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Decimation

Module by: Phil Schniter. E-mail the author

Summary: Introduction to decimation.

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 12⁢M⁢T1 2M T Hz and x b ⁢t x b t is a bandpass component with energy between 12⁢M⁢T1 2M T and 12⁢T⁢Hz1 2 THz. If sampling x c ⁢t x c t with interval TT yields an unaliased discrete representation x⁢mx m, then decimating x⁢mx m by a factor MM will yield y⁢ny n, an unaliased M⁢TM 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 X⁢ej⁢ω=1T⁢∑k X l ⁢j⁢ω−2⁢π⁢kT+1T⁢∑k X b ⁢j⁢ω−2⁢π⁢kT 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 ⁢j⁢Ω X b Ω is the removed by πM M -lowpass filtering, giving V⁢ej⁢ω=1T⁢∑k X l ⁢j⁢ω−2⁢π⁢kT V ω 1 T k k X l ω 2 k T Finally, downsampling yields

Y⁢ej⁢ω=1M⁢T⁢∑p=0M−1∑k X l ⁢j⁢ω−2⁢π⁢pM−2⁢π⁢kT=1M⁢T⁢∑p=0M−1∑k X l ⁢j⁢ω−(2⁢π)⁢(k⁢M+p)M⁢T=1M⁢T⁢∑l X l ⁢j⁢ω−2⁢π⁢lM⁢T 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 M⁢TM 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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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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