Signal Processing 82 (2002) 1127 – 1138
www.elsevier.com/locate/sigpro
Adaptive multi-channel least mean square and Newton
algorithms for blind channel identi&cation
Yiteng Arden Huanga; ∗ , Jacob Benestyb
a Bell
b Bell
Laboratories, Lucent Technologies, Room 2D-526, 600 Mountain Avenue, Murray Hill, NJ 07974, USA
Laboratories, Lucent Technologies, Room 2D-518, 600 Mountain Avenue, Murray Hill, NJ 07974, USA
Received 28 September 2001; accepted 28 March 2002
Abstract
The problem of identifying a single-input multiple-output FIR system without a training signal, the so-called blind system
identi&cation, is addressed and two multi-channel adaptive approaches, least mean square and Newton algorithms, are
proposed. In contrast to the existing batch blind channel identi&cation schemes, the proposed algorithms construct an error
signal based on the cross relations between di4erent channels in a novel, systematic way. The corresponding cost (error)
function is easy to manipulate and facilitates the use of adaptive &ltering methods for an e5cient blind channel identi&cation
scheme. It is theoretically shown and empirically demonstrated by numerical studies that the proposed algorithms converge
in the mean to the desired channel impulse responses for an identi&able system. ? 2002 Published by Elsevier Science B.V.
Keywords: Blind channel identi&cation; Adaptive &ltering; Least mean square; Newton’s method; Multi-channel system
1. Introduction
The desire for blind channel identi&cation and estimation technique arises from a variety of potential
applications in signal processing and communications,
e.g. dereverberation, separation of speech from multiple sources, time-delay estimation, speech enhancement, image deblurring, wireless communications, etc.
In all these applications, a priori knowledge of the
source signal is either inaccessible or very expensive
to acquire, making the blind method a necessity.
Blind channel identi&cation and equalization techniques have gained extensive attention since the innovative idea was &rst proposed by Sato [12]. Early
studies [4,16,2] of blind channel identi&cation and
∗
Corresponding author.
E-mail addresses: [email protected]
Huang), [email protected] (J. Benesty).
(Y.A.
equalization focused primarily on higher (than second) order statistics-based schemes. These schemes
su4er from slow convergence and local minima,
and therefore, are unsatisfactory in tracking a fast
time-varying system. In 1991, Tong et al. [15] demonstrated the possibility of using only second-order
statistics of multi-channel system outputs to solve
the channel identi&cation problem. Since then, many
second order statistics-based approaches have been
proposed, such as the subspace (SS) algorithm [11],
the cross relation (CR) algorithm [8], [17], the least
squares component normalization (LSCN) algorithm
[1], the linear prediction-based subspace (LP-SS) algorithm [13], and the two-step maximum likelihood
(TSML) algorithm [6]. These batch methods can
accurately estimate an identi&able multi-channel system (or equivalently an oversampled single-channel
system [8]) using a &nite number of samples when
the signal-to-noise ratio (SNR) is high. However,
0165-1684/02/$ - see front matter ? 2002 Published by Elsevier Science B.V.
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1128
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
while these algorithms are able to yield a good estimate of the channel impulse responses, they are in
general computationally intensive and are di5cult to
implement in an adaptive mode [14].
For blind channel identi&cation to be practically
useful in real-time applications, it is imperative that
the algorithm should be computationally simple and
can be adaptively implemented. One e4ort was made
to develop an adaptive algorithm using a neural network [3]. Another attempt was based on the least
squares smoothing (LSS) algorithm [18], which is recursive in order and can be implemented in part using
a lattice &lter. To the best of our knowledge, existing
adaptive blind channel identi&cation algorithms are
algebraically complicated in development and computationally demanding in operation. These characteristics not only obstruct research e4orts for performance
improvement but also make these algorithms unattractive in practical implementations.
In this paper, we approach the problem by formulating a new error function for the outputs of
a multi-channel system based on the CR between
di4erent channels. The proposed error signal then
allows use of many traditional and e5cient adaptive
&lters, in both the time and the frequency domains, in
blind channel identi&cation. For a clear presentation
and an easy performance analysis, we present in this
paper a multi-channel least mean square (MCLMS)
algorithm, which is a generalization of the adaptive eigenvalue decomposition algorithm [5], and a
multi-channel Newton (MCN) algorithm.
2. Problem formulation
The problem addressed in this paper is to determine the impulse responses of a single-input
multiple-output (SIMO) FIR system in a blind way,
i.e. only the observed system outputs are available
and used without assuming knowledge of the speci&c
input signal.
2.1. Channel model
For a multi-channel FIR system as presented in
Fig. 1, the ith observation xi (n) is the result of a linear convolution between the source signal s(n) and
Input
Channels
.
s(n)
Additive Observations
Noise
b 1 (n)
x 1 (n)
+
H1 (z)
b 2 (n)
x 2 (n)
+
H2 (z)
.
.
.
..
.
bM (n)
x M (n)
+
HM (z)
Fig. 1. Illustration of the relationships between the input s(n) and
the observations xi (n) in a single-input multi-channel FIR system.
the corresponding channel response hi , corrupted by
an additive noise bi (n):
xi (n) = hi ∗ s(n) + bi (n);
i = 1; 2; : : : ; M;
(1)
where the ∗ symbol is the linear convolution operator
and M is the number of channels. In a vector form,
the relationship of the input and the observation for
the ith channel is written as
xi (n) = Hi · s(n) + bi (n);
(2)
where
xi (n) = [xi (n) xi (n − 1) · · · xi (n − L + 1)]T ;

