Principal Component Filter Banks Research Articles

Insights on coding gain and its properties for principal component filter banks

Insights on coding gain and its properties for principal component filter banks

A Pilot Study on Visually Stimulated Cognitive Tasks for EEG-Based Dementia Recognition

In the status quo, dementia is yet to be cured. Precise diagnosis prior to the onset of the symptoms can prevent the rapid progression of the emerging cognitive impairment. Recent progress has shown that Electroencephalography (EEG) is the promising and cost-effective test to facilitate the detection of neurocognitive disorders. However, most of the existing works have been using only resting-state EEG. The efficiencies of EEG signals from various cognitive tasks, for dementia classification, have yet to be thoroughly investigated. In this study, we designed four cognitive tasks that engage different cognitive performances: attention, working memory, and executive function. We investigated these tasks by using statistical analysis on both time and frequency domains of EEG signals from three classes of human subjects: Dementia (DEM), Mild Cognitive Impairment (MCI), and Normal Control (NC). We also further evaluated the classification performances of two features extraction methods: Principal Component Analysis (PCA) and Filter Bank Common Spatial Pattern (FBCSP). We found that the working memory related tasks yielded good performances for dementia recognition in both cases using PCA and FBCSP. Moreover, FBCSP with features combination from four tasks revealed the best sensitivity of 0.87 and the specificity of 0.80. To our best knowledge, this is the first work that concurrently investigated several cognitive tasks for dementia recognition using both statistical analysis and classification scores. Our results yielded essential information to design and aid in conducting further experimental tasks to early diagnose dementia patients.

Robust Transmit Nulling in Wideband Arrays

The ability to create nulls in the transmit pattern of a phased array antenna has many applications in communication and radar systems, including interference and clutter mitigation. This paper describes the implementation of transmit nulls in wideband arrays through the use of a filter bank inserted behind each array element. The filter bank decimates and partitions the transmit signal into independent subbands and utilizes a tapped delay line (TDL) in each subband to form frequency invariant spatial nulls in the array's transmit pattern. New contributions developed in this paper include an algorithm for determining the TDL coefficients and a novel band partitioning scheme based on principal component filter banks (PCFBs), which is shown to be optimal for minimizing spectral errors if the TDL coefficients are quantized. Numerical techniques are presented for approximating ideal PCFBs using practical paraunitary filter banks (PUFBs) and perfect reconstruction filter banks (PRFBs).

FIR Approximation of GTD Filter Banks and their Multiresolution Optimality

The theory and design of signal adapted filter banks in the coding gain objective as well as the multiresolution objective are of great interest in many signal processing applications. The role of the generalized triangular decomposition (GTD) filter banks in optimizing perfect reconstruction filter banks has been proposed recently by Ching-Chih Weng et al. They have proposed the GTD filter bank as a subband coder for optimizing the theoretical coding gain. In this paper, we show that the design of the GTD filter bank via the singular value decomposition (SVD) will be reduced to the principal component filter bank (PCFB) and it gives optimal performance in the multiresolution objective. The FIR approximation of the optimal GTD filter banks is also discussed in this paper. This is done by using the iterative greedy algorithm.

Adaptive loading with Principal Component Filter Banks in MIMO MultiCarrier Modulation System

We discuss adaptive loading scheme in wireless MultiCarrier Modulation Systems (MCMS), using Variable-Rate Variable-Power (VRVP), coded, M-ary quadrature amplitude modulation. The Principal Component Filter Bank (PCFB), which is adapted to the channel characteristics, is used for converting the wideband channel into narrow-band sub-channels. PCFB-based adaptive scheme gives better performance in terms of spectral efficiency and bit error rate, compared with the conventional scheme in MCMS. The main advantage of using PCFB in adaptive loading is that constellation size need not be varied frequently on each subcarrier depending on the channel Signal-to-Noise Ratio (SNR). We extend this idea to Multi-Input Multi-Output (MIMO) systems.

