Decimation
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Table of Contents
3.2.1 Decimation
Decimation is the process by which high-frequency information is eliminated from a signal
to reduce the sampling frequency without resulting in aliasing. As shown in Chapter 5, a
sampled signal, seen in Figure 3.3, repeats its spectrum every 
radians. If decimation
without filtering were performed, aliasing would occur, an example of which is seen in
Figure 3.4.
Figure 3.5 shows a block diagram of the decimation process. The operation is composed
of lowpass filtering followed by downsampling. The downsampler picks a subset of
the samples that are passed through the lowpass filter (LPF). The LPF used is designed to
avoid aliasing and has a cutoff of 
the point that allows the non-aliased part of the signal
seen in Figure 3.4 to pass.
The end result of the decimation procedure is the content of the original signal below
but it is sampled at a lower rate. Figure 3.6 shows the expected spectrum of the decimated
signal.
To fully explore the process of signal decimation, an analysis of the algorithm is necessary.
Two processes are used in the decimation procedure: filtering and downsampling.
Equation 3.2 shows filtering performed with filter hD(n) on signal x(n) in the time domain.
Subsampling is the selection of every Dth sample, shown in Equation 3.3.
Figure 3.3: Sampled Signal Spectrum.
Figure 3.4: Aliased Signal Spectrum (D = 3).
Figure 3.5: Decimation Circuit Block Diagram.
Figure 3.6: Decimated Signal Magnitude Spectrum. (Note in this case corresponds to
Fs,x/D in actual frequency.)
The combination of Equations 3.2 and 3.3 yields
and model as v(n) sampled with impulse train p(n), where p(n) is defined in Equation 3.6
using the Fourier series for a sampling impulse of period D. 
is shown in Equation
3.7.
Using Equations 3.3–3.7, the following relationship can be derived:
The discrete time Fourier transform (DTFT) of y(m) can be calculated as
represents the normalized sample rate with respect to signal y.
then Equation 3.11 is simplified to
Decimation filters out the information in the original signal above 
(with respect to the original sample rate). A lowpass
direct mapping is possible from ωy to ωx and vice versa; this relationship is best described as the spectrum spanned by 
is also spanned by 
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