Search results
Results from the WOW.Com Content Network
The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT). The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform: [b]
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...
discrete Fourier transform (DFT): When the input sequence is periodic, the DTFT output is also a Dirac comb function, modulated by the coefficients of a Fourier series [1] which can be computed as a DFT of one cycle of the input sequence. The number of discrete values in one cycle of the DFT is the same as in one cycle of the input sequence.
Conversely, when one wants to compute an arbitrary number () of discrete samples of one cycle of a continuous DTFT, (), it can be done by computing the relatively simple DFT of [], as defined above. In most cases, N {\displaystyle N} is chosen equal to the length of the non-zero portion of s [ n ] . {\displaystyle s[n].}
The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the ...
Let X(f) be the Fourier transform of any function, x(t), whose samples at some interval, T, equal the x[n] sequence.Then the discrete-time Fourier transform (DTFT) is a Fourier series representation of a periodic summation of X(f): [d]
An N-point DFT is expressed as the multiplication =, where is the original input signal, is the N-by-N square DFT matrix, and is the DFT of the signal. The transformation matrix W {\displaystyle W} can be defined as W = ( ω j k N ) j , k = 0 , … , N − 1 {\displaystyle W=\left({\frac {\omega ^{jk}}{\sqrt {N}}}\right)_{j,k=0,\ldots ,N-1 ...
But when the DTFT is only sparsely sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue DTFT, those samples are the outputs of the discrete Fourier transform (DFT). The red DTFT has the same interval of zero-crossings, but ...