Fourier transform [Jaeyoung Wiki]

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====== Fourier transform ======

===== Discrete-time Fourier transform =====

$$ x[n] \leftrightarrow X(e^{j\Omega}) $$

$$ X(e^{j\Omega}) = \sum_{k = -\infty}^{\infty} x[k] e^{-j\Omega k} $$

$$ x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\Omega}) e^{j\Omega n} d\Omega $$

===== Continuous-time Fourier transform =====

$$ x(t) \leftrightarrow X(j\omega) $$

$$ X(j\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t} dt $$

$$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega $$

===== Properties of Fourier transforms =====

  * If the real part of a Fourier transform is even, and the imaginary part is odd, then the time-domain signal is real.
  * If a Fourier transform is continuous, then its corresponding signal is absolutely summable.
  * If a Fourier transform is purely imaginary, then its time-domain signal is odd.