Adaptive paraunitary filter banks for contrast-based ... - Semantic Scholar
ADAPTIVE PARAUNITARY FILTER BANKS FOR CONTRAST-BASED MULTICHANNEL BLIND DECONVOLUTION X . Sun and S.C. Douglas Department of Electrical Engineering Southern Methodist University Dallas, Texas 75275 USA
ABSTRACT In this paper, we present novel algorithms for multichannel blind deconvolution under output whitening constraints. The algorithms are inspired by recently-developed procedures for gradient adaptive paraunitary filter banks. Several algorithms are developed, including one algorithm that successfully deconvolves mixtures of arbitrary non-zero kurtosis source signals. We provide detailed local stability analyses of the proposed methods to verify their capabilities. Simulations show that the methods successfully deconvolve spatio-temporal mixtures of statistically-independent source signals. 1.
,
INTRODUCTION
In multichannel blind deconvolution, an m-dimensional vector sequence s(k) containing statistically-independent samples s,(k), 1 5 i 5 m is mixed by an ( n x m), m 5 n unknown multichannel linear system with impulse response A,, -cm < i < CO, to produce the measured sequence
x(k) =
2
A,s(k - 2 ) .
(1)
1=-W
The goal is to process x(k) by an (m x n) multichannel adaptive linear system to produce estimates of the source sequences {s,(k)} in the adaptive system’s outputs without precise knowledge of {sz(k)} or of {At}. Multichannel blind deconvolution is particularly useful in wireless communications employing smart antennas [I]. Although single-channel blind deconvolution is a wellstudied topic 121, there exist relatively few successful multichannel blind deconvolution algorithms. Two simple multichannel blind deconvolution algorithms are described in [3] and [4], respectively. The former algorithm is an extension of the constant modulus algorithm (CMA) equalizer, and the latter algorithm is an extension of the natural gradient blind signal separation (BSS) algorithm in [5]. Both of these algorithms rely on knowledge of the probability density functions (p.d.f.’s) of each s,(k), and they fail to extract these signals if the chosen density models do not accurately match the p.d.f.’s of each s,(k). More recently, contrast-based criteria for BSS have been extended to the multichannel blind deconvolution task [6, 71. These criteria identify separated and deconvolved sources regardless of their p.d.f.’s. OThis material is based in part upon work supported by the Texas Advanced Technology Program under Grant No. 003613-
0031- 1999.
0-7803-7041-4/01/$10.00 02001 IEEE
/
_ - . - - _ _ _ _ _ _ _ - . 1
Fig. 1: Contrast-based multichannel blind deconvolution. In this paper, we derive novel multichannel blind deconvolution algorithms based on the contrast functions in [6, 71. Our methods are inspired by recent work on gradient adaptive paraunitary filter banks [8]. Unlike the approaches in [3, 41, ours do not require precise knowledge about the underlying source p.d.f.’s; rather, they only require knowledge of the number of positive-kurtosis and negative-kurtosis sources within the mixture. The algorithms are simple, requiring between four and seven multiply/adds per adaptive system coefficient at each time instant. We provide detailed local stability analyses of the proposed methods to verify their extraction capabilities. Simulations of the proposed method show their abilities to extract spatio-temporal mixtures of statistically-independent source signals. 2. STRUCTURES AND CRITERIA Our proposed methods share the common system structure shown in Fig. 1. The (m x n) prewhitening filter, denoted by P(z), calculates the m-dimensional prewhitened signal
v(k) =
5 P a x ( k- 2 ) .
(2)
i=O
where Pi, 0 5 i 5 M is this system’s impulse response. The coefficients of this system are designed so that v(k) is approximately spatially- and temporally-uncorrelated, a. e.
E{v(k)vT(k- j ) }
IS(j),
-A4
<j < M.
(3)
Any one of a number of procedures can be used to calculate Pi, such as the multichannel Levinson algorithm or other adaptive approaches [9].The (m x m) separation filter W(z, IC) calculates the source signal estimates as L
(4) l=O
where Wl(k), 0 5 1 5 L are the adaptive coefficients of the separation system. In this paper, we develop adaptive algorithms for adjusting {W, ( I C ) } to obtain separated and deconvolved source estimates in y ( k ) .
