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Downsampling

Module by: Phil Schniter

Summary: (Blank Abstract)

The operation of downsampling by factor M∈ℕM describes the process of keeping every M t h M t h sample and discarding the rest. This is denoted by " ↓M ↓ M " in block diagrams, as in Figure 1.
m10441fig1.png
Figure 1
Formally, downsampling can be written as yn=xnM y n x n M In the zz domain,
Yz=∑nynz-n=∑nxnMz-n=∑mxm1M∑p=0M-1ⅇⅈ2πMpmz-mM 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 (1)
where ∑p=0M-1ⅇⅈ2πMpm=1ifm is a multiple of M0otherwise p 0 M 1 2 M p m 1 m is a multiple of M 0
Yz=1M∑p=0M-1∑mxmⅇ-ⅈ2πMpz1M-m=1M∑p=0M-1Xⅇ-ⅈ2πMpz1M 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 (2)
Translating to the frequency domain,
Yⅇⅈω=1M∑p=0M-1Xⅇⅈω-2πpM Y ω 1 M p 0 M 1 X ω 2 p M (3)
As shown in Figure 2, downsampling expands each 2π2 -periodic repetition of Xⅇⅈω X ω by a factor of MM along the ωω axis, and reduces the gain by a factor of MM. If xm xm is not bandlimited to πMM, aliasing may result from spectral overlap.
note: When performing a frequency-domain analysis of systems with up/downsamplers, it is strongly recommended to carry out the analysis in the z z-domain until the last step, as done above. Working directly in the ⅇⅈω ω -domain can easily lead to errors.
m10441fig2.png
Figure 2

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