Oversampled Filter Banks as Error Correcting Codes: Theory and ...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 12, DECEMBER 2005

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Oversampled Filter Banks as Error Correcting Codes: Theory and Impulse Noise Correction Fabrice Labeau, Member, IEEE, Jui-Chiu Chiang, Michel Kieffer, Pierre Duhamel, Fellow, IEEE, Luc Vandendorpe, Senior Member, IEEE, and Benoît Macq, Senior Member, IEEE

Abstract—Oversampled filter banks (OFBs) provide an overcomplete representation of their input signal. This paper describes how OFBs can be considered as error-correcting codes acting on real or complex sequences, very much like classical binary convolutional codes act on binary sequences. The structured redundancy introduced by OFBs in subband signals can be used to increase robustness to noise. In this paper, we define the notions of code subspace, syndrome, and parity-check polynomial matrix for OFBs. Furthermore, we derive generic expressions for projection-based decoding, suitable for the case when a simple second-order model completely characterizes the noise incurred by subband signals. We also develop a nonlinear hypotheses-test based decoding algorithm for the case when the noise in subbands is constituted by a Gaussian background noise and impulsive errors (a model that adequately describes the action of both quantization noise and transmission errors). Simulation results show that the algorithm effectively removes the effect of impulsive errors occurring with a probability of 10 3 . Index Terms—Error correction coding, impulse noise, multirate filter banks, oversampling.

I. INTRODUCTION

M

ULTIRATE systems based on filter banks [1] often use maximally decimated filter banks. In this paper, we investigate oversampled filter banks (OFBs), i.e., banks composed different filters, but where each filter output is subsamof pled by a factor . Advances in their theory [2], [3] have been motivated by new applications requiring their specific properties. For instance, subband adaptive filtering is an area in which oversampled filter banks can be used [4]–[6], due to their higher attenuation in the stop-band [7], leading to lower levels of aliasing. Related areas of application are echo cancellation in subbands [8] or blind source separation in subbands. In all those cases, the main goal is to reduce the aliasing distortion thanks to oversampling, while still exploiting the computationally attractive subband setting. Manuscript received January 17, 2004. This work was supported in part by the RNRT VIP project, FQRNT, and NSERC. Part of this work was presented at the 2002 International Symposium on Wireless Personal Multimedia Communications, the 2002 Asilomar Conference on Signals, Systems and Computers, and ICASSP’2003. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Karim Drouiche. F. Labeau is with McGill University, Montreal, PQ H3A 2A7, Canada. J.-C. Chiang, M. Kieffer, and P. Duhamel are with the Laboratoire des Signaux et Systèmes, CNRS, Supélec, France. L. Vandendorpe and B. Macq are with Université catholique de Louvain, Belgium. Digital Object Identifier 10.1109/TSP.2005.859314

By construction, oversampled filter banks provide subband signals containing more information than is strictly necessary to represent the input signal. This kind of signal representation is called overcomplete expansion, in the sense that the subband signals form a redundant representation of the input signal. Since OFBs operate linearly on their input signals, it seems reasonable to compare them to linear binary error-correcting codes, but acting on real or complex input and output spaces. This paper expands on this parallelism with error-correcting codes. In the first part, we show the existence of a parity-check matrix for any given perfect reconstruction (PR) OFB, and give a spectral interpretation of the redundancy. We also give general expressions for some particular synthesis stages of importance. In the second part, we outline the noise-correction capabilities of oversampled filter banks in two ways. First, generic geometric properties of the codes created by oversampled filter banks are expressed, leading to projection-based error-correction schemes. Then, we develop a noise detection and correction system in the special case when the channel is a mixture of impulsive and Gaussian noises. This model is known to represent adequately a mixture of quantization noise and transmission errors [9]. Some papers [10], [11] discuss the issue of error-correcting codes for real numbers. The parallelism between these codes and filter bank implementations was sketched in [10]. Recently, a paper [12] has given the first geometric approaches to perfect reconstruction, and we will pursue here, based on another perspective. Note that this paper does not intend to provide simulations in practical situations. The match of the algorithms explained here and practical cases such as image coding robust to transmission errors lead to many questions, which will be addressed at other places. This paper is organized as follows. Section II reviews the basic notations and concepts about oversampled filter banks. A parametrization of all possible PR synthesis filter banks for a given analysis bank is also provided. Section III outlines how OFBs can be considered as error-correcting codes, by defining the notions of code subspace, parity-check filter, syndrome, and decoding by projection onto the code. Section IV studies the problems related to computing the parity-check matrix for a given OFB. Section V first defines the channel of interest and provides a nonlinear decoder for an OFB. Finally, Section VI presents some simulation results.

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Fig. 1.


Generic setting for error correction in oversampled filter banks. Fig. 2. Smith decomposition of the analysis stage of an oversampled filter bank.

