Design Constraints for Image-Reject Frequency-Translating ΔΣ Modulators

This brief derives design constraints for bandpass ΔΣ modulators that use mixers to perform frequency downcon- version 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 sampling 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 noise-shaping 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.