The processing of periodically sampled multidimensional signals
01 Feb 1983-IEEE Transactions on Acoustics, Speech, and Signal Processing (IEEE)-Vol. 31, Iss: 1, pp 188-194
TL;DR: Algorithms for processing multidimensional signals which are sampled on regular, but nonrectangular sampiing lattices, and how generalized decimators and interpolators can be used to convert from one sampling lattice to another are discussed.
Abstract: This paper discusses algorithms for processing multidimensional signals which are sampled on regular, but nonrectangular sampiing lattices. Such sampling lattices are dictated by some applications and may be chosen for others because of their resulting symmetric responses or computational efficiencies. We show that any operation which can be performed on a rectangular lattice can be performed on any regular periodic lattice, including FIR and IIR filtering, DFT calculation, and decimation and interpolation. This paper also discusses how generalized decimators and interpolators can be used to convert from one sampling lattice to another.
Citations
More filters
TL;DR: A directionally oriented 2-D filter bank with the property that the individual channels may be critically sampled without loss of information is introduced and it is shown that these filter bank outputs may be maximally decimated to achieve a minimum sample representation in a way that permits the original signal to be exactly reconstructed.
Abstract: The authors introduce a directionally oriented 2-D filter bank with the property that the individual channels may be critically sampled without loss of information. The passband regions of the component filters are wedge-shaped and thus provide directional information. It is shown that these filter bank outputs may be maximally decimated to achieve a minimum sample representation in a way that permits the original signal to be exactly reconstructed. The authors discuss the theory for directional decomposition and reconstruction. In addition, implementation issues are addressed where realizations based on both recursive and nonrecursive filters are considered. >
911 citations
Book•
11 Sep 2003
TL;DR: In this article, the benefits of channel coding and space time coding in the context of various application examples and features numerous complete system design examples are discussed. But the authors do not discuss the trade-off between channel quality fluctuations and frequency domain spreading codes.
Abstract: From the Publisher:
Orthogonal frequency-division multiplexing (OFDM) is a method of digital modulation in which a signal is split into several narrowband channels at different frequencies.
CDMA is a form of multiplexing, which allows numerous signals to occupy a single transmission channel, optimising the use of available bandwidth. Multiplexing is sending multiple signals or streams of information on a carrier at the same time in the form of a single, complex signal and then recovering the separate signals at the receiving end.
Multi-Carrier (MC) CDMA is a combined technique of Direct Sequence (DS) CDMA (Code Division Multiple Access) and OFDM techniques. It applies spreading sequences in the frequency domain.
Wireless communications has witnessed a tremendous growth during the past decade and further spectacular enabling technology advances are expected in an effort to render ubiquitous wireless connectivity a reality.
This technical in-depth book is unique in its detailed exposure of OFDM, MIMO-OFDM and MC-CDMA. A further attraction of the joint treatment of these topics is that it allows the reader to view their design trade-offs in a comparative context.
Divided into three main parts:
Part I provides a detailed exposure of OFDM designed for employment in various applications
Part II is another design alternative applicable in the context of OFDM systems where the channel quality fluctuations observed are averaged out with the aid of frequency-domain spreading codes, which leads to the concept of MC-CDMA
Part III discusses how to employ multiple antennas at the base station for the sake of supporting multiple users in the uplink
Portrays theentire body of knowledge currently available on OFDMProvides the first complete treatment of OFDM, MIMO(Multiple Input Multiple Output)-OFDM and MC-CDMAConsiders the benefits of channel coding and space time coding in the context of various application examples and features numerous complete system design examplesConverts the lessons of Shannon's information theory into design principles applicable to practical wireless systemsCombines the benefits of a textbook with a research monograph where the depth of discussions progressively increase throughout the book
This all-encompassing self-contained treatment will appeal to researchers, postgraduate students and academics, practising research and development engineers working for wireless communications and computer networking companies and senior undergraduate students and technical managers.
743 citations
21 Apr 1997
TL;DR: The discrete shift-variant 2-D Wiener filter is derived and analyzed given an arbitrary sampling grid, an arbitrary (but possibly optimized) selection of observations, and the possibility of model mismatch to reveal the potential of pilot-symbol-aided channel estimation in two dimensions.