Two-dimensional, two-channel signal-adapted filter banks

The existence of principal component filter banks (PCFBs) has been proved for the class of one-dimensional (1D), two-channel finite impulse response (FIR) filter banks. The existence issues of two-imensional (2D), two-channel PCFBs are studied. The support of the filters will vary with the subsampling matrix used. It is seen that the quincunx matrix is optimum in most of the natural images. Moreover, the existence of non-uniform 2D PCFBs, in which the PCFB is defined by normalised subband variances, for dyadic splitting of the frequency spectrum, in the case of natural images is proved. It is also shown that 2D PCFBs, defined by equivalent uniform variances, can exist in the case of most of the natural images, for octave band filter banks. In addition, 2D FIR signal-adapted filter banks from 1D filters, for separable subsampling, are designed. These 1D filters are designed from the separable components of the 2D power spectral density, using iterative greedy algorithm. The iterative algorithm is extended to two dimensions in order to design non-separable FIR filter banks for different subsampling matrices. The simulated FIR responses match the ideal responses very closely.

Iterative greedy algorithm for solving the FIR paraunitary approximation problem

In this paper, a method for approximating a multi-input multi-output (MIMO) transfer function by a causal finite-impulse response (FIR) paraunitary (PU) system in a weighted least-squares sense is presented. Using a complete parameterization of FIR PU systems in terms of Householder-like building blocks, an iterative algorithm is proposed that is greedy in the sense that the observed mean-squared error at each iteration is guaranteed to not increase. For certain design problems in which there is a phase-type ambiguity in the desired response, which is formally defined in the paper, a phase feedback modification is proposed in which the phase of the FIR approximant is fed back to the desired response. With this modification in effect, it is shown that the resulting iterative algorithm not only still remains greedy, but also offers a better magnitude-type fit to the desired response. Simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank (PCFB)-like filter banks and the FIR PU interpolation problem. Concerning the PCFB design problem, it is shown that as the McMillan degree of the FIR PU approximant increases, the resulting filter bank behaves more and more like the infinite-order PCFB, consistent with intuition. In particular, this PCFB-like behavior is shown in terms of filter response shape, multiresolution, coding gain, noise reduction with zeroth-order Wiener filtering in the subbands, and power minimization for discrete multitone (DMT)-type transmultiplexers.

Algebraic theory of optimal filterbanks

We introduce an optimality theory for finite impulse response (FIR) filterbanks using a general algebraic point of view. We consider an admissible set /spl Lscr/ of FIR filterbanks and use scalability as the main notion based on which performance of the elements in /spl Lscr/are compared. We show that quantification of scalability leads naturally to a partial ordering on the set /spl Lscr/. An optimal solution is, therefore, represented by the greatest element in /spl Lscr/. It turns out that a greatest element does not necessarily exist in /spl Lscr/. Hence, one has to settle with one of the maximal elements that exist in /spl Lscr/. We provide a systematic way of finding a maximal element by embedding the partial ordering at hand in a total ordering. This is done by using a special class of order-preserving functions known as Schur-convex. There is, however, a price to pay for achieving a total ordering: there are infinitely many possible choices for Schur-convex functions, and the optimal solution specified in /spl Lscr/ depends on this (subjective) choice. An interesting aspect of the presented algebraic theory is that the connection between several concepts, namely, principal component filterbanks (PCFBs), filterbanks with maximum coding gain, and filterbanks with good scalability, is clearly revealed. We show that these are simply associated with different extremal elements of the partial ordering induced on /spl Lscr/ by scalability.

Discrete multitone modulation with principal component filter banks

Discrete multitone (DMT) modulation is an attractive method for communication over a nonflat channel with possibly colored noise. The uniform discrete Fourier transform (DFT) filter bank and cosine modulated filter bank have in the past been used in this system because of low complexity. We show in this paper that principal component filter banks (PCFB) which are known to be optimal for data compression and denoising applications, are also optimal for a number of criteria in DMT modulation communication. For example, the PCFB of the effective channel noise power spectrum (noise psd weighted by the inverse of the channel gain) is optimal for DMT modulation in the sense of maximizing bit rate for fixed power and error probabilities. We also establish an optimality property of the PCFB when scalar prefilters and postfilters are used around the channel. The difference between the PCFB and a traditional filter bank such as the brickwall filter bank or DFT filter bank is significant for effective power spectra which depart considerably from monotonicity. The twisted pair channel with its bridged taps, next and fext noises, and AM interference, therefore appears to be a good candidate for the application of a PCFB. This is demonstrated with the help of numerical results for the case of the ADSL channel.

