Departement Elektrotechniek ESAT-SISTA/TR 1997-23
A Relation Between Subband and Frequency-Domain Adaptive Filtering 1
Koen Eneman, Marc Moonen 2 April 1997
Published in the Proceedings of the 13th International Conference on Digital Signal Processing (DSP97),
Santorini, Greece, July 1997
1
This report is available by anonymous ftp from ftp.esat.kuleuven.ac.be in the directory pub/SISTA/eneman/reports/97-23.ps.gz
2
ESAT (SISTA) - Katholieke Universiteit Leuven, Kardinaal Mercier- laan 94, 3001 Leuven (Heverlee), Belgium, Tel. 32/16/321809, Fax 32/16/321970, WWW: http://www.esat.kuleuven.ac.be/sista. E-mail:
[email protected]. Marc Moonen is a Research Associate with
the F.W.O. Vlaanderen (Flemish Fund for Science and Research). This re-
search was carried out at the ESAT laboratory of the Katholieke Universiteit
Leuven and was partly funded by the Concerted Research Action MIPS (Model-
based Information Processing Systems) and F.W.O. project nr. G.0295.97 of
the Flemish Government, and the Interuniversity Attraction Pole (IUAP-nr.02)
initiated by the Belgian State, Prime Minister's Oce for Science, Technology
and Culture. The scienti c responsibility is assumed by its authors.
ADAPTIVE FILTERING
Koen Eneman Marc Moonen ESAT - Katholieke Universiteit Leuven Kardinaal Mercierlaan 94, 3001 Heverlee - Belgium
[email protected] [email protected]
Abstract
- Modelling acoustic impulse responses with lengths up to 250 ms is required for high quality echo can- cellation in strongly reverberating environments, leading to FIR adaptive lters with several thousands of taps.
Classical LMS-based solutions then clearly fail as they exceed the computational capabilities of present-day DSPs. Cheaper alternative solutions have been proposed and are mainly based on either subband or frequency- domain techniques. Subband lter implementations have many interesting properties but their inherent delay and residual errors have made them unattractive for real-time applications up till now. On the other hand, frequency-domain adaptive lters do not su er from these problems despite being (nearly) equivalent to sub- band adaptive lters. In this paper we specify 3 realisation conditions for DFT modulated subband schemes.
Standard subband adaptive lters cannot ful l all conditions. We explain the operation of the frequency-domain techniques in the `subband jargon' and show that the realisation conditions can be ful lled in this case.
1. INTRODUCTION
For high quality echo cancellation long acoustic echoes need to be suppressed. Acoustic echo paths are characterised by FIR lters with lengths up to 250 ms. Filters clocked at a rate of, say, 10 kHz then require several thousands of lters taps to be identi ed. Classical LMS-based echo cancellers are unattractive for real-time processing as their com- putational requirements clearly exceed the capabil- ities of present-day DSPs. Moreover, speech sig- nals have a coloured spectrum and it is well known that the performance of the LMS algorithm is sub- optimal in that case, especially when extremely long FIR lters are being adapted. Therefore alternative solutions have been proposed and they are mainly based on either subband or frequency-domain tech- niques. Such multirate adaptive echo cancellation schemes have been a topic of interest for many years now. Subband adaptive lters and frequency-domain based techniques are mostly considered as being dif- ferent approaches. In this paper we will consider the frequency-domain approach as a special case of sub- band adaptive ltering having some desired proper- ties and point out why frequency-domain techniques are better from certain perspectives, or -at least- are able to compete with subband schemes. It is hoped that this study will eventually lead to improved sub- band techniques, which is a topic of current further research.
2. SUBBAND ADAPTIVE FILTERING
The general setup for a subband acoustic echo can- celler is shown in gure 1. The loudspeaker and mi- crophone signal are fed into identical
M-band anal- ysis lter banks. After subsampling with a factor
L
, (mostly LMS-based) adaptive ltering is done in each subband and nally the outputs of the subband adaptive lters are recombined in the synthesis lter
+
+
+ +
F F
F adaptive filters ...
H H
H
...
H
H analysis filter bank
G
G G
synthesis filter bank near-end signal
...
... ...
0 i=0 1
M-1
1
H 0
M-1 M-1
1
0 0
1
M-1
far-end signal
f
f
f
W(z)
L L
L
L L
L L
L L
+ -
-
+ -
+
e
i=1
i=M-1
Fig. 1. subband adaptive echo canceller
bank. Due to aliasing e ects, this setup will only work for
M L. The ideal frequency amplitude characteristics of the analysis bank lters
Hiand synthesis bank lters
Giare shown (ideal bandpass lters).
