Designing Continuous-Time Modulators

Summary

The best way to choose a loop filter for a CT ΔΣM is to choose a DT loop filter which meets the performance specifications, and then transform it using the impulse-invariant transformation in Matlab to the equivalent CT filter based on the CT ΔΣM DAC pulse shape. Low pass ΔΣMs are built with CT integrators 1/s, and if we wish to build band pass ΔΣMs with purely band pass CT resonators s/(s² + ω²), then we should use two feedback DACs with different pulse shapes to give us full control over the noise-shaping behavior. CT modulators are nice because they sample the signal at the quantizer rather than at the input, which gives them some built-in antialiasing protection that improves with modulator order. Care must be taken with the shape of the DAC pulses: they need not be perfectly rectangular, but they should be as uniform as possible—every DAC pulse should look as much like every other DAC pulse. In a practical circuit with nonideal DAC pulses, there are simulation methods which tell us how to scale the feedback currents to achieve certain noise-shaping behavior; conversely, we can verify that our modulator builds the right equivalent DT filter using the z-domain extraction technique.