II. OVERSAMPLED FILTER BANKS A. General System Overview and Definitions can be In the time domain, the subband samples expressed as decimated convolutions of the filters impulse with the input signal as responses

synthesis bank type-II polyphase matrix [1] be such that (with maybe an additional delay and scaling), so that is a left-inverse of . Namely, a parameterization of all left-inverses of a given polyphase matrix is easily found to be rankrow-permutation matrix [13], [14] as follows: Let be a where the

such that Fig. 1 illustrates the analysis and synthesis filter banks in the time-domain, as well as the noise injection model used throughout this paper. This paper mainly focuses on the polyphase matrix , defined by of the analysis filter bank . In the case of oversampled filter banks, matrix is tall, i.e., it has rows. In order to enable perfect reconstruction, it must have full column normal almost everywhere [2]. In this case, rank, i.e., rank admits a Smith–McMillan decomposition [1], [13] of the form (1) where and are unimodular1 matrices of sizes and , respectively, and is diagonal of size . This factorization clearly shows the various steps of the analysis of a signal by an oversampled filter bank : First operations are carried out on the input ( and ) and resulting sequences enter a multiple-input then the in parallel with inmultiple-output (MIMO) filter puts constituted of all-zero sequences. This process is illustrated in Fig. 2. B. Design Freedom: Choice of Synthesis Filters As opposed to the classical case of critically sampled banks, a given oversampled analysis filter bank enables perfect reconstruction in conjunction with several different reconstruction filter banks (actually an infinite number). As long as the analhas full normal rank, the subband ysis polyphase matrix signals contain enough information about the input signal to enable perfect reconstruction [2]. There is, however, an infinite number of possible PR synthesis filter banks [14], exactly like, e.g., a tall matrix has no unique left inverse. PR implies that the 1Unimodular matrices inverse [1], [13]

are

polynomial

matrices

with

polynomial

matrix

has full normal rank. By straightforward manipulation, it is clear that

is a left inverse of , where is an identity, and is an arbitrary matrix that is bounded on is finite for all ). the unit circle (i.e., Another possible view of the problem of choosing a synthesis for a given analysis bank consists in choosing bank a projection operator along a given subspace (see the following section). III. PARALLELISM WITH ERROR-CORRECTING CODES , and by the vector of signal samples, one can write the time-domain expression of all the is subband coefficients at the same time instant [when finite impulse response (FIR)] Denoting by

the vector

(2) of size where the th row of the rectangular matrix is composed by the time-reversed impulse response of the filter. is the maximum length of a filter in the bank. Altercan be segmented as , where natively, , and each submatrix is given by

We want to show how OFBs can have a subspace-based interpretation as error-correcting codes. The difficulty lies in the fact that the inputs and outputs are constituted of (vectors of) discrete-time sequences. In the polyphase domain, the relationship

LABEAU et al.: OVERSAMPLED FILTER BANKS AS ERROR CORRECTING CODES

between the blocked input with -transform and the with -transform is given by analysis bank output (3) It is clear that we will consider equivalently the set of rational as the space of interest, containing the -transvectors in forms of the signals at hand. Note that (3) is more general than (2), as it is not limited to the FIR case. A. Code Subspace and Parity-Check Matrix A mapping such as the one illustrated in Fig. 2 cannot span being equal to the entire output space. The normal rank of , it is clear that only a subspace of the output space can , which we will call the code be spanned by the columns of generated by (3) belongs to the code subspace. Any vector subspace, or is a code vector. The code subspace is spanned by , or equivalently by the first columns of the columns of matrix defined in (1). Now we can show that there exists a polyphase parity-check which corresponds to the generator filter of size . filter Proposition 1: Given a polyphase filter corresponding to an oversampled analysis filter bank with perfect FIR filter such reconstruction, there exists a that . Proof: See Appendix A. plays the important role of The polynomial matrix span the set of sequences parity-check matrix. The rows of orthogonal to the code sequences generated by the columns of . The syndrome of any vector sequence with -transform is computed by filtering it with the parity-check filter (4)

B. Projections and Syndromes Based on the parity-check matrix and syndrome, one can easily imagine that a projection-based decoding strategy can be used, especially when the noise is supposed to be white or known by its second-order statistics. Another example of decoding with a different noise model is devised in Section V. be a code vector sequence. If a sequence has Let , been corrupted by noise according to then its syndrome will not be zero, but . The decoding principle consists in projecting the onto the code. noisy sequence An orthogonal projection onto the code subspace can be written as

(5) where is composed of the first columns of uniappearing in (1). It is rather easy modular matrix

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to check that the above projection is idempotent (i.e., ) and that the projection is orthogfor any ). onal (i.e., One can point out that an orthogonal projection onto the code is already implemented by a minimum norm perfect reconstruction synthesis stage [2], [12], which in this case amounts to choosing the polyphase synthesis matrix as the parapseudoinverse of the analysis polyphase matrix: . In fact, any projector on the code can be written as the product of the analysis with a perfect reconstruction synthesis polyphase matrix . A generic expression for is given, for infilter stance, in [12]. The above minimum Euclidean distance (i.e., orthogonal) projection onto the code is the maximum-likelihood reconstruction strategy only when the noise affecting subband samples is white. On the other hand, when the noise is colored, one can imagine another decoding strategy, in which the projection onto the code subspace would be made along a noise subspace. By -dimensional subspace noise subspace, we mean the spanned by the most energetic eigendirections of the noise. We now give an alternative general expression for a projection onto the -dimensional code subspace along another subspace of dimension . First we state a general property : of projectors in in onto a Proposition 2: A polynomial projector with rank- subspace with Smith form of dimension has the property that

