4.6 Digital filter design for interpolation and decimation

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

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