= Ratiometric cancellation with matched signal and reference filters
<Ratiometric cancellation with matched signal and reference filters> preserves a resistance <ratio> despite changing excitation when both paths apply the same dynamics to the shared <electric current>. Matching only the steady gain is insufficient during a transient.
Let constant sensor and reference resistances be $R_T,R_{\rm ref}$, and let both paths be <linear time-invariant systems> with <impulse responses> $h_T,h_{\rm ref}$. With consistent initial conditions, their filtered <electric potential differences> are
$$
y_T=R_T(h_T*I),\qquad y_{\rm ref}=R_{\rm ref}(h_{\rm ref}*I),
$$
where $*$ is <convolution>. If $h_T=h_{\rm ref}$, simultaneous sampling gives $y_T/y_{\rm ref}=R_T/R_{\rm ref}$ whenever the denominator is nonzero. Arbitrary unequal stored initial states need not cancel. Nor can a time-varying $R_T$ generally be pulled outside the <convolution>.
For unit steady gains, write $I(t)=I_0(1+\eta(t))$ after settling at $I_0$. Define $q_T=h_T*\eta$ and $q_{\rm ref}=h_{\rm ref}*\eta$. Then
$$
\frac{y_T/y_{\rm ref}}{R_T/R_{\rm ref}}=
\frac{1+q_T}{1+q_{\rm ref}},\qquad
\frac{y_T/y_{\rm ref}}{R_T/R_{\rm ref}}-1
=\frac{q_T-q_{\rm ref}}{1+q_{\rm ref}}.
$$
This displays exactly which excitation component survives. For small fluctuations the fractional residual is $q_T-q_{\rm ref}$, so both amplitude and timing differences between the <transfer functions> matter.
As an original transient example, step the excitation up by ten percent from a settled state. Two <first-order low-pass filters> with <time constants> $\tau_T=1\,\mathrm{ms}$ and $\tau_{\rm ref}=2\,\mathrm{ms}$ give, for $t\ge0$,
$$
\frac{y_T/y_{\rm ref}}{R_T/R_{\rm ref}}=
\frac{1+0.1(1-e^{-t/\tau_T})}{1+0.1(1-e^{-t/\tau_{\rm ref}})}.
$$
At $t=1\,\mathrm{ms}$ the reading is high by approximately $2.2962\%$, although both filters have the same final gain and the error tends to zero. Setting both <time constants> to $1\,\mathrm{ms}$ cancels this particular excitation step at every time. <Sampling skew after matched ratiometric filters> can still break simultaneous cancellation even with identical hardware.
The primary ADS1248 experiment used a resistance decade box, a $1\,\mathrm{mA}$ excitation and a $2\,\mathrm{k\Omega}$ reference. Reference-only filtering produced signal-dependent <noise>; matching the signal and reference paths substantially reduced that dependence. The report provides the circuit and noise-versus-input plots, rather than merely recommending a common nominal cutoff.<RTD Ratiometric Measurements and Filtering Using the ADS1148 and ADS1248 Family of Devices by Luis Chioye>[§§3.1–3.2, printed pp. 6–9, Figures 3-1–3-4]{ref}
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