Independent component analysis P19

Blind separation of convolutive mixtures considers the combined blind deconvolu-tion and instantaneous blind source separadeconvolu-tion problem.. This estimadeconvolu-tion task appears

19 Convolutive Mixtures and

Blind Deconvolution

This chapter deals with blind deconvolution and blind separation of convolutive mixtures

Blind deconvolution is a signal processing problem that is closely related to basic

independent component analysis (ICA) and blind source separation (BSS) In

com-munications and related areas, blind deconvolution is often called blind equalization.

In blind deconvolution, we have only one observed signal (output) and one source signal (input) The observed signal consists of an unknown source signal mixed with itself at different time delays The task is to estimate the source signal from the observed signal only, without knowing the convolving system, the time delays, and mixing coefficients

Blind separation of convolutive mixtures considers the combined blind

deconvolu-tion and instantaneous blind source separadeconvolu-tion problem This estimadeconvolu-tion task appears under many different names in the literature: ICA with convolutive mixtures, mul-tichannel blind deconvolution or identification, convolutive signal separation, and blind identification of multiple-input-multiple-output (MIMO) systems In blind separation of convolutive mixtures, there are several source (input) signals and sev-eral observed (output) signals just like in the instantaneous ICA problem However, the source signals have different time delays in each observed signal due to the finite propagation speed in the medium Each observed signal may also contain time-delayed versions of the same source due to multipath propagation caused typically

by reverberations from some obstacles Figure 23.3 in Chapter 23 shows an example

of multipath propagation in mobile communications

In the following, we first consider the simpler blind deconvolution problem, and after that separation of convolutive mixtures Many techniques for convolutive

mix-355

Independent Component Analysis Aapo Hyv¨arinen, Juha Karhunen, Erkki Oja

Copyright  2001 John Wiley & Sons, Inc ISBNs: 0-471-40540-X (Hardback); 0-471-22131-7 (Electronic)

tures have in fact been developed by extending methods designed originally for either the blind deconvolution or standard ICA/BSS problems In the appendix, certain basic concepts of discrete-time filters needed in this chapter are briefly reviewed

19.1.1 Problem definition

In blind deconvolution [170, 171, 174, 315], it is assumed that the observed discrete-time signalx(t)is generated from an unknown source signals(t)by the convolution model

x(t) =

1 X

k =1 a k

Thus, delayed versions of the source signal are mixed together This situation appears

in many practical applications, for example, in communications and geophysics

In blind deconvolution, both the source signals(t)and the convolution coefficients

a

kare unknown Observingx(t)only, we want to estimate the source signals(t) In other words, we want to find a deconvolution filter

y(t) =

1 X

k =1 h k

which provides a good estimate of the source signals(t)at each time instant This

is achieved by choosing the coefficientsh

k of the deconvolution filter suitably In practice the deconvolving finite impulse response (FIR) filter (see the Appendix for definition) in Eq (19.2) is assumed to be of sufficient but finite length Other structures are possible, but this one is the standard choice

To estimate the deconvolving filter, certain assumptions on the source signals(t)

must be made Usually it is assumed that the source signal valuess(t)at different timestare nongaussian, statistically independent and identically distributed (i.i.d.) The probability distribution of the source signals(t)may be known or unknown The indeterminacies remaining in the blind deconvolution problem are that the estimated source signal may have an arbitrary scaling (and sign) and time shift compared with the true one This situation is similar to the permutation and sign indeterminacy encountered in ICA; the two models are, in fact, intimately related as will be explained

in Section 19.1.4

Of course, the preceding ideal model usually does not exactly hold in practice There is often additive noise present, though we have omitted noise from the model (19.1) for simplicity The source signal sequence may not satisfy the i.i.d condition, and its distribution is often unknown, or we may only know that the source signal is subgaussian or supergaussian Hence blind deconvolution often is a difficult signal processing task that can be solved only approximately, in practice

BLIND DECONVOLUTION 357

If the linear time-invariant system (19.1) is minimum phase (see the Appendix), then the blind deconvolution problem can be solved in a straightforward way On the above assumptions, the deconvolving filter is simply a whitening filter that temporally

whitens the observed signal sequencefx(t)g[171, 174] However, in many appli-cations, for example, in telecommuniappli-cations, the system is typically nonminimum phase [174] and this simple solution cannot be used

