Upsampling (Stretch) Operator

Figure: Upsampling by a factor of $N$: $y \isdef \hbox{\sc Stretch}_N(x)$.

Figure 10.1 shows the graphical symbol for a digital upsampler by the factor $N$. To upsample by the integer factor $N$, we simply insert $N-1$ zeros between $x(n)$ and $x(n+1)$ for all $n$. In other words, the upsampler implements the stretch operator defined in §2.3.9:

$$\begin{eqnarray*} y(n) &=& \hbox{\sc Stretch}_{N,n}(x)\\ &\isdef & \left\{\beg... ...\bf Z} \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array} \right. \end{eqnarray*}$$

In the frequency domain, we have, by the stretch (repeat) theorem for DTFTs:

$$\begin{eqnarray*} Y(z) &=& \hbox{\sc Repeat}_{N,z}(X)\\ &\isdef & X(z^N), \quad z\in{\bf C}. \end{eqnarray*}$$

Plugging in $z=e^{j\omega}$, we see that the spectrum on $[-\pi,\pi)$ contracts by the factor $N$, and $N$ images appear around the unit circle. For $N=2$, this is depicted in Fig.10.2.

Figure: Illustration of $\hbox {\sc Repeat}_2$ in the frequency domain.