General Requirements for a Crossover
The desirable characteristics of a crossover system are easy to state, but not so easy to achieve in practice. There is a general consensus that there are five principal requirements that apply to all crossovers, be they active or passive, and that they should be ranked in order of importance thus:
- Adequate flatness of summed amplitude/frequency response on-axis
- Sufficiently steep rolloff slopes between the output bands
- Acceptable polar response
- Acceptable phase response
- Acceptable group delay behaviour
To some extent, the amount of space devoted in this book to each requirement is dependent on their relative importance as set out in this list.
Adequate Flatness of Summed Amplitude/Frequency Response On-Axis
This requires that the output of each filter is appropriate in both amplitude and phase over a sufficient frequency range so that when summation occurs the overall amplitude/frequency response is flat. Note that this requirement does not place restrictions on the phase of the summed result; in many cases this will show considerable phase shift that varies across the audio band. This issue is addressed in requirement 4 below.
Crossovers that sum to a completely flat amplitude response include the first-order crossover, the second-order Linkwitz–Riley crossover, the third-order Butterworth crossover and the fourth-order Linkwitz–Riley crossover. A new addition to this list is the Neville Thiele Method notch crossover.
There are many crossover types that can be made to sum very nearly flat by tweaking the filter cutoff frequencies. For example, the second-order Bessel crossover can be made flat to within ±0.07 dB by using a frequency offset ratio of 1.45 times, and the third-order Linkwitz–Riley crossover can be made flat to within ±0.33 dB by applying a frequency offset ratio of 0.872 times. There is much more on this in Chapter 4.
Sufficiently Steep Rolloff Slopes between the Filter Outputs
The filter rolloff slopes must be fast enough to prevent driver damage. Even a small amount of LF energy can rapidly wreck a tweeter. The slopes must be steep enough to not excite areas of poor drive unit frequency response, such as resonances outside normal band of usage; this applies only to mid-range drive units and tweeters, as the LF drive unit resonance will always be used. In addition, the linearity of drive units is very often worse outside their intended frequency range, so steeper slopes will give less non-linear distortion, and that has got to be a good thing.
Restricting the frequency range sent to each drive unit will also minimise frequency modulation distortion caused by the Doppler effect. It is also desirable to make the frequency range over which crossover occurs as narrow as possible to minimise the band over which lobing occurs due to two drive units radiating simultaneously.
Unless specially designed drive units are used, the minimum practical slope is usually considered to be 12 dB/octave which requires a second-order crossover. Steeper slopes such as 18 dB/octave (third-order) and 24 dB/octave (fourth-order) are generally considered to be very desirable; 48 dB/octave (eighth-order) slopes are sometimes used in sound reinforcement.
Acceptable Polar Response
An even and well-spread polar response is desirable because it increases the amount of space in which a good sound is obtained. It is also desirable to avoid a large amount of radiated energy from being directed at the floor in front of the loudspeaker, from which it will reflect and cause unwanted comb-filtering effects by interference. Loudspeakers normally give a good polar response in the horizontal on-axis plane, assuming the drive units are mounted in a vertical line as usual. This, however, causes problems in the vertical plane, for in the crossover region two drive units separated in position are radiating simultaneously, and their outputs will interfere, giving reinforcements and cancellations in the radiation pattern at different angles, known as lobing. This is a result of having two drive units separated in space and there is nothing the crossover designer can do about this except make the crossover frequency range as small as possible.
However, it gets worse. If the crossover outputs to each drive unit are in phase, then the main lobe points forward on the horizontal axis, and stays there. If, however, there is a constant phase shift between the outputs, as for first-order crossovers (90° phase shift) or third-order crossovers (270° phase shift) then the main lobe is tilted toward the drive unit that is phase lagging. This is usually the LF unit, so the main energy is being unhelpfully directed towards the floor. This is a frequency-dependent effect because it only occurs in the crossover region and is at its greatest at the crossover frequency itself. It is sometimes simply called “lobing error.”
It is therefore clear that if we are going to keep the main lobe of the summed acoustic output on the axis it is highly desirable that the lowpass and highpass outputs are in phase in the crossover region . This property is possessed only by second-order crossovers (assuming one output is inverted to get a flat response, otherwise the phase-shift is 180°) and fourth-order crossovers with no inversion. This is one reason for the popularity of the fourth-order Linkwitz–Riley crossover.
All the above depends on the drive unit time-delay compensation being correct; the drive units must be either physically mounted or electrically compensated so that the direct sound from each one arrives at the listener’s ear at the same time over the whole of the crossover frequency range. Otherwise, the main lobe will have a frequency-dependent tilt toward the driver with the longest air path to the ear.
A good polar response therefore requires that the crossover outputs be in phase and that the time-delay compensation be correct.
Acceptable Phase Response
An acceptable phase response for the combined output is also required. Most crossovers are not linear-phase or minimum phase but have the phase response of a first-order allpass filter, with the phase changing by 180° over the audio band. The best-known of these are the first-order (inverted), the second-order Linkwitz–Riley, and third-order Butterworth crossovers. The fourth-order Linkwitz–Riley crossover has the phase response of a second-order allpass filter, with the phase changing by 360° over the audio band. These phase responses are generally agreed to be inaudible with music signals so the fourth requirement is not too onerous.
Acceptable Group Delay Behaviour
Group delay is simply a measure of how much a signal is delayed. This is directly connected with an acceptable phase response for the combined output, and in fact the group delay is completely determined by the phase shift. Group delay is mathematically the rate of change of the total phase shift with respect to angular frequency (i.e., frequency measured in radians per second rather than Hertz).
Group delay would be of little interest if it was constant, but as the rate of change of phase varies across the audio band, with the phase response of an allpass filter, the group delay also varies. The change is sometimes smooth, but may show a pronounced peak near the crossover frequency. This variation would, if it was sufficiently severe, cause a time-smearing of acoustical events and would sound truly dreadful, but you must not mistake this with the use of the word “smearing” in hi-fi reviews, where it is purely imaginary.
The thresholds for the perception of group delay variation are well known because of their historical importance on long telephone lines. The most accepted thresholds were given by Blauer and Laws in 1978, and are shown in Table 3.1. These times are given in milliseconds, and a typical group delay for a 1 kHz crossover would be something like 10 times less. The section on phase perception later in this chapter is concerned with the audibility of allpass filters, and the conclusion is firm that neither their phase-shift nor their group delay can be heard on normal musical signals.
The word “group” is derived from “group velocity” in wave-propagation theory, but for our purposes it is simply the amount by which a signal at a given frequency is delayed.
Excerpt from The Design of Active Crossovers by Douglas Self © 2011 Taylor & Francis Group. All Rights Reserved.