Perfect Reconstruction Filter Banks Research Articles

Perfect-reconstruction filtering with unitary operators and projections

Perfect-reconstruction filter banks, wavelets, and multiresolution analysis have generated many new results in harmonic analysis and function theory. Traditionally, these techniques are based on families of convolutions and decimation/interpolation operations. When viewed as linear-algebraic operations, those building blocks can be generalized to unitary operators and related projections. We have developed a general approach to perfect reconstruction and multiresolution analysis that uses polynomial functions of unitary operators and closely related projection operations. This general framework is powerful in that it easily captures much of the known theory as special cases and gives a simple mechanism for constructing arbitrary perfect reconstruction families of operators. Using the general framework, we derive necessary and sufficient conditions for the perfect reconstruction property, even for higher-dimensional non-separable sampling lattices and operators. The necessary conditions can be interpreted in terms of the spectral mapping theorem and, although explicity stated, are not directly computable.

Results on Biorthogonal Filter Banks

For a maximally decimated nonuniform filter bank, the perfect reconstruction (PR) property is equivalent to biorthogonality. We start from this result and derive a number of properties of PR filter banks. For example, no two integer decimators in a biorthogonal system can be coprime; moreover, if all analysis and synthesis filters have unit energy, then perfect reconstruction is equivalent to orthonormality. We also generalize the Nyquist and power complementary properties of orthonormal filter banks, for the biorthogonal case. We then show that whenever the decimation ratios are such that biorthogonality is possible with rational filters, it is, in particular, possible to obtain orthonormality with rational filters. This is done by developing an orthonormalization procedure. While reminiscent of the Gram–Schmidt approach, the procedure converges in a finite number of steps and furthermore preserves the filter-bank-like form of the basis functions. We also show how this technique can be applied for the decorrelation of subband signals. We will consider the problem of alias cancellation and obtain a generalization of a previously known necessary condition called compatibility.

The design of 2-D nonseparable directional perfect reconstruction filter banks

In this paper, we develop a directional 2-D nonseparable filter bank that can perfectly reconstruct the downsampled subband signals. The filter bank represents two powerful image and video processing tools: directional subband decomposition and perfect reconstruction. The directional filter banks consist of (1) the input signal and the subband signals modulation, (2) diamond shape prefilter, and (3) four different parallelogram shape prefilters. This paper addresses the design and implementation of a two-band filter bank that is proved to be able to provide perfect reconstruction of the downsampled subband signals. Finally, we use a conventional 1-D half-band filter as a prototype and then apply the McClellan transform for the specific 2-D diamond shape and parallelogram shape subfilters. This method is extremely simple in designing the analysis/synthesis subfilters for the filter bank.

FIR approximations of inverse filters and perfect reconstruction filter banks

This paper first describes an algorithm that finds the approximate finite impulse response (FIR) inverse of an FIR filter by minimizing the inversion (or reconstruction) error constrained to zero-bias. The generalization of the inverse filtering problem in the two channel case is the design of perfect reconstruction filter banks that use critical sampling. These considerations lead to the derivation of an algorithm that provides a minimum error and unbiased FIR/FIR approximation of a perfect reconstruction IIR/FIR (or FIR/IIR) filter bank. The one-channel algorithm is illustrated with the design of an FIR filter to compute the B-spline coefficients for cubic spline signal interpolation. The two-channel algorithm is applied to the design of a FIR/FIR filter bank that implements the cubic B-spline wavelet transform. Finally, we consider a modification of this technique for the design of modulated-filter banks, which are better suited for subband coding.

Theory of cosine-modulated two-dimensional perfect reconstruction filter banks

This paper considers a cosine-modulated, two-dimensional (2-D) perfect reconstruction (PR) filter banks theory. First, a 2-D digital filter with half-pass-band obtained by the sampling matrix had to be designed. Next, 2-D analysis filter banks are realized by cosine-modulating this prototype 2-D digital filter so that 2-D analyzed signals become real. It is shown that the modulation in the 2-D frequency plane is equivalent to 1-D modulation. A necessary and sufficient condition for 2-D perfect reconstruction filter banks is derived. If the polyphase filter pairs of the prototype filter doubly complement, the resulting 2-D filter bank satisfies the condition of perfect reconstruction.

