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Digital Filter Design for Interpolation and Decimation

Module by: Phil Schniter. E-mail the author

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First we treat filter design for interpolation. Consider an input signal xn x n that is ω 0 ω 0 -bandlimited in the DTFT domain. If we upsample by factor L L to get vm v m , the desired portion of Vⅇⅈω V ω is the spectrum in -πLπL L L , while the undesired portion is the remainder of -ππ . Noting from Figure 1 that Vⅇⅈω V ω has zero energy in the regions

2kπ+ ω 0 L2k+1π− ω 0 L , k∈ℤ 2 k ω 0 L 2 k 1 ω 0 L , k (1)
the anti-imaging filter can be designed with transition bands in these regions (rather than passbands or stopbands). For a given number of taps, the additional degrees of freedom offered by these transition bands allows for better responses in the passbands and stopbands. The resulting filter design specifications are shown in the bottom subplot below.

Figure 1
Figure 1 (m10870fig1.png)

Next we treat filter design for decimation. Say that the desired spectral component of the input signal is bandlimited to ω 0 M<πM ω 0 M M and we have decided to downsample by MM. The goal is to minimally distort the input spectrum over - ω 0 M ω 0 M ω 0 M ω 0 M , i.e., the post-decimation spectrum over - ω 0 ω 0 ω 0 ω 0 . Thus, we must not allow any aliased signals to enter - ω 0 ω 0 ω 0 ω 0 . To allow for extra degrees of freedom in the filter design, we do allow aliasing to enter the post-decimation spectrum outside of - ω 0 ω 0 ω 0 ω 0 within -ππ . Since the input spectral regions which alias outside of - ω 0 ω 0 ω 0 ω 0 are given by

2kπ+ ω 0 L2k+1π− ω 0 L , k∈ℤ 2 k ω 0 L 2 k 1 ω 0 L , k (2)
(as shown in Figure 2), we can treat these regions as transition bands in the filter design. The resulting filter design specifications are illustrated in the middle subplot (Figure 2).

Figure 2
Figure 2 (m10870fig2.png)

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