DFT, DTFT & Z-Transform

This visualisation shows how the discrete Fourier transform (DFT) samples the discrete-time Fourier transform (DTFT) on the unit circle, while the Z-transform extends that idea to the entire complex plane. Rotate the plot to explore the relationships.

The 3D surface represents the magnitude of X(z). The red curve along the unit circle plots the DTFT, and the black markers show the N-point DFT samples. Use the controls below to experiment.

We use the rectangular sequence \(x[k]=\operatorname{rect}_N[k]= \begin{cases}1,&0\le k\le N-1\\0,&\text{otherwise.}\end{cases}\)

Its Z-transform is the finite geometric sum \(X(z)=\displaystyle\sum_{k=0}^{N-1}z^{-k} =\frac{1-z^{-N}}{1-z^{-1}}.\)

Setting \(z=e^{\mathrm{j}\omega}\) gives the DTFT, and sampling that at \(\omega=\tfrac{2\pi k}{N}\) (black dots) yields the DFT.