hi; 0
 0

Hi = 
 ..
 .
0
hi; 1 · · · hi; L−1 0
hi; 0 · · · hi; L−2 hi; L−1
..
.. ..
..
. .
.
.
· · · 0 hi; 0 hi; 1
···
0
···
..
.
0
..
.



;


· · · hi; L−1
s(n) = [s(n) s(n − 1) · · · s(n − L + 1)
· · · s(n − 2L + 2)]T ;
bi (n) = [bi (n) bi (n − 1) · · · bi (n − L + 1)]T ;
and (·)T denotes a vector=matrix transpose.
The additive noise components in di4erent channels are assumed to be uncorrelated with the source
signal even though they might be mutually dependent. The channel parameter matrix Hi is of dimension L×(2L−1) and is constructed from the channel’s
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
impulse response:
hi = [hi; 0
hi; 1
···
hi; L−1 ]T ;
(3)
where L is set to the length of the longest channel
impulse response by assumption.
A global system equation can be constructed by
concatenating the M single-channel outputs of (2) as
follows:
x(n) = H · s(n) + b(n);
(4)
where
x(n) = [x1T (n)
x2T (n)
···
T
xM
(n)]T ;
T T
H = [H1T H2T · · · HM
] ;
T
T
T
(n) :
b(n) = b1 (n) b2T (n) · · · bM
Therefore, mathematically speaking, the blind
multi-channel identi&cation problem is to estimate
the channel parameter matrix H from the observation
x(n) without using the source signal s(n).
2.2. Channel identi8ability
Before we develop approaches to the blind channel
identi&cation problem, it is worthwhile to discuss the
issues of identi8ability, i.e. if the channels are identi&able or whether the channel impulse responses can be
estimated. A multi-channel FIR system can be blindly
identi&ed mainly because of the channel diversity.
As an extreme counter-example, if all channels are
identical, the system reduces to a single-channel case
and therefore, becomes unidenti&able. In addition,
the source signal needs to have su5cient modes to
make the channels fully excited (as evidence to fully
supported channels). According to [7], two inductive
conditions are necessary and su5cient to ensure system identi&ability, which are shared by all second
order statistics-based blind channel identi&cation
methods:
(1) the polynomials formed from hi ; i = 1; 2; : : : ; M ,
are co-prime, i.e. the channel transfer functions
Hi (z) do not share any common zeros;
(2) the autocorrelation matrix Rss = E{s(n)sT (n)}
of the source signal is of full rank, where E{·}
denotes mathematical expectation.
1129
In the rest of this paper, these two conditions are
assumed to hold so that we are dealing with an identi&able multi-channel FIR system.
3. Adaptive MCLMS and Newton algorithms
3.1. Principle
When the input signal is unknown, the CR between
the sensor outputs can be exploited to estimate the
channel impulse responses. By following the fact:
xi ∗ h j = s ∗ h i ∗ h j = x j ∗ h i ;
i; j = 1; 2; : : : ; M; i = j;
(5)
in the absence of noise, we have the following relation
at time n:
xiT (n)hj = xjT (n)hi ;
i; j = 1; 2; : : : ; M; i = j:
(6)
Multiplying (6) by xi (n) and taking expectation
yields,
R x i x i hj = R x i x j hi ;
i; j = 1; 2; : : : ; M; i = j;
(7)
where Rxi xj = E{xi (n)xjT (n)}. Formula (7) speci&es
M (M − 1) distinct equations. If we sum up the M − 1
CR associated with one particular channel hj , we get
M
Rx i x i hj =
i=1; i=j
M
Rx i x j hi ;
j = 1; 2; : : : ; M:
(8)
i=1; i=j
Over all channels, we then have a total of M equations.
In a matrix form, this set of equations is written as:
Rh = 0;
(9)
where

Rxi xi −Rx2 x1 · · · −RxM x1
 i=1


 −Rx1 x2
Rxi xi · · · −RxM x2


i
=
2
R=

..
..
..
..

