Difference between revisions of "Time scale discrete Fourier transform"
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(Created page with "Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a frequency root of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is def...") |
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| − | Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a [[frequency root]] of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is defined by | + | Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a [[frequency roots|frequency root]] of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is defined by |
$$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$ | $$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$ | ||
where $e_{z_n}$ denotes the [[delta exponential]]. | where $e_{z_n}$ denotes the [[delta exponential]]. | ||
Latest revision as of 04:00, 26 February 2018
Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a frequency root of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is defined by $$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$ where $e_{z_n}$ denotes the delta exponential.
