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11: Дискретні перетворення Фур'є
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$\mathrm{DFT}\Big\{f_0, f_1, \dots, f_{N-1}\Big\} = \Big\{F_0, F_1, \dots, F_{N-1}\Big\} \qquad\mathrm{where}\quad F_n = \sum_{m=0}^{N-1} e^{-2\pi i \frac{mn}{N}}\, f_m.$ \[\mathrm{IDFT}\Big\{F_0, F... $\mathrm{DFT}\Big\{f_0, f_1, \dots, f_{N-1}\Big\} = \Big\{F_0, F_1, \dots, F_{N-1}\Big\} \qquad\mathrm{where}\quad F_n = \sum_{m=0}^{N-1} e^{-2\pi i \frac{mn}{N}}\, f_m.$ $\mathrm{IDFT}\Big\{F_0, F_1, \dots, F_{N-1}\Big\} = \Big\{f_0, f_1, \dots, f_{N-1}\Big\} \qquad\mathrm{where}\quad f_m = \frac{1}{N} \sum_{n=0}^{N-1} e^{2\pi i \frac{mn}{N}}\, F_n.$
19.8: Дискретне перетворення Фур'є
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$X_{n}=\sum_{j=0}^{N-1} x_{j} e^{-i 2 \pi j n / N}=\sum_{j=0}^{N-1} \frac{1}{N} \sum_{n^{\prime}=0}^{N-1} X_{n^{\prime}} e^{i 2 \pi n^{\prime} j / N} e^{-i 2 \pi j n / N}$ \[\sum_{j=1}^{N} e^{-i 2 ... $X_{n}=\sum_{j=0}^{N-1} x_{j} e^{-i 2 \pi j n / N}=\sum_{j=0}^{N-1} \frac{1}{N} \sum_{n^{\prime}=0}^{N-1} X_{n^{\prime}} e^{i 2 \pi n^{\prime} j / N} e^{-i 2 \pi j n / N}$ $\sum_{j=1}^{N} e^{-i 2 \pi n j / N} e^{i 2 \pi n^{\prime} j / N}=\sum_{j=1}^{N} e^{i 2 \pi\left(n^{\prime}-n\right) j / N}=N \delta_{n^{\prime} n}$ $\sum_{j=1}^{N} x_{j}^{*} x_{j}=\frac{1}{N^{2}} \sum_{j=1}^{N} \sum_{m=0}^{N-1} X_{m}^{*} e^{-i 2 \pi m j / N} \sum_{n=0}^{N-1} X_{n} e^{i 2 \pi n j / N}$