1.4 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 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