We shall next discuss some popular approaches to blind deconvolution Blind deconvolution is frequently needed in communications applications where it is con-venient to use complex-valued data Therefore we present most methods for this general case The respective algorithms for real data are obtained as special cases Methods for estimating the ICA model with complex-valued data are discussed later

in Section 20.3

Bussgang methods [39, 171, 174, 315] include some of the earliest algorithms [152, 392] proposed for blind deconvolution, but they are still widely used In Bussgang methods, a noncausal FIR filter structure

y(t) =

L X

k =
L w

 k

of length2L+1is used Here

denotes the complex conjugate The weightsw

k (t)of the FIR filter depend on the timet, and they are adapted using the least-mean-square (LMS) type algorithm [171]

w

k

(t + 1) = w

k (t) +
x(t
k)e

 (t) k =
L : : L (19.4) where the error signal is defined by

e(t) = g(y(t))
y(t) (19.5)

In these equations,
is a positive learning parameter,y(t)is given by (19.3), andg(:)

is a suitably chosen nonlinearity It is applied separately to the real and imaginary components ofy(t) The algorithm is initialized by settingw

0 (0) = 1,w

k (0) = 0

k 6= 0

Assume that the filter length2L + 1is large enough and the learning algorithm has converged It can be shown that then the following condition holds for the output

y(t)of the FIR filter (19.3):

Efy(t)y(t
k)g Efy(t)g(y(t
k))g (19.6)

A stochastic process that satisfies the condition (19.6) is called a Bussgang process The nonlinearityg(t)can be chosen in several ways, leading to different Bussgang

type algorithms [39, 171] The Godard algorithm [152] is the best performing

Bussgang algorithm in the sense that it is robust and has the smallest mean-square

error after convergence; see [171] for details The Godard algorithm minimizes the nonconvex cost function

J p (t) =Efjy(t)j

p

p ] 2

wherepis a positive integer and

pis a positive real constant defined by the statistics

of the source signal:

p

=

Efjs(t)j 2p g

Efjs(t)j p g

(19.8)

The constant

pis chosen in such a way that the gradient of the cost functionJ

p (t)

is zero when perfect deconvolution is attained, that is, wheny(t)=s(t) The error signal (19.5) in the gradient algorithm (19.4) for minimizing the cost function (19.7) with respect to the weightw

k (t)has the form

e(t) = y(t)jy(t)j

p
2



p

jy(t)j p

In computinge(t), the expectation in (19.7) has been omitted for getting a simpler stochastic gradient type algorithm The respective nonlinearityg(y(t))is given by [171]

g(y(t)) = y(t) + y(t)jy(t)j

p
2



p

jy(t)j p

Among the family of Godard algorithms, the so-called constant modulus algorithm

(CMA) is widely used It is obtained by settingp = 2in the above formulas The cost function (19.7) is then related to the minimization of the kurtosis The CMA and more generally Godard algorithms perform appropriately for subgaussian sources only, but in communications applications the source signals are subgaussian.1 The CMA algorithm is the most successful blind equalization (deconvolution) algorithm used in communications due to its low complexity, good performance, and robustness [315]

Properties of the CMA cost function and algorithm have been studied thoroughly

in [224] The constant modulus property possessed by many types of communications signals has been exploited also in developing efficient algebraic blind equalization and source separation algorithms [441] A good general review of Bussgang type blind deconvolution methods is [39]

19.1.3 Cumulant-based methods

Another popular group of blind deconvolution methods consists of cumulant-based approaches [315, 170, 174, 171] They apply explicitly higher-order statistics of the observed signal x(t), while in the Bussgang methods higher-order statistics

1 The CMA algorithm can be applied to blind deconvolution of supergaussian sources by using a negative learning parameter in (19.4); see [11]

BLIND DECONVOLUTION 359

are involved into the estimation process implicitly via the nonlinear functiong(
) Cumulants have been defined and discussed briefly in Chapter 2

Shalvi and Weinstein [398] have derived necessary and sufficient conditions and

a set of cumulant-based criteria for blind deconvolution In particular, they intro-duced a stochastic gradient algorithm for maximizing a constrained kurtosis based criterion We shall next describe this algorithm briefly, because it is computationally simple, converges globally, and can be applied equally well to both subgaussian and supergaussian source signalss(t)

Assume that the source (input) signal s(t) is complex-valued and symmetric, satisfying the condition Efs(t)

2

g= 0 Assume that the length of the causal FIR deconvolution filter isM The outputz(t)of this filter at discrete timetcan then be expressed compactly as the inner product

z(t) = w

T

where theM-dimensional filter weight vectorw (t)and output vectory (t)at timet

are respectively defined by

y (t) = y(t)
y(t  1)
: : y(t  M + 1)]

T

(19.12)

w (t) = w(t)
w(t  1)
: :
w(t  M + 1)]

