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A current-sign certificate for an averaged sensor sample bounds the difference between a filtered current reading and the actual electric current throughout a future switching interval. An average is a weighted history, so it cannot automatically be treated as an instantaneous reading at the end of its acquisition window. This matters for current-polarity dead-time compensation, whose current-path model can fail if the actual current crosses zero during the interval. The physical recirculation paths are documented in the manufacturer’s bridge-switching diagrams.^[ref]

Let an averaging window end at $t_b$ and have age coordinate $a\in [0,T_a]$. Suppose the reported reading isAssume $|di/dt|\le L_i$ over the acquisition history and target interval ending at $t_e\ge t_b$. If $k\ge 0$, then for every future $t\le t_e$,where $\bar{a}={\int }_0^{T_a}ak(a)da$ is the mean sample age. The triangle inequality and the derivative bound prove this result; nonnegative normalized weights make it a convex average. A positive sign is certified by $i_m>b$, and a negative sign by $i_m<-b$, with $b$ the boxed quantity.

Uniform averaging gives $\bar{a}=T_a/2$. A triangular weighting favoring recent values, $k(a)=2(1-a/T_a)/T_a$, instead gives $\bar{a}=T_a/3$. For an original comparison, take $T_a=6\mu \mathrm{s}$, $t_e-t_b=2.5\mu \mathrm{s}$, $L_i=5000A/s$ and ${ϵ}_i=0.020\mathrm{A}$. The uniform bound is $0.0475\mathrm{A}$; the triangular bound is $0.0425\mathrm{A}$. A $0.0450\mathrm{A}$ reading certifies positivity only under the triangular model, leaving a $0.0025\mathrm{A}$ lower margin. This comparison holds the error bound fixed; changing a real filter can also change noise and calibration errors.

Timestamp uncertainty enlarges the age budget. If the actual window could end another $\delta t$ earlier than reported, use $t_e-t_b+\delta t$ in the bound. Any uncertain output scheduling or gate delay likewise belongs in the latest possible target end time. These are bounds on the complete timing relation, not just the controller’s nominal sampling period.

A signed-filter current-sign certificate needs a different expression when a reconstruction filter has negative coefficients. The safe bound isNormalization alone does not justify replacing this by a mean age. For the two-tap estimate $i_m=2i(t_b)-i(t_b-T_a)$, the weighted timestamp lies in the future, but the estimate is not a future measurement. At $t=t_b$, the general bound is $L_i{T}_a$ before sensor error, and a current ramp can attain it. A naive negative-age bound would give a false certificate.

A proposed validation applies known ramps and switching waveforms while recording the acquisition weighting, timestamps and independent current trace. Test the claimed derivative and sensor-error bounds over the whole contributing history. Infinite-memory filters require a bounded contribution from the unobserved tail or a justified history assumption. A successful certificate excludes a zero under its assumptions; a failed one leaves the interval undecided rather than proving a sign reversal.

A current-sign certificate with a curvature bound uses a bound on the second derivative of an electric current to certify the sign of an extrapolated current. An affine ramp is predicted exactly by two correctly timed samples, so its uncertainty should depend on curvature rather than counting the entire ramp as error. Current direction matters physically because it selects recirculation paths during switching dead time.^[ref]

Let samples at $-T$ and $0$ produce measurements $y_{-}=i(-T)+e_{-}$ and $y_0=i(0)+e_0$, with $|e_{-}|\le {ϵ}_{-}$ and $|e_0|\le {ϵ}_0$. For a future time $t\ge 0$, use the linear extrapolationIf $i$ is twice continuously differentiable and $|i^{\prime \prime }(s)|\le M$ throughout $[-T,t]$, the polynomial interpolation error givesThe model term follows by applying the remainder to the two-node interpolant: $i(t)-[(1+t/T)i(0)-(t/T)i(-T)]=i^{\prime \prime }(\xi )t(t+T)/2$ for some $\xi \in (-T,t)$. The sample-error terms use absolute coefficients, as required by the signed-filter current-sign certificate. At $t=T$, the model contribution is $M{T}^2$ and equal sample bounds contribute $3ϵ$.

