Phase-Parallel Active Crossover – An Audiophile Quantum Leap

Precise and as non-directional as possible reproduction of high sound frequencies requires light and small diaphragms, while sufficiently loud reproduction of low frequencies is only possible with large diaphragm areas. Therefore, there is no alternative (except for a few specialized designs, all of which have their weaknesses) but to divide the audio frequency range from 20 Hz (wavelength 17 m) to 20 kHz (wavelength 1.7 cm) into at least two to three frequency ranges, each of which is handled by a speaker chassis specialized for that range. This, in turn, presents the challenge of dividing the frequency ranges in such a way that their acoustic addition is perceived by the human ear as the original, broadband sound event. The following options are available for separating the frequency ranges:

1. Passive crossovers made of inductors, capacitors, and resistors, which are placed between the power amplifier and the individual speakers.
2. Active crossovers with analog operational amplifiers.
3. Active crossovers with digital sound processors.

The first option remains the most common and the cheapest because it only requires one power amplifier per stereo channel. However, a "time-correct" – i.e., phase-parallel – acoustic addition of the individual frequency ranges is not possible with passive crossovers, but at best a rough approximation. Contradictory claims are advertising statements that persist as long as they are believed.

The third option will eventually establish itself in the market up to a certain quality level and displace passive systems, which have nothing to do with "high-end." However, only what can already be achieved in practice with the analog active crossover presented here can be replicated on a digital level – approximately equivalent, but not better. For the highest standards of natural music reproduction, the analog LR4 active crossover with Linkwitz transformation and all-pass matrix remains unsurpassed.

Designing a time-correct multi-way system is far more demanding than simply placing an active high-pass filter before the tweeter channel and an active low-pass filter before the woofer channel, then pretending to achieve "phase linearity" (or whatever else). Phase errors that distort the spatial imaging of the music and lead to a colored sound image – which is annoying in conventional active studio monitors and often makes music sound "nicer" than it is in passive home hi-fi speakers – have not one, but three causes:

1. The high- and low-pass filters of the crossover are not Linkwitz-Riley filters but other random filter functions that cannot be added without phase errors.
2. The phase frequency responses generated by the individual speakers due to their respective mass-spring systems are hardly or not at all taken into account.
3. The mechanical arrangement on the baffle causes a phase shift because the acoustic centers of the individual speakers are not on the same plane.

In conventional active systems, usually only the third error source is addressed, while the other two are neglected. As with passive multi-way systems, attempts are made to "compensate" for the characteristic phase frequency responses of the individual speakers by using different filter functions for the high- and low-pass filters. However, the three error sources cannot compensate for each other! Without Linkwitz-Riley filters and without integrating the acoustic transfer functions of the individual speakers into the crossover via Linkwitz transformations, there is no spatial arrangement of the individual speakers that allows for natural sound reproduction.

A phase-parallel 2-way active crossover and a phase-parallel 3-way active crossover that accurately compensate for all three causes of phase errors are explained below without complicated mathematics. The explanation is based on the electronic basic circuits and is therefore intuitively understandable, provided some basic knowledge of electronics (Tietze/Schenk: Semiconductor Circuit Technology / Active Filters) and electroacoustics (Schwamkrug/Römer: Loudspeakers – Truth and Myth) is present.

1. Phase-Parallel 2-Way Active Crossover

Let us first consider a simple Butterworth crossover of 2nd order, where the phase frequency responses of the low-pass, high-pass, and sum are not congruent, and the amplitude addition is therefore only approximate:

TPBu2-HPBu2_1k_plot      TPBu2-HPBu2_1k_1dB_plot

A ripple of the amplitude frequency response of about ±0.5 dB seems insignificant compared to the usual sound pressure fluctuations of real speaker diaphragms, yet the phase difference of up to 43° has dramatic effects on the ear, as the following calculation shows: The addition of logarithmic quantities corresponds to the multiplication of two linear quantities. If the linear quantities A and B are in phase, the product is A*B. If there is a phase difference dPhi between A and B, the product is A*B*cos(dPhi). If we set A*B=1, the error I(dPhi) = 1 – cos(dPhi) results from dPhi, which the ear perceives as intermodulation distortion:

