Exercise set

Sensor Dynamic Response, Filtering and Noise Exercises

Solved sensor dynamic response exercises for first-order lag, bandwidth, phase delay, piezoelectric charge, noise floor, ENBW, quantization and release checks.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises focus on whether a sensor channel can follow the event and preserve useful information. A sensor that is calibrated at steady state may still fail a dynamic measurement because of lag, filtering, resonance, aliasing, saturation, noise, quantization or processing delay.

Assume first-order and simplified noise models unless stated otherwise. Real dynamic measurements should also check mounting resonance, cable effects, anti-alias filtering, clock accuracy, trigger timing, windowing, data processing, environmental noise, calibration bandwidth and event replay evidence.

Release Evidence Notes

Dynamic sensor release should state the required measurement bandwidth, event duration, amplitude range, allowable phase delay, noise floor, sampling rate and anti-alias evidence. Filtered data can be useful, but raw data should be preserved when the measurement supports diagnosis, protection or acceptance.

Engineering Boundary Notes

Dynamic measurements must match the event, not only the sensor nameplate. A pressure spike, vibration burst, biomedical pulse, actuator transient or impact event can be shorter than the sensor response time or hidden by filtering. The release boundary should define the fastest event that must be detected and the amplitude accuracy required at that event duration.

Noise and bandwidth are linked. Lower bandwidth can make a channel look clean while removing the event of interest. Higher bandwidth can preserve the event but expose noise, resonance and electromagnetic pickup. The correct choice depends on the decision: trend monitoring, protection, diagnostic waveform, fatigue load spectrum or final acceptance can require different settings.

Sampling evidence should include anti-alias filtering before the ADC. Once an out-of-band component aliases into the measured band, later filtering or FFT processing cannot prove what the original signal was. For high-value or safety-relevant tests, raw data, filter settings, sample clock, trigger configuration and independent plausibility checks should be retained.

Common Release Mistakes

Do not release a dynamic measurement from steady calibration alone. The steady calibration can be perfect while the channel misses peaks, delays events, filters out content, aliases vibration or saturates on impact. Do not use a smoothed trend as proof of an event unless raw data or an independent fast channel confirms that the event was captured.

Do not compare noise floors unless the bandwidths are the same. A channel with lower reported RMS noise may simply have heavier filtering. Do not increase sampling rate without checking analog anti-alias filtering and sensor bandwidth. High sample rate cannot recover information that never reached the ADC or that already folded into the passband.

Dynamic release should also state what failure looks like. Clipping, overload recovery, cable motion, resonance, filter delay and missed triggers should have review rules. Otherwise the measurement may be accepted because the plotted signal looks clean rather than because the channel is physically valid.

Validation Package Checklist

Minimum evidence should include required event bandwidth, sensor bandwidth, mounted resonance screen, filter settings, sampling rate, anti-alias cutoff, record length, trigger rule, raw-data retention, noise floor, dynamic range and saturation margin. If FFT data are used, include window, record duration and frequency resolution. If time alignment matters, include phase delay or measured timing offset. If the result supports a protection or release decision, keep at least one independent plausibility check from another channel, known input, event recorder or physical limit.

For repeated testing, freeze the acquisition configuration before comparing runs. Changing filter settings, sample rate, window length or averaging can create an apparent improvement that is only a measurement-system change.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Dynamic response	1, 2, 3, 4	first-order lag, rise time and bandwidth	Decide whether the sensor follows the event.
Filtering and phase	5, 6, 7, 8	cutoff, amplitude ratio, phase lag and group delay	Decide whether filtering preserves the signal.
Noise and quantization	9, 10, 11, 12, 13	noise density, ENBW, SNR, ADC count and dynamic range	Decide whether signal is above the measurement floor.
Release gates	14, 15, 16, 17, 18	piezo charge, aliasing, saturation, uncertainty and final acceptance	Release, retest or redesign the channel.