A Review of the Theory and Applications of Optimal Subband and Transform Coders

The problem of optimizing digital filter banks based on input statistics was perhaps first addressed nearly four decades ago by Huang and Schultheiss. These authors actually considered a special case, namely transform coder optimization. Many of the subband coder optimization problems considered in recent years have close similarities to this work, though there are fundamental differences as well. Filter banks are used today not only for signal compression, but have found applications in signal denoising and in digital communications. A recent result is that principal component filter banks (PCFBs) offer an optimal solution to many problems under certain theoretical assumptions. While this result is quite powerful and includes several earlier results as special cases, there still remain some open problems in the area of filter bank optimization. We first give a review of the older classical methods to place the ideas in the right perspective. We then review recent results on PCFBs. The generality of these results is demonstrated by showing an application in digital communications (the discrete multitone channel). We show, for example, that the PCFB minimizes transmitted power for a given probability of error and bit rate. Future directions and open problems are discussed as well.

Results on principal component filter banks: colored noise suppression and existence issues

We have made explicit the precise connection between the optimization of orthonormal filter banks (FBs) and the principal component property: the principal component filter bank (PCFB) is optimal whenever the minimization objective is a concave function of the subband variances of the FB. This explains PCFB optimality for compression, progressive transmission, and various hitherto unnoticed white-noise, suppression applications such as subband Wiener filtering. The present work examines the nature of the FB optimization problems for such schemes when PCFBs do not exist. Using the geometry of the optimization search spaces, we explain exactly why these problems are usually analytically intractable. We show the relation between compaction filter design (i.e., variance maximization) and optimum FBs. A sequential maximization of subband variances produces a PCFB if one exists, but is otherwise suboptimal for several concave objectives. We then study PCFB optimality for colored noise suppression. Unlike the case when the noise is white, here the minimization objective is a function of both the signal and the noise subband variances. We show that for the transform coder class, if a common signal and noise PCFB (KLT) exists, it is, optimal for a large class of concave objectives. Common PCFBs for general FB classes have a considerably more restricted optimality, as we show using the class of unconstrained orthonormal FBs. For this class, we also show how to find an optimum FB when the signal and noise spectra are both piecewise constant with all discontinuities at rational multiples of /spl pi/.

Rate-distortion-optimal subband coding without perfect-reconstruction constraints

We investigate the design of subband coders without the traditional perfect-reconstruction constraint on the filters. The coder uses scalar quantizers, and its filters and bit allocation are designed to optimize a rate-distortion criterion. Using convexity analysis, we show that optimality can be achieved using filterbanks that are the cascade of a (paraunitary) principal component filterbank for the input spectral process and a set of pre and postfilters surrounding each quantizer. Analytical expressions for the pre and postfilters are then derived. An algorithm for computing the globally optimal filters and bit allocation is given. We also develop closed-form solutions for the special case of two-channel coders under an exponential rate-distortion model. Finally, we investigate a constrained-length version of the filter design problem, which is applicable to practical coding scenarios. While the optimal filterbanks are nearly perfect-reconstruction at high rates, we demonstrate an apparently surprising advantage of optimal FIR filterbanks; they significantly outperform optimal perfect-reconstruction FIR filterbanks at all bit rates.

Filterbank optimization with convex objectives and the optimality of principal component forms

This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics. This includes as special cases, many previous results on FB optimization for compression. It also solves problems that have not been considered thus far. FB optimization for coding gain maximization (for compression applications) has been well studied before. The optimum FB has been known to satisfy the principal component property, i.e., it minimizes the mean-square error caused by reconstruction after dropping the P weakest (lowest variance) subbands for any P. We point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of the subband variances produced by the FB. This result has its grounding in majorization and convex function theory and, in particular, explains the optimality of PCFBs for compression. We use the result to show various other optimality properties of PCFBs, especially for noise-suppression applications. Suppose the FB input is a signal corrupted by additive white noise, the desired output is the pure signal, and the subbands of the FB are processed to minimize the output noise. If each subband processor is a zeroth-order Wiener filter for its input, we can show that the expected mean square value of the output noise is a concave function of the subband signal variances. Hence, a PCFB is optimum in the sense of minimizing this mean square error. The above-mentioned concavity of the error and, hence, PCFB optimality, continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of serious practical interest. We prove that certain extensions of this PCFB optimality result to cases where the input noise is colored, and the FB optimization is over a larger class that includes biorthogonal FBs. We also show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs.