T

(19.13) Shalvi and Weinstein’s algorithm is then given by [398, 351]

u(t + 1) = u(t) + sign(

s )jz(t)j 2 z(t)]y

(t)

w (t + 1) = u(t + 1)= k u(t + 1) k (19.14) Here

sis the kurtosis ofs(t),k
kis the usual Euclidean norm, and the unnormalized filter weight vectoru(t)is defined quite similarly asw (t)in (19.13)

It is important to notice that Shalvi and Weinstein’s algorithm (19.14) requires whitening of the output signaly(t)for performing appropriately (assuming thats(t)

is white, too) For a single complex-valued signal sequence (time series)fy(t)g, the temporal whiteness condition is

Efy(t)y

(t  k)g = 

2 y

 tk

= (

 2

t = k

0
t 6= k

(19.15)

where the variance ofy(t)is often normalized to unity:

2

= 1 Temporal whitening can be achieved by spectral prewhitening in the Fourier domain, or by using time-domain techniques such as linear prediction [351] Linear prediction techniques have been discussed for example in the books [169, 171, 419]

Shalvi and Weinstein have presented a somewhat more complicated algorithm for the case Efs(t)

2

g 6= 0in [398] Furthermore, they showed that there exists a close relationship between their algorithm and the CMA algorithm discussed in the previous subsection; see also [351] Later, they derived fast converging but more involved super-exponential algorithms for blind deconvolution in [399] Shalvi and

Weinstein have reviewed their blind deconvolution methods in [170] Closely related algorithms were proposed earlier in [114, 457]

It is interesting to note that Shalvi and Weinstein’s algorithm (19.14) can be derived by maximizing the absolute value of the kurtosis of the filtered (deconvolved) signalz(t)under the constraint that the output signaly(t)is temporally white [398, 351] The temporal whiteness condition leads to the normalization constraint of the weight vectorw (t)in (19.14) The corresponding criterion for standard ICA is familiar already from Chapter 8, where gradient algorithms similar to (19.14) have been discussed Also Shalvi and Weinstein’s super-exponential algorithm [399] is very similar to the cumulant-based FastICA as introduced in Section 8.2.3 The connection between blind deconvolution and ICA is discussed in more detail in the next subsection

Instead of cumulants, one can resort to higher-order spectra or polyspectra [319,

318] They are defined as Fourier transforms of the cumulants quite similarly as the power spectrum is defined as a Fourier transform of the autocorrelation function (see Section 2.8.5) Polyspectra provide a basis for blind deconvolution and more generally identification of nonminimum-phase systems, because they preserve phase information of the observed signal However, blind deconvolution methods based

on higher-order spectra tend to be computationally more complex than Bussgang methods, and converge slowly [171] Therefore, we shall not discuss them here The interested reader can find more information on those methods in [170, 171, 315]

19.1.4 Blind deconvolution using linear ICA

In defining the blind deconvolution problem, the values of the original signals(t)

were assumed to be independent for differenttand nongaussian Therefore, the blind deconvolution problem is formally closely related to the standard ICA problem In fact, one can define a vector

~ s(t) = s(t)
s(t
1)
:::
s(t
n + 1)]

T

(19.16)

by collectingnlast values of the source signal, and similarly define

~

x (t) = x(t)
x(t
1)
:::
x(t
n + 1)]

T

(19.17) Then~

xand~

saren-dimensional vectors, and the convolution (19.1) can be expressed for a finite number of values of the summation indexkas

~

whereAis a matrix that contains the coefficientsa

k of the convolution filter as its rows, at different positions for each row This is the classic matrix representation

of a filter This representation is not exact near the top and bottom rows, but for a sufficiently largen, it is good enough in practice

From (19.18) we see that the blind deconvolution problem is actually (approxi-mately) a special case of ICA The components ofsare independent, and the mixing

is linear, so we get the standard linear ICA model

BLIND SEPARATION OF CONVOLUTIVE MIXTURES 361

In fact, the one-unit (deflationary) ICA algorithms in Chapter 8 can be directly used to perform blind deconvolution As defined above, the inputsx(t)should then consist of sample sequencesx(t)
x(t
1)
:::
x(t
n + 1)of the signalx(t)to be deconvolved Estimating just one “independent component”, we obtain the original deconvolved signal s(t) If several components are estimated, they correspond

to translated versions of the original signal, so it is enough to estimate just one component