M-ary wavelet transform and formulation for perfect reconstruction in M-band filter bank

The binary wavelet transform is generalized and extended to the M-ary biorthonormal case. The computational equivalence between the discrete wavelet analysis and the M-band multirate signal filtering is indicated. The equivalence allows the perfect reconstruction requirement in a filter bank to be investigated from the vector space decomposition/reconstruction in wavelet analysis. From the construction of the biorthonormal wavelet bases, the necessary and sufficient condition for the filters in a perfect reconstruction filter bank is formulated. Under this formulation, an additional optimization procedure is then used to model the frequency domain requirement in filter bank design.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A recursive orthogonal wavelet device

A design for a one-dimensional (1-D), 2-D recursive orthogonal wavelet consisting of a parallel connection of a delay and allpass network is presented. There have been some methods to obtain wavelet functions from perfect reconstruction filter banks following an investigation by Mallat of the relation between wavelet transform and filter banks. These methods are based mostly on an FIR filter. A method based on IIR filters has never been investigated. IIR digital filter has fewer orders than FIR digital filters to realize the same specification and satisfies the orthogonal condition of the orthogonal wavelet by using both parallel connection of some delays and an allpass filter. Furthermore, a maximum number of zeros are placed at aliasing frequencies in the lowpass filter as a way of obtaining the regularity of the wavelet. Finally, some design samples of discrete scaling and wavelet functions are presented.

State space representations of 2-D FIR lossless transfer matrices

This paper examines the properties of state-space realizations of square 2-D FIR lossless transfer matrices. Such matrices have found applications for the design of 2-D orthogonal perfect reconstruction filter banks. It is shown that 2-D FIR lossless matrices admit a minimal first-level realization with respect to one transform variable, where the dynamics are FIR and lossless in the other variable. This result is then employed to prove that 2-D FIR lossless transfer matrices admit a minimal 2-D Roesser realization whose dynamics are parametrized by a single constant orthonormal matrix, which satisfies constraints ensuring that the resulting transfer matrix is FIR. This parametrization is used to compute the number of degrees of freedom available for the synthesis of a 2-D FIR lossless transfer matrix of fixed McMillan degree in each variable. It is also shown that 2-D lossless FIR transfer matrices cannot necessarily be represented as a product of lossless FIR factors of lower degree, and conditions for the existence of such factorizations are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The design of two-channel lattice-structure perfect-reconstruction filter banks using powers-of-two coefficients

An optimization technique is presented for the design of two-channel lattice-structure perfect-reconstruction filter banks with powers-of-two coefficients. The filter coefficients are represented by a canonic signed-digit (CSD) code. The proposed technique requires the original optimal infinite-precision coefficients as the starting point, and searches for the set of CSD coefficients that minimizes the peak stopband ripple. Design examples are given to show that perfect-reconstruction filter banks with good filtering performance can be obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Perfect reconstruction filter banks with rational sampling factors

An open problem, namely, how to construct perfect reconstruction filter banks with rational sampling factors, is solved. Such filter banks have N branches, each one having a sampling factor of p/sub i///sub qi/, and their sum equals one. In this way, the well-known theory of filter banks with uniform band splitting is extended to allow for nonuniform divisions of the spectrum. This can be very useful in the analysis of speech and music. The theory relies on two transforms. The first transform leads to uniform filter banks having polyphase components as individual filters. The other results in a uniform filter bank containing shifted versions of same filters. This, in turn, introduces dependencies in design, and is left for future work. As an illustration, several design examples for the (2/3, 1/3) case are given. Filter banks are then classified according to the possible ways in which they can be built. It is shown that some cases cannot be solved even with ideal filters (with real coefficients).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

FCO sampling of digital video using perfect reconstruction filter banks

Three-dimensional nonseparable perfect reconstruction filter banks using three-dimensional nonseparable sampling by two, FCO, are proposed. Filter structures are derived and applied to digital video. Separation into two bands is obtained, and it is shown to perform better from the perceptual point of view than interlaced sequences resulting from the quincunx sampling of a progressively scanned signal in time-vertical dimensions.

Subband decomposition of signals with generalized sampling

The authors generalize down and up-sampling operations by proposing block sampling. Perfect reconstruction conditions for two-band subband coding with block sampling are derived. By generalizing the sampling operation, new degrees of freedom are introduced and as a result, filter banks which were not previously possible become possible. Generalized down-sampler introduces different aliasing components than that of the traditional down-sampler. This can be used to ease some the requirements of the filter bank design problem. A constructive sampling method is proposed so that coprimeness of the transfer functions of the analysis filter banks is not only a necessary but also sufficient condition for perfect reconstruction. The results are extended to the case where the filter banks are linear periodically time-varying. The multichannel case is analyzed and the relation between unimodular matrices and perfect reconstruction filter banks is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Time-domain design of low delay wavelet transforms

The Letter uses a time-domain design methodology previously used for perfect reconstruction filter banks to design wavelet prototypes which may be used to generate the discrete wavelet transform (DWT). An advantage of this method is the significant reduction in the inherent delay, and consequent reduction in the number of multiplications required to implement the DWT. It is proposed that these low delay wavelets will find applications in time critical systems.

