Tech Design...
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Examination of crossover induced  transient distortion.
The following discussion address how we can look at a multi-way speaker's (the "system")
transient response by decomposing the response into a minimum phase component and an all
pass component.  The minimum phase component of the system represents the best
performance which could be obtained without the used of various types of digital processing. The
all pass component represents the additional phase distortion introduced by the crossover.
Since the all pass response will be, by definition, flat from DC to infinity, if it is not to introduce
any additional phase/transient distortion it must be either minimum (zero) phase or linear phase.
Linear phase reduces to a pure time delay which introduces no transient distortion. Here we shall
limit the discussion to what can be engineered using analog filters. We shall also address the
performance only on the design axis and at the design listing distance. Off axis effects, power
response, polar response, etc are not considered since they not only depend on the crossover
but on driver directionality, spacing, system format (dipole, monopole, line source),  etc.

Before we start a few general comments about crossovers should be presented.

1) All crossovers which sum flat yield an all pass responses.

2) Any crossover which sums flat when connected with one or more sections inverted relative to
the remaining sections can not be minimum or linear phase and therefore, can not be transient
perfect.

3) Any crossover that sums flat in phase (LR type crossovers and the notched variants by Thiele
[1]) can not be minimum phase or transient perfect. (Linear phase crossovers with LR amplitude
response do sum flat in phase, and are transient perfect, but not minimum phase.)

4) Any speaker that uses physical offset of the driver acoustic centers can not be minimum or
linear phase, thus not truly transient perfect.

I make these statements up front because I have seen over the web speaker systems presented
as minimum phase and/or transient accurate which use in phase crossovers and, for example,
have the tweeter connected with inverted polarity. While these speakers may be otherwise well
designed, the claims of minimum phase and/or transient accuracy are exaggerations.

We now begin by observing that every multi-way speaker system is a band pass device with
representative low and high frequency cut offs. For the sake of this discussion we shall consider
a two-way system with 2nd order 40 Hz low frequency cut off and a 25 k Hz high frequency cut off
that  initially follows a nd order roll off and then decays into as steeper 4th order roll off. The
anechoic response of such a system would appear as shown below in Figure 1.
The transient response of such a system is governed by both the amplitude and phase of the
system. Here we shall be concerned mainly with the effect of the system phase on the transient
response.

For any system it is possible to decompose the response into a minimum phase component and
an all pass component. Thus, we can write the system transfer function as

S(s) = MP(s) x AL(s)  

where s is the complex frequency, jω, MP is the minimum phase component and AP is the all
pass component. The minimum phase component of the system response is just that of a band
pass filter having the same amplitude response. The Figure 2, below, shows the phase of the
minimum phase component in red and the associated group delay in blue. The group delay is
zero at frequencies well above the high frequency limit of the system and then increases to a
more or less constant value mid band associated with the low pass nature of the high frequency
limit of the system. (A low pass filter has constant group delay over much of its pass band.) As
the frequency continues to decrease we observe an increase in the group delay becoming
obvious some where around 400 Hz. The Figure 3 shows the square wave response of this
minimum phase system at frequencies between 100 Hz and 10, 000 Hz. Note this over this
frequency range the amplitude response is flat. The 100 Hz response shows a slanted top and
bottom of the wave, This is a result of the varying group delay at low frequency. At 200 Hz the
response is still slanted, but to a lesser degree. Similarly at 400 Hz. As the frequency  rises we
see flatter and flatter tops and bottoms of the square wave because the group delay is becoming
more constant. However, by 1600 Hz wee start to see s slight rounding of the leading edge of the
rise and fall of the wave. The primary reason for this is because the higher frequency
components of the square wave are being attenuated in amplitude by the low pass nature of the
system at high frequency
The square wave responses shown above represent the best possible result for a conventional
speaker system which possesses the band pass response as indicated.

