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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Upsampling and Downsampling
          Upsampling (Stretch) Operator

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Upsampling (Stretch) Operator

Figure: Upsampling by a factor of $ N$: $ y \isdef \hbox{\sc Stretch}_N(x)$.
\begin{figure}\input fig/upsample.pstex_t \end{figure}

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\{\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.11.2.

Figure: Illustration of $ \hbox {\sc Repeat}_2$ in the frequency domain.
\includegraphics[scale=0.8]{eps/upsampspec}

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Next: Downsampling (Decimation) Operator
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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