is a constant diagonal matrix. Proof: See Appendix B. Let us now state the general form of a projector on along some subspace . Proposition 3: For a PR OFB with polyphase matrix , a projection onto the code subspace generated by the along a subspace of dimension can columns of be carried out by the projection operator

where the columns of the polynomial matrix span polynomial matrix the subspace and the rows of the span , the subspace orthogonal to . This is possible . as long as Proof: See Appendix C. Based on the above expression, it is easy to design a reconstruction process in the subband domain that will apply a projection onto the code along the principal eigendirections of the noise affecting the subband signals. Notice that using Propositions 2 and 3 together, one can see that the unimodular matrices appearing in the Smith decompo. sition of the projector span the two subspaces and

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IV. COMPUTATION OF PARITY-CHECK MATRICES A. Generic Solution In order to derive the parity-check matrix for a given , one has to compute a matrix polyphase matrix whose rows span the orthogonal complement of the code subspace. as a submatrix of As mentioned above, one can obtain a matrix that triangularizes or reduces it to Hermite form. Theoretically, this process can be carried out by succes. This procedure sufsive elementary row operations on fers however from the same numerical stability problems as a Gaussian elimination process for real-valued matrices. One then has to resort to other, numerically more stable, . In the rest of this section, we will methods to compute restrict our attention to the FIR case. We use a modified version of the method outlined by Henrion et al. [15]. It is based on a column-permuted Sylvester matrix that is reduced to triangular form through a numerifor cally stable decomposition. The resulting factors can then be remapped into two polynomial matrices. Namely, for a , we use the degree- polyphase matrix Sylvester matrix

sampling. One can then easily check that the following relationship holds: (6) . where Let us also assume that the filter bank used in the oversampled case has perfect reconstruction for critical decimation, i.e., . Then, premultiplying (6) by , one gets that (7) Since the right-hand side of the above equation has dimensions , it clearly cannot have a rank higher than . There exists thus a matrix which can premultiply the above equation and yield an all-zero matrix. An example of such a matrix is

..

.

..

.

.. .

Thus, a possible choice for the parity-check matrix of is . V. IMPULSE NOISE CANCELLATION

.. .

..

.

..

.

..

.

which has 1 rows and 1 columns. For a degree- unimodular matrix , one can form matrix , so that the polynomial matrix triangularization equation can be rewritten as

where for the upper triangular polynomial matrix . Note that the value of in the definition of is . A column permutation matrix is applied to so that all coefficients pertaining to the same entry of are grouped together [15]. The permuted Sylvester matrix is then reduced to upper triangular form by decomposition, and the resulting upper triangular matrix can easily be mapped back to the polynomial domain and yield the actual polynomial triangular matrix . B. Integer Oversampling of a PR Filter Bank The special case of integer oversampling enables a very easy computation of the parity-check matrix. Let us thus consider integer oversampling, i.e., for some positive integer . Let us denote by the polyphase matrix corresponding to the oversampled filter bank, and by the polyphase matrix of the same filter bank when used at critical

Most previous works using OFBs were intending to obtain better rate-distortion tradeoffs, e.g., by alternate projection decoding algorithms. The first part of this paper, based on projections, can also be used in this context. This second part intends to show that the parallelism with channel coders is much stronger, and can be used to correct large errors introduced at random, e.g., modeling transmission errors. Thus, we study an application of the error-correcting capability of OFBs, with a nonlinear decoder based on error detection by syndrome computation. The setting we consider is a model of noise where the subband signals are corrupted by two noise sources: a Gaussian background noise and a Bernoulli Gaussian impulsive noise. This noise model has been shown [9] to accommodate both quantization noise (background noise) as well as transmission errors (impulsive noise). Note that several results already have been obtained for correcting errors in an erasure channel setting [16], [17]. The erasure channel is somewhat different, since an external element (such as the internet channel) provides the information on which samples (or subbands) are in error. The remaining task is thus to reconstruct the missing samples. In our framework such localization information is not available, and the first task of the decoder will be to estimate which sample in which subband is in error. In the case of our Gaussian plus Bernouilli Gaussian noise model, inspection of the syndrome can provide enough information to detect the location of the impulsive errors. As the syndrome is a vector sequence resulting from a vector filtering, it is not obvious to detect which coefficient in which subband was actually altered by impulsive noise just by looking at the value of the syndrome. In order to precisely determine the


location (i.e., time and subband index ) and amplitude of an impulse error, techniques based on hypotheses testing on the syndrome will be developed in the following. Note that several methods were already proposed for correcting transmission errors, based on real-valued Bose–Chaudhuri–Hocquenheim (BCH) codes [9], [18], after the work by Redinbo [19]. Most of these methods were relying on adaptations of the Peterson–Gorensteijn–Zierler decoding algorithm, modified in order to take the background noise into account. Obviously, this class of algorithms is not relevant anymore: strong spectral properties that are imposed by real-valued BCH codes are no longer present in the samples obtained at the output of an OFB.