Abstract: The potential of pilot-symbol-aided channel estimation in two dimensions are explored. In order to procure this goal, the discrete shift-variant 2-D Wiener filter is derived and analyzed given an arbitrary sampling grid, an arbitrary (but possibly optimized) selection of observations, and the possibility of model mismatch. Filtering in two dimensions is revealed to outperform filtering in just one dimension with respect to overhead and mean-square error performance. However, two cascaded orthogonal 1-D filters are simpler to implement and shown to be virtually as good as true 2-D filters.
724 citations
Cites background from "The processing of periodically samp..."
...The sampling theorem also holds for multidimensional processes and signals [1]....
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TL;DR: The pilot-symbol-aided parameter estimation for orthogonal frequency division multiplexing (OFDM) systems is highly robust to Doppler frequency for dispersive fading channels with noise impairment even though it has some performance degradation for systems with lower Dopple frequencies.
Abstract: In this paper, we investigate pilot-symbol-aided parameter estimation for orthogonal frequency division multiplexing (OFDM) systems. We first derive a minimum mean-square error (MMSE) pilot-symbol-aided parameter estimator. Then, we discuss a robust implementation of the pilot-symbol-aided estimator that is insensitive to channel statistics. From the simulation results, the required signal-to-noise ratios (SNRs) for a 10% word error rate (WER) are 6.8 dB and 7.3 dB for the typical urban (TU) channels with 40 Hz and 200 Hz Doppler frequencies, respectively, and they are 8 dB and 8.3 dB for the hilly-terrain (HT) channels with 40 Hz and 200 Hz Doppler frequencies, respectively. Compared with the decision-directed parameter estimator, the pilot-symbol-aided estimator is highly robust to Doppler frequency for dispersive fading channels with noise impairment even though it has some performance degradation for systems with lower Doppler frequencies.
671 citations
Cites background from "The processing of periodically samp..."
...It is demonstrated in [21] and [24] that any regular grid in the 2-...
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TL;DR: In this article, the cardinal series and LCA groups were derived from the L 2 and L p theory and extended to higher dimensions, and the cardinal and allied series were derived.
Abstract: CONTENTS Introduction Story One. Historical notes Story Two. Some methods for deriving the cardinal and allied series Story Three. L 2 and L p theory Story Four. The cardinal series and LCA groups Story Five. Extensions to higher dimensions Conclusion INTRODUCTION
348 citations
References
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Book•
[...]
TL;DR: Geometry of numbers is closely related to other branches of number theory such as algebraic number theory and Diophantine approximation and a flourishing offspring is discrete geometry, developed mainly by Fejes Toth and his school.
Abstract: The geometry. of numbers can be traced back at least to Lagrange [1773], who proved important results about quadratic forms in two variables. The proofs as well as the formulations of results were purely arithmetic. Reviewing a book of Seeber [1831B] on ternary quadratic forms, Gaus [1831] introduced for the first time geometric methods. Geometric methods were predominant in the work of Dirichlet [1850]. On the other hand Hermite [1850] and Korkine and Zolotareff [1872], [1873], [1877] gave arithmetic proofs for their results on quadratic forms in more than three variables. Finally Minkowski [1891] noticed that a simple geometric argument which he used to give a new proof of a theorem of Hermite could be adapted to much more general situations. Then Minkowski [1896B], [7B], [11B] started a systematic study of geometric methods in number theory and called this new branch of number theory geometry of numbers. Many results and most concepts of modern geometry of numbers have their origin in the work of Minkowski. After Minkowski many eminent mathematicians made contributions to this field. In order to avoid controversies I will not mention any of them. Geometry of numbers is closely related to other branches of number theory such as algebraic number theory and Diophantine approximation. A flourishing offspring is discrete geometry, developed mainly by Fejes Toth and his school.
727 citations
TL;DR: The well-known Whittaker-Kotel'nikov-Shannon sampling theorem for frequency-bandlimited functions of time is extended to functions of multidimensional arguments and it is shown that a function whose spectrum is restricted to a finite region of wave-number space may be reconstructed from its samples taken over a periodic lattice having suitably small repetition vectors.