2753
Authorized licensed use limited to: SOUTHERN METHODIST UNIV. Downloaded on November 1, 2008 at 16:10 from IEEE Xplore. Restrictions apply.
sume for the current discussion that all { s i ( k ) ] have the same kurtosis sign, such that ~ [ s i ( k )> ] 0 or ~ [ s , ( k ) ]< 0 for all 1 5 i 5 m. Then, an equivalent formulation t o (5)-(6) is m
maximize
T ( { w ~ ) )=
~{ly;(k)14)
(7)
i=l
L
Wl(k)W&;(k) x Ib(i),
such that
(8)
I=O
where /3 satisfies &[s;(k)]
> 0 for all 1 5 i 5 m.
3. ALGORITHM DERIVATION Multichannel linear systems whose impulse responses obey (8) are called paraunitary filter banks. In (81, the following differential update is proposed to adapt ?he coefficients of a paraunitary filter bank to maximize ~ ( { W I } ) : j = 1 1=0
end end for i = 1 to % do for j = 1 to E do
where GI = ay({W,})/aW, and V' denotes discrete-time convolution of matrix sequences. Eqn. (9) maintains (8) for a doubly-infinite multichannel IIR system. Modifications are required, however, to make the resulting system both causal and numerically-stable [ll]. Applying (9) to (7)-(8), discretizing the resulting updates, and assuming slow adaptation of the system's coefficients results in the first proposed stochastic gradient algorithm for the multichannel blind deconvolution task:
q=o
end
m
L
j=1
I=O
end for i = 1 to m do &(k) = P i € i ( k ) , &(k) = P;y;(k) for j = 1 to 5ii do "ijl(k 1) = W i j l ( k ) k ( k - L)Wj(k - L - 1 ) - (IC - L ) U i j (k - 1 ) end 1) to m do for j = (a+ wiji (k 1) = wijl (IC) r , ( k ) ~(k j - 2~ - 1 ) -y,(k - 2L)Uij(k - L - 1) end end
+
Wi(k
+
u(k) =
(10)
- d)
(11)
+
+
where Do is a diagonal matrix of step sizes {/3;}. This algorithm is simple, requiring 5m2(L 1) 3m multiply/adds per adaptive filter coefficient. Unfortunately, this algorithm fails to maintain the paraunitary constraint in (8) over time due to numerical effects. Similar difficulties have appeared in algorithms for minor subspace analysis and contrastbased BSS [12, 131, and they can often be addressed by modifying the updates to allow numerically-stable behavior. To this end, we propose the following modified algorithms for all p; > 0 and B; < 0, respectively:
+ +
Wi(k + 1) = Wl(k) + D p [f(y(k - L ) ) v T ( k- L - 1 )
+
-
and W l ( k
+ 1)
J({w,)) =
~~[yi(k)ll
E { y ( k ) y T ( k- 1 ) )
M
I6(1), --CO
y ( k - L)uT(k - Z)]
= Wl(k)
+ Do [C(k)vT(k- 2L - 1 )
- y ( k - 2L)uT(k - L - Z)]
(13) (14)
L
(5)
z(k) =
i=l
such that
C wLq(k)f(Y q=o
m
maximize
+ Dp [ + ( k ) v T ( k - L - 1)
L
In [6, 71 a contrast function for the multichannel blind deconvolution task is proposed. A contrast function is a cost function that depends on the source signal estimates whose extrema over the separation system parameters extract all of the source signals [lo]. This formulation assumes that (i) the sources are spatially- and temporally-uncorrelated, such that E { s ( k ) s T ( k- 1 ) ) M D6(Z), --oo < 1 < 03 for an arbitrary nonsingular diagonal scaling matrix D with ,diagonal entries d ; ; , and (ii) each source is spatially- fourthorder uncorrelated, such that E{s;(k)sj(k)sl(k)s,(k)} = ~ [ ~ i ( k ) ] 6 i j l pd i i d l l 6 i j 6 l p d i i d j j [6il6jp bipbjl]. Under these assumptions, the following procedure solves the multichannel blind deconvolution task:
+
= Wi(k)
- y ( k - L)uT(k- Z)]
+
vi
+ 1)
< 1 < 00x6)
W L I ( W ( k - 9)
(15)
W l ( k ) Z ( k- 1).
(16)
q=o
L
where J({Wl)) is the contrast function and ~ [ g ; ( k ) ]= E{lyi(k)I4} - 3E2{1yi(k)12} is the kurtosis of yi(k). As-
C(k) = 1 =o
2754
Authorized licensed use limited to: SOUTHERN METHODIST UNIV. Downloaded on November 1, 2008 at 16:10 from IEEE Xplore. Restrictions apply.
