Raptor codes are known as precoded rateless codes that achieve the capacity of BEC. However the maximum degree of Raptor codes needs to be unbounded to achieve the capacity. In this paper we prove that spatially-coupled precoded rateless codes achieve the capacity with bounded degree under BP decoding.

We study LDPC codes for the channel with input x ∈ Fqmand output y = x + z ∈ Fqm. The aim of this paper is to evaluate decoding performance of qm-ary non-binary LDPC codes for large m. We give density evolution and decoding performance evaluation for regular non-binary LDPC codes and spatially-coupled (SC) codes. We show the regular codes do not achieve the capacity of the channel while SC codes do.

In this paper, we propose a decoding algorithm which lowers decoding erasure rates in the error floor regions for non-binary low-density parity-check codes transmitted over the binary erasure channels. This decoding algorithm is a combination with belief propagation (BP) decoding and maximum a posteriori (MAP) decoding on zigzag cycles, which cause decoding erasures in the error floor region. We show that MAP decoding on the zigzag cycles is realized by means of a message passing algorithm. A simulation result shows that the decoding erasure rates in the error floor regions by the proposed decoding algorithm are lower than those by the BP decoder.

In this paper, we derive the average weight distributions for the irregular non-binary cluster low-density parity-check (LDPC) code ensembles. Moreover, we give the exponential growth rate of the average weight distribution in the limit of large code length. We show that there exist (2, dc)-regular non-binary cluster LDPC code ensembles whose normalized typical minimum distances are strictly positive.

Raptor codes are rateless codes that achieve the capacity on the binary erasure channels. However the maximum degree of optimal output degree distribution is unbounded. This leads to a computational complexity problem both at encoders and decoders. Aref and Urbanke investigated the potential advantage of universal achieving-capacity property of proposed spatially-coupled (SC) low-density generator matrix (LDGM) codes. However the decoding error probability of SC-LDGM codes is bounded away from 0. In this paper, we investigate SC-LDGM codes concatenated with SC low-density parity-check codes. The proposed codes can be regarded as SC Hsu-Anastasopoulos rateless codes. We derive a lower bound of the asymptotic overhead from stability analysis for successful decoding by density evolution. The numerical calculation reveals that the lower bound is tight. We observe that with a sufficiently large number of information bits, the asymptotic overhead and the decoding error rate approach 0 with bounded maximum degree.

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as the blocklength tends to infinity, and the limit is obtained via density evolution. The finite-blocklength correction behaves like α(ε,t)/n+Θ(n-2) as the blocklengthntends to infinity where α(ε,t) denotes a specific constant determined by the code ensemble considered, the numbertof iterations, and the erasure probability ε of the BEC. In this paper, we derive a set of recursive formulas which allows the evaluation of the constant α(ε,t) for standard irregular ensembles. The dominant difference α(ε,t)/ncan be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using α(ε,t) is accurate even for small blocklengths.

We study LDPC codes for the channel with 2m-ary input x ϵ F2mand output y = x + z ϵ F2m. The receiver knows a subspace V ⊂ F2mfrom which z = y - x is uniformly chosen. Or equivalently, the receiver receives an affine subspace y-V where x lies. We consider a joint iterative decoder involving the channel detector and the LDPC decoder. The decoding system considered in this paper can be viewed as a simplified model of the joint iterative decoder over non-binary modulated signal inputs e.g., 2m-QAM. We evaluate the performance of binary spatially-coupled MacKay-Neal codes by density evolution. The iterative decoding threshold is seriously degraded by increasing m. EXIT-like function curve calculations reveal that this degradation is caused by wiggles and can be mitigated by increasing the randomized window size. The resultant iterative decoding threshold values are very close to the Shannon limit.

In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over the general linear group with those for non-binary LDPC codes over finite field transmitted over the q-ary memoryless symmetric channel under belief propagation decoding. To analyze non-binary LDPC codes defined over both general linear group GL(m, F2) and finite field F2m, we investigate non-binary LDPC codes defined over GL(m3, F2m4). We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the optimized non-binary LDPC codes defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels in the edges which satisfy the condition for the optimization.

