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DFT Matrix
In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication. Definition An ''N''-point DFT is expressed as the multiplication X = W x, where x is the original input signal, W is the ''N''-by-''N'' square DFT matrix, and X is the DFT of the signal. The transformation matrix W can be defined as W = \left(\frac\right)_ , or equivalently: : W = \frac \begin 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^ \\ 1&\omega^2&\omega^4&\omega^6&\cdots&\omega^\\ 1&\omega^3&\omega^6&\omega^9&\cdots&\omega^\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega^&\omega^&\omega^&\cdots&\omega^ \end , where \omega = e^ is a primitive ''N''th root of unity in which i^2=-1. We can avoid writing large exponents for \omega using the fact that for any exponent x we have the identity \omega^ = \omega^. This is the Vandermonde matrix for the roots of unity, up to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex number, complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex number, complex Sine wave, sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
