Duelund filter with an additional electrical filter performing the function of

the second section. Similar consideration must be given to the LP response of the woofer and midrange,

taking the acoustic roll off of the midrange driver and woofer into consideration. One important note should

be emphasized. Since the midrange should, in theory, run without a HP filter in a Q = 0.5 sealed enclosure,

there is a danger of over driving the midrange leading to higher distortion and even failure.

Without passing judgment one must wonder as to the benefits of theoretically having all drivers in phase at

all frequencies. The figure below shows an overlay of a 3-way using the Duelund crossover with x-o points at

300 and 2k Hz (γ = 2.582) compared with a 3-way using all LR2 filters. The vertical scale is 5dB/dividion.

Below that are the theoretical 50 Hz square wave response for each, Red = woofer, Green = midrange, Blue

= tweeter and violet = system. Neither shows ringing in the system response while both show some

overshoot in the response of the midrange and tweeter. Additionally, the early time response is not a

monotonic rise in either case.

Over lay of Duelund 3-way target response with LR2 3-way targets for γ = 2.582 with ωo= 1k Hz.

50 Hz square wave response of Duelund 3-way filters. Red = LP, Green = BP, Blue = HP and Violet = summed.

50 Hz square wave response of** LR2** 3-way filters. Red = LP, Green = BP, Blue = HP and Violet = summed.

Below we see the step response for the case of γ = 1.1 on an expanded time scale. Again, Red = LP, green

= BP, Blue = HP and Violet = system. The results are similar to the 50 Hz square wave response. While there

is no overshoot, the rise to the step is not monotonic. There is an initial rise from the tweeter followed by a

dip with contributions from the tweeter and midrange, followed by the rise to the final step amplitude. The

group delay is shown below the step response.

= BP, Blue = HP and Violet = system. The results are similar to the 50 Hz square wave response. While there

is no overshoot, the rise to the step is not monotonic. There is an initial rise from the tweeter followed by a

dip with contributions from the tweeter and midrange, followed by the rise to the final step amplitude. The

group delay is shown below the step response.

Step response of Duelund 3-way crossover. Red = LP, Green = BP, Blue = HP and Violet = summed.

Beyond Duelund

The discussion above demonstrated that for Qd less than or equal to 0.5 the Duelund 3-way crossover

can be constructed from staggered corner frequency, cascaded, 2nd order HP and LP Linkwitz/Riley

transfer functions. All section have two pairs of identical poles; one pair at the low pass and one pair at

the high pass crossover points. I independently found this approach to 3-way crossovers a number of

years ago when investigation the B&O filler driver approach to transient perfect crossover. I did not make

the connection to the Duelund crossover until recently. However, this concept does not stop here. In fact,

3-way crossovers with higher band pass slopes can be developed using the same approach. Not all such

crossovers have the property that all transfer functions have the same phase vs. frequency. The

generalized form of the crossover can be expressed as

The discussion above demonstrated that for Qd less than or equal to 0.5 the Duelund 3-way crossover

can be constructed from staggered corner frequency, cascaded, 2nd order HP and LP Linkwitz/Riley

transfer functions. All section have two pairs of identical poles; one pair at the low pass and one pair at

the high pass crossover points. I independently found this approach to 3-way crossovers a number of

years ago when investigation the B&O filler driver approach to transient perfect crossover. I did not make

the connection to the Duelund crossover until recently. However, this concept does not stop here. In fact,

3-way crossovers with higher band pass slopes can be developed using the same approach. Not all such

crossovers have the property that all transfer functions have the same phase vs. frequency. The

generalized form of the crossover can be expressed as

and is shown below.

Here γ ≥ 1 and n is the order of the filter polynomial, Pn(), which defines the denominator. These

expressions show that the bandpass roll off asymptote is always ½ the order of the HP and LP asymptotic

slopes. N = 1 is, of course, a subset of the generalized, transient perfect, B&O filler driver approach. All

drivers are in phase only when n is an even number. When n is even Pn( ) is of the Linkwitz/Riley type

and the polarity of the band pass is determined by in where i is the square root of -1. Thus for n = 2, as

discussed at length above, the band pass polarity is inverted. When n = 4, the band pass polarity is

positive. When n is odd the band pass sums in quadrature, Butterworth polynomials are used for Pn( ),

and the band pass may be of positive or negative polarity.

In all cases we see that all three sections have two sets of poles, one set each associated with Pn(γs)

and Pn(s/γ).

Example for N = 3

When n = 3 a 3rd order Butterworth polynomial is user for Pn( ). The filters sum in quadrature and a 90

degree phase difference is maintained at all frequencies. It is of interest to note that when γ = 1 the HP

and LP transfer function collapse to LR 6th response. If the HP and LP sections are connected with

normal polarity, a null will occur at the crossover point for any value of γ >1. This null can be “filled” by

the midrange driver. Alternatively, when γ =1 the polarity of the HP or LP section can be inverted yielding

a 2-way, LR6 crossover. The response plot below shows the case for γ = 2.582 (300 Hz and 2k Hz

crossover points). The scale has been changed to 10dB/division to show the change in slope of the HP

and LP section at the LP and HP crossover points, respectively.

The gain for the BP stage is given as

expressions show that the bandpass roll off asymptote is always ½ the order of the HP and LP asymptotic

slopes. N = 1 is, of course, a subset of the generalized, transient perfect, B&O filler driver approach. All

drivers are in phase only when n is an even number. When n is even Pn( ) is of the Linkwitz/Riley type

and the polarity of the band pass is determined by in where i is the square root of -1. Thus for n = 2, as

discussed at length above, the band pass polarity is inverted. When n = 4, the band pass polarity is

positive. When n is odd the band pass sums in quadrature, Butterworth polynomials are used for Pn( ),

and the band pass may be of positive or negative polarity.