.
.
.
.


 −Rx x −Rx x · · ·
Rxi xi
1 M
2 M
i=M
and
h = [h1T
h2T
···
T T
hM
] :













1130
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
In the blind multi-channel identi&cation problem,
Eq. (9) has to be solved for the unknown vector h that
contains all the channel impulse responses. For an
identi&able system in which the two presumptive conditions of Section 2.2 are valid, the matrix R is of rank
ML − 1 and its null space is one-dimensional. Therefore, the channel estimate will be unique if multiplied
by a non-zero constant. In contrast to the CR method
[17] that uses the observed data directly, we construct
the system equation (9) based on the covariance matrices of channel outputs, although the explored subspaces in which the channel impulse response vector
lies are the same.
By examining (9), we see that h is an eigenvector of
R corresponding to a zero eigenvalue. Such a system
equation (9) can be solved for h by direct matrix inversion or eigenvalue decomposition, which lead to a
batch approach. Alternatively, since we are interested
in a more e5cient adaptive algorithm, we will de&ne
an error signal and determine the channel impulse responses by employing a least-squares minimization
strategy.
When observation noise is present, the right-hand
side of (9) is no longer zero and an error vector is
produced,
e = Rh:
(10)
This error can be used to de&ne a cost function
J = e2 = eT e:
(11)
We can then determine a vector ĥ as the solution by
minimizing the cost function (11) in the least-squares
sense:
ĥ = arg min J = arg min hT RT Rh:
h
h
(12)
In this case with the presence of observation noise,
the matrix R is positive de&nite rather than positive
semide8nite and the desired solution ĥ will be the
eigenvector of R corresponding to its smallest eigenvalue.
In order to avoid a trivial estimate with all zero
elements, a constraint has to be imposed on h.
Two constraints have been proposed. One is the
unit-norm constraint, i.e. h = 1. The other is the
component-normalization constraint [1], i.e. cT h = 1,
where c is a constant vector. As an example, if
we know that one element, say hi; k , of the vector
h is equal to which is not zero, then we may
properly specify c = [0; : : : ; 1=; : : : ; 0]T with 1= being the (iL + k)-th element of c. Even though the
component-normalization constraint may be more
robust to noise than the unit-norm constraint [1], a
proper location of the constrained component hi; k
and its value are di5cult to determine a priori in
practice. So a unit-norm constraint will be used in
this paper. Therefore, the blind multi-channel identi&cation is the minimization problem given by (12)
subject to h = 1.
3.2. Multi-channel LMS algorithm
As discussed in the foregoing section and given by
(9), for an identi&able multi-channel FIR system the
unit-norm constraint leads to a solution ĥ that is the
eigenvector of the correlation matrix R corresponding
to a zero eigenvalue in the absence of the observation noise. In practice where noise is always present,
the desired system vector ĥ, which consists of all the
channel impulse responses, is the eigenvector of R
corresponding to the smallest eigenvalue. In order to
estimate the channel impulse responses e5ciently, we
present here an adaptive LMS approach, which is a
generalization of the adaptive eigenvalue decomposition algorithm employed in [7] from a two-channel to
an M -channel (M ¿ 2) system.
3.2.1. Algorithm derivation
To begin, we use (6) to de&ne an error signal based
on the ith and jth observations at time n:
eij (n)
=
xiT (n)hj − xjT (n)hi ; i = j; i; j = 1; 2; : : : ; M;
0;
i = j; i; j = 1; 2; : : : ; M:
(13)
Here, we have (M − 1)M=2 distinct error signals
eij (n), which exclude the case eii (n) = 0 and count the
eij (n) = −eji (n) pair only once. Assuming that these
error signals are equally important, we now de&ne a
cost function as follows:
(n) =
M
−1
M
i=1 j=i+1
eij2 (n):
(14)
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
If the unit-norm constraint h = 1 is enforced at all
times, the error signal becomes:
ij (n)
 T
T

 xi (n)hj =h − xj (n)hi =h;
=
i = j; i; j = 1; 2; : : : ; M;


