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**Introduction to Shannon sampling and interpolation theory.**
*(English)*
Zbl 0729.94001

Springer Texts in Electrical Engineering. New York etc.: Springer-Verlag. xiii, 324 p. DM 114.00 (1991).

The Shannon sampling theorem is a fundamental result for both digital signal processing and mathematical interpolation and approximation theory. It states that a bandlimited signal function can be completely reconstituted from its equally spaced sampled values in terms of the Shannon sampling series (the cardinal series). During the last decades a lot of research has been done on error analysis and generalizations of the sampling series, often inspired by problems arisen in applications. There are some review articles surveying the many results in the field or collecting the vast amount of literature, e.g. that by A. J. Jerri [Proc. IEEE 65, No.11, 1565-1596 (1977; Zbl 0442.94002)], the five short stories by J. R. Higgins [Bull. Am. Math. Soc., New Ser. 12, 45-89 (1985; Zbl 0562.42002)], and the treatise by P. L. Butzer, W. Splettstößer and R. L. Stens [Jahresber. Dtsch. Math.-Ver. 90, 1-70 (1988; Zbl 0633.94002)].

The book by R. J. Marks II is the first text book dealing solely with sampling and interpolation theory based on the Shannon sampling theorem. The contents evolved from lecture notes for a course addressing graduate electrical engineering students. The book consists of seven chapters each of which contains a short introduction, a couple of selected exercises the solutions of which are comprised at the end of the book, and some specific references. The lists of references are far from being complete, but a sequel of this book is already announced, which will contain an extensive bibliography on the subject.

The first chapter introduces the (Shannon) cardinal series and gives a (too) short overview on the history. It is followed by a chapter summarizing fundamentals of Fourier analysis and stochastic processes. The third chapter is entitled ‘The cardinal series’; it contains some of the known proofs of the sampling theorem, properties connected with convergence, trapezoidal integration, as well as an application to bandlimited stochastic processes.

In the fourth chapter a number of generalizations of the sampling theorem is presented, e.g. Kramer’s generalization, generalizations due to Papoulis, oversampling, derivative sampling, nonuniform sampling, bandpass sampling, to mention a few of them. Error sources are the topic of the fifth chapter. It is arranged in three sections dealing with effects of additive noise, jitter error and the truncation error, where both deterministic and random signals are treated. Unfortunately, there is no special section on the aliasing error arising in case the signal function is not exactly bandlimited.

The sixth chapter deals with sampling representations of signals with multidimensional domain. After a short introduction to multidimensional Fourier analysis the multidimensional sampling theorem is investigated, thereby discussing the effects of different shapes of the support of the transform to the sampling matrix or geometry. In this chapter the problems arising with sample decimation and restoring of lost samples are also dealt with.

The last chapter is devoted to so-called continuous sampling, that means, algorithms are discussed by means of which bandlimited functions can be reconstructed, if only one or more parts of them (given on continuous intervals) are known. The well-known Gerchberg-Papoulis extrapolation algorithm is one example considered here.

As already mentioned the book is finished with an appendix containing solutions to selected problems. Also a five-page index is provided. The present book can be recommended for students of electrical engineering and mathematics who have a basic knowledge of Fourier analysis and the theory of random processes. It covers a great deal of the problems and theories connected to the Shannon sampling theorem. Although the presentation does not satisfy all requirements of mathematical strictness the book should be looked at by anybody who is going to prepare a course on sampling theory, or to enlarge his lectures on Fourier analysis or interpolation theory.

The book by R. J. Marks II is the first text book dealing solely with sampling and interpolation theory based on the Shannon sampling theorem. The contents evolved from lecture notes for a course addressing graduate electrical engineering students. The book consists of seven chapters each of which contains a short introduction, a couple of selected exercises the solutions of which are comprised at the end of the book, and some specific references. The lists of references are far from being complete, but a sequel of this book is already announced, which will contain an extensive bibliography on the subject.

The first chapter introduces the (Shannon) cardinal series and gives a (too) short overview on the history. It is followed by a chapter summarizing fundamentals of Fourier analysis and stochastic processes. The third chapter is entitled ‘The cardinal series’; it contains some of the known proofs of the sampling theorem, properties connected with convergence, trapezoidal integration, as well as an application to bandlimited stochastic processes.

In the fourth chapter a number of generalizations of the sampling theorem is presented, e.g. Kramer’s generalization, generalizations due to Papoulis, oversampling, derivative sampling, nonuniform sampling, bandpass sampling, to mention a few of them. Error sources are the topic of the fifth chapter. It is arranged in three sections dealing with effects of additive noise, jitter error and the truncation error, where both deterministic and random signals are treated. Unfortunately, there is no special section on the aliasing error arising in case the signal function is not exactly bandlimited.

The sixth chapter deals with sampling representations of signals with multidimensional domain. After a short introduction to multidimensional Fourier analysis the multidimensional sampling theorem is investigated, thereby discussing the effects of different shapes of the support of the transform to the sampling matrix or geometry. In this chapter the problems arising with sample decimation and restoring of lost samples are also dealt with.

The last chapter is devoted to so-called continuous sampling, that means, algorithms are discussed by means of which bandlimited functions can be reconstructed, if only one or more parts of them (given on continuous intervals) are known. The well-known Gerchberg-Papoulis extrapolation algorithm is one example considered here.

As already mentioned the book is finished with an appendix containing solutions to selected problems. Also a five-page index is provided. The present book can be recommended for students of electrical engineering and mathematics who have a basic knowledge of Fourier analysis and the theory of random processes. It covers a great deal of the problems and theories connected to the Shannon sampling theorem. Although the presentation does not satisfy all requirements of mathematical strictness the book should be looked at by anybody who is going to prepare a course on sampling theory, or to enlarge his lectures on Fourier analysis or interpolation theory.

Reviewer: W.Splettstößer (Kaarst)

### MSC:

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

94A11 | Application of orthogonal and other special functions |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

41A05 | Interpolation in approximation theory |

60G35 | Signal detection and filtering (aspects of stochastic processes) |