{
H
E=Rxx and
{
*
Ed=rx, with
1
T
TT
P
⎡⎤
=
⎣⎦
xx x… ;
1
T
TT
P
⎡⎤
=
⎣⎦
ww w…
and
1
T
pp pL
ww
⎡⎤
=
⎣⎦
w … . In the notation we
are using
a for scalar, a for vector and A for
matrix;
a
, A denotes vector and matrix
respectively in a frequency-domain:
Faa ,
= FAA
.
F
represents the discrete Fourier
transform (DFT) matrix defined as
2jklM
kl
e
π
−
=F ,
with ,0,, 1kl M=−… , 1j =− and
1
F as its
inverse. Of course, in the final implementation, the
DFT matrix is substituted by much more efficient
fast Fourier transforms (FFT). Here
()
.
T
denotes
transpose operator and
() ()
()
*
..
HT
= the
Hermitian operator (conjugate transpose).
The conjugate gradient (CG) method is efficient
to obtain the solution to (2), however, a big delay is
introduced (noted that the system order is
LP LP× ). In order to reduce this convergence
speed problem we propose a new algorithm which
employs much more powerful CG optimization
techniques, but keeping the frequency block
partition strategy to allow computationally realistic
low latency situations. The paper is organized as
follows: Section 2 reviews the Multichannel
PBFDAF approach and its implementation. Section
3 develops the Multichannel Conjugate Gradient
Partitioned Frequency Domain Adaptive Filter
algorithm (PBFDAF–CG). Results of the new
approach are presented in Section 4 and 5 followed
by conclusions.
2 PBFDAF
The PBFDAF was developed to deal efficiently with
such situations. The PBFDAF is a more efficient
implementation of Least Mean Square (LMS)
algorithm in the frequency-domain. It reduces the
computational burden and user-delay bounded. In
general, the PBFDAF is widely used due to be good
trade-off between speed, computational complexity
and overall latency. However, when working with
long impulse response, as the acoustic impulse
responses (AIR) used in MAEC, the convergence
properties provided by the algorithm may not be
enough. Besides, the multichannel adaptive filter is
structurally more difficult, in general, than the single
channel case (Benesty and Huang, 2003).
This technique makes a sequential partition of
the impulse response in the time-domain prior to a
frequency-domain implementation of the filtering
operation. This time segmentation allows setting up
individual coefficient updating strategies concerning
different sections of the adaptive canceller, thus
avoiding the need for disabling the adaptation in the
complete filter. The adaptive algorithm is based on
the frequency-domain adaptive filter (FDAF) for
every section of the filter (Shink, 1992).
The main idea of frequency-domain adaptive
filter is to frequency transform the input signal in
order to work with matrix multiplications instead of
dealing with slow convolutions. The frequency-
domain transform employs one or more DFTs and
can be seen as a pre-processing block that generates
decorrelated output signals.
In the more general FDAF case, the output of the
filter in the time domain (1) can be seen as a direct
frequency-domain translation of the block LMS
(BLMS) algorithm. In the PBFDAF case, the filter is
partitioned transversally in an equivalent structure.
Partitioning
p
w in Q segments ( K length) we
obtain
[] [ ]
()
1
11 0
Q
PK
p
pqK m
pqm
yn x n qK mw
−
+
===
=−−
∑∑∑
(3)
Where the total filter length
L , for each channel,
is a multiple of the length of each segment
LQK
, KL
. Thus, using the appropriate data
sectioning procedure, the
Q linear convolutions
(per channel) of the filter can be independently
carried out in the frequency-domain with a total
delay of
K samples instead of the QK samples
needed in standard FDAF implementations.
Figure 2 shows the block diagram of the
algorithm using the overlap-save method. In the
frequency domain with matrix notation, equation (3)
can be expressed as
⊗YXW.
(4)
Where
FX
represents a matrix of
dimensions
QP
× which contains the Fourier
transform of the
Q partitions and P channels of
the input signal matrix
X
.
ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics
18