Evidence of Intermittent Cascades from Discrete Hierarchical Dissipation in Turbulence

We present the results of a search of log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number (2500) of three-dimensional fully developed turbulence. A simple dynamical representation of the Richardson-Kolmogorov cartoon of a cascade shows that standard averaging techniques erase by their very construction the possible existence of log-periodic corrections to scaling associated with a discrete hierarchy. To remedy this drawback, we introduce a novel ``canonical'' averaging that we test extensively on synthetic examples constructed to mimick the interplay between a weak log-periodic component and rather strong multiplicative and phase noises. Our extensive tests confirm the remarkable observation of statistically significant log-periodic corrections to scaling, with a prefered scaling ratio for length scales compatible with the value gamma = 2. A strong confirmation of this result is provided by the identification of up to 5 harmonics of the fundamental log-periodic undulations, associated with up to 5 levels of the underlying hierarchical dynamical structure. A natural interpretation of our results is that the Richardson-Kolmogorov mental picture of a cascade becomes a realistic description if one allows for intermittent births and deaths of discrete cascades at varying scales.Comment: Latex document of 40 pages, including 18 eps figure

Stabilization of (L,M) shift invariant plant

In this paper, a lifting technique is employed to realize a single input single output linear (L,M) shift invariant plant as a filter bank system. Based on the filter bank structure, a controller is designed so that the aliasing components in the control loop are cancelled and the loop gain becomes a time invariant transfer function. Pole placement technique is applied to stabilize the overall system and ensure the causality of the filters in the controller. An example on the control of a linear (L,M) shift invariant plant with simulation result is illustrated. The result shows that our proposed algorithm is simple and effective

New Evidence of Discrete Scale Invariance in the Energy Dissipation of Three-Dimensional Turbulence: Correlation Approach and Direct Spectral Detection

We extend the analysis of [Zhou and Sornette, Physica D 165, 94-125, 2002] showing statistically significant log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number ($\approx 2500$) of three-dimensional fully developed turbulence. First, we develop a simple variant of the canonical averaging method using a rephasing scheme between different samples based on pairwise correlations that confirms Zhou and Sornette's previous results. The second analysis uses a simpler local spectral approach and then performs averages over many local spectra. This yields stronger evidence of the existence of underlying log-periodic undulations, with the detection of more than 20 harmonics of a fundamental logarithmic frequency $f = 1.434 \pm 0.007$ corresponding to the preferred scaling ratio $\gamma = 2.008 \pm 0.006$.Comment: 9 RevTex4 papes including 8 eps figure

Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

Improving subband spectral estimation using modified AR model

It has already been shown that spectral estimation can be improved when applied to subband outputs of an adapted filterbank rather than to the original fullband signal. In the present paper, this procedure is applied jointly to a novel predictive autoregressive (AR) model. The model exploits time-shifting and is therefore referred to as time-shift AR (TSAR) model. Estimators are proposed for the unknown TS-AR parameters and the spectrum of the observed signal. The TS-AR model yields improved spectrum estimation by taking advantage of the correlation between subseries that after decimation. Simulation results on signals with continuous and line spectra that demonstrate the performance of the proposed method are provided

Design, fabrication, and characterization of deep-etched waveguide gratings

One-dimensional (1-D) deep-etched gratings on a specially grown AlGaAs wafer were designed and fabricated. The gratings were fabricated using state-of-the-art electron beam lithography and high-aspect-ratio reactive ion etching (RIE) in order to achieve the required narrow deep air slots with good accuracy and reproducibility. Since remarkable etch depths (up to 1.5 /spl mu/m), which completely cut through the waveguide core layer, have been attained, gratings composed of only five periods (and, thus, shorter than 6 /spl mu/m) have a bandgap larger than 100 nm. A defect was introduced by increasing the width of the central semiconductor tooth to create microcavities that exhibit a narrow transmission peak (less than 7 nm) around the wavelength of 1530 nm. The transmission spectra between 1460 and 1580 nm have been systematically measured, and the losses have been estimated for a set of gratings, both with and without a defect, for different periods and air slot dimensions. Numerical results obtained via a bidirectional beam propagation code allowed the evaluation of transmissivity, reflectivity, and diffraction losses. By comparing experimental results with the authors' numerical findings, a clear picture of the role of the grating's geometric parameters in determining its spectral features and diffractive losses is illustrated