Downsampling Operator

Downsampling by $L$ (also called decimation 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.7.10. The example is

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

Figure 7.10: Illustration of $\hbox{\sc Downsample}_2(x)$. The white-filled circles indicate the retained samples while the black-filled circles indicate the discarded samples.

Note that the term ``downsampling'' may also refer to the more elaborate process of sampling-rate conversion to a lower sampling rate, in which a signal's sampling rate is lowered by means of bandlimited interpolation (discussed in Appendix D). To distinguish these cases, we can call this bandlimited downsampling, because a lowpass-filter is needed, in general, prior to downsampling so that aliasing is avoided. This topic is address in Appendix D. Early sampling-rate converters were in fact implemented using the $\hbox{\sc Stretch}_L$ operation, followed by an appropriate lowpass filter, followed by $Select_M$, in order to implement a sampling-rate conversion by the factor $L/M$.