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4.1 Downsampling

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(Blank Abstract)

The operation of downsampling by factor M describes the process of keeping every M t h sample and discarding the rest. This is denoted by " ↓ M " in block diagrams, as in [link] .

Formally, downsampling can be written as y n x n M In the z domain,

Y z n n y n z n n n x n M z n m m x m 1 M p 0 M 1 2 M p m z m M
where 1 M p 0 M 1 2 M p m 1 m is a multiple of M 0
Y z 1 M p 0 M 1 m m x m 2 M p z 1 M m 1 M p 0 M 1 X 2 M p z 1 M
Translating to the frequency domain,
Y ω 1 M p 0 M 1 X ω 2 p M

As shown in [link] , downsampling expands each 2 -periodic repetition of X ω by a factor of M along the ω axis, and reduces the gain by a factor of M . If x m is not bandlimited to M , aliasing may result from spectral overlap.

When performing a frequency-domain analysis of systems with up/downsamplers, it is strongly recommended tocarry out the analysis in the z -domain until the last step, as done above. Working directly in the ω -domain can easily lead to errors.
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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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