Exploding the Nyquist barrier misconception
TL;DR: The primary purpose of this contribution is to expose a fundamental misconception regarding the universality of the sampling theorem as taught in most digital signal processing textbooks.
Abstract: The author comments that Smith (IEEE Signal Processing Magazine, Forum Feedback, May 1996) continues to proclaim the novelty of an approach (Smith, 1995) that he purports to "break the Nyquist barrier," in spite of the revelation (Marple Jr., 1996) that his approach is simply a special two-filter case of well known analysis and synthesis filter banks performed with sample-and-hold waveforms. Smith's Fig.4(b) in Smith (1995) can be compared with the conventional filter banks of Fig.4 in Marple Jr. (1996) and it is observed that they are identical. Smith also makes further observations to which the present author responds with additional commentary. However, the primary purpose of this contribution is to expose a fundamental misconception regarding the universality of the sampling theorem as taught in most digital signal processing textbooks. It is this misconception that led Smith to prematurely claim victory over a perceived impenetrable Nyquist barrier.
References
More filters
TL;DR: The sampling of bandpass signals is discussed with respect to band position, noise considerations, and parameter sensitivity, and it is shown that the minimum sampling rate is pathological in that any imperfection in the implementation will cause aliasing.
Abstract: The sampling of bandpass signals is discussed with respect to band position, noise considerations, and parameter sensitivity. For first-order sampling, the acceptable and unacceptable sample rates are presented, with specific discussion of the practical rates which are nonminimum. It is shown that the minimum sampling rate is pathological in that any imperfection in the implementation will cause aliasing. In applying bandpass sampling to relocate signals to a base-band position, the signal-to-noise ratio is not preserved owing to the out-of-band noise being aliased. The degradation in signal-to-noise ratio is quantified in terms of the position of the bandpass signal. For the construction of a bandpass signal from second-order samples, the cost of implementing the interpolant (dynamic range and length) depends on Kohlenberg's sampling factor (1953) k, the relative delay between the uniform sampling streams. An elaboration on the disallowed discrete values of k shows that some allowed values are better than others for implementation. >
864 citations
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01 Jan 1975TL;DR: This paper presents a meta-modelling procedure called “Smart Card” which automates the very laborious and therefore time-heavy and expensive and expensive process of manually cataloging and cataloging the components of a computer.
Abstract: Review of least squares, orthogonality and the Fourier series review of continuous transforms transfer functions and convolution sampling and measurement of signals the discrete Fourier transform the fast Fourier transform the z-transform non-recursive digital systems digital and continuous systems simulation of continuous systems analogue and digital filter design review of random functions correlation and power spectra least-squares system design random sequences and spectral estimation.
286 citations
TL;DR: In this paper, the analysis and design of digital filter banks composed of equally spaced bandpass filters is discussed. And the results are extended to more general filter bank configurations, and it is shown that significant improvement in the composite filter bank response can be achieved by proper choice of the relative phases of the bandspass filters.
Abstract: A bank of bandpass filters is often used in performing short-time spectrum analysis of speech signals. This paper is concerned with the analysis and design of digital filter banks composed of equally spaced bandpass filters. It is shown that significant improvement in the composite filter bank response can be achieved by proper choice of the relative phases of the bandpass filters. The results are extended to more general filter bank configurations.
34 citations
TL;DR: It is shown that significant improvement in the frequency response of the composite filter bank can be achieved by appropriate choice of the relative phases of the bandpass filters.
Abstract: Short‐time spectrum analysis is the basis for many speech analysls systems. Although the fast Fourier transform is generally used to perform spectrum analysis on a general purpose computer, a bank of recursire digital bandpass filters may be the best approach for hardware realizations. This paper discusses the analysis and design of digital filter banks composed of equal‐bandwidth, equally spaced, bandpass filters. It is shown that significant improvement in the frequency response of the composite filter bank can be achieved by appropriate choice of the relative phases of the bandpass filters. Also discussed is an efficient general purpose computer simulation of a bank of recursire digital filters as required, for example, in a phase vocoder analyzer [Flanagan and Golden, Bell Syst. Tech. J. (Nov. 1966), This simulation uses the fast Fourier transform to compute filter outputs at a low sampling rate (approximately 100 Hz). For synthesis, the spectrum parameters are interpolated to a 10‐kHz sampling rate u...
12 citations
TL;DR: Signal processing analysis and simulation software tools should be used knowledgably for purposes of productivity enhancement, and should not be used blindly without the capability to determine when the answer provided by the tool “looks right.”
Abstract: Restoring the Nyquist Barrier “Results of data analyzed by software simulation tools are meaningless.” This was my first impression after reading the SP Lite article “Breaking the Nyquist Bmier” by Lynn Smith in the July 1995 issue [ 11. This article contains a number of fundamental conceptual errors upon which I shall comment. The author has also rediscovered filter banks, despite extensive published art on this topic. However, beyond these conceptual and rediscovery issues, I was most struck by the dependence of the author on the use of a software simulation tool to justify the author’s erroneous conclusions without an apparent full understanding of the graphical results that the tool produced. The Smith article reinforces a concern that I have been expressing to my colleagues in academia regarding the extensive use of DSP software simulation tools in virtual signal environments as a means for teaching signal processing. A selection of DSP software tools were highlighted in the article by Ebel and Younan 171 that appeared in the November 1995 IEEE Signal Processing Magazine, an issue dedicated, coincidentally, to signal processing education. Specifically, there appears to be a growing dependence on these tools with canned experiments that fails to adequately prepare many students for solving real world signal processing problems. This is most manifest during technical interviews that I often conduct with new graduates who are candidates for employment. Without access to software tools during the interview, I have observed with increasing incidence that these graduates, when presented with situations involving typical signal processing applications of importance to my employer, are unable to confidently propose signal processing operations using only knowledge of basic signal processing principles. The most evident difficulty has been their inability to relate properties of continuous time-domain and spatial-domain signals with discrete-domain digital representations of and operations on those signals. Mathematical normalization of parameters (for example, the assumption of an unity sampling rate, or expressing frequency in radian units) often utilized in academic treatments of signal processing operations also handicaps students in forming an intuitive sense of time and frequency scale when confronted with actual signals and their transforms. Signal processing analysis and simulation software tools should be used knowledgably for purposes of productivity enhancement, and should not be used blindly without the capability to determine when the answer provided by the tool “looks right.” This viewpoint is reminiscent of the debate concerning the introduction of hand calculators in public schools, in which it was argued whether hand calculators should be used by students as a substitute before learning the mathematical operations performed by the calculators or should be used only as productivity aids after they had substantial experience with the mathematical operations. I would now like to demonstrate, by use of first principles, “restoration” of the Nyquist bamer of the demonstration signal used in the Smith article [l] by showing that it was never broken in the first place. I will do this armed only with four basic waveforms (depicted in Fig. l), their transforms, and two variants of the convolution theorm. Specifically, if x(t) wX(f) designates the Fourier transform relationship between the temporal waveform x(t) and its Fourier transform X(f), while y(t) -Y(f) designates the Fourier transform relationship between
5 citations