Introduction / Context:
Monotonicity is a key performance attribute of a Digital-to-Analog Converter (DAC). A monotonic DAC guarantees that as the digital input code increases by one or more least significant bits (LSBs), the analog output never decreases. This question probes where a “monotonicity error” appears across input codes.
Given Data / Assumptions:
- A DAC converts N-bit digital codes to corresponding analog levels.
- Monotonic means output never decreases for increasing input codes.
- Real DACs have component mismatches and nonidealities that can create local errors.
- We are not assuming catastrophic failure; rather, occasional code-to-code inversions.
Concept / Approach:
A monotonicity error is a localized violation of the required nondecreasing transfer characteristic. It typically occurs at specific code transitions where element mismatches (for example, in a binary-weighted or segmented architecture) cause the next code step to be smaller than expected or even negative. Therefore, such errors do not necessarily affect every code; they manifest at particular transitions.
Step-by-Step Solution:
Define monotonic behavior: for any code increment, output must not fall.Identify the failure mode: local negative step or insufficiently positive step at certain transitions.Conclude: the error appears only at those specific input codes where mismatch causes inversion.Hence, it shows up only for certain (scattered) inputs.
Verification / Alternative check:
Plot the DAC transfer curve. Nonmonotonicity is visible as local dips at certain code boundaries while adjacent regions may behave correctly.
Why Other Options Are Wrong:
only for higher value inputs: Nonmonotonicity is not confined to large codes.only for lower value inputs: Same reasoning; not restricted to small codes.for all inputs: That would imply a gross failure; monotonicity errors are typically localized.
Common Pitfalls:
Confusing monotonicity error with integral nonlinearity (INL) or gain error; those can be global without inverting direction.Assuming monotonicity implies perfect linearity; it only guarantees direction, not exact step size.
Final Answer:
only for certain (scattered) inputs
Discussion & Comments