##
**The sampling theorem and linear prediction in signal analysis.**
*(English)*
Zbl 0633.94002

In 1979, a standard for the digital storage and the reproduction of audio signals, known as the CD-ROM system, has been defined. The CD-ROM system which forms one of the most remarkable achievements of modern High Tech has become the world standard for achieving fidelity of sound reproduction that far surpasses any other available technique. A metallic disc 120 mm in diameter is used to store the digitized audio waveform. The waveform is sampled at 44.1 kilosamples/s to provide a recorded bandwidth of 20 kHz; each audio sample is uniformly quantized to one of \(2^{16}\) levels (16 bits/sample). Operating a CIRC encoder, the recording equipment stores on a single DM-ROM about \(10^{10}\) bits in the form of a sequence of minute pits that are optically scanned by a laser beam.

The link between an analog waveform and its sampled version is provided by the sampling procedure. This procedure can be implemented in several ways, the most popular being the sample-and-hold operation. The output of the sampling process is called pulse amplitude modulation (PAM) because the successive output intervals can be described as a sequence of pulses with amplitudes derived from the input waveform samples. The analog waveform can be approximately retrieved from a PAM waveform by low-pass filtering. Now the basic question of D/A conversion reads: How closely can a filtered PAM waveform approximate the original input waveform? The question can be answered by reviewing the sampling theorem, which roughly states: A bandlimited signal having no spectral components above \(f_ m\) hertz can be determined uniquely by values sampled at uniform intervals of \(T_ s\) seconds, where \(T_ s\leq 1/(2f_ m).\) This particular statement is also known as the Whittaker-Shannon-Kotel’nikov uniform sampling theorem. Stated another way, the upper limit on \(T_ s\) can be expressed in terms of the sampling rate, denoted \(f_ s=1/T_ s\). The restriction \(f_ s\geq 2f_ m\) is known as the Nyquist criterion. It forms a theoretically sufficient condition to allow an analog signal to be reconstructed completely from a sequence of uniformy spaced discrete- time samples. - For survey articles on the matter see A. J. Jerri: The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE 65, 1565-1596 (1977), the more recent paper by J. R. Higgins: Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12, 45-89 (1985), and finally the important report by R. B. Holmes entitled “Mathematical foundations of signal processing II. The rôle of group theory” (Lincoln Laboratory, Massachusetts Inst. Techn., Lexington, MA 1987) which includes some of the more advanced mathematical developments in this area.

The paper under review presents a detailed survey of the results in the area of the sampling theorem obtained at the Lehrstuhl A für Mathematik of the RWTH Aachen in cooperation with electrical and communication engineers during the last decade and published in about one hundred of papers. Special emphasis is laid on the links to approximation and interpolation theory, commutative harmonic analysis, and linear prediction of signals. Indeed, the cardinal series of the classical Whittaker-Shannon-Kotel’nikov sampling theorem involves the perfect low- pass filter sinc and therefore forms the starting point of cardinal spline approximation theory. For linear prediction in terms of samples from the past, the samples need no longer be uniformly spaced on the whole real line R but only on the (negative) half-line \(R_-\). Another important point of the paper are the various types of error estimates relevant for digital communication. The authors investigates the aliasing error arising when the signal is not exactly bandlimited, the truncation error, the amplitude error due to quantization, the rounding or noise error, and finally the time-jitter error. Most of the results of this carefully written expository paper are provided with proofs, an advantage which is by no means common to papers at the crossroads of mathematics and electrical engineering.

The reconstruction of the analog waveform from its sampled version frozen in the CD-ROM is performed sequentially according to the very nature of the classical Whittaker-Shannon-Kotel’nikov sampling theorem. Another technological disadvantage is the fact that the storing capacity of the CD-ROM is not sufficient for high quality digital image signal processing. A new idea directed towards optical signal processing which is inherently of a parallel nature, depends upon an embedding of the one- dimensional lattice of the classical sampling theorem into a two- dimensional lattice frozen in the holographic plane C. In this way, the CD-ROM may be considered as a one-dimensional hologram and the CD player as a holograph operating serially a CIRC decoder and an oversampling processor. Since the holographic transform includes intrinsically the phase conjugation of optical signals, hence the distortion-correction process or “healing effect” of wavefront reversal, the parallel approach forms an elegant alternative of great promise for application to image transmission and processing. For details of this extension of the sampling procedure, see the forthcoming article by P. Greguss “Towards opto-biological computers”, and the reviewer’s survey paper entitled “The holographic transform and the neural model” (in press).