0;
i = j; i; j = 1; 2; : : : ; M;
=
eij (n)
h(n)
(15)
and the corresponding cost function is given by
J (n) =
M
−1
M
ij2 (n) =
i=1 j=i+1
(k = 1; 2; : : : ; M ) channel impulse response:
M −1 M
2
@(n) @[ i=1
j=i+1 eij (n)]
=
@hk
@hk
=
k−1
=
k−1
2ekj (n)(−xj (n))
j=k+1
M
2eik xi (n) +
i=1
M
M
2eik xi (n) +
i=1
=
2ejk (n)xj (n)
j=k+1
2eik (n)xi (n);
(20)
i=1
(n)
:
h2
(16)
Therefore, the desired solution for h is determined by
minimizing the mean value of the cost function J (n):
ĥ = arg min E{J (n)}:
(17)
h
Direct minimization is computationally intensive and
may be even intractable when the channel impulse responses are long and the number of channels is large.
Here, an LMS algorithm is proposed to solve this minimization problem e5ciently:
ĥ(n + 1) = ĥ(n) − ∇J (n)|h=h(n)
ˆ ;
(18)
where is a small positive step size and ∇ is a gradient
operator.
In order to determine the gradient in (18), we take
a derivative of J (n) with respect to h:
@J (n)
@ (n)
∇J (n) =
=
@h
@h h2
@ (n)
=
@h hT h
1
@(n)
=
− 2J (n)h ;
(19)
h2
@h
where the last step follows from the fact ekk (n)=0. We
may express this equation concisely in matrix form as
follows:
@(n)
= 2X(n)ek (n)
@hk
(21)
= 2X(n)[Ck (n) − Dk (n)]h;
where we have de&ned, for convenience,
X(n) = [x1 (n)
x2 (n)
···
ek (n) = [e1k (n) e2k (n) · · ·
 T
x1 (n)hk − xkT (n)h1
 xT (n)h − xT (n)h
k
2
 2
k
=
..


.
xM (n)]L×M ;
eMk (n)]T



;


T
(n)hk − xkT (n)hM
xM
= [Ck (n) − Dk (n)]h


0 · · · 0 x1T (n) 0 · · · 0
 0 · · · 0 xT (n) 0 · · · 0 


2


 ..
..
..
..
.. 

 . ··· .
.
.
·
·
·
.
Ck (n) = 



T
 0 · · · 0 xM (n) 0 · · · 0 




(k−1)L
(M −k)L
;
M ×ML
T
where
@(n)
=
@h
1131
@(n)
@h1
T @(n)
@h2
T
···
@(n)
@hM
T T
:
Let us now evaluate the partial derivative of (n)
only with respect to the coe5cients of the kth
= [0M ×(k−1)L X (n) 0M ×(M −k)L ]M ×ML ;

 T
xk (n) 0 · · · 0
 0 xT (n) · · · 0 


k

Dk (n) = 
:
..
.. 
 ..
..
.
 .
.
. 
0
0
· · · xkT (n)
M ×ML
1132
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
Continuing, we evaluate the two matrix products in
(21) individually as follows:
Finally, we substitute (26) into (18) and have the update equation
X(n)Ck (n)
ĥ(n + 1)
= [x1 (n)
x2 (n)
···
= ĥ(n) −
xM (n)]L×M
[0M ×(k−1)L XT (n) 0M ×(M −k)L ]M ×ML
M
R̃xi xi (n) 0L×(M −k)L ;
= 0L×(k−1)L
(22)
X(n)Dk (n)
x2 (n)xkT (n)
···
xM (n)xkT (n)]
= [R̃x1 xk (n) R̃x2 xk (n) · · · R̃xM xk (n)];
(23)
where
R̃xi xj (n) = xi (n)xjT (n);
i; j = 1; 2; : : : ; M:
Next, substituting (22) and (23) into (21) yields
@(n)
= 2 −R̃x1 xk (n) − R̃x2 xk (n)
@hk
···

R̃xi xi (n) · · · − R̃xM xk (n) h: (24)
i=k
Thereafter, we incorporate (24) into (19) and obtain
@(n)
= 2R̃(n)h;
@h
∇J (n) =
where
(25)
1
[2R̃(n)h − 2J (n)h];
h2

(28)
If the channel estimate is always normalized after each
update, then we have the simpli&ed algorithm
i=1
= [x1 (n)xkT (n)
2
[R̃(n)ĥ(n) − J (n)ĥ(n)]:
ĥ(n)2
(26)
R̃xi xi (n) −R̃x2 x1 (n) · · · −R̃xM x1 (n)

 i=1





 −R̃ (n)
R̃xi xi (n) · · · −R̃xM x2 (n) 
x1 x2




i=2

:
R̃(n) = 



..
..
.
..
..


