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Decimation

Module by: Phil Schniter

Summary: Introduction to decimation.

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Decimation is the process of filtering and downsampling a signal to decrease its effective sampling rate, as illustrated in Figure 1. The filtering is employed to prevent aliasing that might otherwise result from downsampling.

Figure 1
Figure 1 (m10445fig1.png)

To be more specific, say that x c t= x l t+ x b t x c t x l t x b t where x l t x l t is a lowpass component bandlimited to 12MT1 2M T Hz and x b t x b t is a bandpass component with energy between 12MT1 2M T and 12THz1 2 THz. If sampling x c t x c t with interval TT yields an unaliased discrete representation xmx m, then decimating xmx m by a factor MM will yield yny n, an unaliased MTM T-sampled representation of lowpass component x l t x l t .

We offer the following justification of the previously described decimation procedure. From the sampling theorem, we have Xⅇⅈω=1T∑k X l ⅈω−2πkT+1T∑k X b ⅈω−2πkT 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 ⅈΩ X b Ω is the removed by πM M -lowpass filtering, giving Vⅇⅈω=1T∑k X l ⅈω−2πkT V ω 1 T k k X l ω 2 k T Finally, downsampling yields

Yⅇⅈω=1MT∑p=0M−1∑k X l ⅈω−2πpM−2πkT=1MT∑p=0M−1∑k X l ⅈω−2πkM+pMT=1MT∑l X l ⅈω−2πlMT 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 (1)
which is clearly a MTM T-sampled version of x l t x l t . A frequency-domain illustration for M=2M2 appears in Figure 2.

Figure 2
Figure 2 (m10445fig2.png)

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