The link between an analog waveform and its sampled version is provided by the sampling procedure. This procedure can be implemented in several ways, the most popular being the sample-and-hold operation. The output of the sampling process is called pulse amplitude modulation (PAM) because the successive output intervals can be described as a sequence of pulses with amplitudes derived from the input waveform samples. The analog waveform can be approximately retrieved from a PAM waveform by low-pass filtering. Now the basic question of D/A conversion reads: How closely can a filtered PAM waveform approximate the original input waveform? The question can be answered by reviewing the sampling theorem, which roughly states: A bandlimited signal having no spectral components above \(f_ m\) hertz can be determined uniquely by values sampled at uniform intervals of \(T_ s\) seconds, where \(T_ s\leq 1/(2f_ m).\) This particular statement is also known as the Whittaker-Shannon-Kotel’nikov uniform sampling theorem. Stated another way, the upper limit on \(T_ s\) can be expressed in terms of the sampling rate, denoted \(f_ s=1/T_ s\). The restriction \(f_ s\geq 2f_ m\) is known as the Nyquist criterion. It forms a theoretically sufficient condition to allow an analog signal to be reconstructed completely from a sequence of uniformy spaced discrete- time samples. - For survey articles on the matter see A. J. Jerri: The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE 65, 1565-1596 (1977), the more recent paper by J. R. Higgins: Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12, 45-89 (1985), and finally the important report by R. B. Holmes entitled “Mathematical foundations of signal processing II. The rôle of group theory” (Lincoln Laboratory, Massachusetts Inst. Techn., Lexington, MA 1987) which includes some of the more advanced mathematical developments in this area.

The paper under review presents a detailed survey of the results in the area of the sampling theorem obtained at the Lehrstuhl A für Mathematik of the RWTH Aachen in cooperation with electrical and communication engineers during the last decade and published in about one hundred of papers. Special emphasis is laid on the links to approximation and interpolation theory, commutative harmonic analysis, and linear prediction of signals. Indeed, the cardinal series of the classical Whittaker-Shannon-Kotel’nikov sampling theorem involves the perfect low- pass filter sinc and therefore forms the starting point of cardinal spline approximation theory. For linear prediction in terms of samples from the past, the samples need no longer be uniformly spaced on the whole real line R but only on the (negative) half-line \(R_-\). Another important point of the paper are the various types of error estimates relevant for digital communication. The authors investigates the aliasing error arising when the signal is not exactly bandlimited, the truncation error, the amplitude error due to quantization, the rounding or noise error, and finally the time-jitter error. Most of the results of this carefully written expository paper are provided with proofs, an advantage which is by no means common to papers at the crossroads of mathematics and electrical engineering.

The reconstruction of the analog waveform from its sampled version frozen in the CD-ROM is performed sequentially according to the very nature of the classical Whittaker-Shannon-Kotel’nikov sampling theorem. Another technological disadvantage is the fact that the storing capacity of the CD-ROM is not sufficient for high quality digital image signal processing. A new idea directed towards optical signal processing which is inherently of a parallel nature, depends upon an embedding of the one- dimensional lattice of the classical sampling theorem into a two- dimensional lattice frozen in the holographic plane C. In this way, the CD-ROM may be considered as a one-dimensional hologram and the CD player as a holograph operating serially a CIRC decoder and an oversampling processor. Since the holographic transform includes intrinsically the phase conjugation of optical signals, hence the distortion-correction process or “healing effect” of wavefront reversal, the parallel approach forms an elegant alternative of great promise for application to image transmission and processing. For details of this extension of the sampling procedure, see the forthcoming article by P. Greguss “Towards opto-biological computers”, and the reviewer’s survey paper entitled “The holographic transform and the neural model” (in press).

Reviewer: Walter Schempp

### MSC:

94A05 | Communication theory |

94A15 | Information theory (general) |

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

60G25 | Prediction theory (aspects of stochastic processes) |

62M20 | Inference from stochastic processes and prediction |

93E10 | Estimation and detection in stochastic control theory |

41A45 | Approximation by arbitrary linear expressions |

41A15 | Spline approximation |