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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 otherwiseresult 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 $\frac{1}{2MT}$ Hz and ${x}_{b}(t)$ is a bandpass component with energy between $\frac{1}{2MT}$ and $\frac{1}{2T}\mathrm{Hz}$ . 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 $MT$ -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 $$X(e^{i\omega})=\frac{1}{T}\sum_{k} {X}_{l}(i\frac{\omega -2\pi k}{T})+\frac{1}{T}\sum_{k} {X}_{b}(i\frac{\omega -2\pi k}{T})$$
The bandpass component ${X}_{b}(i\Omega )$ is the removed by $\frac{\pi}{M}$ -lowpass filtering, giving $$V(e^{i\omega})=\frac{1}{T}\sum_{k} {X}_{l}(i\frac{\omega -2\pi k}{T})$$ Finally, downsampling yields
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