# Multirate Signal Processing: Downsampling

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This Demonstration shows the effect of filtering followed by downsampling (sometimes also called subsampling or decimation), on a discrete-time sequence and its discrete-time Fourier transform (DTFT) spectrum. Downsampling by a factor of removes every sample from a discrete-time sequence. This contraction in time results in an expansion in frequency, necessitating filtering prior to downsampling.

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Contributed by: Jelena Kovacevic (August 2012)

Open content licensed under CC BY-NC-SA

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Multirate signal processing is at the heart of most modern compression systems and standards, including JPEG, MPEG, and so on. Multirate refers to the fact that different sequences may have different time scales. One of the basic operations in multirate signal processing is downsampling.

Given a sequence , its downsampled-by- version is

.

Downsampling is only periodically shift invariant, since shifting the sequence by will not necessarily lead to the downsampled output being shifted by .

If is the DTFT spectrum of , then

is the DTFT spectrum of . The shifted versions of present in the output spectrum, are called "aliased" versions (ghost images). The shifting of the input spectrum as well as stretching of frequencies can create frequency content not present in the original spectrum. Filtering prior to downsampling ensures that the filtered part of the input spectrum is preserved after downsampling.

References

[1] M. Vetterli, J. Kovačević, and V. K. Goyal, *Foundations of Signal Processing*, Cambridge: Cambridge University Press, 2014. www.fourierandwavelets.org.

[2] M. Vetterli and J. Kovačević, *Wavelets and Subband Coding*, Englewood Cliffs: Prentice Hall, 1995. http://waveletsandsubbandcoding.org/.

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