⚙️ ENGINEER LEVEL: IIR vs FIR Crossover Design
IIR (Infinite Impulse Response) Crossovers
Standard DSP crossovers in almost all consumer car audio DSPs use IIR (recursive) filters. These are the direct digital equivalents of analog filter circuits.
Properties: - Minimum phase — phase shift is tied to amplitude response - Computationally efficient (biquad sections: 5 multiply-accumulates per sample) - Any slope achievable with sufficient sections in series - Non-linear phase response — different frequencies arrive at different times
Linkwitz-Riley implementation:
LR4 = two cascaded Butterworth 2nd-order sections at same frequency. LR8 = two cascaded Butterworth 4th-order sections.
H_LR4(s) = H_BW2(s) × H_BW2(s)
At crossover frequency:
|H_HPF(-6dB)| + |H_LPF(-6dB)| = flat (both 0.5, sum = 1.0 in linear)
Phase: HPF = −360°, LPF = −360° (identical, in phase)
The phase problem:
Below and above crossover, IIR filter phase rotates differently for different filter orders. A perfect LR4 crossover is flat in magnitude AND phase at the crossover frequency, but the phase of each individual path rotates through several hundred degrees across the full bandwidth.
This means an impulse played through an LR4-crossed system does not arrive with perfect timing at all frequencies — different frequencies in the tweeter range arrive slightly after different frequencies in the midrange range.
For most music listening this is inaudible. For critical SQ competition, this is why FIR crossovers exist.
FIR (Finite Impulse Response) Crossovers
FIR filters can achieve linear phase — all frequencies in the passband are delayed by exactly the same amount (constant group delay).
Linear phase HPF implementation:
h[n] = δ[n - N/2] - h_LPF[n] (spectral subtraction)
Where h_LPF[n] is a linear-phase lowpass FIR.
Key properties: - All frequencies delayed equally (constant group delay = N/2 samples) - No pre-ringing in minimum-phase sense — but has symmetric pre- and post-ringing (inherent to linear phase filters) - Computationally expensive: N multiply-accumulates per sample, N typically 256–8192 for audio crossovers - Latency: N/2 samples at sample rate → at N=4096, 48kHz: 42.7ms latency per filter
DSP FIR support:
At 48 kHz, a 4096-tap FIR provides 42.7ms of group delay (latency). All channels must be delayed equally to maintain coherence — the non-FIR channels receive a matching time delay.
Platforms supporting FIR: miniDSP C-DSP 8×12 DL, Helix DSP Pro MK3, Alpine Bit One HD.