The Discrete-Time Fourier Transform

2010 ◽  
pp. 151-182
Keyword(s):  
Fourier Transform ◽  
Discrete Time

scholarly journals Analytical and Numerical Investigation on the Duffing Oscilator Subjected to a Polyharmonic Force Excitation

2015 ◽  
Vol 45 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Svetlin Stoyanov
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Discrete Time ◽  
Amplitude Modulation ◽  
Duffing Oscillator ◽  
Vibration Signal ◽  
Vibration Diagnostics ◽  
Significant Difference ◽  
Excitation Force ◽  
Force Excitation

Abstract An analytical solution for a specific case of the forced Duffing oscillator is proposed. The excitation force contains two harmonics with significant difference frequencies. This case corresponds to a presence of a defect in the machinery and is in the art of the machinery vibration diagnostics. The results obtained show an amplitude modulation. Therefore, the presence of an amplitude modulation in the vibration signal may be used as an indicator for a malfunction. Analytical solution derived clarifies how the amplitude modulation occurs. Also, a numerical solution is realized and compared with the analytical one. For this, the Duffing equation is solved numerically and then, the spectrograms of vibrations are obtained through a Discrete-time Fourier Transform.

Deterministic Discrete-Time Signal Transforms

2021 ◽  
pp. 714-796
Author(s):  
Stevan Berber
Keyword(s):  
Fourier Transform ◽  
Discrete Time ◽  
Linear Time ◽  
Power Spectra ◽  
Time Signal ◽  
Linear Time Invariant ◽  
Energy Spectral Density ◽  
Analysis And Synthesis ◽  
Spectral Density Functions ◽  
Linear Time Invariant Systems

Chapter 15 presents a detailed analysis of discrete-time signals and systems in the frequency domain, including the theory of the discrete Fourier series, the discrete-time Fourier transform, and the discrete Fourier transform, and key examples relevant for the analysis and synthesis of signals processed in the discrete transceiver blocks of a communication system. Amplitude spectra, magnitude spectra, phase spectra, and power spectra are defined and calculated for typical signals. Using a unique notation that distinguishes between energy signals and power signals, the correlation function and power or energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of linear-time-invariant systems, using the notions of impulse responses, correlation functions, and power spectral densities for both power and energy signals, is presented. The basic theory of the z-transform is also presented.

The Spectra of Filters

2021 ◽  
pp. 204-268
Author(s):  
Victor Lazzarini
Keyword(s):  
Fourier Transform ◽  
Impulse Response ◽  
Discrete Time ◽  
Filter Design ◽  
Finite Impulse Response ◽  
Infinite Impulse Response ◽  
Main Body ◽  
Analysis Tool ◽  
The Fourier Transform ◽  
Definition Of

This chapter now turns to the discussion of filters, which extend the notion of spectrum beyond signals into the processes themselves. A gentle introduction to the concept of delaying signals, aided by yet another variant of the Fourier transform, the discrete-time Fourier transform, allows the operation of filters to be dissected. Another analysis tool, in the form of the z-transform, is brought to the fore as a complex-valued version of the discrete-time Fourier transform. A study of the characteristics of filters, introducing the notion of zeros and poles, as well as finite impulse response (FIR) and infinite impulse response (IIR) forms, composes the main body of the text. This is complemented by a discussion of filter design and applications, including ideas related to time-varying filters. The chapter conclusion expands once more the definition of spectrum.

scholarly journals Gabor frames with trigonometric spline dual windows

2015 ◽  
Vol 08 (04) ◽  
pp. 1550072 ◽  
Author(s):  
Inmi Kim
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Higher Dimensions ◽  
Gabor Frames ◽  
One Dimension ◽  
Trigonometric Spline

A dual Gabor window pair has two functions that can reconstruct any function in [Formula: see text] using certain systems of their modulated and translated forms. Few explicit examples of dual Gabor window pairs are known. This paper constructs explicit examples with trigonometric form in one dimension as well as higher dimensions. Also, in the discrete time domain, the trigonometric form allows us to evaluate the Gabor coefficients efficiently using the Discrete Fourier Transform. The windows have fixed support and arbitrary smoothness.

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