In Mixed-Signal Circuits, Part 1: From *s* to *z* - The Discrete Complex-Frequency Domain, discrete-time analog circuit analysis introduced the *z*-domain, a simple shortcut from discrete time-domain equations to the sampled *s*, or *s**-domain. In this part, we push forth to transfer functions in *s* and *z* of ADCs and DACs, then consider briefly how to design feedback loops that span the analog-digital boundary.

Part 1 left us hanging as to how impulse trains relate to discrete-time waveforms. The development continues, and in the concrete context of ADCs, so that the abstraction of impulses can have a clearer meaning.

**ADC Dynamics in ***z*

To demonstrate how the *z*-domain can be applied in circuit engineering, a 12-bit ADC transfer function is

where *v*_{ADC} is the input voltage, *V*_{R} is the full-scale 3.3 V ADC reference voltage, and *w*_{ADC} is the 12-bit ADC output. This ADC transfer function is quasistatic; it applies at 0+ Hz. The ADC dynamics of the clock-sampled *zero-order hold* (ZOH) output accounts for the sampling behavior of the ADC. The sampler, which produces the impulse trains from Part 1, is followed conceptually by a hold function,

The right-side expression can be interpreted as the gain-normalized integration (1/*s* x *T*_{s}) of an impulse train in the time domain, resulting in a sequence of step functions, each turned off *T*_{s} later (− exp(−*s* x *T*_{s})) so that their effect only occurs over one sample interval. At each sample time, a new value of ADC output is acquired and held. Plotted against time, this waveform has the stair-step look of a sample-and-hold waveform described by *T*_{ZOH}(*s*), as shown below.

The acquired continuous voltage waveform, *v*_{i}(*t*), is plotted along with its *piecewise-continuous* or “stepped” digital output, approximated as the continuous *v*_{o}(*t*), shown as a dotted curve. This approximation of the acquired waveform is reconstructed by shifting the continuous waveform, *v*_{i}, by *T*_{s}/2 to the right, a delay of a half sampling period. It passes through the centers of the vertical steps of the acquired ZOH waveform.

The ADC sample-hold frequency response is the magnitude and phase of *T*_{ZOH}(*j
ω
*);

The main step in this derivation is to multiply numerator and denominator by * e*^{-jω Ts}/2, then apply (from Euler’s equation, *e*^{j
θ
} = cos*
θ
* + sin*
θ
*, where *
θ
* = *
ω
* x *T*_{s}/2),

The amplitude of *T*_{ZOH}(*j
ω
*) is thus

This function begins with a value of one at zero frequency (
*ω
* = 0 s^{–1}) and decreases to zero at the sampling frequency. Then it “bounces” (because of the absolute value) between zero values at harmonics of *f*_{s}, outside the Nyquist interval. The rolloff of magnitude with frequency can be compensated with an inverse ZOH (ZOH^{-1}) function and can be included following the ADC block as code in the
μ
C. It can be omitted if the ADC sample rate is enough above the analog-circuit bandwidth to make amplitude roll-off and phase delay insignificant so that the sampled result is approximately continuous.

The phase is

In degrees, the phase is −180^{o}(*f*/*f*_{s}), where *f*_{s} = 1/T_{s}. The T_{s}/2 delay in the phase is the half-step of delay observed graphically in the plot above of *v*_{o} delayed from *v*_{i}.

At the Nyquist frequency, *f*_{s}/2, the magnitude will have decreased to (sin(
π
/2))/(
π
/2) = 2/
π
≈
0.637 with a phase delay of −180^{o}/2 = −90^{o}. A sampling rate of 10 times *f*_{s} results in a normalized amplitude of 0.9836, with about 6 bits of accuracy. Some values are given in the following table.

We arrive at the final step, the ZOH function, which completes the transformation from *z* to *s*:

Thus, to transform from *z* to *s*, substitute *z*(*s*) = exp(*sT*_{s}) for *z*, then multiply the resulting impulse-train or sampled expression in *s* by the ZOH function.