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# IIR filters
# Filters are used to make some frequencies louder or quieter in order
# to change a sound's tone.
# Each one takes some parameters for its left argument and a signal on
# the right.
⟨
# 1-pole filters (f:frequency)
Lp1, Hp1 # f low/high-pass
# 2-pole filters (g:gain, w:width, Q decreases with width)
Lp2, Hp2 # f Butterworth low/high-pass
Lp2q,Hp2q,Bp2q # f Q Butterworth low/high/band-pass with resonance
Peak # g f Q
LShelf, HShelf # g f Q? low/high-shelf
Notch # f w
AllPass # a‿b a+bi is complex with magnitude <1
# Utilities
NtoQ, QtoN # Bandwidth ←→ Q for peak filters
# Use Lp2⍟2 and Hp2⍟2 (4-pole Linkwitz-Riley) for crossovers
# Other 2-pole low/high-pass filters (I never use these)
# Filter2 generates Lp2 and Hp2 filters except the resonant ones
# If the second argument is n then the filter should be called n
# times and then gives a (2×n)-pass filter
Filter2
# Critically damped and Bessel filters
Lp2_CD,Hp2_CD, Lp2_BS,Hp2_BS
⟩⇐
"filter.bqn takes a single option namespace, or no arguments" ! 1≥≠•args
o ← ≠◶⟨•Import∘"options.bqn", ⊑⟩ •args
Sin‿Cos‿Tan ← •math
# Generalized filtering
# 𝕩 is the signal to filter
# 𝕨 is ⟨result coefficients , 𝕩 coefficients⟩.
Filter ← {
⟨i0⟩‿⟨o0⟩ 𝕊𝕩:
0 (×⟜o0+i0⊸×)` 𝕩
;⟨i0,i1⟩‿⟨o0⟩ 𝕊𝕩:
a1←a0←0
0 { (o0×𝕨)+(i1×a1↩𝕩)+i0×a0↩a1 }` 𝕩
;⟨i0,i1,i2⟩‿⟨o0,o1⟩ 𝕊𝕩:
b0←a2←a1←a0←0
0 { (o1×b0↩𝕨)+(o0×b0)+(i2×a2↩𝕩)+(i1×a1↩a2)+i0×a0↩a1 }` 𝕩
;coeff 𝕊𝕩:
a‿b ← 0×coeff # accumulators for input and result
c ← ∾coeff
{
a«˜↩𝕩
r←+´c×a∾b
b«˜↩r
r
}¨ 𝕩
}⎉1
_f ← { !∘0⊘(𝔽⊸Filter) }
# Compute the frequency response from coefficients
#Response ← ⋈⟜(1∾-)○⌽´ ⊸ ((|·÷´{+⟜(𝕩⊸×)´𝕨}¨)⎉∞‿0) ⟜ (⋆0i1×Om)
Om ← (2×π)×{𝕩÷o.freq}
Tom ← Tan Om÷2˙
# 1-pole low-pass and high-pass filters
Lp1 ← (⋈¨ ·⋈⟜¬ 1+⌾÷Om) _f
Hp1 ← ((-⊸⋈ ⋈ ⋈) 1÷∘+Om) _f
# 2-pole *-pass
f2types ← "bw"‿"cd"‿"bessel"
Filter2 ← { type‿nPasses‿isHighpass:
t ← f2types⊸⊐⌾< type
! 3 > t
c ← t◶⟨4√-⟜1 , √1-˜√ , √3×0.5-˜·√-⟜0.75⟩ nPasses√2 # cutoff frequency correction
x ← t ⊑ ⟨√2‿1 , 2‿1 , 3‿3⟩ # p and g
Clp2 ← {
e ← ¯1 ⊑ a ← 𝕨 × ≍⟜(ט) Tom 𝕩
b ← (1‿2‿1 ∾˜ 2×1-˜÷e) × e ÷ 1++´a
¯3 (↑⋈↓) (1-+´)⊸∾ b
}
(isHighpass ⊑ ⟨
x Clp2 ÷⟜c
⟨1‿¯1‿1, 1‿¯1⟩ × x Clp2 ·{(o.freq÷2)-𝕩}×⟜c
⟩) _f
}
⟨Lp2‿Hp2, Lp2_CD‿Hp2_CD, Lp2_BS‿Hp2_BS⟩ ← <˘f2types Filter2∘{𝕨‿1‿𝕩}⌜ ↕2
# Once more, with resonance
# Lp2q and Hp2q are identical to Lp2 and Hp2 when Q=÷√2
Resfilt ← {
Om⊸(Sin⊸÷⟜(2⊸×) (+⟜1 ÷˜ 𝕏 ⋈ -⟜1≍2⊸×) Cos∘⊣)´ _f
}
Lp2q‿Hp2q‿Bp2q ← Resfilt¨ ⟨ 1⊸-÷2‿1‿2˙ , 1⊸+÷2‿¯1‿2˙ , ×⟜1‿0‿¯1 ⟩
# Other 2-pole filters
Peak ← { g‿f‿q:
k ← Tom f
ab ← 1‿4‿0‿2‿3‿0 ⊏ +⟜(k⊸×)´¨ ¯2‿0‿2 <⊸∾ {1‿𝕩‿1}¨ ⥊≍⟜- q ÷˜ 10⋆0⌈-⊸≍g÷20
3 (↑ ⋈ -∘↓) (1⊸↓ ÷ ⊑) ab
} _f
# Q to bandwidth and vice-versa for peaking filters
NtoQ ← {𝕊𝕩: (√÷-⟜1)𝕩 ; 𝕊⁼𝕩: -⟜1⌾(ט)⊸+ 1+÷2×ט𝕩} 2⊸⋆
QtoN ← NtoQ⁼
Shelf ← {
# t is type: 1 for bass and ¯1 for treble shelf
t‿g‿f‿q ← ∾⟜(√2)⍟(3=≠) 𝕩
k ← Tom f
v1 ← 10 ⋆ t × 0⌈-⊸≍g÷40
ab ← ⥊ 1‿0‿2⊸⊏˘ (ט1⌊v1) ÷˜ (k×v1) {+⟜(𝕨⊸×)´𝕩}⌜ ¯2‿0‿2 <⊸∾ {1‿𝕩‿1}¨ ≍⟜- q
2 (↓ ⋈○⌽ -∘↑) (1⊸↓ ÷ ⊑) ab
} _f
# Shortcuts for low- and high-shelf
LShelf ← 1⊸∾⊸Shelf
HShelf ← ¯1⊸∾⊸Shelf
Notch ← {
# Filter with no gain outside of the notch due to A.G. Constantinides
f‿w ← 𝕩 # w is width of ≥3dB attenuation in Hz.
c←2×Cos Om f ⋄ t←Tom w
⟨1,-c,1⟩‿⟨t-1,c⟩ ÷ 1+t
} _f
AllPass ← (⋈⟜(-∘⌽1⊸↓) ⟨1,¯2×⊑,+´×˜⟩{𝕎𝕩}¨<) _f
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