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be illustrated below that the method meets most practical situations. B. Testing Whether Impulses are Present In order to decide whether impulse errors are present at a given time instant, a hypothesis test has been established. This test, based on the norm of the syndrome, is an extension of that presented in [21] to a more general context. Suppose that impulse errors occur at the same time in some subbands. These errors have an impact in each polyphase , where component of the syndrome between time and is the order of the filters in the parity-check matrix

A. Channel Model Here, all subbands are assumed to be available but some of them may be corrupted by impulse errors. This situation can be modeled as a communication channel corrupted by the sum of Gaussian plus Bernoulli Gaussian noises. Under these assumpand , the input and output tions, the relation between signals of a memoryless channel corrupted by impulse noise and Gaussian noise can be written as

(12) Thus, in order to gather the influence of channel impulses at time in a vector, one has to consider the syndrome (11) over a 1. However, contime-domain observation window of size versely, this syndrome window depends on a number of subband samples, which may also contain some additional errors. This is modeled as follows:

(8) (13) The Gaussian noise has zero mean and variance , while the impulse noise is modeled as Bernoulli Gaussian [20]

where

(9) where stands for a Bernoulli process, an i.i.d. sequence of zeros and ones with Prob , and represents a Gaussian noise with zero mean and variance , such that . The probability density function (pdf) of the channel noise can be expressed as

and

(10) with denoting a Gaussian density having mean and variance . Expressed in the domain, the syndrome (4) becomes

(11) where and are the polyphase representations of impulse and Gaussian noises, respectively. Thus, our intent is to correct the impulse errors, despite the presence of the background noise. The strategy we have chosen for detecting and correcting impulse errors in oversampled filter banks involves the following steps. First, a test is built to assess whether impulse errors are present at a given time instant and optionally to estimate their number. Then one has to determine in which subband they occurred and to evaluate their amplitude before correcting them. In what follows, all quantities (time, subband location, amplitude) will be expressed in the subband domain. Obviously, this method will be efficient only if it is very unlikely that many (say, more than two) errors can be observed at the same time in the syndrome. This states an upper bound on , the probability of occurrence of the transmission errors. It will

.. . which is a tion matrix. In

..

.

..

.

..

.

..

.

..

.

..

.

.. .

(14)

vector convolu-

contains the impulse errors that occur in the time intervals [ 1] and [ 1 ] which also affect the previously defined syndrome. The total number of impulse errors which . affect the syndrome (13) is thus Two hypotheses can now be formulated. 1) : There is no impulse error at time , but there may be impulse errors at time instants belonging to (impulse errors occurring at these time instants affect the syndrome). The is a priori probability for

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2)


: There are impulse errors at the time , and maybe also at time instants belonging to . The a priori probability for is

where is defined in Section V-A [see (9) and (10)]. and , the norm of the syndrome To choose between (13) is employed. Appendix D shows a derivation of the conconditioned on different values ditional probabilities of of and . The likelihood ratio for the test is defined as

whether

can then be compared to a threshold or is true

to decide

(15) is the cost associated with the decision given that where is true. If has a unique solution , the hypothesis testing can be made on

It is then possible to derive the probability of detection and of this test. Extension to multiple decision tests false alarm to determine the number of impulse errors is presented in [22]. This extension, being of combinatorial nature, has a practical usefulness only for reasonable number of “simultaneous” errors. Practically, we have seen that this procedure was efficient up to , which encompasses most practical situations. C. Characteristics of Impulse Errors If the test in Section V-B determines that impulse errors are present at time , then, for each impulse error, two parameters have to be estimated: 1) : the subband where it occurs; 2) : its amplitude. For the sake of brevity, only the determination of the characteristics of a single impulse error will be presented. Assume thus that a single impulse error with amplitude occurs in subband at time . Assume also that there is no other impulse error afsamples fecting the syndrome (13). The observation of of the syndrome in the subband domain starting at time

will provide information about the impulse error as

.. .

(16)

for . In (16), corresponds to the th row and th column of the matrix defined by (14) and are the time domain components of . The pdf of the noise in the th subband

is easily proved Gaussian with pdf

(17) , where can be expressed using s defined in (12), and denotes the the coefficients of the available a priori information, including the time at which the impulse error has been detected. In each subband, the syndrome (16) can be written in vector form (18) , . with The estimation of and will require the evaluation of the marginal a posteriori pdfs of and from their joint pdf, built using all available information. Estimates and for and correspond to the argument of the maximum of these pdfs. Detailed derivations can be found in Appendix E. Remark 1: Once the subband location of the error has been estimated, its amplitude may be obtained using (34); see Appendix E. An alternative approach is to use the estimate of the subband to solve (18) in a least squares sense to obtain . When the estimate is correct, this performs better than maximizing (34), which does not take into account any specific knowledge about . Note that the effective operation of the algorithm only requires a subset of the above computations at each time instant: first the test (15) to detect whether impulse errors are present is carried out. It only requires computing a few scalar products to and to compare it to a threshold; then, get the value of only if the test result is positive should the maximization of (33) in Appendix E and the solution of (18) take place. D. Impulse Correction Scheme Fig. 3 summarizes the impulse error detection and correction algorithm. At each time instant in the subband domain, a syndrome is computed. If the test on the norm of the syndrome described in Section V-B concludes that there are errors, an estimation of the characteristics of these errors is performed (Section V-C) before correction.


