Columbia ELEN E4896 - Filters and Reverb

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E4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Lecture 7:Filters & ReverbDan EllisDept. Electrical Engineering, Columbia [email protected] http://www.ee.columbia.edu/~dpwe/e4896/11. Filters & EQ2. Time delay effects3. ReverbELEN E4896 MUSIC SIGNAL PROCESSINGE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /171. Filters & EQ•EQ is a critical tool in audio mixingboost/cut on single controleach instrument has its own “space”•Different formatsLow/Mid/High, Parametric2Graphic EQE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17EQ filters•How to get boost + cut from a single filter?use allpassthen +/- to get phase cancellation/reinforcement30.2π 0.4π 0.6π 0.8ππlevel / dB0-20-10010201 + g A(z)g = 1g = 0g = -100.2π 0.4π 0.6π 0.8π π 0.2π 0.4π 0.6π 0.8ππ-20-10level / dB01020-1 0 1-1-0.50.51Re{z}Im{z}0-π-0.8π-0.6π-0.4π-0.2π0|A(z)|A(z)∠{A(z)}A(z)+g = -1...1x[n] y[n]s03-allpass1.pdE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Allpass Filters•Allpass filters have flat gain: |A(ej)| = 1 from mirror-image numerator and denominator:e.g. forslope governs phase interactions, group delay4A(z)=zmDm(z1)Dm(z)Dm(z)=1 0.6z1 A(z)=0.6+z11  0.6z100.2π 0.4π 0.6π 0.8π π 0.2π 0.4π 0.6π 0.8ππ-20-10level / dB01020-1 0 1-1-0.50.51Re{z}Im{z}0-π-0.8π-0.6π-0.4π-0.2π0|A(z)|A(z)∠{A(z)}g(c)=d()d=cE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Group Delay•Local phase change of H(z) governs effective delay of that frequency region: “group delay”5TimeFrequency0 0.5 1 1.5 2 2.5 302000400060008000TimeFrequency0 0.5 1 1.5 2 2.5 302000400060008000g(c)=d()d=cE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Parametric EQ•2nd order Allpass → slope & place θ(ω) = πangle → frequencyradius → slope → bandwidth•Same structure for EQ600.2π 0.4π 0.6π 0.8π-30-20-100100 0.2π 0.4π 0.6π 0.8πωω-2π-1.5π-π-0.5π0|A(z)| ∠{A(z)}dB-1 -0.5 0 0.5-1-0.500.5Re{z}Im{z}A(z)+g = -1...1x[n] y[n]0.2π 0.4π 0.6π 0.8ππlevel / dB0-20-10010201 + g A(z)g = 1g = 0g = -1s06-allpass2.pdE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Time-Varying Filters•Classic Wah-Wah ?•Iterated Filters...7s07-wahwah.pdE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /172. Time Delays•Delays correspond to sound propagation340 m/s ≈ 1 foot / ms •Delays are a simple kind of filtercan analyze from Fourier perspective...8h[n] = δ[n – τ]⇒H(ejω) = e–jωτnω01τ|H(ejω)|ω∠{H(ejω)}1−jωτx[n] y[n]delay τE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Fractional Delays•For short delays, one sample quantization may be too coarse 1 sample @ 44.1 kHz = 22.7 µs•Fractional delay can be recovered from Fourier domaintruncated FIR implementation9ejsin (n   )(n   )= sinc(n   )0 1 2 3 4 5 6 7 8 9 10-0.4-0.200.20.40.60.81h[n]time / samplessinc(n-τ)E4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Comb Filters•Delay added to direct path causes “comb”.. from phase interactions•Range of perceptual effects< 10 ms - phasing (spectral structure)20-100 ms - chorus/doubling> 100 ms - echo10+gx[n] y[n] h[n]n01gτdelay τ001π/τ 2π/τ 3π/τ 4π/τ 5π/τ ...1+g1- g-1 -0.5 0 0.5 1-1-0.500.51Re{z}Im{z}s10-comb.pdE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17IIR Comb Filters•Feedback delay spreads out more in time•Poles can give unbounded gain1101-1 -0.5 0 0.5 1-1-0.500.51π/τ 2π/τ 3π/τ 4π/τ 5π/τ1+g11- g...1Re{z}Im{z}+x[n] y[n]h[n]n01ggτdelay τg2τg3τg4τ...s11-iircomb.pdE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Time Varying Delay•Periodic variation over large range󲰛 Pitch modulationanalogous to Doppler shift•Random (but smooth) shift over short delay󲰛 Choruspattern of cancellation “notches” like detuned voices12x[n]y[n]delay lines12-vardelay.pdE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /173. Reverberation•Source→Ear is ~ LTIcan use impulse response, convolution 13tl(t)tR/cRr(t)directreflected0 0.2 0.4 0.6 0.8-0.1-0.0500.050.10 0.01 0.02 0.03-0.1-0.0500.050.1time / stime / stime / sfreq / Hzhlwy16 - 128pt window0 0.1 0.2 0.3 0.4 0.5 0.6 0.702000400060008000-70-60-50-40-30-20-10•Received sound is direct path + reflectionsdelayed relative to direct pathdifferent at each earDirect-to-Reverberant...E4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Early Echoes & Late Reverb•Reflected paths are like “virtual sources”first part of reverberant IR is sparse14source listenervirtual (image) sourcesreflected path0 0.2 0.4 0.6 0.8-0.1-0.0500.050.10.0050 0.01 0.02 0.03-0.1-0.0500.050.1.40.2 0.3 0.4 0.5 0.6-0.01-0.00500.0050.01•Reflections quickly build up & mergelater part of reverb is like decaying noiseE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Nested Allpass•Allpass efficiently creates decaying responsemultiple, combined filters for complex patterns15z-k+ +-ggg,kx[n]y[n] nk 2k 3k-g1-g2g(1-g2)g2(1-g2)h[n] z-k - g1 - g·z-kH(z) =20,0.3AllpassNested+Cascade AllpassSynthetic Reverb30,0.750,0.5AP0+AP1AP2LPFga0a1a2+ +s15-gardnerverb.pdFIR CombiIR Comb−1 −0.5 0 0.5 1−1−0.500.51Real PartImaginary Part0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.511.520 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10246E4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Feedback Delay Network•Matrix of feedbacks gives even more complex patterns“Unitary” matrix ensures decay16from Stauter & Puckette 1982s16-reverb.pdE4896 Music Signal Processing (Dan Ellis) 2013-03-04 - /17Summary•Filters:EQ used to balance mixesVarying filters gives effects e.g. Wah-wah•DelaysWide range of effects: phasing ... echoFractional delays•ReverbJust a complex pattern of echoesDiscrete early echoes → reverberant


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