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Discrete-Time Fourier Series (DTFS)
Published in Samir I. Abood, Digital Signal Processing, 2020
The representation of periodic signals becomes the discrete-time Fourier series (DTFS), and for aperiodic signals, it becomes the discrete-time Fourier transform(DTFT). The motivation for representing discrete-time signals as a linear combination of complex exponentials is identical in both continuous-time and discrete-time. The complex exponentials are eigenfunctions of linear, time-invariant systems, and consequently, the effect of an LTI system on each of these basic signals is simply the amplitude change. An LTI system is completely considered by a spectrum applies at each frequency. In representing discrete-time periodic signals through the Fourier series, use harmonically related complex exponentials with fundamental frequencies. In this chapter will discuss the discrete-time Fourier transform and its application in digital signal processing.
Fourier Series, Fourier Transforms and the DFT
Published in Wai-Kai Chen, Mathematics for Circuits and Filters, 2000
The DTFT is defined so that the time domain is discrete and the frequency domain is continuous. This is in contrast to the CTFT that is defined to have continuous time and continuous frequency domains. The mathematical dual of the DTFT also exists, which is a transform pair that has a continuous time domain and a discrete frequency domain. In fact, the dual concept is really the same as the Fourier series for periodic CT signals presented earlier in the chapter, as represented by (4.5a) and (4.5b). However, the classical Fourier series arises from the assumption that the CT signal is inherently periodic, as opposed to the time domain becoming periodic by virtue of sampling the spectrum of a continuous frequency (aperiodic time) function [8]. The dual of the DTFT, the discrete frequency Fourier transform (DFFT), has been formulated and its properties tabulated as an interesting and useful transform in its own right [5]. Although the DFFf is similar in concept to the classical CT Fourier series, the formal properties of the DFFT [5] serve to clarify the effects of frequency domain sampling and time domain aliasing. These effects are obscured in the classical treatment of the CT Fourier series because the emphasis is on the inherent “line spectrum” that results from time domain periodicity. The DFFT is useful for the analysis and design of digital filters that are produced by frequency sampling techniques.
Frequency Domain Analysis
Published in Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala, Digital and Statistical Signal Processing, 2018
Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala
The discrete-time Fourier transform of a discrete-time signal, x(n), is the representation of this signal as a combination of complex exponential sequences of the form e–jωn, where ω is the angular frequency in rad/sec. The original signal can be computed when its DTFT is given, by means of the inverse discrete-time Fourier transform.
Finite frequency fault estimation observer design for T-S Fuzzy discrete-time descriptor systems
Published in International Journal of Systems Science, 2020
Yimin Liu, Jianliang Chen, Hongjun Chu, Juan Li
For simplicity, . denotes the discrete-time Fourier transform (DTFT) of . Using the inverse discrete-time Fourier transform and Parseval's theorem, we obtain Let , then When , in (14) represents the low-frequency domain, i.e. |, can be written as , , according to the property of the discrete-time Fourier transform, we know . Using the similar approach, can be given in middle and high-frequency domain. Consequently, inequality (16) implies that is in . This explains why inequality (15) is the finite frequency robust performance index.
A case study on the use of data mining for detecting and classifying abnormal power system modal behaviors
Published in Quality Engineering, 2019
Tianzhixi Yin, Shaun S. Wulff, John W. Pierre, Timothy J. Robinson
It is common in practice to identify power system events in the frequency domain or as a function of frequency components (Undrill and Trudnowski 2008). The frequency domain was chosen because when checking the historical data, the data analysts determined that the recorded events could not be readily detected in the time domain. A PSD was estimated to identify the events of interest in the frequency domain. The PSD represents the distribution of power into frequency components composing the PMU signal. A random signal usually has finite average power, so that it can be characterized by an average PSD (Stoica and Moses 2005). The PSD shows the distribution of the signal power versus frequency. The power might be strong at some frequency ranges, but weak at others. There are parametric and nonparametric methods for estimating the spectral density. In this study, a nonparametric method was used because it makes fewer underlying assumptions about the data. Calculating the PSD of a signal usually involves the discrete-time Fourier transform (DTFT). For a signal with f samples per unit time, the periodogram is defined as where is the sampling interval. In this study, the PSD was estimated with the commonly used Welch estimator. This estimator is based on an overlapped segment averaging estimator in MATLAB (MATLAB 2014) with a Von Hann window (Welch 1967).
Stochastic Simulation of Vertical Components of Near-field Pulse-less Ground Motions
Published in Journal of Earthquake Engineering, 2022
Zakariya Waezi, M. Javad Hashemi
and denote the expected value of ZC and PMNM cumulative count of the evolutionary processes, respectively. Also, , , and represent the EPSD of the nonstationary process, time-step, and its Nyquist frequency, respectively. It should be noted that in the PMNM formula, is the continuous frequency of discrete-time Fourier transform.