This slide introduces the spectral characteristics of the sinusoid contained within the map. Because the map has finite amplitude resolution, D-bits to be precise, its output values are not an exact reproduction of an ideal sinusoid. This should be readily apparent from the equation that was used to calculate the D-bit values for the map, as it relied on a rounding function. The rounding operation means that some precision is lost in the calculation, which translates directly to a loss of precision in terms of reproducing an ideal sinusoid. This loss of precision has a direct impact on the spectral purity of the map. These diagrams represents the output spectrum assuming the x input to the map increments by 1 at a constant rate (i.e., the sample rate). Because the map contains 2 to the P samples, the spectrum spans 2 to the P frequency points with 0 representing DC and 2 to the P corresponding to the sample rate. Furthermore, the map contains 1 cycle of a sinusoid, so it takes all 2 to the P samples to complete 1 cycle. This means that the fundamental frequency appears at position 1 in the spectrum. Remember, though, that the map output constitutes a sampled sinusoid, so there appears a Nyquist image at frequency position 2 to the P minus 1, as well. If the map had infinite precision (that is, D equals infinity), then the map output values would reproduce a sinusoid perfectly. As such, the spectral content of the sinusoid would be pure resulting in the spectrum in the upper diagram. But D is not infinite, so there are quantization errors in the output values of the map. Hence, the map output does not follow an exact sinusoidal path. The result is the appearance of quantization spurs in the output spectrum as shown in the lower diagram. The spur pattern is symmetric about the frequency point 2 raised to the P minus 1 power, which corresponds to the Nyquist frequency (i.e., one half of the sample rate). The magnitude of the spurs depends on the size of the quantization errors. Specifically, as D increases the spur magnitude decreases. This constitutes one of the design considerations in a DDS. Large D means smaller quantization spurs, but at the cost of a larger map and a more complex DAC because the D-bit output of the map is intended to drive a D-bit DAC.