Exercise 1: First-Order Step Response

A sensor has time constant $\tau=0.20\ \text{s}$ . What fraction of a step is reached after $0.40\ \text{s}$ ?

Solution

f=1-e^{-t/\tau}=1-e^{-0.40/0.20}=1-e^{-2}=0.865

Engineering Comment

After two time constants the sensor still has about $13.5\%$ residual error.

Plausibility Check

First-order systems reach about $86\%$ after two time constants.

Exercise 2: Time to 95 Percent

For the same sensor, estimate time to reach $95\%$ of a step.

Solution

0.95=1-e^{-t/\tau}

t=-\tau \ln(0.05)=0.20(2.996)=0.599\ \text{s}

Plausibility Check

Faster rise time requires higher bandwidth.

Exercise 5: Low-Pass Amplitude Ratio

A first-order low-pass has $f_c=100\ \text{Hz}$ . Find amplitude ratio at $f=50\ \text{Hz}$ :

\displaystyle |H|=\frac{1}{\sqrt{1+(f/f_c)^2}}

Solution

\displaystyle |H|=\frac{1}{\sqrt{1+0.5^2}}=0.894

Engineering Comment

Even below cutoff, amplitude is reduced. Acceptance measurements should include correction or margin.

Plausibility Check

At half the cutoff, attenuation is noticeable but not severe.

Exercise 6: Filter Phase Lag

For the same first-order filter, estimate phase lag:

\phi=-\tan^{-1}(f/f_c)

at $f=50\ \text{Hz}$ .

Solution

\phi=-\tan^{-1}(0.5)=-26.6^\circ

Engineering Comment

Phase lag matters for timing, control and event alignment.

Plausibility Check

At half cutoff, phase lag should be less than $45^\circ$ .

Exercise 7: Equivalent Time Delay

Convert phase lag $26.6^\circ$ at $50\ \text{Hz}$ to equivalent time delay.

Solution

Period:

\displaystyle T=\frac{1}{50}=0.020\ \text{s}

Delay:

\displaystyle t_d=\frac{26.6}{360}(0.020)=1.48\ \text{ms}

Engineering Comment

Small millisecond delays can matter in protection, control and synchronized measurements.

Plausibility Check

The delay is a small fraction of the $20\ \text{ms}$ period.

Exercise 8: Moving-Average Delay

A moving average uses $N=11$ samples at $f_s=1000\ \text{Hz}$ . Estimate group delay:

\displaystyle t_d=\frac{N-1}{2f_s}

Solution

\displaystyle t_d=\frac{10}{2000}=0.005\ \text{s}

So:

t_d=5\ \text{ms}

A dynamic channel has peak output $4.8\ \text{V}$ , output limit $5.0\ \text{V}$ and RMS noise $0.010\ \text{V}$ . Is there at least $10\sigma$ peak headroom?

Solution

Headroom:

H=5.0-4.8=0.2\ \text{V}

Ten-sigma noise:

10\sigma=10(0.010)=0.10\ \text{V}

Since:

0.2>0.10

the headroom passes.

Engineering Comment

Headroom should include real transient overshoot, not only steady sine amplitude.

Plausibility Check

The margin is twice the ten-sigma noise band.

Exercise 18: Dynamic Sensor Release Gate

A channel has:

Check	Result	Gate
Rise-time bandwidth	$70\ \text{Hz}$	$\ge 100\ \text{Hz}$
Filter amplitude at signal frequency	$0.894$	$\ge 0.95$
SNR	$56.5\ \text{dB}$	$\ge 40\ \text{dB}$
Alias frequency risk	present	absent
Saturation headroom	pass	pass

Decide release status.

Solution

Bandwidth fails:

70<100\ \text{Hz}

Filter amplitude fails:

0.894<0.95

SNR and saturation pass, but alias risk fails. The channel should not be released for dynamic acceptance.

Engineering Comment

Clean amplitude at low noise is not enough when bandwidth and aliasing fail.

Plausibility Check

Three dynamic validity gates fail, so a hold decision is consistent.

REF