ELEN E4896 MUSIC SIGNAL PROCESSING
Lecture 7:
Filters & Reverb
1. Filters & EQ
2. Time delay effects
3. Reverb
Dan Ellis
Dept. Electrical Engineering, Columbia University
dpwe@ee.columbia.edu
E4896 Music Signal Processing (Dan Ellis)
http://www.ee.columbia.edu/~dpwe/e4896/
2013-03-04 - 1 /17
1. Filters & EQ
• EQ is a critical tool in audio mixing
boost/cut on single control
each instrument has its own “space”
• Different formats
Low/Mid/High, Parametric
Graphic EQ
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 2 /17
EQ filters
• How to get boost + cut from a single filter?
1
A(z)
-0.5
-0.2π
-0.4π
0
-0.6π
-10
-1
0
1 Re{z}
-20
∠{A(z)}
0
10
0.5
-1
|A(z)|
20
level / dB
Im{z}
use allpass
-0.8π
0
0.2π
0.4π
0.6π
0.8π
-π
π
0
0.2π
0.4π
0.6π
0.8π
π
then +/- to get phase cancellation/reinforcement
+
A(z)
y[n]
level / dB
x[n]
1 + g A(z)
20
g = -1
10
g=0
0
g = -1...1
-10
-20
0
g=1
0.2π
0.4π
0.6π
0.8π
π
s03-allpass1.pd
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 3 /17
Allpass Filters
•
Allpass filters have flat gain: |A(ej)| = 1
from mirror-image numerator and denominator:
A(z) =
z
1
A(z)
)
0.6 + z
A(z) =
1 0.6z
1
0
-0.5
0
1 Re{z}
-20
1
∠{A(z)}
-0.4π
0
-0.6π
-10
-1
1
-0.2π
10
0.5
-1
0.6z
1
|A(z)|
20
level / dB
Im{z}
e.g. for Dm (z) = 1
Dm (z
Dm (z)
m
-0.8π
0
0.2π
0.4π
0.6π
0.8π
-π
π
0
0.2π
0.4π
0.6π
0.8π
π
slope governs phase interactions, group delay
g ( c)
E4896 Music Signal Processing (Dan Ellis)
=
d ( )
d
=
c
2013-03-04 - 4 /17
Group Delay
• Local phase change of H(z) governs effective
delay of that frequency region: “group delay”
g ( c)
d ( )
d
=
=
c
8000
Frequency
6000
4000
2000
0
0
0.5
1
1.5
Time
2
2.5
3
0
0.5
1
1.5
Time
2
2.5
3
8000
Frequency
6000
4000
2000
0
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 5 /17
Parametric EQ
Im{z}
• 2nd order Allpass → slope & place θ(ω) = π
|A(z)|
10
dB
0.5
0
0
0
-0.5π
-10
-π
-20
-0.5
-1.5π
-30
-1
-1
-0.5
0
0.5
0
Re{z}
∠{A(z)}
0.2π
0.4π
0.6π
0.8π
-2π
ω
0
0.2π
0.4π
0.6π
0.8π
ω
angle → frequency
radius → slope → bandwidth
x[n]
+
A(z)
y[n]
1 + g A(z)
20
level / dB
• Same structure for EQ
g=1
10
g=0
0
g = -1...1
-10
-20
E4896 Music Signal Processing (Dan Ellis)
g = -1
0
0.2π
0.4π
0.6π
0.8π
π
s06-allpass2.pd
2013-03-04 - 6 /17
Time-Varying Filters
• Classic Wah-Wah ?
• Iterated Filters...
s07-wahwah.pd
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 7 /17
2. Time Delays
• Delays correspond to sound propagation
340 m/s ≈ 1 foot / ms
• Delays are a simple kind of filter
can analyze from Fourier perspective...
x[n]
delay τ
y[n]
1
|H(ejω)|
h[n] = δ[n – τ]
ω
⇒H(ejω) = e–jωτ
∠{H(ejω)}
1
0
τ
E4896 Music Signal Processing (Dan Ellis)
ω
−jωτ
n
2013-03-04 - 8 /17
Fractional 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 domain
e
sin (n
(n
j
)
)
= sinc(n
)
1
0.8
sinc(n-τ)
0.6
0.4
h[n]
0.2
0
-0.2
-0.4
0
1
2
3
4
5
6
truncated FIR implementation
E4896 Music Signal Processing (Dan Ellis)
7
8
9
10
time / samples
2013-03-04 - 9 /17
Comb Filters
x[n]
y[n]
+
delay τ
g
h[n]
1
g
0
τ
• Delay added to direct path causes “comb”
n
Im{z}
.. from phase interactions
1+g
1
0.5
1
0
-0.5
1- g
0
0
-1
π/τ
2π/τ
3π/τ
4π/τ
5π/τ
...
