4. Amplitude Scaling and Offset Correction for Correlation

Why Amplitude and Offset Errors Matter in Test–CAE Correlation

Time synchronization and filtering bring two signals into the same time base and frequency band, but they do not guarantee that the signals share the same vertical scale or the same zero reference. A CAE response can be 15 % too stiff, or a test channel can carry a 0.4 g DC bias from an accelerometer's signal conditioner. Both conditions leave the waveform shape intact while shifting the CORA metrics in ways that are easy to misattribute to modeling error. This case isolates the two scalar mismatches — a multiplicative gain and an additive offset — and shows how to quantify, correct, and document them.

The working model is a simple affine map applied to the CAE signal:

y_corrected(t) = a · y_CAE(t) + b

where a is the amplitude scale factor and b is the DC offset. The task is to estimate a and b from the data, apply them, and record why the correction is defensible rather than a curve-fit that hides genuine model error.

Separating Gain from Offset

Least-squares estimation on the paired samples gives both parameters in one step. With x = y_CAE and y = y_test over the evaluation window:

a = cov(x, y) / var(x)
b = mean(y) − a · mean(x)

This is the ordinary linear regression of test on CAE. It is the right estimator when the mismatch is genuinely affine, but it will absorb real shape error into a and b if the two curves diverge in phase or in higher-frequency content. That is why the correction must be applied after time alignment and filtering, and why the residual must be inspected before the parameters are accepted.

A more robust decomposition uses peak-to-peak amplitude and mean value independently:

• Gain from amplitude: a = (max(y_test) − min(y_test)) / (max(y_CAE) − min(y_CAE)) over the same window.
• Offset from means: b = mean(y_test) − a · mean(y_CAE).

The peak-to-peak route is less sensitive to a few outlier samples than the regression route, but it is more sensitive to noise spikes. In practice, compute both and compare; a large disagreement between the two estimates of a is itself a diagnostic that the signals differ in shape, not just in scale.

Worked Example: A Single Acceleration Channel

Consider a body-mounted accelerometer channel evaluated over a 0–120 ms window. The test signal has a mean of 0.42 g and a peak-to-peak of 18.6 g. The CAE signal, after filtering and lag correction, has a mean of −0.05 g and a peak-to-peak of 21.4 g. These are illustrative values chosen to make the arithmetic transparent, not measured results.

Applying y_corrected = 0.869 · y_CAE + 0.463 brings the CAE mean to 0.42 g and the peak-to-peak to 18.6 g, matching the test on both first-order statistics. The shape, phase, and frequency content are untouched — which is exactly the point: this correction removes only the affine mismatch and leaves genuine modeling error visible for the CORA metrics to score.

How the Correction Propagates into CORA

The two parameters act on different CORA sub-metrics. Offset correction primarily affects the DC offset or mean-deviation term and, through it, the magnitude-phase composite. Gain correction primarily affects the magnitude term and the shape term when the scorecard normalizes by amplitude. The diagram below shows the signal path and where each parameter enters.

Documenting the Rationale

A correction that is applied but not justified is indistinguishable from a fudge factor. The record should state, for each channel, the estimated parameters, the estimator used, the window over which they were fitted, and the physical explanation. Typical defensible rationales include:

• Sensor sensitivity or calibration: the test channel's scale factor differs from nominal, so the gain corrects a known instrumentation property.
• Signal conditioner bias: a DC offset from the amplifier or a high-pass corner set too low explains a constant shift.
• Unit or normalization convention: CAE output in a different unit or normalized to a different reference than the test channel.

If none of these applies and the parameters are large, the mismatch is more likely a genuine modeling discrepancy — for example, an over-stiff mount producing higher peaks — and correcting it would mask the very error the correlation study exists to find. In that case, record the parameters as observed but not applied, and let the CORA score reflect the raw mismatch.

Verification and Impact on Metrics

After applying the correction, verify three things before accepting it:

1. First-order statistics match: the corrected CAE mean and peak-to-peak equal the test values to within rounding.
2. Residual is zero-mean and trend-free: plot y_test − y_corrected; a visible slope or curvature indicates the affine model is insufficient.
3. CORA sub-metrics move as expected: the DC offset term should improve markedly, the magnitude term should improve, and the phase term should be essentially unchanged. If the phase term shifts, the correction has disturbed the time alignment and should be re-derived.

Record the before-and-after sub-metric values in the scorecard's correction log so that a reviewer can see exactly how much of the final composite score is attributable to the affine correction versus the underlying model quality. That separation is what makes the correlation defensible in a design review.