From an information-theoretic point of view, it is well known that the capacity of relay channel comprising of three terminals is much more than that of two terminal direct channel especially for low SNRs. Previously invented relay coding strategies have not been designed to achieve this enormous capacity occurring in the low SNR region. In this paper, we propose a simple coding strategy for a relay channel with low SNRs or, equivalently, for a very noisy relay channel. The multiplicative repetition is utilized to design this simple coding strategy. We claim that the proposed strategy is simple since the destination and the relay can decode with almost the same computational complexity by sharing the same structure of decoder. An appropriate static power allocation which yields the maximum throughput close to the optimal one in low SNRs is suggested. Under practical constraints such as equal time-sharing etc., the asymptotic performance of this simple strategy is within 0.5 dB from the achievable rate of relay channel. Furthermore, the performance at short code lengths enjoys a relaying gain by approximately 1.4 dB.

While it is known that spatially-coupled low-density lattice codes (SC-LDLC) have better decoding performance than conventional (non-coupled) LDLC lattices, in this paper it is shown that their encoding complexity is also lower. Since nonzero elements are mainly in lower triangular entries of the sparse inverse generator matrix of SC-LDLC, iterative encoding with the Gauss-Seidel method performs well. The convergence speed of iterative encoding is evaluated by both the mean square error (MSE) and the symbol error rate between a given integer vector b and the inversely generated integer vector from the codeword of b. Numerical experiments show that the convergence of encoding for SC-LDLC is 3 times faster than that of the conventional LDLC, at an MSE of 10-10for dimension n = 10000.

Spatially-coupled low-density parity-check codes attract much attention due to their capacity-achieving performance and a memory-efficient sliding-window decoding algorithm. On the other hand, the encoder needs to solve large linear equations to terminate the encoding process. In this paper, we propose modified spatially-coupled codes. The modified (dl, dr, L) codes have less rate-loss, i.e., higher coding rate, and have the same threshold as (dl, dr, L) codes and are efficiently terminable by using an accumulator.

We study fountain codes transmitted over the binary-input symmetric-output channel. For channels with small capacity, receivers in fountain coding systems needs to collects many channel outputs to recover information bits. Since a collected channel output yields a check node in the decoding Tanner graph, the channel with small capacity leads to large decoding complexity. In this paper, we introduce a novel fountain coding scheme with non-binary LDPC codes. The decoding complexity of the proposed fountain code does not depend on the channel. Numerical experiments show that the proposed codes exhibit better performance than conventional fountain codes, especially for moderate number of information bits.

In this paper, we consider quantum error correction over depolarizing channels with nonbinary low-density parity-check codes defined over Galois field of size 2p. The proposed quantum error correcting codes are based on the binary quasi-cyclic Calderbank, Shor, and Steane (CSS) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., 2p, the error floors are considerably improved.

The scaling law developed by Amraoui et al. is a powerful technique to estimate the block erasure probabilities of finite- length low-density parity-check (LDPC) codes. Solving a system of differential equations called covariance evolution, one can obtain the scaling parameter. However, the covariance evolution has not been analytically solved. In this paper, we present the analytical solution of the covariance evolution for irregular LDPC code ensembles.

For decoding non-binary low-density parity-check (LDPC) codes, logarithm-domain sum-product (Log-SP) algorithms were proposed for reducing quantization effects of SP algorithm in conjunction with FFT. Since FFT is not applicable in the logarithm domain, the computations required at check nodes in the Log-SP algorithms are computationally intensive. What is worse, check nodes usually have higher degree than variable nodes. As a result, most of the time for decoding is used for check node computations, which leads to a bottleneck effect. In this paper, we propose a Log-SP algorithm in the Fourier domain. With this algorithm, the role of variable nodes and check nodes are switched. The intensive computations are spread over lower-degree variable nodes, which can be efficiently calculated in parallel. Furthermore, we develop a fast calculation method for the estimated bits and syndromes in the Fourier domain.

