Upsampling (Stretch) Operator

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

Figure 11.1 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:

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

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

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

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.11.2.

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