Discrete Fourier Transform of Discrete-Time Signals

2013 ◽  
pp. 103-136
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Time Signals

Single-Rate Discrete-Time Signals and Systems

2009 ◽  
pp. 1-22
Author(s):  
Ljiljana Milic
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Transform Domain ◽  
Concise Review ◽  
Discrete Signals ◽  
Time Signals ◽  
The Time Domain

This chapter is a concise review of time-domain and transform-domain representations of single-rate discrete-time signals and systems. We consider first the time-domain representation of discrete-time signals and systems. The representation in transform domain comprises the discrete-time Fourier transform (DTFT), the discrete Fourier transform (DFT), and the z-transform. The basic realization structures for FIR and IIR systems are briefly described. Finally, the relations between continuous and discrete signals are given.

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.

scholarly journals A Parallel Framework with Block Matrices of a Discrete Fourier Transform for Vector-Valued Discrete-Time Signals

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Pablo Soto-Quiros
Keyword(s):  
Fourier Transform ◽  
Parallel Computing ◽  
Discrete Time ◽  
Parallel Implementation ◽  
Multicore Processors ◽  
Block Matrix ◽  
Mathematical Framework ◽  
Matrix Operations ◽  
Time Signals ◽  
Vector Valued

This paper presents a parallel implementation of a kind of discrete Fourier transform (DFT): the vector-valued DFT. The vector-valued DFT is a novel tool to analyze the spectra of vector-valued discrete-time signals. This parallel implementation is developed in terms of a mathematical framework with a set of block matrix operations. These block matrix operations contribute to analysis, design, and implementation of parallel algorithms in multicore processors. In this work, an implementation and experimental investigation of the mathematical framework are performed using MATLAB with the Parallel Computing Toolbox. We found that there is advantage to use multicore processors and a parallel computing environment to minimize the high execution time. Additionally, speedup increases when the number of logical processors and length of the signal increase.

scholarly journals There Is Only One Fourier Transform

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Periodic Functions ◽  
Distributional Sense ◽  
Integral Fourier Transform ◽  
The Fourier Transform

Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.

scholarly journals Four Particular Cases of the Fourier Transform

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

Fundamentals of Fourier Analysis

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Fourier Transform ◽  
Discrete Time ◽  
Continuous Time ◽  
Fourier Integral ◽  
Time Signal ◽  
Continuous Sampling ◽  
Common Reference ◽  
Multidimensional Signals ◽  
Time Signals ◽  
The Fourier Transform

This chapter contains foundational material for modelling of signals and systems. Section 2.2 introduces classes of functions useful in signal processing and analysis. The Fourier transform, in Section 2.3, begins with the Fourier integral and develops the Fourier series, the discrete time Fourier transform and the discrete Fourier transform as special cases. The following material in this chapter can be skipped on a first reading. † denotes material relevant to multidimensional signals in Chapters 8 and 11. ‡ denotes material relevant to probability and stochastic processes in Chapter 4. ¶ denotes material used in continuous sampling in Chapter 10. There are a number of signal classes to which we will make common reference. Continuous time signals are denoted with their arguments in parentheses, e.g., x(t). Discrete time signals will be bracketed, e.g., x[n]. A continuous time signal, x(t), is periodic if there exists a T such that x(t) = x(t − T) for all t. The function x(t) = constant is periodic. A discrete time signal, x[n], is periodic if there exists a positive integer N such that x[n] = x[n − N] for all n. The function x[n] = constant is periodic.

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