Downsampling Operator

Downsampling by $L$ is defined for $x\in{\bf C}^N$ as taking every $L$th sample, starting with sample zero:

$$\begin{eqnarray*} \hbox{\sc Downsample}_{L,m}(x) &\isdef & x(mL),\\ m &=& 0,1,2,\ldots,M-1\\ N&=&LM. \end{eqnarray*}$$

The $\hbox{\sc Downsample}_L()$ operator maps a length $N=LM$ signal down to a length $M$ signal. It is the inverse of the $\hbox{\sc Stretch}_L()$ operator (but not vice versa), i.e.,

$$\begin{eqnarray*} \hbox{\sc Downsample}_L(\hbox{\sc Stretch}_L(x)) &=& x \\ \hb... ...L(\hbox{\sc Downsample}_L(x)) &\neq& x\quad \mbox{(in general).} \end{eqnarray*}$$

The stretch and downsampling operations do not commute because they are linear time-varying operators. They can be modeled using time-varying switches controlled by the sample index $n$.

An example of $\hbox{\sc Downsample}_2(x)$ is shown in Fig. 7.8. The example is

$$\hbox{\sc Downsample}_2([0,1,2,3,4,5,6,7,8,9]) = [0,2,4,6,8]$$

Figure: Illustration of $\hbox{\sc Downsample}_2(x)$. The white-filled circles indicate the retained samples while the black-filled circles indicate the discarded samples.