Theory of rate-distortion-optimal, constrained filterbanks-application to IIR and FIR biorthogonal designs

We design filterbanks that are best matched to input signal statistics in M-channel subband coders, using a rate-distortion criterion. Previous research has shown that unconstrained-length, paraunitary filterbanks optimized under various energy compaction criteria are principal-component filterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of rate-distortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfect-reconstruction (PR) filterbanks take the form of the cascade of principal-component filterbanks and a bank of pre- and post-conditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented.

A state space approach to the design of globally optimal FIR energy compaction filters

We introduce a new approach for the least squared optimization of a weighted FIR filter of arbitrary order N under the constraint that its magnitude squared response be Nyquist(M). Although the new formulation is general enough to cover a wide variety of applications, the focus of the paper is on optimal energy compaction filters. The optimization of such filters has received considerable attention in the past due to the fact that they are the main building blocks in the design of principal component filter banks (PCFBs). The newly proposed method finds the optimum product filter F/sub opt/(z)=H/sub opt/(Z)H/sub opt/(z/sup -1/) corresponding to the compaction filter H/sub opt/(z). By expressing F(z) in the form D(z)+D(z/sup -1/), we show that the compaction problem can be completely parameterized in terms of the state-space realization of the causal function D(z). For a given input power spectrum, the resulting filter F/sub opt/(z) is guaranteed to be a global optimum solution due to the convexity of the new formulation. The new algorithm is universal in the sense that it works for any M, arbitrary filter length N, and any given input power spectrum. Furthermore, additional linear constraints such as wavelets regularity constraints can be incorporated into the design problem. Finally, obtaining H/sub opt/(z) from F/sub opt/(z) does not require an additional spectral factorization step. The minimum-phase spectral factor H/sub min/(z) can be obtained automatically by relating the state space realization of D/sub opt/(z) to that of H/sub opt/(z).

Signal-adapted recursive interpolating wavelet packets

In this work, recursive interpolating wavelet packets with shape parameter vectors are constructed and the parameter-adaptive decomposition algorithms are presented, which are based on the subband coding gain. Unlike the previous structure-adaptive one in which every node employs the same pair of filters, the shape parameter vectors of every node are signal-adaptively selected in our approaches, thus an adaptive decomposition with the optimal parameter vectors is obtained. Moreover, the approach resolves the design of two-band interpolating principal component filter banks. The experiment results show that the presented approaches can markedly improve the coding gain.

FIR principal component filter banks

Two-dimensional (2-D) principal component filter banks (PCFBs) of finite impulse response (FIR) are proposed. For 2-D signals, among all uniform paraunitary FIR analysis/synthesis filter banks, the FIR PCFBs have the most energy compaction and maximize the arithmetic mean to geometric mean ratio (AM/GM ratio) of subband variances, which is the theoretic coding gain (TCC) of the systems under proper assumptions. The theoretic proof and design techniques are provided. Several special cases are discussed. Experimental results show the potential power of the FIR PCFBs.

Multidimensional, paraunitary principal component filter banks

In this correspondence, the one-dimensional (1-D) principal component filter banks (PCFB's) derived by Tsatsatsanis and Giannakis (1995) are generalized to higher dimensions. As presented by Tsatsatsanis and Giannakis, PCFB's minimize the mean-squared error (MSE) when only Q out of P subbands are retained. Previously, two-dimensional (2-D) PCFB's were proposed by Tirakis et al. (1995). The work by Tirakis et al. was limited to 2-D signals and separable resampling operators. The formulation presented here is general in that it can easily accommodate signals of arbitrary (yet finite) dimension and nonseparable sampling. A major result presented in this paper is that in addition to minimizing MSE when reconstructing from Q out of p subbands, the PCFB's result in maximizing theoretical coding gain (TCG) thereby performing optimally in a energy compaction sense.

Principal component filter banks for optimal multiresolution analysis

An important issue in multiresolution analysis is that of optimal basis selection. An optimal P-band perfect reconstruction filter bank (PRFB) is derived in this paper, which minimizes the approximation error (in the mean-square sense) between the original signal and its low-resolution version. The resulting PRFB decomposes the input signal into uncorrelated, low-resolution principal components with decreasing variance. Optimality issues are further analyzed in the special case of stationary and cyclostationary processes. By exploiting the connection between discrete-time filter banks and continuous wavelets, an optimal multiresolution decomposition of L/sub 2/(R) is obtained. Analogous results are also derived for deterministic signals. Some illustrative examples and simulations are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>