.
.
.




 −R̃x x (n) −R̃x x (n) · · ·
R̃xi xi (n) 
1 M
2 M
i=M
(27)
ĥ(n + 1) =
ĥ(n) − 2[R̃(n)ĥ(n) − (n)ĥ(n)]
:
ĥ(n) − 2[R̃(n)ĥ(n) − (n)ĥ(n)]
(29)
The MCLMS adaptive algorithm for blind channel
identi&cation is summarized in Table 1.
3.2.2. Convergence analysis
Assuming that the independence assumption [5]
holds, it can be easily shown that the LMS algorithm
converges in the mean if the step size satis&es
1
0¡¡
;
(30)
max
where max is the largest eigenvalue of the matrix
E{R̃(n) − J (n)IML×ML }.
After convergence, taking the expectation of (28)
gives
R
ĥ(∞)
ĥ(∞)
= E{J (∞)}
;
ĥ(∞)
ĥ(∞)
(31)
which is the desired result: ĥ converges in the mean
to the eigenvector of R corresponding to the smallest
eigenvalue E{J (∞)}.
3.3. MCN algorithm
In the previous section, a MCLMS algorithm was
developed to blindly identify an SIMO FIR system
by minimizing the power of the error signal given by
(13). While the MCLMS algorithm has been shown
to converge in the mean to the desired channel impulse responses, one of the di5culties in the design
and implementation of the LMS adaptive &lter is the
selection of the step size . In selecting the step size
in an LMS algorithm, there is a tradeo4, as pointed
out in many studies, between the rate of convergence,
the amount of excess mean-square error, and the ability of the algorithm to track the system as its impulse
responses change.
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
1133
Table 1
The multi-channel LMS adaptive algorithm for the blind identi&cation of a SIMO FIR system
T
T
T
Parameters:
ĥ = [ĥ1 ĥ2 · · · ĥM ]T , adaptive &lter coe5cients;
, step size
Initialization:
ĥi (0) = [1 0 · · · 0]T ; i = 1; 2; : : : ; M
√
ĥ(0) = ĥ(0)= M (normalization)
Computation:
For n = 0; 1; : : : compute
(a) eij (n) =
(b) (n) =
xiT (n)hj − xjT (n)hi ; i = j; i; j = 1; 2; : : : ; M
0;
i = j; i; j = 1; 2; : : : ; M
M −1 M
i=1
2
j=i+1 eij (n);
(c) Construct the matrix R̃(n) given by (27);
(d) ĥ(n + 1) =
Aiming to achieve a good balance of the three
competing design objectives, we present here a MCN
algorithm (see [9] for the Newton method) with a
variable step size during adaptation:
ĥn+1 = ĥn − E −1 {∇2 J (n)}∇J (n)|h=hˆn ;
(33)
we obtain
∇2 J (n)
=
−
.
Taking mathematical expectation of (35) and invoking
the independence assumption [5] produces
E{∇2 J (n)} = 2R − 4hhT R − 4RhhT
− 2E{J (n)}[IML×ML − 4hhT ]:
(32)
where ∇2 J (n) is the Hessian matrix of J (n) with respect to h. Taking derivative of (26) with respect to h
and using the formula
T
@
@J (n)
+ J (n)IML×ML
[J (n)h] = h
@h
@h
= h[∇J (n)]T + J (n)IML×ML ;
ĥ(n) − 2[R̃(n)ĥ(n) − (n)ĥ(n)]
ĥ(n) − 2[R̃(n)ĥ(n) − (n)ĥ(n)]
(36)
In practice, R and E{J (n)} are not known such that
we have to estimate their values. Since J (n) decreases
as adaptation proceeds and is relative small particularly after convergence, we can neglect the term
E{J (n)} in (36) for simplicity and with appropriate
accuracy, as suggested by simulations. The matrix
R is estimated recursively in a conventional way as
follows:
M
x T xi
i
R̂(0) =
IML×ML ;
L
i=1
2h2 {R̃(n) − [h(∇J (n))T + J (n)IML×ML ]}
h4
4[R̃(n)h − J (n)h]hT
:
h4
(34)
With the unit-norm constraint h = 1, Eq. (34) can
be simpli&ed as follows:
∇2 J (n) = 2{R̃(n) − h[∇J (n)]T − J (n)IML×ML }
T
− 4[R̃(n)h − J (n)h]h :
(35)
R̂(n) = R̂(n − 1) + R̃(n)
for n ¿ 1;
(37)
where (0 ¡ ¡ 1) is an exponential forgetting
factor.
By using these approximations, we &nally get an
estimate W(n) for the mean Hessian matrix of J (n)
and hence deduce the MCN algorithm:
W(n) = R̂(n) − 2ĥ(n)ĥ(n)T R̂(n) − 2R̂(n)ĥ(n)ĥ(n)T ;
(38)
1134
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
Table 2
The multi-channel Newton algorithm for the blind identi&cation of a SIMO FIR system.
T
T
T
T
Parameters:
ĥ = ĥ1 ĥ2 · · · ĥM
, step size
, adaptive &lter coe5cients;
Initialization:
ĥi (0) = [1 0 · · · 0]T ; i = 1; 2; : : : ; M
√
ĥ(0) = ĥ(0)= M (normalization)
Computation:
For n = 0; 1; : : : compute
(a) eij (n) =
(b) (n) =
xiT (n)hj − xjT (n)hi ; i = j; i; j = 1; 2; : : : ; M
0;
i = j; i; j = 1; 2; : : : ; M
M −1 M
j=i+1
i=1
eij2 (n);
(c) Construct the matrix R̃(n) given by (27);
(d) R̂(n) =
 M
x T xi