In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels. We clarify a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the BP decoder. We expurgate non-binary LDPC code ensemble to analyze and to lower the error floor by using the above condition. Finally, we show upper and lower bounds of the error floors of the expurgated LDPC code ensemble over the MBIOS channel.

For designing low-density parity-check (LDPC) codes for quantum error-correction, we desire to satisfy the conflicting requirements below simultaneously. 1) The row weights of parity-check “should be large”: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices “should not be large”: The performance of the sum-product decoding algorithm at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled (SC) LDPC codes exhibit capacity-achieving performance for classical channels. SC LDPC codes have both large row weight and capacity-achieving error-floor and water-fall performance. In this paper, we propose a new class of quantum LDPC codes based on spatially coupled quasi-cyclic LDPC codes. The performance outperforms that of quantum “non-coupled” quasi-cyclic LDPC codes.

In this paper, we propose a construction method for two-level quantum error-correcting codes via non-binary LDPC codes over an extended field of order 2p, p an integer p >; 1. The proposed quantum error-correcting codes are based on binary quasi-cyclic LDPC codes which have almost achieved a “Bounded Distance Decoding (BDD)” limit but have not surpassed the limit yet. Quantum codes constructed from the proposed method surpass the BDD limit. Furthermore the codes outperform the efficiently-decodable state-of-the-art quantum codes.

Kudekar et al. recently proved that for transmission over the binary erasure channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the coupled ensemble to the MAP threshold of the underlying LDPC codes. One major drawback of the capacity-achieving spatially coupled LDPC codes is that one needs to increase the column and row weight of parity-check matrices of the underlying LDPC codes.

The fixed points of the belief propagation decoder for non-binary low-density parity-check (LDPC) codes are referred to as stopping constellations. In this paper, we give the stopping constellation distributions for the irregular non-binary LDPC code ensembles defined over the general linear group. Moreover, we derive the exponential growth rate of the average number of the stopping constellation distributions in the limit of large code length.

We consider spatially-coupled LDPC codes for the three terminal erasure relay channel. It is observed that BP threshold value of spatially-coupled LDPC codes, in particular spatially-coupled MacKay-Neal code, is close to the theoretical limit for the relay channel. Empirical results suggest that spatially-coupled LDPC codes have great potential to achieve theoretical limit of a general relay channel.

We propose nonbinary LDPC codes concatenated with multiplicative repetition codes. By multiplicatively repeating the (2,3)-regular nonbinary LDPC mother code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9, 1/12,.... Surprisingly, such simple low-rate nonbinary LDPC codes outperform the best low-rate binary LDPC codes so far. Moreover, we propose the decoding algorithm for the proposed codes, which can be decoded with almost the same computational complexity as that of the mother code.

Rate-compatible asymmetric Slepian-Wolf coding with non-binary LDPC codes of moderate code length is presented.The proposed encoder and decoder use only one single mother code.With the proposed scheme, better compressed rate and lower error rate than those ofconventional scheme are achieved with even smaller source length.

We propose non-binary LDPC codes concatenated with multiplicative repetition codes. To the best of the authors' knowledge, for the transmissions over the memoryless binary-input output-symmetric channels, 2m-ary the (2,dc)-regular LDPC code for m ~ 8 and dc≥ 3 is the best code so far among codes with moderate code length. By multiplicatively repeating the 2m-ary (2,3)-regular LDPC code of rate 1/3, we construct rate-compatible codes of lower rates 1/6,1/9,1/12,.... Surprisingly, such simple low-rate codes outperform the best low-rate binary codes so far.

In this paper, we analyze (2, k)-regular non-binary low-density parity-check codes over the binary erasure channels. We propose a method to improve the error floors by optimizing labels in zigzag cycles in the Tanner graph. We analyze the error floors for codes designed by the proposed optimization method and show that the error floors are decreasing in the size of Galois field.

