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Inverse DT Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\mathcal{F}^{-1}(\mathcal{X}(2\pi f))=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(2\pi f)e^{i2\pi ft} d2\pi f= \int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $

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