Downsampling 2.11 2002/01/09 2005/09/20 21:07:59.017 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Adan Galvan jago@rice.edu Erin Maloney emaloney@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu decimation downsampler downsampling (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 .

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 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 , 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 xm 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 to carry 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.