Abstract: The well-known Whittaker-Kotel'nikov-Shannon sampling theorem for frequency-bandlimited functions of time is extended to functions of multidimensional arguments. It is shown that a function whose spectrum is restricted to a finite region of wave-number space may be reconstructed from its samples taken over a periodic lattice having suitably small repetition vectors. The most efficient lattice (i.e., requiring minimum sampling points per unit hypervolume) is not in general rectangular, nor is a unique reconstruction function associated with a given sampling lattice. The above results also apply to homogeneous wave-number-limited stochastic processes in the sense of a vanishing mean-square error. It is also found that, given a particular sampling lattice, the optimum (mean-square) presampling filter for nonwave-number-limited processes effects an ideal wave-number cutoff appropriate to the specified sampling lattice. Particular attention is paid to isotropic processes: minimum sampling lattices are specified up to eight-dimensional spaces, and a number of typical reconstruction functions are calculated.
648 citations
"The processing of periodically samp..." refers background or methods in this paper
...Then, if x(n) = X, (Vn), it can be shown using techniques enumerated in [ 3 ] that...
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...While most of this material appears in the key paper by Petersen and Middleton [ 3 ], a review of the major results is unavoidable since the sampling operation plays such a central role in what is to follow....
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01 Jun 1973
TL;DR: In this article, the authors examined the relative merits of finite-duration impulse response (FIR) and infinite duration impulse response(IIR) digital filters as interpolation filters and showed that FIR filters are generally to be preferred for interpolation.
Abstract: In many digital signal precessing systems, e.g., vacoders, modulation systems, and digital waveform coding systems, it is necessary to alter the sampling rate of a digital signal Thus it is of considerable interest to examine the problem of interpolation of bandlimited signals from the viewpoint of digital signal processing. A frequency dmnain interpretation of the interpolation process, through which it is clear that interpolation is fundamentally a linear filtering process, is presented, An examination of the relative merits of finite duration impulse response (FIR) and infinite duration impulse response (IIR) digital filters as interpolation filters indicates that FIR filters are generally to be preferred for interpolation. It is shown that linear interpolation and classical polynomial interpolation correspond to the use of the FIR interpolation filter. The use of classical interpolation methods in signal processing applications is illustrated by a discussion of FIR interpolation filters derived from the Lagrange interpolation formula. The limitations of these filters lead us to a consideration of optimum FIR filters for interpolation that can be designed using linear programming techniques. Examples are presented to illustrate the significant improvements that are obtained using the optimum filters.
643 citations
Additional excerpts
...[ 6 ] J. M. Wozencraft and M. I. Jacobs, Principles of Communication...
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01 Jun 1979
TL;DR: Methods for the processing of two- dimensional signals which have been sampled as two-dimensional hexagonal arrays are presented and some comparisons between the two methods for representing planar data will also be presented.
Abstract: Two-dimensional signals are normally processed as rectangularly sampled arrays; i.e., they are periodically sampled in each of two orthogonal independent variables. Another form of periodic sampling, hexagonal sampling, offers substantial savings in machine storage and arithmetic computations for many signal processing operations. In this paper, methods for the processing of two-dimensional signals which have been sampled as two-dimensional hexagonal arrays are presented. Included are methods for signal representation, linear system implementation, frequency response calculation, DFT calculation, filter design, and filter implementation. These algorithms bear strong resemblances to the corresponding results for rectangular arrays; however, there are also many important differences. Some comparisons between the two methods for representing planar data will also be presented.
393 citations
TL;DR: The Cooley-Tukey fast Fourier transform (FFT) algorithm is generalized to the multidimensional case in a natural way which allows for the evaluation of discrete Fourier transforms of rectangularly or hexagonally sampled signals or of signals which are sampled on an arbitrary periodic grid in either the spatial or Fourier domain.
Abstract: In this paper the Cooley-Tukey fast Fourier transform (FFT) algorithm is generalized to the multidimensional case in a natural way which allows for the evaluation of discrete Fourier transforms of rectangularly or hexagonally sampled signals or of signals which are sampled on an arbitrary periodic grid in either the spatial or Fourier domain. This general algorithm incorporates both the traditional rectangular row-column and vector-radix algorithms as special cases. This FFT algorithm is shown to result from the factorization of an integer matrix; for each factorization of that matrix, a different algorithm can be developed. This paper presents the general algorithm, discusses its computational efficiency, and relates it to existing multi-dimensional FFT algorithms.
73 citations