We must next consider what degradation in the response we might see when we include the
contributions form the all pass component of the system. Since an all pass response is flat from
DC to infinity by definition, the all pass component can only introduce phase distortion. First we
must ask, where does the all pass component come form? There are two sources for a multi-way
speaker. The first is the propagation delay from the sources to the listening position. If the
drivers are correctly aligned, with their acoustic centers the same distance from the listening
position, then this aspect of the all pass response is nothing more than a constant time delay.
SInce a constant time delay does not introduce any transient distortion we can place this aside
for the moment. The second aspect of the all pass response arises from the crossover. Any
2-way crossover which has a low pass and high pass section that sums flat in amplitude at the
listening position is, again by definition, and all pass response. Thus, all odd order Butterworth
crossovers are all pass crossovers as are the even order Linkwitz Riley crossovers. In addition to
these common crossovers there are also any number of transient perfect crossovers such as the
constant voltage (subtractive) crossovers discussed by Small [2], the transient perfect 2nd order
crossovers discussed by Kreskovsky [3,4], and the family linear phase crossovers defined by
subtraction and delay as originally discussed by Lipshitz and Vanderkooy[5].

Before we begin to look at the different possibilities for the all pass component of the multi-way
speaker's response we must emphasize that when the term crossover is used it is taken to mean
that the acoustic output of the drivers has been shaped to exactly match the transfer function
defined by the target response. That is, if an LR4 crossover is referred to, it means that at the
listening position the acoustic output of the driver would exactly match the LR4 HP or LP target,
except for the roll offs associated with the system response at the frequency extremes. We must
also accept that these idealized crossovers sum perfectly flat only if the phase variation due to
the high and low frequency cut off of the system response does not interfere with the targeted
crossover phase. This is predominantly a problem with lower order crossovers which will result in
small amplitude errors in the response. These small amplitude variations should be included as
part of the minimum phase component of the decomposition, but for convenience, here we shall
assume the minimum phase component is always given by that presented above where the
amplitude in the pass band is perfectly flat. As such, the all pass components to be presented
may also include a small amplitude variation to account for equalizing the system response to the
flat minimum phase component.

To obtain the characteristic of the all pass response we divide the system response S(s) byt the
response of the minimum phase component;

AP(s) = S(s) / MP(s)

This all pass response then defines the additional phase distortion introduced by the crossover,
relative to the listening position. The ideal all pass response would be a linear phase response
representative of a pure time delay of finite magnitude. Note that zero phase vs frequency is just
a linear phase response associated with a zero time delay. Now let's look at the effects of
different crossover. In all cases we shall look at a crossover at 1.5 K Hz. To demonstrate the
effect of the crossover on transient response we shall examine reproduction of an 800 Hz square
wave since the frequency components of such a wave will span the crossover frequency.

1st order Butterworth:

The all pass component for the B1 crossover is shown in Figure 4 with the amplitude (red) and
phase(green) at the left and the 800 Hz square wave reproduction at the left. The all pass
response shows a small error in amplitude and phase at high frequency which is due to the
effects discussed above. Other than that, it is apparent the square wave reproduction is near
perfect as would be expected for a B1 crossover since it introduce no amplitude or phase
distortion, as is well known. That is, the response is transient perfect. In this regard, a speaker
system using a correctly implemented B1 crossover with band pass response as shown in Figure
1 would exhibit the square wave response shown for the minimum phase band pass system
response shown in Figure 3 without further degradation by the crossover.
Figure 1.
Figure 2.
Figure 3.
Figure 4.
3rd order Butterworth, in phase:

The all pass component for the B3 crossover is shown in Figure 5.  It is apparent that the all
pass response introduces significant phase distortion and the square wave response is highly
degraded compare to the 800 Hz square wave shown in Figure 3 for the minimum phase system
component. The square wave response shows the initial pulse (from the tweeter) going in the
correct direction but the woofer pulse is slow to response and tweeter output has decayed to
near zero before the woofer output rises to complete the response. The phase response of the
all pass component is clearly neither minimum phase nor linear phase. The minimum phase for a
flat, all pass response is zero phase vs frequency. If the all pass were linear phase then the
square wave reproduction would be perfect, but shifted slightly to the right due to the constant
time delay linear phase represents.

