To obtain the signal from the coefficients of the discrete-time Fourier transform we can use the [[discrete Fourier series (DFS)]] synthesis equation: $ \tilde{x}(n)=\frac{1}{N}\sum_{k=0}^{N-1} \tilde{X}(k) e^{j\frac{2\pi}{N}kn} $ and the relationship between the Fourier series coefficients and the Fourier transform: $ \tilde{X}(k) = X(e^{j\omega})|_{w=k \frac{2\pi}{N}} $ and noting that $\omega_0=2\pi/N$: $ \begin{aligned} \tilde{x}(n) &= \frac{1}{N}\sum_{k=0}^{N-1} X(e^{jk\omega_0}) e^{jk\omega_0 n}\\ &=\frac{1}{2\pi} \sum_{k=0}^{N-1} X(e^{jk\omega_0}) e^{jk\omega_0 n} \omega_0 \end{aligned}$ when $N \rightarrow \infty$ and noting the periodicity of $X(e^{j\omega})$: $x(n) = \frac{1}{2\pi} \int_{2\pi} X(e^{j\omega}) e^{j\omega n} d\omega$