Nyquist frequency

Abstract: We present a straightforward and robust algorithm for periodicity detection, working in the lag (autocorrelation) domain. When it is tested for periodic signals and for signals with additive noise or jitter, it proves to be several orders of magnitude more accurate than the methods commonly used for speech analysis. This makes our method capable of measuring harmonics-to-noise ratios in the lag domain with an accuracy and reliability much greater than that of any of the usual frequency-domain methods. By definition, the best candidate for the acoustic pitch period of a sound can be found from the position of the maximum of the autocorrelation function of the sound, while the degree of periodicity (the harmonics-to-noise ratio) of the sound can be found from the relative height of this maximum. However, sampling and windowing cause problems in accurately determining the position and height of the maximum. These problems have led to inaccurate timedomain and cepstral methods for pitch detection, and to the exclusive use of frequency-domain methods for the determination of the harmonics-to-noise ratio. In this paper, I will tackle these problems. Table 1 shows the specifications of the resulting algorithm for two spectrally maximally different kinds of periodic sounds: a sine wave and a periodic pulse train; other periodic sounds give results between these. Table 1. The accuracy of the algorithm for a sampled sine wave and for a correctly sampled periodic pulse train, as a function of the number of periods that fit in the duration of a Hanning window. These results are valid for pitch frequencies up to 80% of the Nyquist frequency. These results were measured for a sampling frequency of 10 kHz and window lengths of 40 ms (for pitch) and 80 ms (for HNR), but generalize to other sampling frequencies and window lengths (see section 5).

Abstract: Wideband analog signals push contemporary analog-to-digital conversion (ADC) systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the band limit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its band limit in hertz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W hertz. In contrast to Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system's performance that supports the empirical observations.

Abstract: We introduce a powerful semi-numeric modeling tool, 21cmFAST, designed to efficiently simulate the cosmological 21-cm signal. Our code generates 3D realizations of evolved density, ionization, peculiar velocity, and spin temperature fields, which it then combines to compute the 21-cm brightness temperature. Although the physical processes are treated with approximate methods, we compare our results to a state-of-the-art large-scale hydrodynamic simulation, and find good agreement on scales pertinent to the upcoming observations (>~ 1 Mpc). The power spectra from 21cmFAST agree with those generated from the numerical simulation to within 10s of percent, down to the Nyquist frequency. We show results from a 1 Gpc simulation which tracks the cosmic 21-cm signal down from z=250, highlighting the various interesting epochs. Depending on the desired resolution, 21cmFAST can compute a redshift realization on a single processor in just a few minutes. Our code is fast, efficient, customizable and publicly available, making it a useful tool for 21-cm parameter studies.

Abstract: The sampling theorem for bandlimited signals of finite energy can be interpreted in two ways, associated with the names of Nyquist and Shannon. 1) Every signal of finite energy and bandwidth W Hz may be completely recovered, in a simple way, from a knowledge of its samples taken at the rate of 2W per second (Nyquist rate). Moreover, the recovery is stable, in the sense that a small error in reading sample values produces only a correspondingly small error in the recovered signal. 2) Every square-summable sequence of numbers may be transmitted at the rate of 2W per second over an ideal channel of bandwidth W Hz, by being represented as the samples of an easily constructed band-limited signal of finite energy. The practical importance of these results, together with the restrictions implicit in the sampling theorem, make it natural to ask whether the above rates cannot be improved, by passing to differently chosen sampling instants, or to bandpass or multiband (rather than bandlimited) signals, or to more elaborate computations. In this paper we draw a distinction between reconstructing a signal from its samples, and doing so in a stable way, and we argue that only stable sampling is meaningful in practice. We then prove that: 1) stable sampling cannot be performed at a rate lower than the Nyquist, 2) data cannot be transmitted as samples at a rate higher than the Nyquist, regardless of the location of sampling instants, the nature of the set of frequencies which the signals occupy, or the method of construction. These conclusions apply not merely to finite-energy, but also to bounded, signals.