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,

Fig. 3. Error correction scheme.

E. Improving the Impulse Error Localization Technique The decision threshold in the hypotheses test (15) realizes a compromise between a high possibility of detection ( ) and a low probability of false alarm ( ). In practice, false alarms are unavoidable and the experience has shown that for an impulse actually occurring at time , the hypotheses test not only detects the impulse at time but sometimes also concludes that there are impulses at time instants around . A solution would be to increase the decision threshold , but this would imply too many undetected impulse errors. This is why we have chosen to tolerate false alarms and to use the algorithm described below to detect false alarms. The main idea is to recompute the norm of the syndrome after correction of the impulse error. It is likely that the reestimated syndrome norm will be the smallest after a correction for the actual subband, time, and amplitude. Assume that an impulse error occurs at time and that the hypotheses test detects impulse errors at time , . Let

be the vector containing all syndromes that have been evaluated larger than the decision threshold . For each , evaluate and using (33) and (34). Carry out the impulse error correction and evaluate the new syndrome vector

after correction of an impulse estimated at time . The error location estimate is chosen as the one that miniafter correction. mizes the norm of the syndrome VI. EXAMPLE AND SIMULATION RESULTS Here, a two-times oversampled filter bank has been built from a critically sampled time-domain aliasing cancellation (TDAC) filter bank. It will be shown that the paraunitary property is preserved. Besides, a polynomial parity check matrix can be computed without using the Smith–McMillan decomposition. The critically sampled TDAC filter bank [23] can be defined by a transform matrix

, where

Then, can be rewritten as , where and are matrices. Hence, the critically sampled filter bank analysis paraunitary matrix is expressed as , where is a counter identity matrix. If the sampling rate is increased by an integer factor ( ), an oversampled filter bank is obtained and its is [6] analysis polyphase matrix (19) If is paraunitary, we obtain

, then, from (19),

(20) thus is also paraunitary. In Section IV-B, it has been shown that for an OFB with integer oversampling factor, a parity-check polynomial matrix can be defined without using any Smith–McMillan decomposition . by times oversampled filter bank with In our example, an subbands and a decimation factor has been built is from a critically sampled TDAC filter bank. The order of , and an observation window of size four has thus been chosen. The filter bank input is a correlated noise sequence of unit variance. All subbands have been corrupted with Gaussian and impulse noise with impulse-to-Gaussian noise ratio INR dB and , and the impulse error localization and correction algorithm has been applied. and For the test defined in Section V-B, . We computed that ; thus, to compute , only the cases ( , ), ( , ), and ( , ) have been considered. On the other hand, . , , and are taken As a result, only the cases . The obtained theinto account for computing oretical and simulated receiver operating characteristics (ROC) curves have been represented in Fig. 4. There is a very good match between the two curves. Table I presents the signal-to-noise ratio (SNR) after impulse noise cancellation for different thresholds . These numbers have to be compared to the values of the SNR without impulse dB (the SNR in the noise, which is our target value: SNR case of Gaussian channel only) and to the initial SNR, before dB, which contains Gaussian as well as correction, SNR Bernoulli Gaussian noise. If, after correction, the SNR is close to 30 dB, we shall have completed the task of correcting impulse errors. Globally, the SNR are very close for various tunings of the threshold, and very close to 30 dB as well. The best performance . This corresponds to , is obtained when meaning that some impulse errors with low amplitudes have not been detected. Missing the low amplitude impulse errors that

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to find the parity-check matrix is more than of the filter necessary: one simply has to find a polynomial matrix that below the row. zeroes out all entries of Remark 2: Putting in column Hermite form [13] is already more than enough. Actually, going through a complete Smith form would also lead to a parity-check matrix with a higher degree than its minimal value. This can easily be impleby series of elementary row-opmented by premultiplying eration matrices, which directly leads to the result.

Fig. 4. Theoretical and simulated ROC curves for test (15).

PERFORMANCES

OF IMPULSE

TABLE I NOISE CANCELLATION

VALUES OF

FOR

APPENDIX B PROOF OF PROPOSITION 2

DIFFERENT

could not be distinguished from Gaussian noise does not impair the SNR. This robustness can be attributed to the existence of the improved impulse localization technique, which removes most of the incorrect estimations of the impulse errors due to false alarms. Globally, it is seen that the SNR after correction is only 0.05 dB beyond the original signal.

The basic definition of a projector [13] is that it has to be . Plugging the Smith form in idempotent: , this expression, premultiplying by the left-inverse of , one gets and postmultiplying by the right-inverse of . Since and are that portions of unimodular matrices, these two inverses are guaris determined by the anteed to exist. Since the rank of has full normal rank, so that dimension of , we know that . Based on the fact that both and are polynomial matrices, it results that must be unimodular and diagonal, which means that it has to be a constant diagonal matrix. APPENDIX C PROOF OF PROPOSITION 3

VII. CONCLUSION In this paper, we have formalized the error correcting capabilities of oversampled filter banks. We have shown how the structured redundancy introduced by OFBs creates a code subspace, and how projection onto this subspace can be used to cancel noise affecting subband signals. Moreover, we have developed a nonlinear decoding system in the special case of a noise modeled as a mixture of Gaussian background noise and Bernoulli Gaussian impulsive noise. Simulation results demonstrate the performance of the proposed error detection and correction method. Further work involves the application of this framework to image coding systems and the study of the impact of the oversampling ratio on the error-correcting performance of the proposed scheme.