For an original example, take $T=6\mu \mathrm{s}$, $M=10^{8}A/{\mathrm{s}}^2$ and each sample error bounded by $0.002\mathrm{A}$. At $t=T$,If $y_0=0.030\mathrm{A}$ and $y_{-}=0.040\mathrm{A}$, the predicted current is $0.020\mathrm{A}$, certifying actual current at least $0.0104\mathrm{A}$ at that future instant. To certify the entire interval $0\le t\le T$, check $\hat{i}(t)-B(t)>0$ everywhere. In this example the prediction decreases linearly and $B$ increases, so the minimum margin occurs at $T$ and the same lower bound applies throughout.

The curvature limit is a physical assumption. A switching event, sensor clipping or a slope discontinuity can invalidate it even when a fitted ramp looks smooth at the sample times. A bounded first derivative alone does not imply any finite $M$. Timing uncertainty in a curvature-based current-sign certificate also matters: two accurate current values at incorrectly modeled times need not extrapolate an affine ramp correctly.

A proposed test supplies known smooth ramps and curved waveforms, records sample timestamps and checks the independently measured future waveform against the enclosure. Test transitions separately. The result is a deterministic sign guarantee only when the sample and curvature bounds are deterministic; substituting one standard uncertainty for an absolute error bound changes the claim.

A signed-filter current-sign certificate guarantees the sign of an actual load current from a filtered estimate whose coefficients can be negative. Such a filter may extrapolate or sharpen a waveform, but its output is not necessarily a convex average of past currents. The sign must hold over the intended switching interval before it can justify current-polarity dead-time compensation; the relevant recirculation paths depend on current direction.^[ref]

Suppose $\hat{i}={\sum }_j{a}_{j}i(t_j)+\eta$, with ${\sum }_j{a}_j=1$, $|\eta |\le ϵ$ and the Lipschitz bound $|i(t)-i(s)|\le L_i|t-s|$ over all relevant times. ThenThe absolute coefficients are essential. To certify an interval, maximize the right-hand side over its times. A positive sign follows if the estimate exceeds that bound; a negative sign follows if it is less than the negative bound. If individual sample errors are bounded by ${ϵ}_j$, their contribution is at most ${\sum }_j|a_j|{ϵ}_j$ rather than a signed cancellation.

The generic bound can be conservative because all samples come from the same current history. For the two-tap predictor $\hat{i}=2i(0)-i(-T_a)+\eta$ and future time $\delta \ge 0$, rewrite the error asThis gives the sharper resultinstead of the direct absolute-weight bound $ϵ+L_i(3\delta +T_a)$. It is sharp under only a Lipschitz current assumption: a history proportional to $|t|$ changes slope at the central sample and attains the positive error limit. The Lipschitz assumption permits that corner even though a derivative need not exist at the corner itself.

For an original example, take $L_i=5000A/s$, $T_a=6\mu \mathrm{s}$, a target interval $0\le t\le 2\mu \mathrm{s}$, and aggregate estimate error $ϵ=0.020\mathrm{A}$. The sharper interval bound is $0.060\mathrm{A}$; the direct absolute-weight bound is $0.080\mathrm{A}$. A reading $\hat{i}=0.065\mathrm{A}$ therefore certifies at least $0.005\mathrm{A}$ throughout the interval using the sharper proof. A signed timestamp average would misleadingly place the predictor at time $+T_a$; it is an extrapolation, not a measured future sample.

A current-sign certificate with a curvature bound can exploit stronger information. For the target time $T_a$ and a twice differentiable current with $|i^{\prime \prime }|\le M_i$ on $[-T_a,T_a]$,Taylor’s theorem, or integrating the second derivative against its triangular weight, proves the inequality. At $T_a=6\mu \mathrm{s}$ and stipulated $M_i=10^{8}A/{\mathrm{s}}^2$, the model error bound is $0.0036\mathrm{A}$ before sample errors. This smaller number is justified only if the curvature bound also covers switching transitions; an affine-ramp fit alone does not establish it.

A proposed validation injects known ramps and waveforms with changing slope, records the actual filter coefficients and sampling times, and checks the resulting enclosures against an independent current trace. Account for sensor saturation, timing uncertainty and the history used by the filter. A sharper mathematical bound and a stronger physical assumption are different ways of obtaining a smaller certificate margin.