Discrete Time Fourier Transform (DTFT)

Formula:

Analysis/Direct Transform

$$X(\omega )=\sum _{n=-\infty }^{\infty }x(n)e^{-j\omega n}$$

Synthesis or Inverse

$$x(n)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X(\omega )e^{j\omega n}d\omega$$ $$=\frac{1}{2\pi }{\int }_{-0.5}^{0.5}X(f)e^{j2\pi fn}df$$

where $f$ is the normalised frequency