, let us consider a vector To prove that Ker . This means that can be expressed in a basis of as . Furtherimplies that , more, the definition of . Thus, the projection of is

Notice that the construction of based on is easily carried out by using the result of Proposition 1. , let us consider a vector . To prove that can be expressed in a basis of This means that as . Thus, the projection of is

APPENDIX A PROOF OF PROPOSITION 1 Since the polyphase matrix corresponds to a perfect rein (1) has full normal construction system, the matrix rank . This Smith–McMillan form can be rewritten as . Since is unimodular, its inverse exists and is polynomial (i.e., FIR). In this case, one can rewrite and partition

as two matrices

,

and are of size and , where respectively. Evaluating the partitioned equation completes the proof. Actually, going through the Smith form decomposition

APPENDIX D DERIVATION OF THE CONDITIONAL PROBABILITY OF THE SYNDROME MAGNITUDE The aim of this section is to evaluate the pdf of the norm of the syndrome vector


observed between time and , under the hypotheses that errors occur at time and errors occur at time instants different . from but also affect Let

be the vector of time instants at which these impulse errors occur, if and if . The vectors containing the amplitude and subband location of each impulse error may be written as

The syndrome (13) then reduces to

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Now, since is Hermitian, it is diagonalizable and there exists a unitary matrix and a diagonal matrix such that

Hence

where . Since is normalized Gaussian, is also normalized Gaussian. As a result, is a linear combination of squared Gaussian variables , where are the entries of and are the diagonal entries of

(21) where

is a matrix of size obtained by considering only the columns , . of indexed by At this point, a hypothesis test for determining whether impulse errors are present at time could be built using the pdf of the syndrome with and without impulses at time . This would, however, be mathematically quite intricate (and computationally cumbersome). This is why it has been chosen to build the hypothesis test on the norm of the syndrome. Two hypotheses and . have then to be considered, namely, A.

(22)

, There Are Impulse Errors at Time

This section aims at finding the pdf of the norm of the syndrome when impulse errors are present at time ( ). In (21), the Gaussian and impulse noises do not have the same variance. It is thus convenient to multiply onal matrix

where

by a diag-

to normalize it; this makes computations easier. Then (21) can be written as

where

The pdf of each term of the sum is a central chi-square with one degree of freedom and variance . Therefore

The pdf of the norm of the syndrome in presence of impulse errors at time is then obtained by averaging (22) for all possible , , and subband and time locations for the impulse errors as shown in (23) at the bottom of the page, where is the a priori probability of having impulses at time and impulse errors at other time instants, at time instants and subbands . This probability can be computed from the noise model. B.

, There Is No Impulse Error at Time

In this case, the same computations as before can be per, but . formed. The main difference is that now One thus obtains (24) shown at the top of the next page.

(23)

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(24)

APPENDIX E COMPUTATION OF THE MARGINAL A POSTERIORI PDF OF

Now, after inserting (29) into (28) and combining (25), (26), and (28), one gets

AND

Using Bayes’ rule, the joint a posteriori pdf of written as

and can be (30)

(25)

The denominator of (25) is a positive constant that does not depend on and . The first term of its numerator is such that

with

and , where is independent with and . Now, (30) has to be marginalized in order to get the marginal a posteriori pdfs for and . The a priori pdfs for and are

(31) (26) as impulse error and background noise are independent and amplitude and subband location also. Using the chain rule, the second term of the numerator of (25) can be written as

and (32) The a posteriori pdfs for and are obtained by marginalizing (30) after introduction of (31) and (32). After some simple calculations, one obtains

(33) (27) and As gets

provide no information about

, one

(34) Therefore, (27) becomes

where depend on

and

,

are constants that do not

and . REFERENCES

(28) The pdf of the th syndrome subband is obtained by combining (17) and (18)

(29)

[1] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, N.J.: Prentice-Hall, 1993. [2] Z. Cvetkovic´ and M. Vetterli, “Oversampled filter banks,” IEEE Trans. Signal Process., vol. 46, pp. 1245–1255, May 1998. [3] H. Bölcskei, F. Hlawatsch, and H. G. Feichtinger, “Frame-theoretic analysis of oversampled filter banks,” IEEE Trans. Signal Process., vol. 46, pp. 3256–3268, Dec. 1998. [4] K. Eneman and M. Moonen, “DFT modulated filter bank design for oversampled subband systems,” Signal Process., vol. 81, pp. 1947–1973, 2001.