[
Table 2: Local stability analysis results for the multichannel blind deconvolution algorithms. 1) Hilt, i < j Hiil, # 0 hiio
Algorithm
+ +
+
where Al are ( m x m ) matrices with IlAlll$ 0 and
tri[F]
=
{
ifizj otherwise.
for an ( m x m) matrix F with,entries { f i j } . Assuming slow adaptation, these algorithms can be combined and simplified to produce the algorithm listed in Table 1, where choosing E = 0 and E = m yields updates identical to (17) and (18), respectively, as lpil -+ 0. These algorithms have the same computational complexities as (13) and (14). In addition, when E corresponds to the number of negativekurtosis sources in s(lc),the combined algorithm in Table 1 can potentially separate arbitrary source mixtures, as will be shown. 4. STABILITY ANALYSES We now provide stability analyses of (lo), (13), (14), (17), and (18). These analyses determine the constraints on the step size parameters { p i } to guarantee stability of the algorithms about separating and deconvolving solutions. The procedure used is the ordinary differential equation (ODE) method, in which the dynamics of the linearized averaged ODE of the coefficient updates is elucidated. For a similar analysis of orthonormal-update BSS algorithms, see [13]. Our analyses use a simplified notation whereby time indices are suppressed, such that y(k p ) = yp. Our analyses also ignore truncation of the prewhitening and deconvolution filters. Without loss of generality, we describe the evolutionary behaviors of the algorithms near a separating solution
+
CL = Wi * P i *Ai = IS(Z)+Al,
where the H;jl, Hi;[, and hi;o depend on p;, pj, and the statistics of sE(IF)and s j ( k ) . Shown in Table 2 are the expressions for these quantities for the five proposed algorithms; detailed derivations for each case are omitted for brevity. Because (23)-(25) are linear ODES, local stability is guaranteed if Hijl, Hiil, and hiio have negative eigenvalues, producing constraints on the values of {pi} and { ~ i } . From these results, a number of conclusions can be drawn: 1. Because h;io = 0 for the analysis of (lo), this algorithm fails to maintain a paraunitary filter impulse response in the multichannel blind deconvolution task. 2 . Eqns. (13) and (17) are locally stable for positive-kurtosis source mixtures ( ~ >i 0) if pi > 0 for all i. Similarly, Eqns. (14) and (18) are locally stable for negative-kurtosis source mixtures ( ~
ABSTRACT In this paper, we present novel algorithms for multichannel blind deconvolution under output whitening constraints. The algorithms are inspired by recently-developed procedures for gradient adaptive paraunitary filter banks. Several algorithms are developed, including one algorithm that successfully deconvolves mixtures of arbitrary non-zero kurtosis source signals. We provide detailed local stability analyses of the proposed methods to verify their capabilities. Simulations show that the methods successfully deconvolve spatio-temporal mixtures of statistically-independent source signals. 1.
,
INTRODUCTION
In multichannel blind deconvolution, an m-dimensional vector sequence s(k) containing statistically-independent samples s,(k), 1 5 i 5 m is mixed by an ( n x m), m 5 n unknown multichannel linear system with impulse response A,, -cm < i < CO, to produce the measured sequence
x(k) =
2
A,s(k - 2 ) .
(1)
1=-W
The goal is to process x(k) by an (m x n) multichannel adaptive linear system to produce estimates of the source sequences {s,(k)} in the adaptive system’s outputs without precise knowledge of {sz(k)} or of {At}. Multichannel blind deconvolution is particularly useful in wireless communications employing smart antennas [I]. Although single-channel blind deconvolution is a wellstudied topic 121, there exist relatively few successful multichannel blind deconvolution algorithms. Two simple multichannel blind deconvolution algorithms are described in [3] and [4], respectively. The former algorithm is an extension of the constant modulus algorithm (CMA) equalizer, and the latter algorithm is an extension of the natural gradient blind signal separation (BSS) algorithm in [5]. Both of these algorithms rely on knowledge of the probability density functions (p.d.f.’s) of each s,(k), and they fail to extract these signals if the chosen density models do not accurately match the p.d.f.’s of each s,(k). More recently, contrast-based criteria for BSS have been extended to the multichannel blind deconvolution task [6, 71. These criteria identify separated and deconvolved sources regardless of their p.d.f.’s. OThis material is based in part upon work supported by the Texas Advanced Technology Program under Grant No. 003613-
0031- 1999.
