Ideal Nth-Band Discrete Filters

This Demonstration shows impulse and magnitude responses of ideal -band lowpass discrete filters for . While such filters are not realizable in practice, they serve as a desired template for passing certain frequencies while blocking others. The impulse response of an ideal lowpass filter is a sinc sequence (of unit norm in the figure), while its magnitude response is constant in the passband.

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An ideal -band discrete filter is a non-realizable filter whose magnitude response takes a single nonzero value in its passband. For example, an ideal lowpass filter passes frequencies below some cut-off frequency and blocks the others; its passband is thus the interval .
The impulse response of an ideal lowpass -band filter is a sinc sequence; its unit-norm version is
for .
The impulse response at is thus (see figure).
The magnitude response of a unit-norm is (see figure)
for , and 0 otherwise.
References
[1] M. Vetterli, J. Kovačević, and V. K. Goyal, Foundations of Signal Processing, Cambridge: Cambridge University Press, 2014. www.fourierandwavelets.org.
[2] Wikipedia. "Sinc Filter." (Nov 10, 2014) en.wikipedia.org/wiki/Sinc_filter.
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