[5] M. Harteneck, S. Weiss, and R. Stewart, “Design of near perfect reconstruction oversampled filter banks for subband adaptive filters,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, pp. 1081–1085, Aug. 1999. [6] J. Kliewer and A. Mertins, “Design of paraunitary oversampled cosinemodulated filter banks,” in IEEE Int. Conf. Acoustics, Speech, Signal Processing, Munich, Germany, Apr. 1997, pp. 2073–2076. [7] T. Tanaka and Y. Yamashita, “The generalized lapped pseudo-biorthogonal transform,” in IEEE Int. Conf. Acoustics, Speech, Signal Processing, vol. II, Orlando, FL, 2002, pp. 1273–1276. [8] Q.-G. Liu, B. Champagne, and D. K. Ho, “Simple design of oversampled uniform DFT filter banks with applications to subband acoustic echo cancellation,” Signal Process., vol. 80, pp. 831–847, 2000. [9] A. Gabay, O. Rioul, and P. Duhamel, “Joint source-channel coding using structured oversampled filter banks applied to image transmission,” in IEEE Int. Conf. Acoustics, Speech, Signal Processing, vol. 4, 2001, pp. 2581–2584. [10] T. G. Marshall, “Coding of real-number sequences for error correction: A digital signal processing problem,” IEEE J. Select. Areas Commun., vol. 2, pp. 381–392, Mar. 1984. [11] J. K. Wolf, “Redundancy, the discrete Fourier transform, and impulse noise cancellation,” IEEE Trans. Commun., vol. COM-31, pp. 458–461, Mar. 1983. [12] H. Bölcskei and F. Hlawatsch, “Noise reduction in oversampled filter banks using predictive quantization,” IEEE Trans. Inform. Theory, vol. 47, pp. 155–172, Jan. 2001. [13] I. Gohdberg, P. Lancaster, and L. Rodman, Polynomial Matrices. New York: Academic, 1982. [14] H. Bölcskei and F. Hlawatsch, “Oversampled filter banks: Optimal noise shaping, design freedom, and noise analysis,” in IEEE Int. Conf. Acoustics, Speech, Signal Processing, vol. 3, Munich, Germany, Apr. 1997, pp. 2453–2456. ˇ [15] D. Henrion and M.Sebek, “Reliable numerical methods for polynomial matrix triangularization,” IEEE Trans. Autom. Control, vol. 44, pp. 497–508, Mar. 1999. [16] P. L. Dragotti, J. Kovacevic, and V. K. Goyal, “Quantized oversampled filter banks with erasures,” in IEEE Data Compression Conf., 2001, pp. 173–182. [17] J. Kovacevic, P. L. Dragotti, and V. K. Goyal, “Filter bank frame expansions with erasures,” IEEE Trans. Inform. Theory, vol. 48, pp. 1439–1450, 2002. [18] G. Rath and C. Guillemot, “Characterization of a class of error-correcting frames and their application to image transmission,” in Proc. PCS, St Malo, France, 2003. [19] R. Redinbo, “Decoding real block codes: Activity detection, Wiener estimation,” IEEE Trans. Inform. Theory, vol. 46, pp. 609–623, 2000. [20] M. Ghosh, “Analysis of the effect of impulse noise on multicarrier and single carrier QAM systems,” IEEE Trans. Commun., vol. 44, pp. 145–147, Feb. 1996. [21] F. Abdelkefi, P. Duhamel, and F. Alberge, “A posteriori control of complex Reed Solomon decoding with application to impulse noise cancellation in HIPERLAN/2,” in IEEE Int. Conf. Communications, vol. 2, 2002, pp. 659–663. [22] F. Abdelkefi, “Les Codes Reed-Solomon Complexes Pour la Correction des Erreurs Impulsives Dans Les Systèmes Multiporteuses,” Ph.D., ENST, Paris, 2002. [23] J.-P. Princen and A. B. Bradley, “Analysis/Synthesis filterbank design based on time domain aliasing cancellation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1153–1161, Oct. 1986.

Fabrice Labeau (M’00) received the Ingénieur civil degree, Diplôme d’études avancées en Télécommunications, and Ph.D. degree in electrical engineering from the Université catholique de Louvain, Belgium, in 1995, 1996, and 2000, respectively. Since 2000, he has been with the Electrical and Computer Engineering Department, McGill University, Montreal, QC, Canada, as an Assistant Professor. His current research interests include joint source/channel coding, multirate systems and multimedia transmission. Prof. Labeau was a TPC member for the IEEE Vehicular Technology Conference in Fall 2004. He was a TPC member for the EUSIPCO 2004 conference and part of the Organizing Committee of ICASSP 2004 in Montreal.

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Jui-Chiu Chiang received the B.S. degree from National Taiwan Uiversity, Taiwan, R.O.C., in 1997, the M.S. degree from National Chiao-Tung University, Taiwan, in 1999, and the Ph.D degree from Paris-Sud University, Paris, France, in 2004. Her current research interests include joint souce channel coding, video coding, and wireless communication.

Michel Kieffer was born in Sarreguemines, France, in 1972. He received the Agrégation degree in physics from Ecole Normale Supérieure de Cachan, France, in 1996 and the Ph.D. degree in control theory from the University of Paris-Sud XI, Orsay, France, in 1999. He is currently a Lecturer in signal processing and source and coding at the University Paris-Sud XI. His main research activity is on joint source-channel coding techniques for the robust transmission of multimedia contents. Methods studied are soft decoding of variable length codes and overcomplete signal representations by real or complex frame expansions or oversampled filterbanks. Applications are still-image and video transmission. He is also interested in robust state and parameter estimation using interval analysis. He is a coauthor (with L. Jaulin, O. Didrit, and E. Walter) of Applied Interval Analysis (Berlin, Germany: Springer-Verlag, 2001).