The slow woofer response has lead to speculation that "time aligning" the system by moving the
tweeter back to off set the woofer delay could improve the response. This intensional
miss-alignment can also result in introducing response irregularities. The amplitude irregularities
can be correct using appropriate equalization, but it remains to be seen if the resulting all pass
response is minimum or linear phase.  This will be discussed further later on.
Figure 5
2nd Order Linkwitz/Riley

The all pass component for the LR2 crossover is shown in Figure 6.  As with the B3 crossover, it
is apparent that the all pass response introduces phase distortion and the square wave
response is highly degraded compare to the 800 Hz square wave shown in Figure 3. The phase
response of the all pass component is still neither minimum phase nor linear phase. However, we
must be careful when interpreting the result for the LR2 crossover. We note that the initial spike
in the square wave response is in the inverted direction. Thus, when the polarity of the input
changes suddenly the output yields a spike in the wrong direction with the appearance of an
exponential decay to what should be the level of the top or bottom of the square wave. If we
examine this at a low frequency the result appears as shown in Figure 7 where the frequency is
400 Hz. Again, this is only the response of the all pass component of the system response. I
present this figure because I have seen such responses presented claiming that it shows good
square wave reproduction and transient accuracy. They say a picture is worth 1000 words, and
in such a case, believe the picture, not the words. Any system that shows the initial part of a step
or square wave response going in the wrong direction can not be transient accurate regardless
of what other virtues the system may posses.
Figure 6
Figure 7. 400 Hz square wave reproduction.
4nd Order Linkwitz/Riley

The all pass component for the LR crossover is shown in Figure 8. Close examination of the
plots will reveal that there is great similarity between the all pass response of the LR4 crossover
and that of the B3 crossover. In fact, this is the case with the exception that the phase rotation
near the crossover point is somewhat more gradual.
Figure 8
2nd Order Transient Perfect Crossover:

The 2nd order transient perfect crossover is described by Kreskovsky [3,4] and is based on the
concepts presented by Vanderkooy and Lipshtiz [6] where the corner frequencies of the HP and
LP sections are overlapped so that the summed response, while no longer flat, is minimum
phase. Then, minimum phase equalization is applied to obtain flat response. The result is an all
pass response with flat amplitude and zero phase shift; transient perfect, in which the HP and LP
sections have 2nd order asymptotic slopes. The all pass response is shown in Figure 9. A
speaker system build with this type of crossover will produce the transient response of the band
pass system of Figure 3 with no additional crossover induced transient distortion.
Figure 9
Transient Perfect crossover Derived by Subtraction:

Transient perfect crossovers derived by subtraction define an infinite family of all pass responses
which are either of the minimum (zero) phase or linear phase. The fundamental idea is that if the
summed response is to be all pass and either zero or linear phase then it is given as

AP(s) = exp(-s x Td)

where Td is a time delay which. The phase vs frequency is given as

Phi(ω) = - ω x Td.

It is obvious that such a relationship yields a phase shift which varies linearly with frequency. In the
case where Td = 0, AP(s) = 1., flat amplitude and zero phase. In this case we can assure AP(s) will
remain 1.0 if we define high pass and low pass responses as

HP(s) = 1 - LP(s)

for a given, arbitrary LP response. When the HP response is initially specified  we have

LP(s) = 1 - HP(s)

Thus by construction

HP(s) + LP(s) = 1 = AP(s)

Such crossovers are representative of the Constant Voltage crossover discussed by Small [2].

When Td is non-zero useful results can be obtained by specifying the low pass response and
subtracting it from an all pass response with linear phase, thus defining the high pass section:

HP(s) = AP(s) = LP(s)
= exp(-s x Td) - LP(s)

The caveat is that in these types of crossovers Td must be equal to (or very close to) the DC
group delay of the specified low pass section. The usefulness of this approach is limited to the
choice of the low pass section. The basic concept was first introduced by Lipshitz and Vanderkooy  
[5]. It is important to realize that this form of "Subtractive-Delayed" crossover is in
no way
equivalent to physically offsetting the tweeter to compensate for the group delay of the woofer
response. As discussed above, such offsetting will never yield an all pass response with zero or
linear phase and flat amplitude. There are cases where it may approximate such a response but
there is always some error.

CV crossover using an LR4 LP response:

Figure 10 shows the all pass response of a CV crossover using an LR4 low pass response. To
within the small errors arising from processing, as discussed previously, it is observed that the  all
pass response is flat with zero phase. The 800 Hz square wave reproduction shows no transient
degradation. A speaker system build with this type of crossover will produce the transient response
of the band pass system of Figure 3 with no additional crossover induced transient distortion.
Figure 10
SD crossover using an LR4 LP response:

Figure 11 shows the all pass response and 800 Hz square wave reproduction for a SD type
crossover implemented using an LR4 LP response. That the phase varies linear with frequency (a
pure time delay) is shown more clearly in Figure 12 using a linear frequency scale. A speaker
system build with this type of crossover will produce the transient response of the band pass
system of Figure 3 with no additional crossover induced transient distortion.
Figure 11
Figure 12. Phase with linear frequency scale.
Approximately Transient Perfect Crossovers using Time Alignment

