Crossovers and frequency bands

A crossover is not merely a device that “sends bass to one driver and treble to another”. It is a frequency-dependent complex transfer system. The target is the combined acoustic response - magnitude, phase and directivity - after the drivers, enclosure, geometry and filters have all interacted.

1. Electrical and acoustic crossover

The electrical filter is only one part of the final slope. The driver already has a natural magnitude and phase response; enclosure loading and diffraction add more. The acoustic crossover is the measured result of the electrical network multiplied by the driver’s own transfer function.

2. Filter order and slope

An ideal nth-order filter has an asymptotic slope of roughly 6n dB/octave: first order 6, second 12, third 18, fourth 24 dB/octave. Real acoustic slopes can be steeper or shallower because the transducer response is part of the system.

3. Butterworth and Linkwitz-Riley

Different alignments optimise different objectives. The fourth-order Linkwitz-Riley (LR4) is especially common in sound reinforcement because each branch is -6 dB at crossover and the two acoustic outputs sum flat when correctly aligned. LR4 can be formed by cascading two second-order Butterworth sections. The branches differ by an integer 360° in phase - equivalent phase at the crossover summation - so the main lobe need not tilt purely because of crossover phase when acoustic centres are aligned.

4. Phase and delay

A pure time delay τ produces a phase slope proportional to frequency:

Symbols: φ(f) is delay-induced phase shift in radians, f frequency in hertz, τ delay in seconds and ω=2πf angular frequency. The minus sign indicates that a delayed signal lags in phase.

This is why a fixed physical offset cannot be represented by a constant “phase degrees” number across frequency. For true constant delay:

Symbols: τ is a constant time delay and dφ/dω is the derivative of unwrapped phase with respect to angular frequency. For a pure delay the phase slope is constant, so this expression returns the same τ at every frequency.

5. Polarity is not the same as phase

Polarity reversal multiplies a waveform by -1. For a single sine this is equivalent to 180° inversion, but for a broadband signal it is not a time delay and does not create a frequency-proportional phase slope. Confusing polarity with delay is a common alignment error.

6. Lobing and vertical directivity

Two drivers separated in space create interference around crossover. Path differences vary with listening angle, producing lobes and nulls. Vertical spacing is especially important in conventional multi-way boxes. Even a perfectly flat on-axis response can hide poor off-axis summation.

7. Passive crossover

Passive networks operate at loudspeaker level using inductors, capacitors and resistors. Their response depends on the driver’s complex impedance, so textbook R or RC formulas are only a starting point. Component losses, power handling and impedance interaction matter.

8. Active DSP crossover

DSP allows precise filter shapes, delay, polarity, EQ, all-pass functions and protection. It also makes mistakes easier: a visually tidy filter graph does not guarantee correct acoustic summation. Measurement remains essential.

9. Transfer functions and complex summation

The outputs must be summed as complex quantities:

Symbols: H₁(f) and H₂(f) are the complex frequency responses of the two branches, while H_sum(f) is their complex sum. Magnitude and phase must both be retained; adding magnitudes alone cannot predict cancellation correctly.

Magnitude-only addition ignores phase and therefore cannot predict cancellations or lobing. All-pass filters are sometimes used to adjust phase without intentionally changing magnitude; first- and second-order structures provide different phase-rotation ranges.

10. Group delay

Group delay is defined as:

Symbols: τ_g is group delay in seconds, φ(ω) unwrapped phase in radians and dφ/dω its local slope versus angular frequency. Group delay therefore describes phase slope as a time quantity.

It describes the local slope of phase versus angular frequency. Steep IIR/analogue filter phase rotation can therefore create local group-delay peaks. Group delay is not automatically audible or harmful; its significance depends on magnitude, bandwidth and frequency.

11. FIR versus IIR

IIR filters are computationally efficient and naturally resemble analogue filter behaviour, with coupled magnitude and phase. FIR filters can realise linear-phase or deliberately shaped phase responses, but at low frequencies they may require long tap lengths and therefore substantial latency. The “better” choice is application-dependent.

12. Acoustic centre and driver offset

The apparent acoustic origins of LF and HF drivers are rarely on the same physical plane. A geometric offset d corresponds approximately to τ=d/c and therefore a frequency-dependent phase difference. DSP delay can correct a pure offset, but not every phase difference is a simple propagation delay.

13. Directivity matching

A strong crossover design tries to make the radiation patterns of adjacent drivers compatible near crossover. A large mid/woofer begins to beam as wavelength becomes comparable with diaphragm diameter; in dimensionless form, narrowing becomes significant as ka approaches and exceeds order unity, with pronounced beaming by around ka≥2. Crossing to a waveguide whose coverage is similar at that frequency produces smoother off-axis behaviour.

14. Why spinorama-like data matter

On-axis response alone is insufficient. Families of off-axis curves, listening-window data, early-reflection estimates and sound-power/directivity information reveal whether a loudspeaker changes timbre consistently around the room. Smooth directivity often makes equalisation more predictable.

15. A practical crossover measurement workflow

1. Measure each driver independently with timing/reference preserved.
2. Verify polarity and acoustic delay.
3. Inspect usable bandwidth and directivity limits.
4. Choose target acoustic slopes, not merely electrical filter labels.
5. Apply delay/all-pass only when the measured phase relationship calls for it.
6. Measure the combined response on and off axis.
7. Check maximum level, distortion and protection after small-signal alignment.