On Reverse Waterfilling in Closed-Loop LPC with Noise Shaping
Hauke Krüger, Bernd Geiser, Peter Vary
Institute of Communication Systems and Data Processing ( ), RWTH Aachen University, 52056 Aachen, Germany
Email: {krueger,geiser,vary}@ind.rwth-aachen.de
Web: www.ind.rwth-aachen.de
Abstract
A new criterion is derived for the design of the noise weighting
filter in linear predictive coding (LPC) that accounts for the issue
of noise propagation. It will be shown that the new filter design
constitutes an approximation of the reverse waterfilling proce-
dure known from rate-distortion theory and audio transform cod-
ing. It improves the overall coding SNR of closed-loop LPC with
noise shaping. A practical application within the 3GPP AMR-
WB codec is presented. Objective test results indicate notable
quality improvements.
1 Introduction
For a long time linear predictive coding (LPC) [1] has been (and
still is) the method of choice for low-rate speech coding with ei-
ther scalar (e.g. ADPCM codecs) or vector quantization (CELP
codecs [2]). For all approaches, to actually turn the LPC predic-
tion gain (at least partially) into an overall SNR gain, the (filtered)
quantization error e
f
(k) must be fed back and subtracted from the
(unquantized) prediction residual d(k), cf. Figure 1. However,
so far, the surprisingly intricate interaction between the quantizer
Q and the error feedback has not been adequately addressed in
codec design.
1.1 Noise Propagation in Closed-Loop LPC
In a previous paper [3], we have proposed a new quantization
noise production and propagation model for open and closed-loop
LPC to generalize the conventional high rate theory of LPC [4]
towards lower bit rates. In LPC, quantization noise is effectively
processed by the cascade of an error weighting filter 1 − F (z) and
autoregressive synthesis with H
−1
A
(z)=(1 − A(z))
−1
. In [3], we
used a noise propagation network to model the effect of this pro-
cessing on the quantization noise. In this model, the signal x(k)
is generated by an autoregressive process (all-pole filter H
0
(e
jΩ
))
driven by a spectrally white excitation signal d
0
(k). The quanti-
zation noise Δ(k)=
˜
d(k) − d
′
(k) is assumed to be generated by a
power-controlled additive noise source:
E {Δ
2
(k)} =
E {d
′2
(k)}
SNR
0
⇒ G
d
′
,Δ
.
=
E {Δ
2
(k)}
E {d
′2
(k)}
= SNR
−1
0
, (1)
motivated by the fact that practical quantizers for LPC are oper-
ate with a nearly constant quantization SNR
0
[5]. Based on this
model, the overall coding SNR of LPC was derived:
SNR
lpc
= E {x
2
(k)}/E {( ˜ x(k) − x(k))
2
}
=
G
d
0
,x
G
Δ, ˜ x−x
· G
d
0
,d
1 −
G
Δ,e
f
SNR
0
· SNR
0
(2)
comprising the following set of filter gains:
G
Δ, ˜ x−x
=
1
2π
π
−π
1 − F (e
jΩ
)
H
A
(e
jΩ
)
2
dΩ (3)
G
Δ,e
f
=
1
2π
π
−π
F (e
jΩ
)
2
dΩ (4)
G
d
0
,d
=
1
2π
π
−π
H
A
(e
jΩ
)
H
0
(e
jΩ
)
2
dΩ (5)
G
d
0
,x
=
1
2π
π
−π
1
H
0
(e
jΩ
)
2
dΩ (6)
H
A
(z)
F(z)
H
−1
A
(z)
x(k) d(k) d
′
(k)
Δ(k) e
f
(k)
˜
d(k)
Q
˜ x(k)
Figure 1: Closed-Loop LPC with noise feedback.
with the “noise gain” G
Δ, ˜ x−x
and the “feedback gain” G
Δ,e
f
.
Compared to the well-known high rate approximation of (2),
i.e. SNR
lpc,hr
= G
d
0
,x
· SNR
0
for F (z)= A(z), it can be observed
that SNR
lpc
is significantly smaller at low bit rates which is con-
sistent with measurement results. As another important finding,
the noise feedback in closed-loop LPC can lead to encoder insta-
bilities and overall performance losses. The formal condition for
stable encoder operation is given as
G
Δ,e
f
!
< SNR
0
. (7)
1.2 Paper Overview
In this paper, we propose a new design for the noise feedback fil-
ter F (z) based on a revised LPC optimization criterion. This de-
sign constitutes a time-domain approximation of the reverse wa-
terfilling principle known from rate-distortion theory and audio
transform coding. In accord with rate-distortion theory, the new
design leads to a higher coding SNR than the conventional error
weighting filter F
conv
(z)= A(z/γ ) [6]. We will then briefly dis-
cuss the applicability of the new findings to CELP speech codecs.
An example implementation of the proposed noise feedback fil-
ter within the AMR-WB encoder [7, 8] is presented and objective
evaluation results are given. In the final discussion, we will ad-
dress in how far several techniques encountered in modern speech
codecs (although they have initially been introduced for other rea-
sons) already help to mitigate the observed adverse effects to a
certain extent.
2 New LPC Optimization Criterion
An important observation is that d(k) and d
′
(k) in Figure 1 can
significantly differ. The quantizer input is given as
d
′
(k)= x(k) −
N
lpc
∑
i=1
a
i
· x(k − i)
d(k)
−
N
F
∑
i=1
b
i
· Δ(k − i), (8)
whereby, in the new filter design, we consider separate coefficient
sets for the LP analysis filter H
A
(z)= 1 − ∑
N
lpc
i=1
a
i
· z
−i
and for the
error weighting filter F (z)= ∑
N
F
i=1
b
i
· z
−i
. Assuming a constant
SNR
0
, the noise power in the decoded output is always propor-
tional to the power of d
′
(k), i.e.
E {( ˜ x(k) − x(k))
2
} = G
Δ, ˜ x−x
· E {Δ
2
(k)}
(1)
=
G
Δ, ˜ x−x
SNR
0
· E {d
′2
(k)}. (9)
ITG-Fachbericht 252: Speech Communication, 24. – 26. September 2014 in Erlangen
ISBN 978-3-8007-3640-9 1 © VDE VERLAG GMBH ∙ Berlin ∙ Offenbach