Design Constraints for Image-Reject Frequency-Translating $\Delta\Sigma$ Modulators
Bibliographic record
Abstract
This brief derives design constraints for bandpass ΔΣ modulators that use mixers to perform frequency downconversion inside their ΔΣ loop. Such systems, which are referred to as frequency-translating ΔΣ modulators, facilitate direct analog-to-digital conversion (ADC) of high-frequency signals that cannot adequately be processed using classical bandpass ΔΣ modulator architectures. The derived constraints are required for the correct design of frequency-translating ΔΣ modulators: 1) The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sampling</i> constraints maintain the stability of the ΔΣ feedback loop and prevent the mixing of the undesired signal content into the input-signal band, thereby ensuring that the time-varying behavior of the mixers does not affect the ADC resolution; and 2) the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">noise</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">shaping</i> constraints minimize performance loss during the recombination of the in-phase and quadrature feedback paths. This brief analyzes frequency-translating ΔΣ modulators that are designed with image-reject (quadrature) mixing and that are implemented using continuous- or discrete-time lowpass or complex-bandpass inner-loop ΔΣ modulators. Thus, the derived constraints offer a valuable reference for the design of image-reject frequency-translating ΔΣ ADCs.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".