i

IML×ML ; n = 0





i=1
R̂(n − 1) + R̃(n);
(f) W(n) = R̂(n) −
(g) ĥ(n + 1) =
ĥ(n + 1)
=
ĥ(n) − W−1 (n)[R̃(n)ĥ(n) − (n)ĥ(n)]
;
ĥ(n) − W−1 (n)[R̃(n)ĥ(n) − (n)ĥ(n)]
(39)
where is a new step size, close to but ¡ 1. The MCN
algorithm for blind channel identi&cation is summarized in Table 2.
4. Simulations
We have proposed two adaptive algorithms, the
MCLMS and the MCN algorithms, for the blind channel identi&cation problem and have shown that they
would converge in the mean to the desired channel impulse responses of an identi&able multi-channel system. In this section, we evaluate their performance via
Monte Carlo simulations and demonstrate how they
behave using short channel impulse responses that are
common in digital communication problems. For comparison, the CR algorithm is also studied.
L
2ĥ(n)ĥ(n)T R̂(n)
n¿1
− 2R̂(n)ĥ(n)ĥ(n)T ;
ĥ(n) − W−1 (n)[R̃(n)ĥ(n) − (n)ĥ(n)]
.
ĥ(n) − W−1 (n)[R̃(n)ĥ(n) − (n)ĥ(n)]
The normalized root mean square error (NRMSE)
in dB is used as a performance measure in this paper
and is given by,


