The Discrete Fourier Transform
نویسنده
چکیده
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you can find more thorough discussions in Oppenheim's book. Please let me know if you find any typos. In this lecture, we discuss the Discrete Fourier Transform (DFT), which is a fourier representation for finite length signals. The main practical importance of this new representation is that (unlike the DTFT), it can be computed very efficiently, for arbitrary inputs. This makes it the primary tool for performing frequency-domain analysis of signals on a computer. The discrete-time Fourier transform is an extremely useful tool for analyzing signal processing systems. However, for practical calculation, it has the disadvantage that X(e jω) is defined on ω ∈ R, and hence the full transform cannot be directly calculated. It is also worth noting that the discrete-time Fourier transform is defined via an infinite summation X(e jω) = ∞ n=−∞ x[n] exp(−jωn), (1.1) while, in practice we always work with signals x[n] of finite length. The discrete Fourier transform (DFT) is a Fourier representation for finite-length signals We say that such an x has length N .
منابع مشابه
A general construction of Reed-Solomon codes based on generalized discrete Fourier transform
In this paper, we employ the concept of the Generalized Discrete Fourier Transform, which in turn relies on the Hasse derivative of polynomials, to give a general construction of Reed-Solomon codes over Galois fields of characteristic not necessarily co-prime with the length of the code. The constructed linear codes enjoy nice algebraic properties just as the classic one.
متن کاملDetection of high impedance faults in distribution networks using Discrete Fourier Transform
In this paper, a new method for extracting dynamic properties for High Impedance Fault (HIF) detection using discrete Fourier transform (DFT) is proposed. Unlike conventional methods that use features extracted from data windows after fault to detect high impedance fault, in the proposed method, using the disturbance detection algorithm in the network, the normalized changes of the selected fea...
متن کاملSampling Rate Conversion in the Discrete Linear Canonical Transform Domain
Sampling rate conversion (SRC) is one of important issues in modern sampling theory. It can be realized by up-sampling, filtering, and down-sampling operations, which need large complexity. Although some efficient algorithms have been presented to do the sampling rate conversion, they all need to compute the N-point original signal to obtain the up-sampling or the down-sampling signal in the tim...
متن کاملThere is only one 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 distributiona...
متن کاملp-adic Shearlets
The field $Q_{p}$ of $p$-adic numbers is defined as the completion of the field of the rational numbers $Q$ with respect to the $p$-adic norm $|.|_{p}$. In this paper, we study the continuous and discrete $p-$adic shearlet systems on $L^{2}(Q_{p}^{2})$. We also suggest discrete $p-$adic shearlet frames. Several examples are provided.
متن کاملFractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform
Conventional Fourier analysis has many schemes for different types of signals. They are Fourier transform (FT), Fourier series (FS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). The goal of this correspondence is to develop two absent schemes of fractional Fourier analysis methods. The proposed methods are fractional Fourier series (FRFS) and discrete-time fract...
متن کامل