It was mentioned above in the 3rd order butterworth discussion that it has been contemplated that,
due to the slow response of the woofer in a 2-way system, some semblance of transient perfect
response may be obtainable by physically offsetting the tweeter to account for this slow woofer
response. First we ask what the origin of the slow woofer response is? In fact, it is due to the nature
of any low pass filter. All minimum phase low pass filters introduce a frequency dependent group
delay. As we move somewhat below the cut off frequency of the filter this group delay become
constant and is equivalent to a linear phase or constant time delay. This observation is what lead to
the higher order, SD type filters originally proposed by Lipshitz and Vanderkooy  [5]. However, this
same observation also lead to consideration of physically offsetting the tweeter to account for the
woofer low pass response delay. The idea was that by offsetting the tweeter a crossover could be
constructed which, while not necessarily having flat response, would hopefully result is a response
which was composed of a linear phase component and a minimum phase component with the
minimum phase component representing the deviation from flat response. If this were possible then
minimum phase equalization of the response deviation from flat would result in a linear phase
crossover. Unfortunately, Vanderkooy and Lipshitz [6] showed that for conventional all pass
crossovers higher than 1st order (odd order Buterworth and even oder Linkwitz/Riley) this is not
possible. That is, they showed that after removal if the linear phase component of the response, the
remaining phase component is not minimum phase. Thus minimum phase equalization would not
result in a linear phase all pass response.

However, Vanderkooy and Lipshtiz did not consider filter other than conventional all pass
crossovers. In the early 80's Spica, for example, introduce a series of speakers which employed
tweeter offset to time align the system and these speakers were said to have very good transient
response. These speakers used a 1st order electrical high pass filter with impedance compensation
for the tweeter and a 2nd order electrical low pass for the woofer with an additional notch filter for.
When combined with the acoustic response of the drivers the results was a intended to be a 4th
order Bessel low pass and a quasi 1st order high pass acoustic response. Quasi 1st order because
ultimately the 1 st order electrical filter combined with the tweeter acoustic output would yield a 3rd
order roll off at frequencies below the crossover point. The tweeter off set was then adjusted to yield
optimal time response. To see how well the potential performance of such a system would be we
shall examine the acoustic crossover response alone. In Figure 13, however, it is assumed that the
tweeter roll off remains 1st order and the excess phase associated with the tweeter offset has been
removed. The phase of the speaker is shown in green. The lower red line is the minimum phase for
the system amplitude. From this and from the square wave response to the right, while an
improvement over standard crossovers, the response is not truly linear phase.
Figure 13.
When the roll off of the tweeter is included in the result the response appears as shown in Figure
14. The dip in the response could be compensated for, but as is indicated buy the comparison of the
actual and minimum phase response, and by the square wave distortion, the response is not
minimum or linear phase.
Thus it appears that the conclusions of Vanderkooy and Lipshitz [6] regarding the used of physical
offset, or "time alignment", to compensate for the GD of the low pass response applies to crossovers
of mixed types as well. However, while not perfect, it is readily apparent that these types of
crossovers do offer an improvement in transient response over the standard crossover; Butterworth
of order greater than 1, and the Linkwits/Riley crossovers. However, with any system there are the
other consideration regarding off axis and polar response, and the complexities introduced by
physically offsetting drivers.

We can analyze the result more deeply. The question is, can a linear (or zero) phase crossover be
developed by introducing physical offset, resulting in delay, to only the high pass section? In
accordance with the analysis of [6] it is not required that the crossover sum flat, but only that the
summed response be composed of a minimum phase component and a linear phase component.
The minimum phase component accounts for any response irregularities and application of minimum
phase equalization would render the summed response flat and linear phase, thus transient perfect.
We have seen this failed above, but what about in general?

We again start with the definition of the system response, ignoring the low and high frequency cutoffs.

