Amplitude Errors in the Summed Response of Audio Crossover Filters
 
An informal analysis by Jerry Martin, Senior Engineer JRM Engineering

The interaction of adjacent filter sections must be considered when implementing 3-way or higher crossovers.   Consider the case where a 4-way system is implemented with 1st order filters.   The summed signal has an all-pass response with zero ripple and no phase shift.
 
* Zero ripple / zero phase shift is only true for 1st order filters.   All higher order filters exhibit increasing ripple with increasing order and closeness of adjacent sections.
* Cascading low pass or high pass 1st order sections yields identical results.
* Cascading low pass or high pass higher order sections yields similar results, the position of the ripple being below crossover for cascaded low pass, and the position of the ripple being above crossover for cascaded high pass.
* Magnitude of ripple increases as adjacent sections that are not cascaded move closer together.
* For active crossovers, it is desirable to cascade low pass sections so that the total number of gain stages for the high frequency sections is minimum.
 
> The math program separates the real and imaginary components of the equations to calculate on-axis power, total power, wrapped phase, and group delay.
> "Normal" output for the math program has alternating section polarity as required for 2nd order Linkwitz-Riley and 3rd order Butterworth crossovers.
> Except for Fig. 10, low pass sections are cascaded, crossover frequencies are 100, 600 and 7K Hz.
> Except for Fig. 13, the ultrasonic filter is set to 500KHz, the subsonic filter is set to 1 Hz; thus they serve only as simple inverting stages to yield correct relative polarity.
> Fig. 3 and 4 show undesirable result of paralleling sections.

From the above diagram :

  Tweeter   = OutD = Out(LP-D) + Out(HP-D)
  Mid-Range = OutC = Out(LP-D) + Out(LP-C) + Out(HP-C)
  Mid-Bass  = OutB = Out(LP-D) + Out(LP-C) + Out(LP-B) + Out(HP-B)
  Bass      = OutA = Out(LP-D) + Out(LP-C) + Out(LP-B) + Out(LP-A) + Out(HP-A) 
  OutSum    = OutA + OutB + OutC + OutD

For 1st order high pass, HP1(s) = s / (s + 1), s = jF/Fo
For 1st order low pass,  LP1(s) = 1 / (s + 1)
Drivers must have same absolute polarity.

For 2nd order high pass, HP2(s) = s^2 / (s^2  + 2*s + 1); Linkwitz-Riley
For 2nd order low pass,  LP2(s) = 1 / (s^2 + 2*s + 1)
Drivers must have alternating absolute polarity.

For 3rd order high pass, HP3(s) = s^3 / (s^3 + 2*s^2 + 2*s + 1); Butterworth
For 3rd order low pass,  LP3(s) = 1 / (s^3 + 2*s^2 + 2*s + 1)
Drivers must have alternating absolute polarity.

Fig. 1   SPICE simulation of cascaded 1st order high or low pass sections.
 
Response is virtually flat with no phase shift.

Fig. 2   SPICE simulation cascaded 1st order, drivers have alternating polarity.   (Inverted from normal)

Fig. 3   SPICE simulation 1st order Parallel connection with drivers having same polarity.

Fig. 4   SPICE simulation 1st order Parallel drivers have alternating polarity.   (Inverted from normal)

Fig. 5   SPICE simulation of cascaded 2nd order Linkwitz-Riley sections.
 
Again, cascading high or low pass sections yields similar results with only a small shift in magnitude and position of the peaks / dips.
This result is virtually identical to the graph below that solves the transfer function equations.

Fig 6.   Plot from math program for 2nd order Linkwitz-Riley filters.
 
Note the significant drop in overall total power, a consequence of the Linkwitz-Riley alignment having -3dB total power at crossover.

Fig. 7   Math plot 2nd order Linkwitz-Riley with drivers having same polarity.   (Inverted from normal)
 
Note the discontinuities in group delay where the phase changes direction and returns to 0°.
A phase shift change of 360° is not the same as returning to 0°.

Fig. 8   SPICE simulation 2nd order Linkwitz-Riley with drivers having same polarity.   (Inverted from normal)

Fig. 9   SPICE simulation of 3rd order Butterworth.
 
Note position of peaks is below crossover frequency.

Fig. 10   SPICE simulation of 3rd order Butterworth with cascaded high pass filters.
 
Note position of peaks is above crossover frequency.

Fig. 11   Plot from math program of 3rd order Butterworth.
 
Note constant total power.

Fig. 12   Math plot 3rd order Butterworth with drivers having same polarity.   (Inverted from normal)

Fig. 13   Optimized 3rd order Butterworth with crossover frequencies slightly staggered.
 
This alignment yields only ~ 0.5 dB error in the summed response for both on-axis and total power.   Also includes 8.2 Hz subsonic and 134 KHz ultrasonic sections.

Fig. 14   SPICE simulation as in Fig. 13, optimized 3rd order Butterworth including drivers for Sec. B and Sec C.
 
Driver modeled as 2nd order high-pass filter :
HP2(s) = s^2 / (s^2 + k*s + 1);
 
Driver Sec. B (mid-bass) :
F = 92 Hz, Q = 0.43.
Driver Sec. C (mid-range) :
F = 500 Hz, Q = 1.0

Fig. 15   Spice simulation as in Fig. 14, optimized 3rd order Butterworth including drivers and LFEQ for Sec. B and C.
 
LFEQ implemented as inverse 2nd order high-pass filter with F and Q same as driver.
InvHP2(s) = (s^2 + k*s + 1) / s^2;
 
MLSSA analysis in a semi-anechoic environment has validated the efficacy of the LFEQ compensation.