19.2.1 The convolutive BSS problem

In several practical applications of ICA, some kind of convolution takes place simul-taneously with the linear mixing For example, in the classic cocktail-party problem,

or separation of speech signals recorded by a set of microphones, the speech signals

do not arrive in the microphones at the same time This is because the sound travels in the atmosphere with a very limited speed Moreover, the microphones usually record echos of the speakers’ voices caused by reverberations from the walls of the room

or other obstacles These two phenomena can be modeled in terms of convolutive mixtures Here we have not considered noise and other complications that often appear in practice; see Section 24.2 and [429, 430]

Blind source separation of convolutive mixtures is basically a combination of standard instantaneous linear blind source separation and blind deconvolution In the convolutive mixture model, each element of the mixing matrixAin the modelx(t)

=As(t)is a filter instead of a scalar Written out for each mixture, the data model

for convolutive mixtures is given by

x i (t) = n X

j=1 X

k a

ik j s j (t
k)
fori = 1
:::
n (19.19)

This is a FIR filter model, where each FIR filter (for fixed indicesiandj) is defined by the coefficientsa

ik j Usually these coefficients are assumed to be time-independent constants, and the number of terms over which the convolution indexkruns is finite Again, we observe only the mixturesx

i (t), and both the independent source signals

s

i

(t)and all the coefficientsa

ik j must be estimated

To invert the convolutive mixtures (19.19), a set of similar FIR filters is typically used:

y i (t) = n X

j=1 X

k w

ik j x j (t
k)
fori = 1
:::
n (19.20)

The output signals y

1 (t)
: : y (t) of the separating system are estimates of the source signalss

1

(t)
: : s (t)at discrete timet Thew

ik j give the coefficients of the FIR filters of the separating system The FIR filters used in separation can be

either causal or noncausal depending on the method The number of coefficient in each separating filter must sometimes be very large (hundreds or even thousands) for achieving sufficient inversion accuracy Instead of the feedforward FIR structure, feedback (IIR type) filters have sometimes been used for separating convolutive mixtures, an example is presented in Section 23.4 See [430] for a discussion of mutual advantages and drawbacks of these filter structures in convolutive BSS

At this point, it is useful to discuss relationships between the convolutive BSS problem and the standard ICA problem on a general level [430] Recall first than in standard linear ICA and BSS, the indeterminacies are the scaling and the order of the estimated independent components or sources (and their sign, which can be included

in scaling) With convolutive mixtures the indeterminacies are more severe: the order

of the estimated sourcesy

i (t)is still arbitrary, but scaling is replaced by (arbitrary) filtering In practice, many of the methods proposed for convolutive mixtures filter the estimated sourcesy

i (t)so that they are temporally uncorrelated (white) This follows from the strong independence condition that most of the blind separation methods introduced for convolutive mixtures try to realize as well as possible The temporal whitening effect causes some inevitable distortion if the original source signals themselves are not temporally white Sometimes it is possible to get rid of this by using a feedback filter structure; see [430]

Denote by

y (t) = y

1 (t)
y 2 (t)
: : y (t)]

T

(19.21) the vector of estimated source signals They are both temporally and spatially white if

Efy (t)y

H (t
k)g = 

tk

I = ( I
t = k

0
t 6= k

(19.22)

whereH denotes complex conjugate transpose (Hermitian operator) The standard spatial whitening condition Efy (t)y

H (t)g =I is obtained as a special case when

t = k The condition (19.22) is required to hold for all the lag valueskfor which the separating filters (19.20) are defined Douglas and Cichocki have introduced a simple adaptive algorithm for whitening convolutive mixtures in [120] Lambert and Nikias have given an efficient temporal whitening method based on FIR matrix algebra and Fourier transforms in [257]

Standard ICA makes use of spatial statistics of the mixtures to learn a spatial blind

separation system In general, higher-order spatial statistics are needed for achieving this goal However, if the source signals are temporally correlated, second-order spatiotemporal statistics are sufficient for blind separation under some conditions,

as shown in [424] and discussed in Chapter 18 In contrast, blind separation of

convolutive mixtures must utilize spatiotemporal statistics of the mixtures to learn a

spatiotemporal separation system.