In all cases we see that all three sections have two sets of poles, one set each associated with Pn(γs)

and Pn(s/γ).

Example for N = 3

When n = 3 a 3rd order Butterworth polynomial is user for Pn( ). The filters sum in quadrature and a 90

degree phase difference is maintained at all frequencies. It is of interest to note that when γ = 1 the HP

and LP transfer function collapse to LR 6th response. If the HP and LP sections are connected with

normal polarity, a null will occur at the crossover point for any value of γ >1. This null can be “filled” by

the midrange driver. Alternatively, when γ =1 the polarity of the HP or LP section can be inverted yielding

a 2-way, LR6 crossover. The response plot below shows the case for γ = 2.582 (300 Hz and 2k Hz

crossover points). The scale has been changed to 10dB/division to show the change in slope of the HP

and LP section at the LP and HP crossover points, respectively.

The gain for the BP stage is given as

Gain vs. Gamma for N =3

Below the plot of the amplitude response the 50 Hz square wave response when the bands pass is

connected with normal and inverted polarity is shown. While the three section now sum in phase

quadrature it should be noted that when the band pass polarity is inverted the system square wave (and

step) response is similar to the case for n = 2.

connected with normal and inverted polarity is shown. While the three section now sum in phase

quadrature it should be noted that when the band pass polarity is inverted the system square wave (and

step) response is similar to the case for n = 2.

Amplitude response for 3-way crossover when n = 3. The band pass

slopes are 3rd order The HP and LP slopes are initially 3rd order

increasing to 6th order outside the band pass corner frequencies.

slopes are 3rd order The HP and LP slopes are initially 3rd order

increasing to 6th order outside the band pass corner frequencies.

50 Hz square wave response for n = 3, in phase band pass.

Note overshoot in system response (violet).

Note overshoot in system response (violet).

50 Hz square wave response for n = 3, inverted band pass.

Note the lack of overshoot in system response (violet).

Note the lack of overshoot in system response (violet).

Example for N = 4.

When n = 4, Pn( ), must be of the form of a 4th order Linkwitz/Riley polynomial. The result is that the

asymptotic slope of the HP and LP section is 8th order while the band pass roll off symmetrically at 4th

order. All filters are connected in phase. All driver are filters are connected in phase and all driver have

the same phase at all frequencies. The gain for γ >1 is given as

When n = 4, Pn( ), must be of the form of a 4th order Linkwitz/Riley polynomial. The result is that the

asymptotic slope of the HP and LP section is 8th order while the band pass roll off symmetrically at 4th

order. All filters are connected in phase. All driver are filters are connected in phase and all driver have

the same phase at all frequencies. The gain for γ >1 is given as

and is shown below.

Gain vs. Gamma for n = 4.

This filter, like the Duelund 3-way, has all drivers in phase at all frequencies. Also like the Duelund

filter, for = 1 Pn( ) can take on other forms with appropriate adjustment to the BP gain. For example, if

Pn( ) is taken as a 4th order Butterworth polynomial the HP and LP sections default to LR8 filters which

sum flat in phase without the addition of a BP filter. That is, we have a 2-way LR8 crossover. Here we

are concerned only with the case of γ >1.0

The response plot below again shows the case for γ = 2.582 (300 Hz and 2k Hz crossover points). The

scale has again been changed to 10dB/division to show the change in slope of the HP and LP section

at the LP and HP crossover points, respectively. Below the plot of the amplitude response the 50 Hz

square wave response when the bands pass is connected with normal polarity is shown.

filter, for = 1 Pn( ) can take on other forms with appropriate adjustment to the BP gain. For example, if

Pn( ) is taken as a 4th order Butterworth polynomial the HP and LP sections default to LR8 filters which

sum flat in phase without the addition of a BP filter. That is, we have a 2-way LR8 crossover. Here we

are concerned only with the case of γ >1.0

The response plot below again shows the case for γ = 2.582 (300 Hz and 2k Hz crossover points). The

scale has again been changed to 10dB/division to show the change in slope of the HP and LP section

at the LP and HP crossover points, respectively. Below the plot of the amplitude response the 50 Hz

square wave response when the bands pass is connected with normal polarity is shown.

Amplitude response for 3-way crossover when n = 4. The band pass

slopes are 4rd order. The HP and LP slopes are initially 4rd order

increasing to 8th order outside the band pass corner frequencies.

slopes are 4rd order. The HP and LP slopes are initially 4rd order

increasing to 8th order outside the band pass corner frequencies.

50 Hz square wave response for n = 4. Note the slight

overshoot in system response (violet).

overshoot in system response (violet).

1. Duelund, Steen. The art of building a loudsepaker to the end, Part II - The crossover

network. Web publication, circa 2004?

network. Web publication, circa 2004?

Group delay for Duelund 3-way crossover with 300 and 2k hz crossover points.

Group delay for n = 3, 3-way crossover with 300 and 2k hz crossover points.

The group delay is identical to that of the Duelund (n=2) crossover.

The group delay is identical to that of the Duelund (n=2) crossover.

Group delay for n = 4, 3-way crossover. Note the higher DC group delay and

non-monotonic rise with bump at 200 Hz, typical of higher order crossovers.

non-monotonic rise with bump at 200 Hz, typical of higher order crossovers.