We study the information reconciliation (IR) scheme for quantum key distribution (QKD) protocols. The IR for the QKD can be seen as the asymmetric Slepian-Wolf problem, which low-density parity-check (LDPC) codes can solve with efficient algorithms, i.e., the belief propagation. However, the LDPC codes are needed to be chosen properly from a collection of codes optimized for multiple key rates, which leads to complex decoder devices and performance degradation for unoptimized key rates. Therefore, it is desired that establish an IR scheme with a single LDPC code which supports multiple rates. To this end, in this paper, we propose an IR scheme with a rate-compatible non-binary LDPC code. Numerical results show the proposed scheme achieves IR efficiency comparable to the best know conventional IR scheme with lower decoding error rates.

Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.

For a (lambda(x); rho(x)) standard irregular LDPC code ensemble, the growth rate of the average weight distribution for small relative weight omega is given by log(lambda'(0)rho'(1))omega + O(omega2) in the limit of code length n. If lambda'(0)rho'(1) < 1, there exist exponentially few code words of small linear weight, as n tends to infinity. It is known that the condition coincides with the stability condition of density evolution over the erasure channels with the erasure probability 1. In this paper, we show that this is also the case with multi-edge type LDPC (MET-LDPC) codes. MET-LDPC codes are generalized structured LDPC codes introduced by Richardson and Urbanke. The parameter corresponding lambda'(0)rho'(1) appearing in the conditions for MET-LDPC codes is given by the spectral radius of the matrix defined by extended degree distributions.

The covariance evolution is a system of differential equations with respect to the covariance of the number of edges connecting to the nodes of each residual degree. Solving the covariance evolution, we can derive distributions of the number of check nodes of residual degree 1, which helps us to estimate the block error probability for finite-length LDPC code. Amraoui et al. resorted to numerical computations to solve the covariance evolution. In this paper, we give the analytical solution of the covariance evolution.

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. Furthermore, a gap between the bit error probability after finite number of iterations for finite block length n and that for infinite block length is asymptotically α/n, where α denotes a speci..c constant determined by a degree distribution, a number of iterations and erasure probability. Our main result is to derive an ef..cient algorithm for calculating α for regular ensembles.

Multi-Edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles has better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. We derive the condition corresponding to Orlitsky’s condition for two-edge type LDPC code ensembles which is simple example of Multi-Edge type LDPC code ensembles.

Luby et al. derived evolution of degree distributions in residual graphs for irregular LDPC code ensembles. Evolution of degree distributions in residual graphs is an important characteristic which is used for finite-length analysis of the expected block and bit error probabilities over the binary erasure channel. In this paper, we derive detailed evolution of degree distributions in residual graphs for irregular LDPC code ensembles with joint degree distributions.

Upper bounds of minimum distance distributions of Gallger codes and irregular LDPC codes were derived by Callage and Di, respectively. Di's bounds are tight for irregular LDPC codes which have variable nodes of degree two, however, it is not tight for irregular LDPC codes which do not. In this paper, we derive tight lower and upper bounds of minimum distance distributions of irregular LDPC code ensembles without variable nodes of degree two.

In this paper, we present a new class of low-density MDS array codes for tolerating double disk failures in disk arrays. The proposed MDS array code has lower encoding and decoding complexity than the EVENODD code of Blaum et al

The support weight distribution of a code is the number of unique subspaces of the code with specified dimension and support weight. In this paper, we formulate the average second support weight distribution and its asymptotic exponent of regular LDPC code ensembles

In this paper, we have derived the upper bound of the average block error probability of a given detailedly represented irregular low-density parity-check (LDPC) code ensemble under maximum likelihood decoding.