N
1
1
U(i) 2  ; (40)
NRMSE , 20 log10 
h N
i=1
where N is the number of Monte Carlo runs, (·)(i)
denotes a value obtained for the ith run, and
hT ĥ
U = h − T ĥ
ĥ ĥ
is a projection error vector. By projecting h onto ĥ and
de&ning a projection error, we take only the misalignment of the channel estimate into account [10].
In all simulations, the source signal was an uncorrelated binary phase shift keying sequence and the
additive noise is i.i.d. zero mean Gaussian. The speci&ed SNR is de&ned as:
2 h2
SNR , 10 log10 s 2 ;
(41)
Mb
where s2 and b2 are the signal and noise powers,
respectively.
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
20
0
+
o
x
NRMSE (dB)
− 20
− 30
− 40
− 50
0
− 10
− 20
− 30
− 40
− 60
− 70
MCLMS
MCN
10
Cost Function J(n) (dB)
− 10
CR
MCLMS
MCN
1135
− 50
0
10
20
30
40
50
SNR (dB)
1]T ;
h2 = [1 − 2 cos( + ) 1]T ;
(42)
where is the absolute phase value of the zeros of the
&rst channel and speci&es the angular distance between the zeros of the two channels on the unit circle.
The use of such a second-order two-channel system
was &rst introduced in [6] and was also employed in
[18,1] This system allows us to investigate the robustness of a blind channel identi&cation algorithm to the
channel identi&ability, conceptually described by the
parameter .
In this simulation, N = 100 Monte Carlo runs were
performed. For the CR method, 50 samples of observations from each channel were used. For the proposed MCLMS and MCN algorithms, the step sizes
= 0:01 and = 0:95 were &xed, respectively. First, a
well-conditioned system is considered with = =10
and = . A comparison of the NRMSEs among the
CR, the MCLMS, and the MCN algorithms is presented in Fig. 2. In this case, the NRMSEs of all
studied algorithms decrease steadily as the SNR increases. The trajectories of the cost function J (n) for
one typical run of the MCLMS and MCN algorithms
at 20 and 50 dB SNRs are presented, respectively, in
100
150
Time n (sample)
200
250
MCLMS
MCN
10
Cost Function J(n) (dB)
h1 = [1 − 2 cos()
50
20
Fig. 2. Comparison of NRMSE vs. SNR among the CR, MCLMS,
and MCN algorithms for the well-conditioned two-channel system
( = =10; = ).
The &rst simulation is concerned with a two-channel
FIR system whose impulse responses in each channel
are second order and are given by:
0
(a)
0
− 10
− 20
− 30
− 40
− 50
(b)
0
50
100
150
Time n (sample)
200
250
Fig. 3. Comparison of convergence between the MCLMS and
MCN adaptive algorithms for the well-conditioned two-channel
system ( = =10; = ). Trajectories of the cost function J (n)
vs. time for one typical run of the MCLMS and MCN algorithms
at 20 dB (a), and 50 dB (b) SNRs.
Fig. 3 (a) and (b) to compare their convergence. As
clearly shown in these &gures, both the MCLMS and
the MCN algorithms can converge quickly to the desired channel impulse responses at these SNRs for
such a well-conditioned system. Second, we studied
an ill-conditioned system with = =10 and = =10.
The resulting NRMSEs of these algorithms are plotted
in Fig. 4. Remarkablely, the adaptive algorithms are
still able to identify the channels. The MCLMS algorithm is stable since no matrix inversion is performed.
However, it cannot be concluded that the proposed
MCLMS and MCN algorithms perform signi&cantly
better than the CR method particularly when the SNR
1136
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
20
0
+
o
x
NRMSE (dB)
− 20
− 30
− 40
− 50
0
− 10
− 20
− 30
− 40
− 60
− 70
MCLMS
MCN
10
Cost Function J(n) (dB)
− 10
CR
MCLMS
MCN
− 50
0
10
20
30
40
50
SNR (dB)
Fig. 4. Comparison of NRMSE vs. SNR among the CR, MCLMS,
and MCN algorithms for the ill-conditioned two-channel system
( = =10; = =10).
0
100
200
300
Time n (sample)
400
500
Fig. 6. Comparison of convergence between the MCLMS and
MCN adaptive algorithms for the ill-conditioned two-channel system ( = =10; = =10) at 50 dB SNR.
20
SNR=0dB
10
Cost Function J(n) (dB)
SNR=10dB
0
SNR=20dB
− 10
SNR=30dB
− 20
SNR=40dB
− 30
SNR=50dB
− 40
− 50
0
0.4
(a)
0.8
1.2
Time n (×104 sample)
1.6
2
20
SNR=0dB
Cost Function J(n) (dB)
10
− 10
SNR=20dB
− 20
SNR=30dB
− 30
SNR=40dB
− 40
SNR=50dB
− 50
(b)
SNR=10dB
0
0
100
200
300
Time n (sample)
400
500
Fig. 5. Trajectories of the cost function J (n) vs. time for one typical
run of the MCLMS (a) and MCN (b) algorithms at di4erent SNRs
for the ill-conditioned two-channel system ( = =10; = =10).
is ¿35 dB because the adaptive algorithms uses more
samples in this simulation. The trajectories of the cost
function J (n) for one typical run of the NRMSE and
MCN algorithms at di4erent SNRs are presented in