A discrete-time multiresolution theory

Multiresolution analysis and synthesis for discrete-time signals is described. Concepts of scale and resolution are first reviewed in discrete time. The resulting framework allows one to treat the discrete wavelet transform, octave-band perfect reconstruction filter banks, and pyramid transforms from a unified standpoint. This approach is very close to previous work on multiresolution decomposition of functions of a continuous variable, and the connection between these two approaches is made. It is shown that they share many mathematical properties such as biorthogonality, orthonormality, and regularity. However, the discrete-time formalism is well suited to practical tasks in digital signal processing and does not require the use of functional spaces as an intermediate step.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of 2‐D perfect reconstruction filter banks for arbitrary sampling lattices—design of 2‐D N‐th‐band digital filters

AbstractThis paper presents a design method of two‐dimensional (2‐D) perfect reconstruction filter banks for arbitrary sampling lattices. First, a design of 2‐D N‐th‐band digital filters with arbitrary sampling lattices is shown. Next, the relation between the sampling matrix and the number of paths is presented and the band‐limited region and the condition of delays are defined. From these conditions, the approximation problem is shown of the 2‐D allpass filter whose 2‐D N‐th‐band filter is composed and the characteristics of 2‐D N‐th‐band filters are explained. Then 2‐D analysis banks consist of 2‐D discrete Fourier transform (DFT) and 2‐D N‐th‐band filter which consists of some delays and 2‐D interpolated allpass networks. To achieve the perfect reconstruction, by evaluating the aliasing component matrix, it is shown that the synthesis banks can also be constructed the same as the analysis bank.

Linear phase paraunitary filter banks: theory, factorizations and designs

M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. The authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. They begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, they develop a minimal factorization for a large class of such systems. This factorization will be proved to be complete for even M. Further, they structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. They then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory. >

Cosine-modulated FIR filter banks satisfying perfect reconstruction

The authors obtain a necessary and sufficient condition on the 2M (M=number of channels) polyphase components of a linear-phase prototype filter of length N=2 mM (where m=an arbitrary positive integer), such that the polyphase component matrix of the modulated filter is lossless. The losslessness of the polyphase component matrix, in turn, is sufficient to ensure that the analysis/synthesis system satisfies perfect reconstruction (PR). Using this result, a novel design procedure is presented based on the two-channel lossless lattice. This enables the design of a large class of FIR (finite impulse response)-PR filter banks, and includes the N=2M case. It is shown that this approach requires fewer parameters to be optimized than in the pseudo-QMF (quadrature mirror filter) designs and in the lossless lattice based PR-QMF designs (for equal length filters in the three designs). This advantage becomes significant when designing long filters for large M. The design procedure and its other advantages are described in detail. Design examples and comparisons are included.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R/sup n/

New results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented. Among the topics discussed are sampling in multiple dimensions, multidimensional perfect reconstruction filter banks, the two-channel case in multiple dimensions, the synthesis of multidimensional filter banks, and the design of compactly supported wavelets. >

Adaptive filtering in subbands with critical sampling: analysis, experiments, and application to acoustic echo cancellation

An exact analysis of the critically subsampled two-band modelization scheme is given, and it is demonstrated that adaptive cross-filters between the subbands are necessary for modelization with small output errors. It is shown that perfect reconstruction filter banks can yield exact modelization. These results are extended to the critically subsampled multiband schemes, and important computational savings are seen to be achieved by using good quality filter banks. The problem of adaptive identification in critically subsampled subbands is considered and an appropriate adaptation algorithm is derived. The authors give a detailed analysis of the computational complexity of all the discussed schemes, and experimentally verify the theoretical results that are obtained. The adaptive behavior of the subband schemes that were tested is discussed. >

Commutativity of up/down sampling

A technique for the construction of perfect reconstruction filter banks with division into non-uniform channels has been introduced. This construction uses the commutativity of subsampling and upsampling. Explicit conditions for commutativity have been presented only for the 1-D and 2-D cases. The Letter generalises the conditions for commutativity of subsampling and upsampling to arbitrary dimensions.