Decimation 2.10 2002/01/10 2003/01/15 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 aliasing anti-aliasing decimation decimator downsampling Introduction to decimation. Decimation is the process of filtering and downsampling a signal to decrease its effective sampling rate, as illustrated in . The filtering is employed to prevent aliasing that might otherwise result from downsampling.

To be more specific, say that x c t x l t x b t where x l t is a lowpass component bandlimited to 1 2M T Hz and x b t is a bandpass component with energy between 1 2M T and 1 2 THz. If sampling x c t with interval T yields an unaliased discrete representation x m, then decimating x m by a factor M will yield y n, an unaliased M T-sampled representation of lowpass component x l t . We offer the following justification of the previously described decimation procedure. From the sampling theorem, we have X ω 1 T k k X l ω 2 k T 1 T k k X b ω 2 k T The bandpass component X b Ω is the removed by M -lowpass filtering, giving V ω 1 T k k X l ω 2 k T Finally, downsampling yields Y ω 1 M T p 0 M 1 k k X l ω 2 p M 2 k T 1 M T p 0 M 1 k k X l ω 2 k M p M T 1 M T l l X l ω 2 l M T which is clearly a M T-sampled version of x l t . A frequency-domain illustration for M2 appears in .