I(47°) = 31.8%
I(33°) = 16.1%
I(22°) = 7.3%
I(15°) = 3.4%
I(10°) = 1.52%

In the crossover range, which extends ±2 octaves around the crossover frequency – and where the ear is most sensitive in a 2-way system – the sound event is radiated simultaneously by two separate sound sources. However, they cannot do this "simultaneously" if there is a phase difference between the two sound sources. "Phase-parallel" means that the phase difference is zero for at least ±2 octaves around the crossover frequency. The absolute phase frequency response of the sum signal remains inaudible as long as it does not "slope too steeply" or contain abrupt phase jumps. The much-touted "phase linearity" is therefore of secondary importance to the listening experience. The ear, however, is extremely sensitive to phase differences, as otherwise, we would not be able to locate a sound source in space. This now makes it clear why various multi-way systems behave very differently – or differently flawed – in terms of spatial imaging, even when using the best individual speakers and achieving a balanced amplitude frequency response.

Active filters, where low-pass and high-pass filters add without phase difference, were described by Siegfried Linkwitz and Russ Riley in the late 1970s and have since been known as Linkwitz-Riley filters. The best filters for phase-parallel active systems are Linkwitz-Riley low- and high-pass filters of 4th order (abbreviated: LR4) with a slope of 24dB/octave. Crossovers of higher order only increase the circuit complexity and at best offer gradual improvements in the PA area, where maximum efficiency is important; 3rd order crossovers are unsuitable due to their "odd" phase shift of 270°; and 2nd order crossovers do not have enough attenuation, so a tweeter will distort, or in the worst case, be destroyed before reaching even a fraction of its actual power capability. Although the sound pressure drops by 12dB/octave below the crossover frequency, the diaphragm displacement does not, leading to early mechanical and thermal overload.

An LR4 crossover has a phase shift of one wavelength, i.e., 360°, which the ear perceives as in-phase. The phase frequency responses of the low-pass and high-pass filters are congruent, and the amplitude addition is mathematically exact:

TPLR4-HPLR4_1k_plot

But we are not done yet, as the LR4 filters of the crossover multiply with the acoustic transfer functions of the individual speakers. Let us consider a sealed 2-way speaker with a net volume of 24.5 liters and the following drivers:

The speakers are mass-spring systems, characterized by a resonance frequency fc and a quality factor Qtc. The acoustic transfer functions, each consisting of amplitude and phase frequency response, correspond approximately to 2nd order high-pass filters, so the overall result in the high-frequency range is a 6th order high-pass filter, and in the low-frequency range, a band-pass filter is created:

TW030_SW223_1kHz_asc      TW030_SW223_1kHz_plot

The phase difference between the tweeter and woofer is 55° at the crossover frequency and decreases above 1 kHz, but increases below 1 kHz, causing a dip in the amplitude frequency response at the lower end of the crossover range. The result sounds as bad as it looks. Raising the crossover frequency by an octave makes the result look somewhat better:

TW030_SW223_2kHz_asc      TW030_SW223_2kHz_plot

The phase error becomes smaller when the crossover frequency is placed well above the tweeter’s resonance frequency fc. This is why tweeters in conventional multi-way systems are usually crossed over too high. However, a too-high crossover frequency leads to non-homogeneous radiation. In the midrange, the sound is no longer radiated in a single lobe, but instead, three radiation lobes form due to interference: one directed forward, the second diagonally upwards, and the third diagonally downwards.

To properly couple the tweeter at 1 kHz, a Linkwitz transformation is required, which shifts its resonance frequency fc to the crossover frequency and adjusts the overall Qtc to 0.707. The tweeter's acoustic transfer function now replaces one of the two 2nd order Butterworth high-pass filters, together forming a 4th order Linkwitz-Riley high-pass filter:

TW030-LiTr_SW223_1kHz_asc      TW030-LiTr_SW223_1kHz_plot

This already looks much better and also sounds good, but the bass reproduction is still a bit thin. For an extended low bass range, the woofer is also driven by a Linkwitz transformation and tuned to a 4th order Butterworth high-pass filter with a cutoff frequency of 30 Hz:

TW030-LiTr_SW223-Bu4_1kHz_asc      TW030-LiTr_SW223-Bu4_1kHz_plot

The amplitude frequency response is perfect, but due to the phase shift caused by the bass correction, the phase frequency responses of the woofer and tweeter are now shifted by exactly one wavelength (360°) relative to each other. To correct this, a 2nd order all-pass filter is inserted into the tweeter path:

TW030-LiTr-AP2_SW223-Bu4_1kHz_asc      TW030-LiTr-AP2_SW223-Bu4_1kHz_plot

Of course, this is only the theoretical basic principle for the simplest case of a phase-parallel 2-way active crossover. In practice, additional filter circuits are required to achieve a balanced overall amplitude frequency response without jeopardizing the exact phase parallelism in the crossover range. In this case, the woofer still needs an active gyrator notch filter to damp the diaphragm resonance at the upper end of its transmission range, and the tweeter's frequency response must be linearized with an additional shelving low-pass filter.

