Talk:Linkwitz–Riley filter

Definite article

Do we need a definite article in The Linkwitz-Riley filter? Definite articles refer to particular or specfic things, and here we are talking about L-R filters in general. So I suggest using A Linkwitz-Riley filter instead. This would be consistent with Elliptical filter, Comb filter and Bessel filter. They used the definite article for Butterworth filter tho.

"compared to Butterworth crossovers, whose summed output has a +3 dB peak around the crossover frequency. "

I've encountered this statement before, and it seems to me that this cannot possibly be true since the only consistent definition of crossover frequency is the frequency where the high-pass and low-pass filter yield equal magnitude, and as long as both filters yield -3dB at that frequency, they must add up to 0 dB, not +3 dB. It is therefore a question of the frequency spacing of the high-pass and low-pass sections. There is evidently some rationale by which the two sections should be separated in frequency, such that some other criteria is satisfied, and when this criteria is satisfied, the consequence is that at the frequency where the two sections yield equal magnitude, they each yield less than -3 dB of attenuation. So rather than just make a statement of this sort, it would be much nicer if the criteria is explained, i.e., the criteria for the frequency spacing of the high-pass and low-pass sections that results in this circumstance of the two sections summing to 3 dB at the crossover point. Perhaps this has to do with this being the one point where the two sections are aligned in phase. If so, this should be explained.

72.150.130.231 (talk) 18:40, 21 February 2010 (UTC)

Phase considerations

Saying that the Sum of two Butterworth-filters results in "+3dB" is misleading. That would only be true if the filters are (of the same) linear phase (e.g. digital FIR filter), but crossovers are typically made from analog components, or implemented as their digital equivalent, IIR.

The phase changes by IIR filters will lead to unexpected cancellations when adding filtered and (un-)filtered signals (cf. Comb filter).

In the second order butterworth case ($Q=1/{\sqrt {2}}$):
LPF: $H_{LPF}(s)={\frac {1}{s^{2}+{\sqrt {2}}s+1}}$,
HPF: $H_{HPF}(s)={\frac {s^{2}}{s^{2}+sqrt{2}s+1}}$.

The sum LPF+HPF is: $H_{LPF+HPF}(s)={\frac {s^{2}+1}{s^{2}+{\sqrt {2}}s+1}}$, which is known as Notch filter, i.e. the gain at 1 is -∞ dB, because an analog/IIR LPF adds -90° phase shift at frequency 1, and a HPF adds +90° at that frequency.

To prevent the cancellation, either the LPF or the HPF must be negated, i.e. the Butterworth difference HPF-LPF (or LPF-HPF) is taken instead of the sum:
$H_{HPF-LPF}(s)={\frac {s^{2}-1}{s^{2}+{\sqrt {2}}s+1}}$; this has the plotted +3dB response.

Linkwitz-Riley filters also use the difference (and not the sum!) in typical IIR/analog implementations (this is mentioned in the article).

Example with Second order LR ($Q=1/2$):
$H_{LPF}(s)={\frac {1}{s^{2}+2s+1}}$, $H_{HPF}(s)={\frac {s^{2}}{s^{2}+2s+1}}$.
$H_{HPF-LPF}(s)={\frac {s^{2}-1}{s^{2}+2s+1}}={\frac {(s-1)(s+1)}{(s+1)(s+1)}}={\frac {s-1}{s+1}}$, i.e. the recombined signal is not exactly the original, but "first order all-pass"-filtered.

--77.11.42.207 (talk) 14:04, 28 May 2015 (UTC)

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