2. Sampling theory essentials for crash signals

Every crash pulse that reaches a CFC filter has already been sampled by an ADC, and every CFC filter that runs in software operates on that sampled sequence. This makes sampling theory a prerequisite rather than a background topic: the Nyquist criterion determines whether the high-frequency content that CFC filtering is supposed to remove can even be represented, and aliasing determines whether that content has already been folded back into the passband before any digital filter sees it. A calibration engineer who selects CFC 1000 without checking the sample rate can produce a result that looks numerically stable yet is physically meaningless.

The Nyquist criterion in the crash-data context

For a signal sampled at a uniform interval Δt, the sampling frequency is f_s = 1/Δt. The Nyquist–Shannon sampling theorem states that a band-limited signal containing no energy above f_max can be reconstructed exactly only if f_s > 2·f_max. The quantity f_s/2 is the Nyquist frequency, f_N. In practice, equality is never sufficient because real anti-alias filters have finite roll-off, so the usable band is placed below f_N with margin.

Crash signals are not naturally band-limited. A frontal barrier pulse contains a low-frequency rigid-body deceleration component (roughly DC to 200–300 Hz) plus structural ringing, sensor resonance, and cable/harness microphonics that can extend into the kilohertz range. The CFC specification itself defines a passband and a stopband: for CFC 60, the passband is flat to 100 Hz and the attenuation reaches −30 dB at 189 Hz; for CFC 1000, the corresponding points are approximately 1650 Hz and 3000 Hz. These numbers, not the sample rate, define what the filter is expected to preserve and reject.

A practical rule follows directly: the sample rate must be high enough that the CFC stopband corner lies comfortably below f_N. A common working margin is f_s ≥ 5–10 × f_stopband. For CFC 1000 with a 3000 Hz stopband corner, that implies at least 15–30 kHz, which is why crash DAQ systems typically sample at 20 kHz, 50 kHz, or 100 kHz per channel rather than at the 10 kHz that would merely satisfy 2 × 3000 Hz.

Aliasing: how out-of-band energy becomes in-band error

Aliasing occurs when a component at frequency f > f_N is sampled and appears at the folded frequency |f − k·f_s| for integer k. The folded component is indistinguishable from a genuine in-band signal; no post-processing can separate them. This is the critical point for crash analysis: a digital CFC filter applied after sampling cannot remove aliased content, because the aliased content now sits inside the passband.

Consider a concrete example. A vehicle channel is sampled at f_s = 10 kHz, so f_N = 5000 Hz. A sensor resonance at 6200 Hz with modest amplitude is present in the analog signal. It folds to |6200 − 10000| = 3800 Hz. If the analysis uses CFC 1000, whose passband extends to about 1650 Hz, the 3800 Hz artifact is attenuated but not eliminated; if the analysis uses CFC 6000 or an unfiltered peak read, the artifact appears as a real oscillation and can shift the reported peak by several g. The same resonance sampled at 50 kHz would fold to |6200 − 50000| = 43800 Hz, far outside any CFC passband, and would be removed cleanly by the digital filter.

Two further aliasing mechanisms deserve attention in crash work. First, non-uniform sampling caused by DAQ timing jitter or dropped samples produces broadband noise that is not describable by a single folding frequency; it appears as elevated noise floor and can corrupt zero-crossing-based metrics. Second, resampling — for example, decimating a 100 kHz acquisition to 10 kHz for storage — is itself a sampling operation and requires its own anti-alias filter before decimation. Decimating without a pre-filter is one of the most common causes of "mysterious" high-frequency content in archived crash data.

What this means for filter design and channel setup

The sampling analysis translates into three design decisions that must be made together, not independently.

Anti-alias filtering belongs in the analog domain. The ADC's own sample-and-hold provides some aperture filtering, but it is not a substitute for a deliberate analog low-pass stage. A practical choice is a passive or active RC/Butterworth filter with a corner between the highest CFC stopband of interest and f_N. For a 20 kHz channel intended for CFC 1000, a 4th-order Butterworth at 8 kHz gives roughly −24 dB at 16 kHz and more than −60 dB by 25 kHz, which keeps the 3 kHz CFC stopband well clear of the fold-back region. The filter corner must not intrude on the CFC passband: for CFC 1000 the passband edge is about 1650 Hz, so an 8 kHz analog corner leaves more than two octaves of margin.

The digital CFC filter must be designed for the actual sample rate. CFC filters are specified in the continuous-time frequency domain, but implemented as IIR or FIR approximations. If the implementation uses a bilinear transform without pre-warping, the corner frequency shifts; at f_s = 10 kHz and a 1650 Hz corner, the warping error is small but not negligible, and at lower sample rates it becomes significant. Always verify the realized −3 dB and −30 dB points against the CFC table by sweeping a sine through the implemented filter, not by trusting the coefficient derivation alone.

Resampling and decimation need their own filter. When a 100 kHz acquisition is reduced to 10 kHz for archiving, insert a digital low-pass with a stopband below the new Nyquist frequency (5 kHz) before dropping samples. A half-band decimator or a polyphase FIR with a 4 kHz corner is typical. Skipping this step is equivalent to sampling the original signal at 10 kHz with no anti-alias filter, and it reintroduces exactly the folding shown in Case A of the figure.

Verifying the sampling chain before trusting the CFC result

A short verification sequence catches most sampling-related errors before they contaminate an analysis. First, confirm the sample rate from the file header rather than from the acquisition software's nominal setting; dropped samples and clock drift change the effective rate. Second, compute the ratio f_s / f_stopband for the highest CFC you intend to apply and confirm it is at least 5. Third, inject a known sine at a frequency just above f_N (for example, 1.2 × f_N) into the full chain and confirm that it does not appear as a stable in-band tone; if it does, the analog anti-alias stage is inadequate. Fourth, sweep a sine through the implemented digital CFC filter and tabulate the measured attenuation at the CFC passband and stopband frequencies against the specification table.

These checks are inexpensive and directly protect the validity of every downstream metric — peak deceleration, pulse duration, and the filtered traces used for occupant injury criteria. Sampling theory is not an abstract preliminary; it is the boundary condition that determines whether a CFC-filtered crash signal represents the physical event or an artifact of the measurement chain.