Pierre Duhamel (F’98) was born in France in 1953. He received the Eng. degree in electrical engineering from the National Institute for Applied Sciences (INSA) Rennes, France, and the Dr.Eng. and Doctorat ès Sciences degrees from Orsay University, Orsay, France, in 1978 and 1986, respectively. From 1975 to 1980, he was with Thomson-CSF, Paris, France, where his research interests were in circuit theory and signal processing, including digital filtering and analog fault diagnosis. In 1980, he joined the National Research Center in Telecommunications (CNET), Issy les Moulineaux, France, where his research activities were first concerned with the design of recursive CCD filters. Later, he worked on fast algorithms for computing Fourier transforms and convolutions, and applied similar techniques to adaptive filtering, spectral analysis and wavelet transforms. From 1993 to 2000, he was a Professor at the National School of Engineering in Telecommunications (ENST), Paris with research activities focused on signal processing for communications. He was Head of the Signal and Image Processing Department from 1997 to 2000. He is now with CNRS, Laboratoire de Signaux et Systemes, Gif sur Yvette, France, where he is developing studies in signal processing for communications (including equalization, iterative decoding, multicarrier systems) and signal/image processing for multimedia applications, including source coding, joint source/channel coding, watermarking, and audio processing. Dr. Duhamel was an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1989 to 1991, an Associate Editor of IEEE SIGNAL PROCESSING LETTERS, and a Guest Editor for the Special Issue of IEEE TRANSACTIONS ON SIGNAL PROCESSING on wavelets. He was an IEEE Distiguished Lecturer in 1999. He was a corecipient of the Best Paper Award from the IEEE TRANSACTIONS ON SIGNAL PROCESSING in 1998. He received the Grand Prix France Telecom from the French Science Academy in 2000. He was Co-General Chair of the 2001 International Workshop on Multimedia Signal Processing, Cannes, France. He will be Co-Technical Chair of ICASSP’06, Toulouse, France. He was Chairman of the DSP committee from 1996 to 1998 and a member of the SP for Com committee until 2001.

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Luc Vandendorpe (SM’99) was born in Mouscron, Belgium, in 1962. He received the Electrical Engineering degree (summa cum laude) and the Ph.D. degree from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1985 and 1991, respectively. Since 1985, he has been with the Communications and Remote Sensing Laboratory of UCL, where he first worked in the field of bit-rate reduction techniques for video coding. From March to August 1992, he was a Visiting Scientist and Research Fellow with the Telecommunications and Traffic Control Systems Group, Delft Technical University, The Netherlands, where he worked on spread spectrum techniques for personal communications systems. From 1992 to 1997, he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor. Presently he is a Professor. He is the Belgian Delegate to COST 273 and 289. He is mainly interested in digital communication systems: equalization, joint detection/synchronization for CDMA, OFDM (multicarrier), MIMO and turbo-based communications systems (UMTS, xDSL, WLAN, etc.), and joint source/channel (de)coding. In 1990, he was a Corecipient of the Biennal Alcatel-Bell Award from the Belgian NSF for a contribution in the field of image coding. In 2000, he was a Corecipient (with J. Louveaux and F. Deryck) of the Biennal Siemens Award from the Belgian NSF for a contribution about filter bank based multicarrier transmission. In 2004, he was a Cowinner (with J. Czyz) of the Face Authentication Competition (FAC 2004). He is or has been a TPC member for IEEE VTC Fall 1999, IEEE Globecom 2003 Communications Theory Symposium, the 2003 Turbo Symposium, IEEE VTC Fall 2003, and IEEE SPAWC 2005. He is Co-Technical Chair (with P. Duhamel) for IEEE ICASSP 2006. He is an Associate Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and a member of the Signal Processing Committee for Communications. He was an Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS for Synchronization and Equalization between 2000 and 2002 and Chair of the IEEE Benelux joint chapter on Communications and Vehicular Technology between 1999 and 2003.

Benoît Macq (M’89–SM’01) was born in 1961. He received the electrical engineering and Ph.D. degrees from the Université catholique de Louvain (UCL), Louvain, Belgium, in 1984 and 1989, respectively. His Ph.D. dissertation was on perceptual coding for digital TV (under the supervision of Prof. P. Delogne). He has been a Professor in the Telecommunications Laboratory, UCL, since 1996. From 1992 to 1996, he was a Senior Researcher with the Belgian NSF at UCL and an invited Assistant Professor in the Telecommunications Laboratory. From 1990 to 1991, he worked on network planning for Tractebel S.A., Brussels, Belgium. He was a Visiting Scientist at the Ecole Nationale Superieure des Telecommunications, Paris, France, and at the Université de Nice Sophia Antipolis, France, from 1999 to 2000. He was a Guest Editor of Signal Processing and he has been a member of the program committees of several SPIE conferences. His main research interests are image compression, image watermarking, and image analysis for medical and immersive communications. Prof. Macq was a Guest Editor of the PROCEEDINGS OF THE IEEE. He has been a member of the Program Committees of several IEEE conferences.