Adaptive Feedback Cancellation for Audio signals:
an Application of Prediction Error Identification with Cascaded Noise Models
Toon van Waterschoot and Marc Moonen
Katholieke Universiteit Leuven, ESAT-SCD, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium toon.vanwaterschoot@esat.kuleuven.be http://homes.esat.kuleuven.be/∼tvanwate/
1 Acoustic Feedback Problem
G
music
u (t)
v (t) y (t)
forward path feedback path
acoustic electroacoustic
loudspeaker
microphone
x (t) F
• closed-loop system instability if loop gain satisfies Nyquist criterion
|G(ω, t)F (ω, t)| ≥ 1
∠G(ω, t)F (ω, t) = n2π, n ∈ Z
• poor sound quality (ringing, reverberation) if gain margin
< 3 dB
2 Adaptive Feedback Cancellation
F
d [t, ˆ f (t)]
ˆ
y [t|ˆ f (t)]
G
u (t)
F ˆ
y (t) v (t)
x (t)
• closed-loop transfer function after feedback cancellation:
U (ω, t)
V (ω, t) = G(ω, t)
1 − G(ω, t)[F (ω, t) − ˆF (ω, t)]
• adaptive feedback cancellation objectives:
1. to increase the maximum stable gain MSG(t) = −20 log10h
maxω |G(ω, t)[F (ω, t) − ˆF (ω, t)]|i 2. to improve sound quality
• closed-loop identification using the direct method:
– no reference signal (sound quality)
– noise model is required to obtain consistent estimate
• adaptive feedback cancellation challenges:
– find suitable noise model structure (audio signals) – noise model is unknown and time-varying
– persistence of excitation is not guaranteed
3 Cascaded noise models
• Noise (audio) signal is known to admit a sinusoids+noise representation:
v(t) =
XN n=1
βn cos(ωnt + φn)
| {z }
tonal components
+ 1
C(q, t)e(t)
| {z }
noise components
0 0.5 1 1.5 2
x 104
−40
−30
−20
−10 0 10 20 30 40
f (Hz) 20log 10|X(ej2πf/fs )|(dB)
• Noise model is chosen as cascade of two linear models:
v(t) = H1(q, t)
| {z }
tonal components model
· H2(q, t)
| {z }
noise components model
·e(t)
– noise components model H2(q, t) = 1 C(q, t) – tonal components model H1(q, t) = B(q, t)
A(q, t) with
Model structure Prediction error filter Conventional LP model (LP) A(q, t) = 1 +
nA
X
i=1
ai(t)q−i
Pole-zero LP model (PZLP) A(q, t) B(q, t) =
nA/2
Y
i=1
1 − 2νi cos θiq−1 + νi2q−2 1 − 2ρi cos θiq−1 + ρ2iq−2 Pitch prediction model (PLP) A(q, t) = 1 −
X1
i=−1
αi(t)q−K−(l/D)−i
Warped LP model (WLP) A(q, t) = D0−1(q, λ)[1 +
nA
X
i=1
αi(t)Di(q, λ)]
Selective LP model (SLP) A(q, t) = 1 +
nA
X
i=1
αi(t)q−iΓ
4 Prediction Error Identification
• Prediction error identification criterion:
minξ(t)
1 2N
Xt k=1
ε2[k, ξ(t)]
ε[t, ξ(t)] = H2−1(q, t)H1−1(q, t)[y(t) − F (q, t)u(t)]
ξ(t) ,
fT(t) cT(t) aT(t)T
• Prediction error identification is decoupled in three stages:
1. Identification of tonal components model
e(t)
ε[t, ˆξ(t − 1)]
w[t, ˆf(t − 1), ˆc(t − 1)]
Hˆ1−1 Hˆ2−1
F
x(t) G
u(t)
v(t) y(t)
Fˆ
H1 H2
2. Identification of noise components model
e(t)
r[t, ˆf(t − 1), ˆa(t − 1)]
Hˆ2−1 Hˆ1−1
ε[t, ˆξ(t − 1)]
F
x(t) G
u(t)
v(t) y(t)
Fˆ
H1 H2
3. Identification of acoustic feedback path
ε[t, ˆξ(t − 1)]
˜
u[t, ˆa(t − 1), ˆc(t − 1)]
˜
y[t, ˆa(t − 1), ˆc(t − 1)]
Fˆ
Fˆ Hˆ2−1 Hˆ1−1 Hˆ2−1
Hˆ1−1 F
x(t) G
u(t)
v(t)
y(t) H1 H2
e(t)
• Identifiability conditions:
– electro-acoustic forward path G(q, t) contains delay d1 – acoustic feedback path F (q, t) contains delay d2
– d1 + d2 ≥ nA + nC + 1
5 Simulation Results
Public Address application
0 10 20 30 40 50 60
−4
−2 0 2 4 6 8 10
t (s) MAF(dB)
NLMSPEM-AFROW H1 = LP
H1 = PZLP H1 = PLP H1 = WLP H1 = SLP
←
'
&
$
%
F (q, t) = room acoustic impulse response fs = 44100 Hz
nF = 4410 (= 100 ms)
v(t) = Partita No. 2 in D minor (Allemande) for solo violin by J. S. Bach
NLMS: no noise models (nA = nC = 0)
PEM-AFROW: single noise model (nA = 0, nC = 30) H1 = ·LP: cascaded noise models (nA = nC = 30)
Performance measure: MAF(dB) = 20 log10 kˆf(t) − f k kf k
'
&
$
%
F (q, t) = hearing aid acoustic impulse response fs = 16000 Hz
nF = 200 (= 12.5 ms)
v(t) = Kyrie from Mass in C minor (“Grosse Messe”) by W. A. Mozart
→
Hearing Aid application
0 2 4 6 8 10 12 14 16
−8
−6
−4
−2 0 2 4
t (s) MAF(dB)
NLMSPEM-AFROW H1 = LP
H1 = PZLP H1 = PLP H1 = WLP H1 = SLP
References:
1 T. van Waterschoot and M. Moonen, “Comparison of linear prediction models for audio signals,” EURASIP J. Audio, Speech, Music Process., submitted for publication, ESAT-SISTA Technical Report TR 07-29, Dec. 2007.
2 T. van Waterschoot and M. Moonen, “Adaptive feedback cancellation for audio applications,” IEEE Trans. Audio, Speech, Language Process., to be submitted for publication, ESAT-SISTA Technical Report TR 07-30, Nov. 2008.
