SAMPLED-DATA SIMULATION OF OVER-SAMPLED SIGMA-DELTA MODULATORS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 196

with an unclocked quantizer, the output of the linear block must be
evaluated in every time step.
The output of the linear block is fed to the quantizer. For sigma-delta
modulators with a clocked quantizer, the output of the quantizer
changes only at the edge of the clock phases and remains unchanged
until the arrival of the edge of the next clock phase. Simulation pro-
ceeds by calculating the output of the linear block after each clocking
instant, updating the state of the quantizer based on the output of
the linear block, and repeating the process for the next clock phase.
For sigma-delta modulators with an unclocked quantizer, the exact
time instant at which the output of the quantizer changes must be
determined and the consistent initial conditions of the modulators
after the quantizer changes its state must be calculated. Simulation
then proceeds from the consistent initial conditions.
Because sampled-data simulation yields the exact time-domain re-
sponse of linear circuits regardless of the step size, also because
clocked quantizers update their output only at the clocking instants,
over-sampled sigma-delta modulators with clocked quantizers can
therefore be analyzed effectively using the sampled-data simulation
of periodically switched linear circuits presented in Chapter 6. The
step size of sampled-data simulation can be set to the width of clock
phases to speed up simulation.
7. Examples
The first example is the single-bit second-order continuous-time over-
sampled sigma-delta modulator shown in Fig.9.5. The comparator is
clocked at 1.024kHz. The operational amplifiers are considered to be
ideal. The input signal is a 1kHz sinusoid of amplitude 2V. The time-
domain response of the modulator is shown in Fig.9.6 and the out-
put spectrum of the modulator obtained from transient-FFT analysis is
shown in Fig.9.7. The noise shaping characteristics of the second-order
sigma-delta modulator are evident.
The second example is a single-bit second-order switched capacitor
over-sampled sigma-delta modulator shown in Fig.9.8 [73, 74]. It is