If
Lis chosen equal to
Ma critically downsampled subband adaptive lter is being implemented. Such subband systems are attractive because optimal com- putational savings can be obtained when
Lis as high as possible. In [1] it is shown that critically down- sampled subband systems lead to a residual mod- elling error which is considerable unless cross lters are included between neighbouring subbands. Cross lters again increase the complexity, which is un- wanted. Furthermore, cross lters fail to converge quickly to the optimum solution. This suggests the use of oversampled subband schemes where
M>L.
2.1. DFT modulated Subband Schemes
Oversampled subband acoustic echo cancellers are
mainly based on DFT modulated lter banks.
Msubband lters are derived by frequency shifting a well-designed prototype lowpass lter
h0(
k). DFT modulated lter banks lead to ecient implemen- tation by using polyphase decomposition and fast signal transforms. In [2] a general framework for oversampled DFT modulated subband systems was proposed. A DFT modulated lter bank with
L-fold downsampling can be implemented as a tapped delay line of size
Lfollowed by a structured
ML-matrix
B (
z), containing polyphase components of the pro- totype
h0, and an
MM-DFT matrix F . In case of DFT modulated lter banks, gure 1 can be redrawn resulting in gure 2. It can be shown that element (
i;j) of B (
z) is given by :
B
ij
(
z) =
z;lE(j+lL):K(
zJ) (1)
i;j
0, (
j+
lL) mod
M=
i, (
i;j) mod
g= 0
;J
=
Mg,
g= gcd(
M;L),
K=
MLgE
k :K
(
z) is the
k-th
K-th order polyphase component of the prototype lter
h0. The synthesis bank is con- structed in a similar fashion with matrix C (
z).
A DFT modulated analysis/synthesis lter bank set is (preferably) designed such that the following 3 re- alisation conditions are met :
the analysis lters are frequency selective. This prevents inserting too much inter-subband alias- ing components by downsampling as it has an in- hibitive e ect on the convergence of the adaptive lters.
the analysis/synthesis lter bank set is (nearly) perfect reconstructing i.e. a near-end source signal is not distorted by the analysis/synthesis system.
A condition ensuring perfect reconstruction is
C (
z) B (
z) =
I(2)
the acoustic path can be modelled by nite-length adaptive subband lters. In general, the lower branch in g. 2 models a time-varying periodic system (see [3]). Only when
C (
z) F
;1diag
fFi(
z)
gFB (
z) =
2
6
4
W0 W1 ::: W
L;1
z
;1
W
L;1
W0 ::: W
... ... ... ...
L;2 37
5
| {z }
=pseudo-circulant
(3)
the following time-invariant path is being mod- elled:
W
(
z) =
L;1Xl=0 z
;l
W
l
(
zL) (4) The above conditions are necessary conditions to en- sure complete modelling with a set of nite-length adaptive lters.
2.2. Subband Echo Cancellation : Performance
Splitting signals into subbands seems very promis- ing since for coloured input spectra, fullband conver-
z
z -1
-1
-1
z
z
+
+
-1 z
+
-1 z (z)
B
(z) B
-1 ... ...
d
x
... ...
-1 F
F
F 0
1
M-1 ...
e F
F
... ...
F C(z)
L
L
L L
L
L
L
L
L
Fig. 2. DFT modulated subband adaptive echo canceller gence is slow due to ill-conditioned covariance matri- ces. In the subband case, each subband signal will have a atter spectrum after appropriate subsam- pling, leading to improved convergence. Instead of a single fullband
N-taps FIR lter,
Msubband l- ters of, say,
NLtaps are used to model the acoustic path. As the adaptive computations as well as the lter bank convolutions can be done at a reduced sampling rate, this subband approach is supposed to give a better performance at a lower cost.
It is clear that this picture is certainly too optimistic.
The assumption of having
Msubband lters with re- duced length
NLseems to be quite wrong. It appears that in the case of
M-band,
L-fold downsampled ide- ally frequency selective lter banks the adaptive l- ters should converge to an
L-fold downsampled con- volution of the acoustic path and a double-sided sinc.
