Dynamic Element Matching
# Dynamic Element Matching: Share DAC Mismatch Across Time
Dynamic element matching (DEM) changes which nominally equal physical elements represent a converter code over successive updates. It reduces the dependence of conversion error on a fixed assignment of mismatched elements. Depending on the algorithm, mismatch energy can be scrambled or shaped in frequency rather than appearing as strongly code-dependent distortion.
DEM is used with unit elements such as current sources or capacitors, including multibit feedback DACs in delta-sigma converters. It changes selection, not the physical matching of the devices.
## 1. Separate code value from element identity
*Illustrative four-cell unary DAC with levels corresponding to zero through four selected cells. It is not a four-code binary DAC. Current values are chosen examples, not measurements. The drawing omits unselected steering destinations, biasing and switch-drive detail.*
For a code requesting k unit elements, the ideal output is k times the unit value. A static decoder repeatedly chooses the same subset for that code. Its output error then depends on the particular mismatch of those selected cells.
A DEM selector chooses a different valid subset while preserving the requested count. The same nominal code can therefore produce slightly different instantaneous analog outputs. Averaging and spectral redistribution can reduce harmful code-correlated effects, but the mismatch has not physically disappeared.
## 2. Work through an equal-use example
*Calculated constant-code example. The four chosen deviations sum to zero around a nominal 1-mA unit. Equal use over a complete cycle gives the array’s mean unit current. This does not establish broadband distortion or noise performance.*
With fixed selection of cell A, code 1 always gives 1.03 mA. Rotating A, B, C and D gives 1.03, 0.99, 1.01 and 0.97 mA, whose mean is 1.00 mA. If all cells were 2% high, equal use would preserve that 2% mean gain error. DEM primarily addresses deviations between elements; absolute reference and gain accuracy need their own treatment.
The displayed sequence is periodic, so its error can create discrete spectral components. A good average alone is not proof of a clean output spectrum. Behavior must be evaluated with changing codes as well as the constant-code illustration.
## 3. Distinguish scrambling from shaping
| Selection method | Basic idea | Main verification concern |
|---|---|---|
| Static mapping | Reuse the same subset for each code | Code-dependent mismatch distortion |
| Randomized selection | Vary valid subsets using a random or pseudorandom rule | Added mismatch-noise spectrum and sequence correlation |
| Data-weighted averaging | Rotate a pointer by the number of selected units | Periodic patterns, pointer behavior and in-band tones |
| Higher-order mismatch shaping | Use stateful selection to shape mismatch error | Stability, overload/state behavior and out-of-band energy |
In a simple cyclic data-weighted-averaging scheme, code k selects k consecutive cells starting from a pointer; the pointer advances by k modulo the array size. For constant code 1 and four cells, this can produce the illustrated rotation. More general inputs require analysis of the complete sequence.
Mismatch scrambling aims to reduce deterministic correlation; mismatch shaping aims to move error energy away from a desired band. Neither guarantees lower noise at every frequency. Reconstruction or decimation filtering must handle the redistributed energy, and the algorithm’s actual spectral behavior matters.
## 4. Include switching errors and physical limits
Selection changes can increase switching activity. Timing skew, charge injection, finite output resistance, settling and intersymbol interference may limit the benefit of a technique designed for static amplitude mismatch. The unit-element model used in analysis must reflect the error sources that dominate the real converter.
Careful layout, larger devices or calibration can complement DEM. They address different aspects of mismatch and implementation error. In a delta-sigma feedback DAC, include the mismatch-error path through the loop; do not assume it follows the same transfer function as quantization noise.
## 5. Verify the converter, not only the average
Model or measure element mismatch and run the actual selector across DC values, ramps, small signals and representative dynamic inputs. Inspect INL/DNL as applicable, spectra, integrated in-band noise, spurs and settling. Record sample rate, signal band, sequence initialization, mismatch realization and filter settings.
Test pointer wraparound and valid element counts, and compare static versus dynamic selection with the same physical element set. Check for delta-sigma idle tones or other pattern-related spurs where stateful algorithms interact with repeated codes. ADC dither changes quantization statistics; DEM changes the physical element assignment, so the mechanisms should be evaluated separately.
## References
- [IEEE: Why Dynamic-Element-Matching DACs Work](https://ieeexplore.ieee.org/document/5420027/) — mismatch scrambling and mismatch shaping principles.
- [Analog Devices: Current-Steering DACs](https://wiki.analog.com/university/courses/tutorials/cmos-dac-chapter) — unit-element architectures, mismatch and switching errors.
- [Sharifi and Hashemipour: Mismatch Error Shaping of DAC Unit Elements](https://journals.tubitak.gov.tr/elektrik/vol29/iss2/5/) — DEM in multibit delta-sigma conversion.