DAC nonlinearity is a consequence of the inability to design a perfect DAC. There will always be an error associated with the expected DAC output level for a given input code and the actual
output level. DAC manufacturers express this error as DNL (differential nonlinearity) and INL (integral nonlinearity). The net result of DNL and INL is that the relationship between the DAC’s expected output and its actual output is not perfectly linear. This means that an input signal will be transformed through some nonlinear process before appearing at the output. If a perfect digital sine wave is fed into the DAC, the nonlinear process causes the output to contain the desired sine wave plus harmonics. Thus, a distorted sine wave is produced at the DAC output. This form of error is known as harmonic distortion. The result is harmonically related spurs in the output spectrum. The amplitude of the spurs is not readily predictable as it is a function of the DAC linearity. However, the location of such spurs is predictable, since they are harmonically related to the tuning word frequency of the DDS. For example, if the DDS is tuned to 100kHz, then the 2nd harmonic is at 200kHz, the 3rd at 300kHz, and so on. Generally, for a DDS output frequency of fo, the nth harmonic is at nfo. Remember, however, that a DDS is a
sampled system operating as some system sample rate, Fs. So, the Nyquist criteria are applicable. Thus, any harmonics greater than ½Fs will appear as aliases in the frequency range between 0 and ½Fs (also known as the first Nyquist zone). The 2nd Nyquist zone covers the range from ½Fs to Fs. The 3rd Nyquist zone is from Fs to 1.5Fs, and so on. Frequencies in the ODD Nyquist zones map directly onto the 1st Nyquist zone, while frequencies in the EVEN Nyquist zones map in mirrored fashion onto the 1st Nyquist zone. This is shown pictorially in Figure 4.10.

The procedure, then, for determining the aliased frequency of the Nth harmonic is as follows:

1. Let R be the remainder of the quotient (Nfo)/Fs, where N is an integer.
2. Let SPURN be the aliased frequency of the Nth harmonic spur.
3. Then SPURN = R if (R £ ½Fs), otherwise SPURN = Fs - R.

The above algorithm provides a means of predicting the location of harmonic spurs that result from nonlinearities associated with a practical DAC. As mentioned earlier, the magnitude of the spurs is not predictable because it is directly related to the amount of non-linearity exhibited by a particular DAC (i.e., non-linearity is DAC dependent).