0-7803-7041-4/01/$10.00 02001 IEEE
/
_ - . - - _ _ _ _ _ _ _ - . 1
Fig. 1: Contrast-based multichannel blind deconvolution. In this paper, we derive novel multichannel blind deconvolution algorithms based on the contrast functions in [6, 71. Our methods are inspired by recent work on gradient adaptive paraunitary filter banks [8]. Unlike the approaches in [3, 41, ours do not require precise knowledge about the underlying source p.d.f.’s; rather, they only require knowledge of the number of positive-kurtosis and negative-kurtosis sources within the mixture. The algorithms are simple, requiring between four and seven multiply/adds per adaptive system coefficient at each time instant. We provide detailed local stability analyses of the proposed methods to verify their extraction capabilities. Simulations of the proposed method show their abilities to extract spatio-temporal mixtures of statistically-independent source signals. 2. STRUCTURES AND CRITERIA Our proposed methods share the common system structure shown in Fig. 1. The (m x n) prewhitening filter, denoted by P(z), calculates the m-dimensional prewhitened signal
v(k) =
5 P a x ( k- 2 ) .
(2)
i=O
where Pi, 0 5 i 5 M is this system’s impulse response. The coefficients of this system are designed so that v(k) is approximately spatially- and temporally-uncorrelated, a. e.
E{v(k)vT(k- j ) }
IS(j),
-A4
<j < M.
(3)
Any one of a number of procedures can be used to calculate Pi, such as the multichannel Levinson algorithm or other adaptive approaches [9].The (m x m) separation filter W(z, IC) calculates the source signal estimates as L
(4) l=O
where Wl(k), 0 5 1 5 L are the adaptive coefficients of the separation system. In this paper, we develop adaptive algorithms for adjusting {W, ( I C ) } to obtain separated and deconvolved source estimates in y ( k ) .
2753
Authorized licensed use limited to: SOUTHERN METHODIST UNIV. Downloaded on November 1, 2008 at 16:10 from IEEE Xplore. Restrictions apply.
sume for the current discussion that all { s i ( k ) ] have the same kurtosis sign, such that ~ [ s i ( k )> ] 0 or ~ [ s , ( k ) ]< 0 for all 1 5 i 5 m. Then, an equivalent formulation t o (5)-(6) is m
maximize
T ( { w ~ ) )=
~{ly;(k)14)
(7)
i=l
L
Wl(k)W&;(k) x Ib(i),
such that
(8)
I=O
where /3 satisfies &[s;(k)]
> 0 for all 1 5 i 5 m.
3. ALGORITHM DERIVATION Multichannel linear systems whose impulse responses obey (8) are called paraunitary filter banks. In (81, the following differential update is proposed to adapt ?he coefficients of a paraunitary filter bank to maximize ~ ( { W I } ) : j = 1 1=0
end end for i = 1 to % do for j = 1 to E do
where GI = ay({W,})/aW, and V' denotes discrete-time convolution of matrix sequences. Eqn. (9) maintains (8) for a doubly-infinite multichannel IIR system. Modifications are required, however, to make the resulting system both causal and numerically-stable [ll]. Applying (9) to (7)-(8), discretizing the resulting updates, and assuming slow adaptation of the system's coefficients results in the first proposed stochastic gradient algorithm for the multichannel blind deconvolution task:
q=o
end
m
L
j=1
I=O
end for i = 1 to m do &(k) = P i € i ( k ) , &(k) = P;y;(k) for j = 1 to 5ii do "ijl(k 1) = W i j l ( k ) k ( k - L)Wj(k - L - 1 ) - (IC - L ) U i j (k - 1 ) end 1) to m do for j = (a+ wiji (k 1) = wijl (IC) r , ( k ) ~(k j - 2~ - 1 ) -y,(k - 2L)Uij(k - L - 1) end end
+
Wi(k
+
u(k) =
(10)
- d)
(11)
+
+
where Do is a diagonal matrix of step sizes {/3;}. This algorithm is simple, requiring 5m2(L 1) 3m multiply/adds per adaptive filter coefficient. Unfortunately, this algorithm fails to maintain the paraunitary constraint in (8) over time due to numerical effects. Similar difficulties have appeared in algorithms for minor subspace analysis and contrastbased BSS [12, 131, and they can often be addressed by modifying the updates to allow numerically-stable behavior. To this end, we propose the following modified algorithms for all p; > 0 and B; < 0, respectively:
