Discrete Fourier Series

	Discrete Fourier Series In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. ''Fourier'') discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. A specific example is the inverse Discrete Fourier transform (general), discrete Fourier transform (inverse DFT). Definition The general form of a DFS is: which are harmonics of a fundamental frequency 1/N, for some positive integer N. The practical range of k, is [0,\ N-1], because periodicity causes larger values to be redundant. When the X[k] coefficients are derived from an N-length DFT, and a factor of 1/N is inserted, this becomes an inverse DFT. And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series. A common practice is to create a sequence of length N from a longer x[n] sequence by partitioning it into N-length segments and adding them together, pointwise.(see ) That ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Digital Signal Processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor. Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include audio and speech processing, sonar, radar and other sensor array processing, spectral density estimation, statistical signal processing, digital image processing, data compression, video coding, audio coding, image compression, signal processing for telecommunications, control systems, biomedical engineering, and seismology, among others. DSP can involve linear or nonli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Discrete Fourier Transform (general) In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. Definition Let R be any ring, let n\geq 1 be an integer, and let \alpha \in R be a principal ''n''th root of unity, defined by:Martin Fürer,Faster Integer Multiplication, STOC 2007 Proceedings, pp. 57–66. Section 2: The Discrete Fourier Transform. : \begin & \alpha^n = 1 \\ & \sum_^ \alpha^ = 0 \text 1 \leq k < n \qquad (1) \end The discrete Fourier transform maps an ''n''-tuple $(v_0,\ldots,v_)$ of elements of $R$ to another ''n''-tuple $(f_0,\ldots,f_)$ of elements of $R$ according to the following formula: : $f_k = \sum_^ v_j\alpha^.\qquad (2)$ By convention, the tuple $(v_0,\ldots,v_)$ is said to be in the ''time domain'' and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Periodic Summation In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called periodic summation: :s_P(t) = \sum_^\infty s(t + nP) = \sum_^\infty s(t - nP). When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or ''samples'') of the continuous Fourier transform, S(f) \triangleq \mathcal\, at intervals of \tfrac. That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of s(t) at constant intervals (''T'') is equivalent to a periodic summation of S(f), which is known as a discrete-time Fourier transform. The periodic summation of a Dirac delta function is the Dirac comb. Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb. Quotient spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Discrete-time Fourier Transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see ), which is by far the most common method of modern Fourier analysis. Both transforms are invertible. The inverse DTFT is the origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Poisson Summation Formula In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation. Forms of the equation Consider an aperiodic function s(x) with Fourier transform S(f) \triangleq \int_^ s(x)\ e^\, dx, alternatively designated by \hat s(f) and \mathcal\(f). The basic Poisson summation formula is: Also consider periodic functions, where parameters T>0 and P>0 are in the same units as x: :s_(x) \triangleq \sum_^ s(x + nP) \quad \text \quad S_(f) \triangleq \sum_^ S(f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]