In fact, this corresponds to an interpolation opera- tion. The adaptive identi cation process therefore has to track more than
NLsamples and due to the spreading out in both directions of the time axis, an extra delay has to be inserted in the near-end signal path [4]. Neglecting the additional subband lter length due to these sinc-e ects strongly limits the convergence of the adaptive lters and leads to a residual under-modelling error.
Furthermore, delay constraints also make subband schemes unattractive. Selective lter banks are need- ed to avoid aliasing distortion within subbands. They introduce a substantial processing delay and thus put a constraint on the downsampling factor
L. However, the implementation cost is more or less inversely pro- portional to
L.
Filter bank sets can be designed such that the rst 2 realisation conditions are met. As indicated, con- dition 3 can not be met with nite-length lters in this standard subband approach.
3. FREQUENCY-DOMAIN ADAPTIVE FILTERS
As a cheaper alternative to LMS, the frequency-
domain adaptive lter (FDAF) was introduced, which
is a direct translation of Block LMS in frequency do-
main [5]. Correlation (weight updating) and convo-
lution ( ltering) operations are expensive but in the
case of block processing, they may be implemented
more eciently in frequency domain. Instead of a
linear convolution/correlation a circular operation is
performed. This requires some `restore' operations which can be of the overlap-save or overlap-add type.
If only the convolution operation is corrected a so- called unconstrained FDAF is obtained requiring 3 FFTs. Two more FFTs are needed for the gradient estimate correction resulting in a constrained FDAF.
A major drawback concerning standard frequency- domain adaptive lters is the inherent delay.
3.1. Partitioned Block FDAF
By splitting the acoustic impulse response in equal parts, a kind of mixed time and frequency convolu- tion canceller is obtained, called Partitioned Block Frequency-Domain Adaptive Filter (PBFDAF) [6].
Here block lengths can be adjusted, resulting in a cheap echo canceller with acceptable processing de- lay.
3.2. PBFDAF as a special case of Subband Adaptive Filtering
The PBFDAF scheme can be put into the oversam- pled subband framework proposed in [2]. Call
X(
z) the far-end signal and
D(
z) the near-end signal, so
D
(
z) =
S(
z) +
W(
z)
X(
z) (5)
S
(
z) is the contribution of a near-end source. The acoustic impulse response
W(
z) can be split up in its
P
-th order polyphase components
D
(
z) =
S(
z) +
P;1Xp=0 z
;p
W
p
(
zP)
X(
z) (6) This leads to equation 7 (
M=
P+
L;1), which is rewritten as
D (
z) = S (
z) + M (
z) X (
z) (8) Transfer matrix M (
z) was made circulant so that it can be transformed into a diagonal matrix by means of DFT operations, i.e., FM (
z) F
;1= diag
fW^
i(
zP)
g.
^
W
i
(
z) are related to the DFT coecients of the rst column of M (
z) and therefore they are of nite length.
Instead of identifying a \half-full" matrix M (
z), a di-
agonal matrix can be tracked in frequency domain.
An adaptive identi cation process trying to match
W
(
z) in frequency domain based on the above for- mulas is depicted in gure 3.
Looking closer, g. 3 can be cast in the oversampled subband framework of g. 2, i.e. with size
L(in- stead of size
M) tapped delay lines. The B (
z)-matrix for an
M-band
L-fold oversampled DFT modulated analysis lter bank is a structured matrix satisfying Eq. 1. The lter bank used here is a simple DFT lter bank for which only
Ek :K(
z) = 1,
k <M, are non-zero i.e.
B (
z)
|{z}
ML
=
2
6
6
6
6
6
4
I
Lz
;1
I
L...
z
;(b M;1
L
c;1)
I
LR (
z)
3
7
7
7
7
7
5
(9)
z z
IDFT
-1 -1
z
-1-1
z
-1z
-1...
...
0 0
...
...
z z
-1z
-1-1
z
-1z
-1z
-1z
-1z
-1z
-1F
LF
M-1+
+
+ IDFT
+
D(z) z
-(M-1)-(M-1)
E(z) z +
E(z) z
-(L-1)z
z
-1z
-1z
-1IDFT
IDFT
F F
01
i=0
i=0 ...
...
...
DFT DFT
DFT DFT
... ...
...
...
...
...
... ...
0 0
...
... ...
F ...
...
...
...
i=L-1
i=M-1 L
L L L L
...