In this paper, we propose a method that recovers a smooth high-resolution image from several blurred and roughly quantized low-resolution images. For compensation of the quantization effect we introduce a measurement of smoothness originally used for suppression of block noises in a JPEG compressed image [Schultz & Stevenson '94]. With a simple operator that approximates to the convex projection onto constraint set defined for each quantized image [Hasegawa et al. '05], we propose a method that minimizes these cost functions, which are smooth convex functions, over the intersection of all constraint sets, i.e. the set of all images satisfying all quantization constraints simultaneously, by using hybrid steepest descent method [Yamada & Ogura '04]. Finally in the numerical example we compare images derived by the proposed method, POCS based conventional method, and generalized proposed method minimizing smoothed total variation and energy of output of Laplacian

We present a novel adaptive super-resolution of videos based on an embedded constraint version of adaptive projected subgradient method (Yamada & Ogura, Numerical Functional Analysis and Optimization, vol 25, no.7&8, p.593-617, 2004). The super-resolution image recovery problem is formulated as an estimation of linear time-varying systems, which is a modified version of (Elad & Feuer, IEEE Trans. on Image Proc., vol.8, no.3, p.387-395, 1999). Our method efficiently improves the estimation accuracy by simple iterative operations which can be processed on parallel systems. Robustness to additive noise as well as inaccurate estimation of degradation parameters, is realized by incorporating stochastic information of the noise.

An efficient scheme is presented to the color super-resolution problem for recovery of a color high-resolution image with knowledge of multiple Bayer filtered low-resolution images. To recover a visually natural high-resolution image, we restrict fairly smooth initial candidates to all images satisfying all bounds imposed on the several nonsmooth convex color total variations as well as a non-smooth convex inter cross correlation measure among color channels. In the proposed scheme, the data-fidelity is optimized in a systematic way, over all initial candidates, with the hybrid steepest descent method for quasi-nonexpansive mappings [Yamada & Ogura 2004], by minimizing successively an weighted average of pure mean square errors between the low-resolution transforms of the high-resolution estimate and the multiple low-resolution images. Numerical examples show that the proposed scheme recovers visually natural high resolution images by resolving the tradeoff between noise suppression and edge preservation of the recovered image while keeping fair inter channel cross correlation among color channels.

In this paper we propose an algebraic construction of efficiently encodable irregular LDPC codes. The proposed irregular LDPC codes have not only an efficient encoding algorithm but also guaranteed minimum distances. Simulation results show that the proposed codes perform well compared to randomly constructed irregular LDPC codes

In this paper, we explicitly formulate the average stopping set distributions and their asymptotic exponents of two instances of two-edge type LDPC code ensembles. Further we investigate the relation between the asymptotic exponents of those two code ensembles

Low-density parity-check (LDPC) codes are one of the most promising next generation error correcting codes and many investigations shows that LDPC codes suitable for many hardware implementation. Although randomly constructed LDPC codes are usually encoded by using generator matrix, this method requires quadratic time complexity and is not easy to implement. This work presents the encoding of array-type LDPC codes and a special class of Sridhara-Fuja-Tanner (SFT) codes by division circuits as cyclic codes, which are very easy to implement.

This work presents an ensemble of irregular low-density parity check (LDPC) codes based on an ensemble of bipartite graphs, which formulated the average weight distribution. An analysis of the error performance of LDPC codes over a binary erasure channel together with an iterative decoding algorithm based on belief propagation has been clarified that a notion of stopping sets and their distribution with the asymptotic expression are derived. An irregular LDPC code ensemble that exhibits better performance in the sense of threshold is obtained by density evolution and explicitly formulates weight and stopping set distributions are defined.

In this paper, we present a novel iterative MPEG super-resolution method based on an embedded constraint version of adaptive projected subgradient method [Yamada & Ogura 2003]. We propose an efficient operator that approximates convex projection onto a set characterizing framewise quantization, whereas a conventional method can only handle a convex projection defined for each DCT coefficient of a frame. By using the operator, the proposed method generates a sequence that efficiently approaches to a solution of super-resolution problem defined in terms of quantization error of MPEG compression.

In this paper, we propose a novel higher order time-frequency distribution (GDH) for discrete time signals. This distribution is defined over the original discrete time-frequency grids through a delicate discretization of an equivalent expression of a higher order distribution, for continuous time signals, in [Fonollosa & Nikias 1993]. We also present a constructive design method, for the kernel of the GDH, by which the distribution satisfies (i) the alias free condition as well as (ii) the marginal conditions. A numerical example shows that the proposed distribution reasonably suppresses the artifacts which are observed severely in a simple higher order generalization of the Wigner distribution.

