Upsampling

Upsampling

The operation of upsampling by factor $L\in \mathbb{N}$ describes the insertion of $L-1$ zeros between every sample of the input signal. This is denoted by" $\uparrow (L)$ " in block diagrams, as in .

Formally, upsampling can be expressed in the time domain as $$y(n)=\begin{cases}x(\frac{n}{L}) & \text{if $\frac{n}{L}\in \mathbb{Z}$}\\ 0 & \text{otherwise}\end{cases}$$ In the $z$ -domain, $$Y(z)=\sum y(n)z^{-n}=\sum_{\frac{n}{L}\in \mathbb{Z}} x(\frac{n}{L})z^{-n}=\sum x(k)z^{(-kL)}=X(z^L)$$ and substituting $z=e^{i\omega }$ for the DTFT,