Stationarity of the sources has a decisive role in separating convolutive mixtures, too If the sources have nonstationary variances, second-order spatiotemporal statis-tics are enough as briefly discussed in [359, 456]

BLIND SEPARATION OF CONVOLUTIVE MIXTURES 363

For convolutive mixtures, stationary sources require higher than second-order statistics, just as basic ICA, but the following simplification is possible [430] Spa-tiotemporal second-order statistics can be used to decorrelate the mixtures This step returns the problem to that of conventional ICA, which again requires higher-order spatial statistics Examples of such approaches are can be found in [78, 108, 156] This simplification is not very widely used, however

Alternatively, one can resort to higher-order spatiotemporal statistics from the beginning for sources that cannot be assumed nonstationary This approach has been adopted in many papers, and it will be discussed briefly later in this chapter

19.2.2 Reformulation as ordinary ICA

The simplest approach to blind separation of convolutive mixtures is to reformulate the problem using the standard linear ICA model This is possible because blind deconvolution can be formulated as a special case of ICA, as we saw in (19.18) Define now a vector~ sby concatenatingM time-delayed versions of every source signal:

~

s(t) = s

1

(t)
s

1 (t
1)
:::
s

1 (t
M + 1)
s

2 (t)
s 2 (t
1)
:::
s

2 (t
M + 1)

: : s (t)
s (t
1)
:::
s (t
M + 1)]

T

(19.23) and define similarly a vector

~

x(t) = x

1

(t)
x

1 (t
1)
:::
x

1 (t
M + 1)
x

2 (t)
x 2 (t
1)
:::
x

2 (t
M + 1)

: : x n (t)
x n (t
1)
:::
x

n (t
M + 1)]

T

(19.24) Using these definitions, the convolutive mixing model (19.19) can be written

~

x =

~

where ~

Ais a matrix containing the coefficientsa

ik j of the FIR filters in a suitable order Now one can estimate the convolutive BSS model by applying ordinary ICA methods to the standard linear ICA model (19.25)

Deflationary estimation is treated in [108, 401, 432] These methods are based on finding maxima of the absolute value of kurtosis, thus generalizing the kurtosis-based methods of Chapter 8 Other examples of approaches in which the convolutive BSS problem has been solved using conventional ICA can be found in [156, 292]

A problem with the formulation (19.25) is that when the original data vectorx

is expanded tox ~, its dimension grows very much The numberM of time delays that needs to be taken into account depends on the application, but it is often tens

or hundreds, and the dimension of model (19.25) grows with the same factor, to

nM This may lead to prohibitively high dimensions Therefore, depending on the application and the dimensionsnandM, this reformulation can solve the convolutive BSS problem satisfactorily, or not

In blind deconvolution, this is not such a big problem because we have just one signal to begin with, and we only need to estimate one independent component, which

is easier than estimating all of them In convolutive BSS, however, we often need

to estimate all the independent components, and their number isnMin the model (19.25) Thus the computations may be very burdensome, and the number of data points needed to estimate such a large number of parameters can be prohibitive in practical applications This is especially true if we want to estimate the separating system adaptively, trying to track changes in the mixing system Estimation should then be fast both in terms of computations and data collection time

Regrettably, these remarks hold largely for other approaches proposed for blind separation of convolutive mixtures, too A fundamental reason of the computational difficulties encountered with convolutive mixtures is the fact that the number of the unknown parameters in the model (19.19) is so large If the filters have lengthM, it is

M-fold compared with the respective instantaneous ICA model This basic problem cannot be avoided in any way

19.2.3 Natural gradient methods

In Chapter 9, the well-known Bell-Sejnowski and natural gradient algorithms were derived from the maximum likelihood principle This principle was shown to be quite closely related to the maximization of the output entropy, which is often called the information maximization (infomax) principle; see Chapter 9 These ICA estimation criteria and algorithms can be extended to convolutive mixtures in a straightforward way Early results and derivations of algorithms can be found in [13, 79, 121, 268,

363, 426, 427] An application to CDMA communication signals will be described later in Chapter 23

Amari, Cichocki, and Douglas presented an elegant and systematic approach for deriving natural gradient type algorithms for blind separation of convolutive mixtures and related tasks It is based on algebraic equivalences and their nice properties Their work has been summarized in [11], where rather general natural gradient learning rules have been given for complex-valued data both in the time domain and z -transform domain The derived natural gradient rules can be implemented in either batch, on-line, or block on-line forms [11] In the batch form, one can use a noncausal FIR filter structure, while the on-line algorithms require the filters to be causal

In the following, we represent an efficient natural gradient type algorithm [10, 13] described also in [430] for blind separation of convolutive mixtures It can be implemented on-line using a feedforward (FIR) filter structure in the time domain The algorithm is given for complex-valued data

The separating filters are represented as a sequence of coefficient matricesW

k (t)

at discrete timet and lag (delay) k The separated output with this notation and causal FIR filters is

y (t) =

L X

k =0 W k

Herex(t
k)isn-dimensional data vector containing the values of thenmixtures (19.19) at the time instant , and is the output vector whose components are