• Range of perceptual effects
< 10 ms - phasing (spectral structure)
20-100 ms - chorus/doubling
> 100 ms - echo
E4896 Music Signal Processing (Dan Ellis)
-1
-0.5
0
0.5
1
Re{z}
s10-comb.pd
2013-03-04 - 10/17
IIR Comb Filters
x[n]
• Feedback delay spreads out more in time
y[n]
+
h[n] 1
delay τ
g
0
g
τ
τ
g2
τ
g3
τ
g4 ...
n
Im{z}
• Poles can give unbounded gain
1
1+g
1
0.5
0
-0.5
1
1
1- g
0
-1
π/τ
2π/τ
3π/τ
4π/τ
5π/τ
...
-1
-0.5
0
0.5
1
Re{z}
s11-iircomb.pd
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 11/17
Time Varying Delay
• Periodic variation over large range
Pitch modulation
analogous to Doppler shift
x[n]
delay line
y[n]
• Random (but smooth) shift over short delay
Chorus
pattern of cancellation “notches” like detuned
voices
s12-vardelay.pd
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 12/17
3. Reverberation
• Received sound is direct path
+ reflections
l(t)
delayed relative
to direct path
different at
each ear
Direct-toReverberant...
direct
reflected
t
r(t)
R
R/
t
c
• Source→Ear is ~ LTI
can use impulse response, convolution
freq / Hz
hlwy16 - 128pt window
0.1
0.05
0
-10
0.1
-20
6000
0.05
-30
0
-40
-0.05
-50
4000
-0.05
-0.1
8000
2000
0
0.2
0.4
0.6
0.8
time / s
E4896 Music Signal Processing (Dan Ellis)
-0.1
0
0
-60
0
0.01
0.1
0.2
0.02
0.3
0.4
time 0.5
/s
0.03
0.6
0.7 time / s
-70
2013-03-04 - 13/17
Early Echoes & Late Reverb
• Reflected paths are like “virtual sources”
virtual (image) sources
reflected
path
source
listener
first part of reverberant IR is sparse
• Reflections quickly
build up & merge
later part of
reverb is like
decaying noise
0.1
0.05
0
-0.05
0 05
0.1
-0.1
0.05
0.01
0
0.2
0.4
0.
.4 0.005 0.6
0
0
-0.05
-0.005
-0.1
E4896 Music Signal Processing (Dan Ellis)
0
0.01
0.02
0.03
-0.01
0.2
0.8
0.3
0.4
0.5
0.6
2013-03-04 - 14/17
Nested Allpass
• Allpass efficiently creates decaying response
multiple, combined filters for complex patterns
FIR Comb
2
1.5
Allpass
z-k
1
-g
H(z) =
1 - g·z-k
-g
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
6
4
x[n]
y[n]
+
z-k
+
2
h[n]
1-g2
g(1-g2) 2
g (1-g2)
k
g
g,k
3k
iIR Comb
n
1
Imaginary Part
-g
Nested+Cascade Allpass
50,0.5
20,0.3
2k
0
Synthetic Reverb
+
30,0.7
a0
+
AP0
g
E4896 Music Signal Processing (Dan Ellis)
AP1
LPF
0
−0.5
−1
+
a1
0.5
a2
−1
−0.5
0
0.5
Real Part
1
AP2
s15-gardnerverb.pd
2013-03-04 - 15/17
Feedback Delay Network
• Matrix of feedbacks gives even more
complex patterns
from Stauter & Puckette 1982
“Unitary” matrix ensures decay
s16-reverb.pd
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 16/17
• Filters:
Summary
EQ used to balance mixes
Varying filters gives effects e.g. Wah-wah
• Delays
Wide range of effects: phasing ... echo
Fractional delays
• Reverb
Just a complex pattern of echoes
Discrete early echoes → reverberant tail
E4896 Music Signal Processing (Dan Ellis)
2013-03-04 - 17/17