+ +
Wi(k + 1) = Wl(k) + D p [f(y(k - L ) ) v T ( k- L - 1 )
+
-
and W l ( k
+ 1)
J({w,)) =
~~[yi(k)ll
E { y ( k ) y T ( k- 1 ) )
M
I6(1), --CO
y ( k - L)uT(k - Z)]
= Wl(k)
+ Do [C(k)vT(k- 2L - 1 )
- y ( k - 2L)uT(k - L - Z)]
(13) (14)
L
(5)
z(k) =
i=l
such that
C wLq(k)f(Y q=o
m
maximize
+ Dp [ + ( k ) v T ( k - L - 1)
L
In [6, 71 a contrast function for the multichannel blind deconvolution task is proposed. A contrast function is a cost function that depends on the source signal estimates whose extrema over the separation system parameters extract all of the source signals [lo]. This formulation assumes that (i) the sources are spatially- and temporally-uncorrelated, such that E { s ( k ) s T ( k- 1 ) ) M D6(Z), --oo < 1 < 03 for an arbitrary nonsingular diagonal scaling matrix D with ,diagonal entries d ; ; , and (ii) each source is spatially- fourthorder uncorrelated, such that E{s;(k)sj(k)sl(k)s,(k)} = ~ [ ~ i ( k ) ] 6 i j l pd i i d l l 6 i j 6 l p d i i d j j [6il6jp bipbjl]. Under these assumptions, the following procedure solves the multichannel blind deconvolution task:
+
= Wi(k)
- y ( k - L)uT(k- Z)]
+
vi
+ 1)
< 1 < 00x6)
W L I ( W ( k - 9)
(15)
W l ( k ) Z ( k- 1).
(16)
q=o
L
where J({Wl)) is the contrast function and ~ [ g ; ( k ) ]= E{lyi(k)I4} - 3E2{1yi(k)12} is the kurtosis of yi(k). As-
C(k) = 1 =o
2754
Authorized licensed use limited to: SOUTHERN METHODIST UNIV. Downloaded on November 1, 2008 at 16:10 from IEEE Xplore. Restrictions apply.
[
Table 2: Local stability analysis results for the multichannel blind deconvolution algorithms. 1) Hilt, i < j Hiil, # 0 hiio
Algorithm
+ +
+
where Al are ( m x m ) matrices with IlAlll$ 0 and
tri[F]
=
{
ifizj otherwise.
for an ( m x m) matrix F with,entries { f i j } . Assuming slow adaptation, these algorithms can be combined and simplified to produce the algorithm listed in Table 1, where choosing E = 0 and E = m yields updates identical to (17) and (18), respectively, as lpil -+ 0. These algorithms have the same computational complexities as (13) and (14). In addition, when E corresponds to the number of negativekurtosis sources in s(lc),the combined algorithm in Table 1 can potentially separate arbitrary source mixtures, as will be shown. 4. STABILITY ANALYSES We now provide stability analyses of (lo), (13), (14), (17), and (18). These analyses determine the constraints on the step size parameters { p i } to guarantee stability of the algorithms about separating and deconvolving solutions. The procedure used is the ordinary differential equation (ODE) method, in which the dynamics of the linearized averaged ODE of the coefficient updates is elucidated. For a similar analysis of orthonormal-update BSS algorithms, see [13]. Our analyses use a simplified notation whereby time indices are suppressed, such that y(k p ) = yp. Our analyses also ignore truncation of the prewhitening and deconvolution filters. Without loss of generality, we describe the evolutionary behaviors of the algorithms near a separating solution
+
CL = Wi * P i *Ai = IS(Z)+Al,
where the H;jl, Hi;[, and hi;o depend on p;, pj, and the statistics of sE(IF)and s j ( k ) . Shown in Table 2 are the expressions for these quantities for the five proposed algorithms; detailed derivations for each case are omitted for brevity. Because (23)-(25) are linear ODES, local stability is guaranteed if Hijl, Hiil, and hiio have negative eigenvalues, producing constraints on the values of {pi} and { ~ i } . From these results, a number of conclusions can be drawn: 1. Because h;io = 0 for the analysis of (lo), this algorithm fails to maintain a paraunitary filter impulse response in the multichannel blind deconvolution task. 2 . Eqns. (13) and (17) are locally stable for positive-kurtosis source mixtures ( ~ >i 0) if pi > 0 for all i. Similarly, Eqns. (14) and (18) are locally stable for negative-kurtosis source mixtures ( ~