As with any multi-way speaker with a flat baffle, the sound radiation of the tweeter must also be delayed because its voice coil is a few centimeters in front of the woofer’s voice coil. A digital delay is not required for this, as a frequency-independent signal delay corresponds to a frequency-proportional phase shift, which only needs to work for ±2 octaves around the crossover frequency (in this case from 250 Hz to 4 kHz) to maintain the phase parallelism that matters. If the acoustic center of the woofer is 6 cm behind that of the tweeter, the phase of the tweeter must be shifted by 16°/32°/63°/127°/253° at 250Hz/500Hz/1kHz/2kHz/4kHz, respectively. This function is performed by a 2nd order Bessel all-pass filter (Q=0.58), which is inserted into the tweeter path:

APBe2_6cm_HT_asc

The plot illustrates what happens to the frequency response when the sound radiation of the tweeter in a flat baffle is not delayed:

APBe2_6cm_HT_plot

2. Phase-Parallel 3-Way Active Crossover

Even with Linkwitz-Riley filters, the 3-way crossover does not yet allow for an exact addition of the individual frequency ranges. This is because, with 4th order filters, the midrange path is a bandpass with a phase shift of up to 720°, while the phase shift in the high- and low-frequency paths remains at 360°:

LR4_3-Wege_asc      LR4_3-Wege_plot

Only when the phase frequency responses in the high- and low-frequency paths are compensated with additional 2nd order all-pass filters does the desired result appear:

LR4_3-Wege_2AP2_asc      LR4_3-Wege_2AP2_plot

Even in the middle between the crossover frequencies, i.e., at sqr(300Hz*1.6kHz) = 693Hz, the level of the midrange path remains 0.6 dB below the sum level because both the high-frequency path and the low-frequency path still contribute a – practically audible! – level. Therefore, exact phase parallelism must be maintained over the entire frequency range from 75 Hz (2 octaves below 300 Hz) to 6.4 kHz (2 octaves above 1.6 kHz). If this is not achieved, 2-way systems are preferable to conventional 3-way systems. This is because, with passive or simple active crossovers without Linkwitz transformations, a rough approximation to phase parallelism is at least somewhat achievable with two ways, rather than three. However, with Linkwitz transformations and 2nd order all-pass filters (which are not realizable with passive crossovers), it is relatively simple to integrate the acoustic transfer functions of the individual speakers into the 3-way active crossover,...

…if you first understand the basic principle:

LR4_3-Wege_3AP2_3LiTr_asc      LR4_3-Wege_3AP2_3LiTr_plot

The speaker chassis corresponds to the current configuration of the Audio Optimum MS-10, where an actively filtered and sealed 4th order bass system is not used, but rather an actively filtered 6th order bass reflex system with a 12-inch passive diaphragm to extend the low bass range down to 24 Hz. Explaining this would go beyond the scope of this text and is not necessary for understanding phase parallelism. In any case, two more 2nd order Bessel all-pass filters are required to delay the midrange driver relative to the bass in the frequency range from 75 Hz to 1.2 kHz, and to delay the tweeter relative to the midrange driver in the frequency range from 400 Hz to 6.4 kHz, aligning the acoustic centers of all three individual speakers.

With the phase-parallel 3-way active crossover, it is also possible to build proper subwoofer-satellite systems, where the subwoofers are no longer locatable, even if the crossover frequency is chosen relatively high (e.g., 100 Hz) to keep the size of the satellites small. However, the satellites must be designed as sealed systems. Bass reflex boxes are generally unsuitable as satellites.

Using this method, multi-way systems can be built for the first time in such a way that they are no longer audible as separate systems, but are perceived by the ear as "ideal broadband systems," projecting the music event into the listening room without coloration and in three dimensions. Even with "just" two phase-parallel multi-way speakers, the localization is significantly better than with conventional so-called surround systems.