It is well-known that the unbiased estimate of the covariance matrix of a real vector wide sense stationary process is not necessarily positive semidefinite. By defining the real Hilbert space of all symmetric matrices, the conditions for a symmetric matrix to be positive definite, block Toeplitz, as well as to satisfy other design constraints, are formed as closed convex sets. This paper demonstrates that the problem of approximating the unbiased estimate of the covariance matrix of a real vector wide sense stationary process over the intersection of those closed convex sets in an optimal way can be resolved by the Hybrid Steepest Descent Method. An optimal solution is also provided even when inconsistent constraints are met, i.e., whenever the intersection of the closed convex sets is empty. The numerical results exhibit significant improvement of the proposed method over the standard estimates of the covariance matrix.

We propose a simple set-theoretic blind deconvolution scheme based on a previously developed convex projection technique called hybrid steepest descent methods. The scheme is essentially motivated by Kundur and Hatzinakos's (see IEEE Signal Processing Magazine, vol.13, no.3, p.43-63, 1996 and IEEE Trans. Signal Processing, vol.46, p.375-90, 1998) idea that minimizes a certain cost function uniformly reflecting all a priori information such as the (i) nonnegativity of the true image and the (ii) support size of the original object. The most remarkable feature of the proposed scheme is that one can utilize each a priori information separately from other ones, where some partial information are treated in a set theoretic sense while the others are incorporated in a cost function to be minimized.

Recently, a novel set-theoretic linear blind deconvolution scheme was developed by applying hybrid steepest descent method to Kundur and Hatzinakos' simple a priori information on the original object, where the performance of the scheme seems relatively sensitive to the additive measurement noise. In this paper, we remark some well-known nonlinear filtering techniques which realize immediate effect to suppress the influence of the additive measurement noise in the input to the scheme. Numerical examples show ϵ-separating nonlinear pre-filtering techniques work suitably to this noisy blind deconvolution problem.

The convex set feasibility framework has been widely applied to signal and image processing problems including signal deconvolution, tomographic reconstruction, band limited extrapolation, image restoration and image synthesis. In this paper, we consider convex constrained versions of inconsistent signal feasibility problems. First we derive some new properties of variational nonexpansive operators and convex projections. Based on these properties and fixed point theorems, we propose some types of algorithms called constrained parallel projection methods (CPPM) that solve the convex constrained versions of inconsistent signal feasibility problems.

Discrete linear time-varying (TV) systems have been playing an important role in many real-time signal processing and control systems. The stability criteria of these TV systems require information of the impulse response sequence of the state transition matrices in an infinite time interval which is often difficult to obtain in practice. This paper reports some results on this problem. We first presents sufficient conditions for exponential, l/sup p/ and l/sup p/-BIBO stabilities. A detailed analysis is then given on this condition and the distribution domain of the instantaneous poles meeting the condition.>

Necessary and sufficient conditions for t-unidirectional error correcting and d-unidirectional error detecting (t-UEC and d-UED) codes are shown. In addition, an error in a theorem previously published on t-UEC and d-UED codes (see D.J. Lin and B. Bose, IEEE Trans. Comput., vol.37, p.433-39, Apr. 1988) is corrected.>

This paper presents a hierarchical classification of signals based on their smoothness. By this hierarchical classification, we can obtain the class of bandlimited signals as an innermost signal class and the class of signals composed of differentiable and square integrable functions as the outermost class. Moreover, for each class of signals, we can define "minimum norm signal". The minimum norm signal is defined as the signal of minimum norm which takes specified sample values on a set of given sampling points. By making use of the minimum norm signal, we can construct a unified and efficient approximation method for all these classes of signals. The method has the following special features: i) it is free from numerical integration error, ii) the sequence of approximate signals is guaranteed to uniformly converge to the desired signal as the number of sampling points is increased infinitely.