...
i=M-1
i=L-1
L-1L
L L
L L
L L L L L L L L L
L D(z)
X(z) W(z)
Fig. 3. Partitioned block frequency-domain adaptive ltering
−3 −2 −1 0 1 2 3
0 2 4 6 8 10 12
pulsation (rad/sec
Fig. 4. analysis and synthesis bank prototype lters and R (
z) is an (
M;LbM;1L c)
L-matrix :
R (
z) =
2
6
4 z
;b M;1
L c
:::
0
... ... ...
::: z
;b M;1
L c
::: 3
7
5
(10) The prototype frequency response has a sinc-like shape with a low frequency selectivity. The analysis proto- type frequency amplitude response in shown in gure 4 in full line for
M=12 and
L=6.
Also the synthesis part can be t into the subband lter approach. The synthesis bank C (
z)-matrix is given by :
C (
z)
|{z}
LM
=
I
L0
L :::0
L0
(11)
The synthesis lters are time-reversed and complex conjugated versions of a DFT modulated lter bank with prototype polyphase components
Ek :K(
z) = 1,
k < L
. The other polyphase components are zero.
The synthesis prototype frequency response for
M=12 and
L=6 is shown in g. 4 in dashed dotted line. Its frequency response is twice as wide as the analysis equivalent.
It is easily veri ed that the perfect reconstruction
condition (Eq. 2) holds. The additional condition
(Eq. 3) is also satis ed for
Fi(
z) = ^
Wi(
zPL) if
Pis
a multiple of
L. It is known that in this case extra
savings can be made since signal bu ers can be re-
cuperated from previous block cycles (see [6]).
6
6
6
6
6
6
6
6
6
6
6
6
4
D(z)
z
;1
D(z)
.
.
.
z
;(L;1)
D(z)
.
.
.
7
7
7
7
7
7
7
7
7
7
7
7
5
= 6
6
6
6
6
6
6
6
6
6
6
6
4
S(z)
z
;1
S(z)
.
.
.
z
;(L;1)
S(z)
0
.
.
.
0 7
7
7
7
7
7
7
7
7
7
7
7
5 +
6
6
6
6
6
6
6
6
6
6
6
6
4 W0(z
P
) ::: WP;1(z P
) 0 ::: 0
0 W
0 (z
P
) ::: W
P;1 (z
P
) ::: 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ::: 0 W
0 (z
P
) ::: W
P;1 (z
P
)
WP;1(z P
) ::: 0 0 W0(z
P
) :::
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
W1(z P
) ::: 0 0 ::: W0(z
P
) 7
7
7
7
7
7
7
7
7
7
7
7
5 2
6
6
6
6
6
6
6
6
6
6
4
X(z)
z
;1
X(z)
.
.
.
z
;(P;1)
X(z)
.
.
.
z
;(M;1)
X(z) 3
7
7
7
7
7
7
7
7
7
7
5
(7)
-1
z
+
z
+-1
z
-1
+
z
0 I 0
0 F
L -1
F error correction
-1
-1
z z
-1
x
... ...
F F
F ...
e
... ...
C(z)
i=0
[I 0]
F F
i=L-1
L
L
L L L L
(z) B
0
1
z I
M-1I
-1 L
-1
L
L
... ...
d
i=0F
i=L-1
L L L
(z) B
z I I
-1 L L
Fig. 5. General oversampled subband scheme The implicit error `restore' or projection operation in frequency-domain adaptive ltering consists of a transformation to time domain, zeroing of certain components and transformation back to frequency domain (see gure 3). A general scheme, shown in gure 5, now depicts the PBFDAF completely in the subband 'jargon'. An extra module called `error cor- rection' was included to do the circular-to-linear con- version as no projection operations are applied in the standard subband approach.
The PBFDAF turns out to be a special case of sub- band adaptive ltering. It implements a simple lter bank with low frequency selectivity. The PBFDAF satis es all 3 realisation conditions, so there won't be any residual error as a complete and unique mod- elling can be done with nite-length lters. It is re- markable how an unselective lter bank can lead to satisfactory results.
In the previous,
Mwas set equal to
P+
L;1. But of course, there is an extra degree of freedom of choos- ing
M >P+
L;1. This can have some advantages.
If
Mis a power of 2 ecient FFT implementation is possible. Now, in this case an extra ambiguity occurs as the subband lters
Fi(
z) are not uniquely de ned anymore.
A random
(
zP) can appear for instance as an ex- tra (
P+ 1)-th polyphase component provided it is compensated for at the rst component (see Eq. 7):
W
0
