# 1.6 Digital filter design for interpolation and decimation

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(Blank Abstract)

First we treat filter design for interpolation. Consider an input signal $x(n)$ that is ${\omega }_{0}$ -bandlimited in the DTFT domain. If we upsample by factor $L$ to get $v(m)$ , the desired portion of $V(e^{i\omega })$ is the spectrum in $\left[\frac{-\pi }{L} , \frac{\pi }{L}\right)$ , while the undesired portion is the remainder of $\left[-\pi , \pi \right)$ . Noting from that $V(e^{i\omega })$ has zero energy in the regions

$\left[\frac{2k\pi +{\omega }_{0}}{L} , \frac{2(k+1)\pi -{\omega }_{0}}{L}\right),k\in \mathbb{Z}$
the anti-imaging filter can be designed with transition bands in these regions (rather than passbands or stopbands). For a givennumber of taps, the additional degrees of freedom offered by these transition bands allows for better responses in thepassbands and stopbands. The resulting filter design specifications are shown in the bottom subplot below .

Next we treat filter design for decimation. Say that the desired spectral component of the input signal is bandlimited to $\frac{{\omega }_{0}}{M}< \frac{\pi }{M}$ and we have decided to downsample by $M$ . The goal is to minimally distort the input spectrum over $\left[\frac{-{\omega }_{0}}{M} , \frac{{\omega }_{0}}{M}\right)$ , i.e. , the post-decimation spectrum over $\left[-{\omega }_{0} , {\omega }_{0}\right)$ . Thus, we must not allow any aliased signals to enter $\left[-{\omega }_{0} , {\omega }_{0}\right)$ . To allow for extra degrees of freedom in the filter design,we do allow aliasing to enter the post-decimation spectrum outside of $\left[-{\omega }_{0} , {\omega }_{0}\right)$ within $\left[-\pi , \pi \right)$ . Since the input spectral regions which alias outside of $\left[-{\omega }_{0} , {\omega }_{0}\right)$ are given by

$\left[\frac{2k\pi +{\omega }_{0}}{L} , \frac{2(k+1)\pi -{\omega }_{0}}{L}\right),k\in \mathbb{Z}$
(as shown in ), we can treat these regions as transition bands in the filter design.The resulting filter design specifications are illustrated in the middle subplot ( ).

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
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for teaching engĺish at school how nano technology help us
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or in general
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in general
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On having this app for quite a bit time, Haven't realised there's a chat room in it.
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