What is Monotonicity?

Digital-to-analog converter (DAC or D-to-A) circuits are devices that convert digital code to analog signal. The conversion can be done the other way, using an analog-to-digital converter (ADC). In both devices, two of the key specifications are linearity and monotonicity. Technological limitations prevent DACs and ADCs from being ideal. Montonicity is one measure that helps to describe how well they function. In general, a non-monotonic DAC is not valuable.

The term monotonicity comes from the field of mathematics. Certain mathematical functions, called monotonic mathematical functions, maintain the same order between x and y, on the one hand, or consistently reverse the order, on the other hand. Two possibilities result: either x and y are both increasing or staying the same or x and y are both decreasing or staying the same.

In electronics, a monotonic DAC is one in which the analog output follows the direction of the digital input, no matter how the digital input varies. In other words, the analog output consistently increases as the input increases, and decreases as the input decreases. If the analog output were not to follow the direction of the digital input at any time — decrease when it increases or increase when it decreases — then the DAC would be considered non-monotonic. There is no such thing as degrees when it comes to monotonicity.

The linearity — a reference to ideal output, which would look like a straight line — is related to monotonicity. The ideal change for a single step is 1 LSB (Least Significant Bit). Missing code is indicated by -1LSB. In order to guarantee monotonic response, Differential Nonlinearity (DNL) error, which is a measure of the difference between the ideal output and the actual output for successive steps, a measure of its precision, needs to be less than 1 LSB. That is, it can be up to, but not including, ±1 LSB. According to some, however, that is only for a guarantee, and it is possible for a converter to be monotonic and have a DNL that reaches a maximum or more than 1 LSB.