14.3 DECIMATOR DEPENDENCE GRAPH

The author and his group provided a z-transform technique for obtaining several decimator structures [96–98]. However, for the case of multirate systems, this approach was not able to provide the rich set of design space exploration that the dependence graph approach could provide. Figure 14.2 shows the dependence graph of the decimator, which was obtained for the two signals in Eqs. 14.1 and 14.2. The horizontal axis is the n-axis and vertical axis is the k-axis. The figure shows the dependence graph of the filter whose output samples are u(n). At the top of the figure, we indicate the decimator output y(n). Note that sample y(n) corresponds to the sample u(M n). In order to conserve space, we used subscripts in the figure to indicate index values for the different samples.

Figure 14.2 General M-to-1 decimator dependence graph for the case when M = 3 and N = 12.

The thick vertical lines indicate the decimator output y(n). The solid circles in the figure indicate useful filtering operations that result in the generation of the output samples u(nM) and y(n), while the empty circles indicate filtering operations that will result in no useful output samples. In a sense, these are wasted operations that consume unnecessary resources. Essentially, the decimator uses a regular low-pass filter to produce some output samples at the high input data rate. Then, we selectively pick the Mth sample to represent the desired decimator outputs.

Figure 14.3 shows the dependence graph of the decimator where only the useful operations and outputs are indicated.

Figure 14.3 M-to-1 decimator dependence graph for the case when M = 3 and N = 12. Unnecessary operations and output samples have been removed from the figure.

This figure is a bit less cluttered compared with Fig. 14.2. We removed the empty circles and only retained the shaded circles that give the desired outputs.