Fig. 5(a) and (b), respectively. A close snapshot of the
trajectories of these adaptive algorithms at 50 dB SNR
for comparison of their convergence is presented in
Fig. 6. We see that the closer the zeros of the two channels approach to each other, the slower the MCLMS
converges. In terms of convergence, the MCN algorithm is more robust to the system identi&ability and
it takes only about 100 iterations to converge at 50 dB
SNR. The convergence is dramatically accelerated by
the MCN algorithm.
In the second experiment, we consider a threechannel system whose impulse responses are longer
(L = 15) than those in the &rst simulation. An FIR
system with more channels is less likely for all of the
channels to share a common zero which would invalidate the system’s identi&ability. This system was &rst
used for performance evaluation in [1]. The coe5cients of the three impulse responses were randomly
selected whose zeros are shown in Fig. 7. The NRMSEs were obtained by averaging the results of N =200
Monte Carlo runs. The CR method used 120 samples
and the step sizes for the MCLMS and MCN algorithms were = 0:001 and = 0:95, respectively. The
results are shown in Figs. 8–10, using the same layout
as that for the ill-conditioned two-channel system. In
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
2.5
30
Channel 1
Channel 2
Channel 3
2.0
1137
SNR=0dB
20
Cost Function J(n) (dB)
Imaginary Part
1.5
1.0
0.5
0
− 0.5
− 1.0
− 1.5
0
SNR=20dB
− 10
SNR=30dB
− 20
SNR=40dB
− 30
SNR=50dB
−3
−2
−1
0
1
2
3
+
o
x
CR
MCLMS
MCN
Cost Function J(n) (dB)
0
− 20
1.5
20
SNR=0dB
10
SNR=10dB
0
SNR=20dB
− 10
SNR=30dB
− 20
SNR=40dB
− 30
− 40
− 30
SNR=50dB
0
(b)
− 40
− 50
0.5
1.0
Time n (×104 sample)
30
Fig. 7. Illustration of the positions of the channel zeros for the
three-channel system in a z-plane.
− 10
0
(a)
Real Part
NRMSE( dB)
SNR=10dB
− 40
− 2.0
− 2.5
10
200
400
600
Time n (sample)
800
1000
Fig. 9. Trajectories of the cost function J (n) vs. time for one
typical run of the MCLMS (a) and MCN (b) algorithms at di4erent
SNRs for the three-channel system.
0
10
20
30
40
50
SNR( dB)
Fig. 8. Comparison of NRMSE vs. SNR among the CR, MCLMS,
and MCN algorithms for the three-channel system.
this case, even though some zeros of the three channels are very close and a small step size was used, the
MCLMS converged faster than for the ill-conditioned
two-channel system. The MCN method performed
still better than the MCLMS algorithm.
As clearly demonstrated by these simulations, the
MCLMS and MCN algorithms are robust to the system
identi&ability and noise. They converge in the mean
to the desired channel impulse responses. Poor convergence rate of the proposed MCLMS algorithm for
ill-conditioned systems is mainly because only an instantaneous gradient estimate is used for each update
in the LMS algorithm. Such a critical drawback has
been overcome by the MCN algorithm which controls
the step size based on the Hessian of the cost function.
In this paper, we formulate the CR between channels
in a systematic way and the &nal concise cost function potentially facilitates the use of adaptive &ltering algorithms, such as the block LMS, the recursive
least squares (RLS), and the frequency-domain LMS
methods, in the future development of e5cient blind
channel identi&cation schemes.
5. Conclusions
The blind channel identi&cation and estimation
problem is studied in this paper. An error function
1138
Y.A. Huang, J. Benesty / Signal Processing 82 (2002) 1127 – 1138
30
MCLMS
MCN
Cost Function J(n) (dB)
20
10
0
− 10
− 20
− 30
− 40
0
200
400
600
Time n (sample)
800
1000
Fig. 10. Comparison of convergence between the MCLMS and
MCN adaptive algorithms for the three-channel system at 50 dB
SNR.
based on the cross relations between di4erent channels
is constructed in a systematic way for a multi-channel
FIR system. The resulting cost function is concise
and potentially facilitates the use of adaptive &ltering techniques in the future development of e5cient
adaptive blind channel identi&cation schemes. As an
example, an LMS approach was proposed and its convergence in the mean to the desired channel impulse
responses was theoretically shown. Furthermore, in
order to accelerate convergence of the LMS adaptive
algorithm, a Newton method was proposed. Simulation results justi&ed our analysis and the proposed
MCLMS and MCN algorithms performed well for an
identi&able system.
Acknowledgements
We would like to thank B.H. Juang and D.R. Morgan for their constructive comments and suggestions
that have improved the clarity of this paper.
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