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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
Even though a source output may be analog to start with, it can be converted to digital with negligible loss of fidelity provided that the analog to digital (A/D) conversion is carried out properly. This conversion is done by sampling and quantization. Sampling theorem states that, if the message signal is bandlimited to W, we can reconstruct it from its samples provided that the samples are taken at a rate greater than or equal to the Nyquist rate, i.e., 2W. For example, a telephone-quality speech signal is bandlimited to W=3.4KHz. Therefore, if we sample the signal using a sampling frequency of 6.8 KHz (i.e., 6800 samples per second) or more, the samples are theoretically sufficient to represent the original message signal. We note that there is no loss in signal fidelity in this step.
Tera Sample-per-Second Time Stretched Analog-to-Digital Conversion
Published in Chi H. Lee, Microwave Photonics, 2017
Bahram Jalali, Ali M. Fard, Yan Han
The validity of time-wavelength representation can be understood through the following argument. Suppose that the chirped pulse is obtained by linearly dispersing a transform-limited pulse of the width, Δto. This represents the system’s time resolution; in other words, time cannot be localized beyond Δto. Hence, the λ-to-t mapping picture is only valid when the time scale of interest is sufficiently larger than this time ambiguity. The time scale of interest depends on the speed of the electrical signal. The well-known Nyquist sampling theorem states that it is enough to sample a band-limited signal with the maximum frequency ωRF at the sample interval of ΔT = π/ωRF without the loss of information. To map the time scale of an electrical signal with frequency ωRF to optical wavelength, it is required that Δto≪π/ωRF. Fundamentally, Δto is related to the optical bandwidth, ωopt, by the uncertainty principle, ωoptΔto ≥ 1/2. Relating optical and electrical bandwidths, the combination of uncertainty and Nyquist principles imply that the time-wavelength mapping representation is valid as long as ωRF≪2πωopt. Typically, ωopt is in the THz range, whereas ωRF is in the GHz range. Therefore, the assumption is readily justified.
Practical Design Considerations for Switched-Capacitor Filters
Published in John T. Taylor, Qiuting Huang, CRC Handbook of ELECTRICAL FILTERS, 2020
Equation 34 shows that a sample-and-hold circuit replicates the input spectrum at frequency intervals equal to the sampling frequency, before filtering the replicated spectrum with a crude lowpass function, sin(x)/x. Signals at frequencies higher than half of the sampling frequency will appear in the baseband and degrade the quality of signals there. This is called aliasing. To prevent aliasing of input signals, the latter is made bandlimited by a continuous-time antialiasing lowpass filter, and the sampling frequency is chosen to be higher than twice the resulting signal bandwidth.
A sampling theorem by perturbing the zeros of a sine-type function
Published in Applicable Analysis, 2021
Hussain Al-Hammali, Adel Faridani
The Shannon sampling theorem, which is also known as the Whittaker–Kotelnikov–Shannon (WKS) theorem, allows the reconstruction of a band-limited function from its sampled values. It reads as follows: if a functionfis band-limited to, it is represented asfor some function, thenfcan be reconstructed from its samples, , . The reconstruction formula isThe series converges absolutely, in the-sense, and uniformly on.
Sub-diffraction focusing of light by aperiodic masks
Published in Journal of Modern Optics, 2022
Seyeddyako Mostafavi, Ferhat Nutku, Yasa Ekşioğlu
In the above-mentioned imaging methods, SNOM is based on the use of evanescent fields, which place a limit on their operating distance. This constraint does not apply to a newly developed class of superoscillation-based sub-diffraction imaging systems [14]. It has long been considered that a bandlimited signal has a maximum oscillation frequency that is naturally governed by the highest frequency of the Fourier spectrum of the signal. For instance, if the Fourier spectrum of a time-dependent signal is bandlimited, such as its Fourier spectrum is given as follows: where is identically zero outside of the range . Accepted Fourier thinking indicates that the signal has no fluctuations with a period less than . However, a signal can have superoscillations or places where the local frequency is higher, in principle arbitrarily higher, than the highest nonzero frequency in the bandwidth [15]. There are several techniques exist for producing superoscillations in optics [16,17]. Before the Bessel beam studies, a variety of techniques had been described. A superoscillating light is created by diffraction throughout a series of nanometer-sized holes arranged in the micro-scale [18,19]. The holes are carefully placed according to an aperiodic tiling. After diffraction, superposed light beam forms a small superoscillating focal point that can be utilized to scan across an object for super-resolved imaging.