Shown here is a diagram of the behavior of the map. Users can think of the map as a file drawer containing 2 to the P files with the files ordered from front to back and indexed by number from 0 to 2 to the P minus 1, as shown in the upper part of the diagram. The files are the vertical segments with the front of the file drawer corresponding to the left side and the back of the file drawer to the right side. The indices appear at the top of each file in the drawer and correspond to a specific angle from 0 to 2-pi radians in linear fashion. That is, the file at the front of the drawer (left) has an index of 0 corresponding to 0 radians, while the file at the back of the drawer (right) has an index of 2 to the P minus 1 corresponding to just shy of 2-pi radians. The index numbers in between 0 and 2 to the P minus 1 correspond to an angle proportional to the index number. Keep in mind that the index numbers are equivalent to the value of x at the input to the map. To obtain the required sinusoidal mapping, each file in the drawer contains the specific D-bit amplitude value associated with its file index number (which corresponds to x). The D-bit value associated with the file index number is the value given by Equation 4 on the previous slide for a given P-bit input value, x. In operation, the map takes whatever P-bit value appears on its input, x, and treats it as the file index number. That index number corresponds to a particular vertical segment in the upper part of the diagram shown here. It then takes the D-bit contents of that particular file (vertical segment), which is y, and delivers it to the map output. In the diagram at the bottom, the map is shown with a sawtooth-like input waveform. This represents starting with an x-value of 0, incrementing by 1 (at a constant rate) and counting up to 2 to the P minus 1. At this point the next increment results in a rollover to 0 and the sequence repeats. Because of the way the files in the map are organized, the sawtooth pattern of x values produces a sinusoidal pattern of y values. It is interesting to note that incrementing the sawtooth by a value greater than 1 yields the same result. The only difference is the number of samples appearing in each cycle of the sinusoidal output. The key connection here is to recall that the output of a DDS accumulator is sawtooth in nature, with the rollover of the accumulator corresponding to the vertical edges of the teeth and with the sawtooth increment relating to the FTW. Because the accumulator drives the input to the map in a DDS, its sawtooth-like output pattern causes a sinusoidal pattern at the map output. Furthermore, the standard DDS equation predicts the average rollover rate of the accumulator, which corresponds to the period of the teeth in the sawtooth input to the map. Hence, users now have the link between the accumulator and how it establishes the sinusoidal output frequency of a DDS.