S(s) = exp(-sTdl) x MP(s)

where exp(-sTdl) is the linear phase component and MP(s) is the minimum phase component. Since
the response deviation form flat is assumed to be contained in the minimum phase component the
system response can be equalized to an all pass response by applying minimum phase equalization,

AP(s) = EQ(s) x S(s) = EQ(s) x exp(-sTdl) x MP(s)

where EQ(s) = 1 / MP(s)

We next consider the high pass function, HP(s) representing the on axis response of the filtered
tweeter. If the tweeter AC is offset from the woofer AC by some distance, d, then it is equivalent to a
time delay of Td = d/c where c is the sound. Thus the offset tweeter response may be expressed as

T(s) = exp(-sTd) x HP(s)

when referenced to the plane containing the woofer AC, as shown in Figure 16.





















Figure 16.

We now need to find the woofer response, W(s) , which when summed to the offset tweeter will yield
the system response, S(s), or,

W(s) = S(s) - T(s)

Using the responses defined above,

W(s) = exp(-sTdl) x MP(s) - exp(-sTd) x HP(s)

We can factor out the linear phase component of the system response to yield,

W(s) = exp(-sTdl) x [MP(s) - exp(-s(Td')) x HP(s)]

where Td' = Td - Tdl

Now, assume that W(s) has the form of some kind of low pass response. As such, the output of the
system as the frequency goes to infinity will be solely that of the tweeter. Thus, as the frequency
goes to infinity the delay would be that of the tweeter response, HP(s), plus the delay due to the
offset. If HP(s) is a high pass, minimum phase response the delay associated with HP(s) will be zero
as the frequency goes to infinity and the delay of the output will be that due to the tweeter offset.
However, this leads to a contradiction because by construction the system output (when Tdl = 0.) will
have zero delay since MP(s), being minimum phase, will have zero delay as f goes to infinity. [ In the
case where Tdl is not zero the system output would be delayed by Tdl while the tweeter response at
high frequency would be delayed by Tdl + Td' = Td. ] So what is wrong? Either MP(s) can not be
minimum phase (as was shown above in the in the 2nd order Bessel / 1st order Butterworth case), or
the assumption that W(s) has the form of a low pass response is incorrect. This proof by
contradiction shows that it is not possible to construct a transient perfect crossover (linear or zero
phase) by offsetting the tweeter as is commonly attempted in so called "time aligned" systems. That
is, there are no possible high pass / low pass filter combinations which will combined to reduce the
system response to a minimum phase component plus a linear phase component when the tweeter is
offset behind the woofer. While the approach taken, for example, by Spica years ago may yield an
acceptable approximation to a linear phase crossover, there will always be some error and deviation
from linear phase in the crossover region and the crossover will not be transient perfect.

There is one last interesting case to consider; What if Td' = 0.0. That is, what if the tweeter offset is
equal to the system delay, Tdl? In this case we have

W(s) = exp(-sTdl) x
[ MP(s) -  HP(s) ]  

The term in the red brackets is just the form the low pass function for Small's [2] CV crossover,
Wcv(s), with result that

W(s) = exp(-sTdl) x Wcv(s).

The linear phase term, exp(-sTdl) can be interpreted as either an electronic delay, or a physical
offset of the woofer. In either case this delay is identical to that of the tweeter since Td' = 0 and
should be interpreted as shifting either the apparent or actual acoustic center of the woofer to lie in
the same plane as the tweeter AC.

In the final analysis what has been shown is that any crossover which is transient perfect must have
the AC of the drivers aligned to lie in the same plane. This can be accomplished by physical offset of
the correct amount or by electronic delay. However, the use of excess offset, positioning the tweeter
AC behind that of the woofer, to attempt to compensate for the GD of the woofer response, will never
yield a crossover which is truly transient perfect.


References

1.  Thiele, N.: Loudspeaker Crossovers with Notched Responses, 108th AES Convention, Feb. 2000.

2. Small, R. H.: Constant-Voltage Crossover Network Design, JAES, V 19, No. 1, 1971.

3. Kreskovsky, J. P.: A Transient-Perfect Second-Order Passive Crossover, Audio Express, May
2001.

4. Kreskovsky, J. P.: Design an Active Transient-Perfect Second-Order Crossover, Audio Express,
Dec. 2002.

5. Lipshitz, S. P. and Vanderkooy, J.: A Family of Linear-Phase Crossover Networks of High Slope
Derived by Time Delay, JAES, B 31, No. 1/2, 1983.

6. Vanderkooy, J. and Lipshitz, S. P., Is Phase Linearization of Loudspeaker Crossover Networks
Possible by Time Offset and Equalization?, JAES, V 32, Dec, 1984.
Figure 14.